Schrödinger model · Example 1
Ground-state probability in an infinite well
Problem & parameters. A particle is confined by infinite walls at x = 0 and L. Find the normalized ground-state probability density.
Solution. The walls select ψ = A sin(πx/L). Normalization gives A = √(2/L); square the wavefunction to obtain the plotted density.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Particle-in-a-box model · Example 1
Ground-state probability in an infinite well
Problem & parameters. A particle is confined by infinite walls at x = 0 and L. Find the normalized ground-state probability density.
Solution. The walls select ψ = A sin(πx/L). Normalization gives A = √(2/L); square the wavefunction to obtain the plotted density.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Dirac model · Example 1
Positive free-particle energy
Problem & parameters. For a free massive Dirac particle, evaluate the positive-energy branch versus momentum.
Solution. Squaring the free Dirac Hamiltonian gives E² = m²c⁴+p²c². Select its positive root.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1.8028 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Born–Oppenheimer approximation · Example 1
Nuclear motion on a harmonic energy surface
Problem & parameters. Approximate one Born–Oppenheimer potential-energy surface near its minimum by a spring of stiffness k.
Solution. Taylor-expand the electronic energy about its minimum. The linear term vanishes; retain the quadratic term.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local harmonic approximation on a single adiabatic surface; electronic crossings and nonadiabatic coupling are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hartree–Fock model · Example 1
One-electron hydrogenic radial probability
Problem & parameters. Use the normalized hydrogen 1s state for one electron in a Coulomb potential. Plot probability per radial interval.
Solution. The 1s density is exp(−2r/a₀)/(πa₀³). Multiply by the spherical volume factor 4πr².
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 0.089235 on the vertical axis. Values are rounded for display.
Scope. Hartree–Fock is exact for this one-electron case. For DFT this is an exact-functional reference; approximate functionals need not reproduce it exactly.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Density functional theory (DFT) · Example 1
One-electron hydrogenic radial probability
Problem & parameters. Use the normalized hydrogen 1s state for one electron in a Coulomb potential. Plot probability per radial interval.
Solution. The 1s density is exp(−2r/a₀)/(πa₀³). Multiply by the spherical volume factor 4πr².
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 0.089235 on the vertical axis. Values are rounded for display.
Scope. Hartree–Fock is exact for this one-electron case. For DFT this is an exact-functional reference; approximate functionals need not reproduce it exactly.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Time-dependent DFT (TDDFT) · Example 1
A coherent two-state population
Problem & parameters. Consider a resonantly driven, noninteracting two-level reference starting in its lower state.
Solution. Solve the resonant two-amplitude system to obtain upper-state amplitude −i sin(Ωt/2), then take its squared magnitude.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Two-level rotating-wave reference for time-dependent electronic calculations; not a general TDDFT solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Tight-binding model · Example 1
Nearest-neighbor chain band
Problem & parameters. An infinite one-orbital chain has nearest-neighbor hopping tₕ and zero on-site energy.
Solution. Insert a Bloch state exp(ikna) into the hopping equation; the two neighbors contribute −tₕ(exp(ika)+exp(−ika)).
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain -2 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hubbard model · Example 1
Hubbard dimer singlet ground energy
Problem & parameters. Find the two-electron singlet ground energy of a two-site Hubbard dimer with hopping tₕ > 0 and repulsion U.
Solution. In the coupled singlet/double-occupancy block the matrix has diagonal 0,U and off-diagonal −2tₕ. Solve its quadratic characteristic equation and select the lower eigenvalue.
Worked evaluation. Substitute the marked horizontal coordinate, 6, into the displayed formula to obtain -0.60555 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Heisenberg spin model · Example 1
Two classical spins
Problem & parameters. Two classical unit spins interact through −J s₁·s₂ with J > 0.
Solution. The dot product of two unit vectors is cos θ. Parallel alignment minimizes the energy.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5708, into the displayed formula to obtain -6.1232e-17 on the vertical axis. Values are rounded for display.
Scope. Classical two-spin special case; quantum spin spectra require a different treatment.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Ising model · Example 1
One Ising spin in a field
Problem & parameters. A single spin s = ±1 has energy −hs at inverse temperature β. Find its thermal mean.
Solution. Its partition function is 2 cosh(βh). The weighted spin sum is 2 sinh(βh); divide to obtain tanh(βh).
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Quantum harmonic oscillator · Example 1
Oscillator ground-state density
Problem & parameters. Use oscillator length ℓ = √(ℏ/mω) and find the normalized ground-state density.
Solution. Substitute a Gaussian into the stationary Schrödinger equation. The ground-state wavefunction is exp(−x²/2ℓ²)/(π¼√ℓ); square it.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Classical molecular dynamics (MD) · Example 1
Isolated harmonic vibration
Problem & parameters. Take one isolated coordinate with potential kq²/2, initial displacement A, and zero initial velocity.
Solution. Newton’s equation reduces to q″+ω²q = 0. The initial data select A cos(ωt).
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Harmonic force benchmark for MD or locally harmonic ab initio dynamics; real many-atom trajectories are not generally sinusoidal.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Ab initio molecular dynamics · Example 1
Isolated harmonic vibration
Problem & parameters. Take one isolated coordinate with potential kq²/2, initial displacement A, and zero initial velocity.
Solution. Newton’s equation reduces to q″+ω²q = 0. The initial data select A cos(ωt).
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Harmonic force benchmark for MD or locally harmonic ab initio dynamics; real many-atom trajectories are not generally sinusoidal.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Lennard–Jones potential · Example 1
Lennard–Jones pair contribution
Problem & parameters. Evaluate an unshifted 12–6 pair potential at reduced separation r/σ.
Solution. Insert the reduced distance into the two inverse powers. Differentiating gives a minimum at r/σ = 2^(1/6), with U/ε = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 1.975, into the displayed formula to obtain -0.066264 on the vertical axis. Values are rounded for display.
Scope. For water and Martini entries, this is only a Lennard–Jones interaction contribution; electrostatics, constraints, and other sites are not included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
SPC/E water model · Example 1
Lennard–Jones pair contribution
Problem & parameters. Evaluate an unshifted 12–6 pair potential at reduced separation r/σ.
Solution. Insert the reduced distance into the two inverse powers. Differentiating gives a minimum at r/σ = 2^(1/6), with U/ε = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 1.975, into the displayed formula to obtain -0.066264 on the vertical axis. Values are rounded for display.
Scope. For water and Martini entries, this is only a Lennard–Jones interaction contribution; electrostatics, constraints, and other sites are not included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
TIP4P water-model family · Example 1
Lennard–Jones pair contribution
Problem & parameters. Evaluate an unshifted 12–6 pair potential at reduced separation r/σ.
Solution. Insert the reduced distance into the two inverse powers. Differentiating gives a minimum at r/σ = 2^(1/6), with U/ε = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 1.975, into the displayed formula to obtain -0.066264 on the vertical axis. Values are rounded for display.
Scope. For water and Martini entries, this is only a Lennard–Jones interaction contribution; electrostatics, constraints, and other sites are not included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Martini coarse-grained model · Example 1
Lennard–Jones pair contribution
Problem & parameters. Evaluate an unshifted 12–6 pair potential at reduced separation r/σ.
Solution. Insert the reduced distance into the two inverse powers. Differentiating gives a minimum at r/σ = 2^(1/6), with U/ε = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 1.975, into the displayed formula to obtain -0.066264 on the vertical axis. Values are rounded for display.
Scope. For water and Martini entries, this is only a Lennard–Jones interaction contribution; electrostatics, constraints, and other sites are not included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Morse potential · Example 1
Morse bond stretching
Problem & parameters. Evaluate a Morse bond with its dissociation limit set to zero.
Solution. At q = 0 the energy is −Dₑ. As q increases, the exponential tends to zero and the energy approaches the dissociation limit.
Worked evaluation. Substitute the marked horizontal coordinate, 1.7, into the displayed formula to obtain -0.33199 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Embedded-atom method (EAM) · Example 1
Illustrative embedding-energy contribution
Problem & parameters. Choose the illustrative embedding function F = −E*√(ρ/ρ*). Plot its density dependence.
Solution. Substitute the normalized local density into the chosen function. This evaluates the embedding contribution before summing pair terms.
Worked evaluation. Substitute the marked horizontal coordinate, 2.005, into the displayed formula to obtain -1.416 on the vertical axis. Values are rounded for display.
Scope. Illustrative EAM-type embedding function; not a fitted material parameterization. MEAM angular screening and density corrections are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Modified embedded-atom method (MEAM) · Example 1
Illustrative embedding-energy contribution
Problem & parameters. Choose the illustrative embedding function F = −E*√(ρ/ρ*). Plot its density dependence.
Solution. Substitute the normalized local density into the chosen function. This evaluates the embedding contribution before summing pair terms.
Worked evaluation. Substitute the marked horizontal coordinate, 2.005, into the displayed formula to obtain -1.416 on the vertical axis. Values are rounded for display.
Scope. Illustrative EAM-type embedding function; not a fitted material parameterization. MEAM angular screening and density corrections are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Tersoff bond-order potential · Example 1
A frozen bond-order pair
Problem & parameters. In a Tersoff-form pair term, hold cutoff and bond order at one and choose two exponential terms with coefficients 1 and 2.
Solution. Substitute the fixed bond order into the repulsive-minus-attractive energy. Differentiate the resulting two exponentials to inspect the force.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain -0.25235 on the vertical axis. Values are rounded for display.
Scope. Toy fixed-environment pair contribution; this excludes environment-dependent bond order and cutoff transitions.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Stillinger–Weber potential · Example 1
Tetrahedral angular penalty
Problem & parameters. Hold the radial factor of a Stillinger–Weber three-body term fixed and vary the included angle.
Solution. The squared angular factor vanishes at cos θ = −1/3, giving the tetrahedral angle.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5708, into the displayed formula to obtain 0.11111 on the vertical axis. Values are rounded for display.
Scope. Angular contribution only, with fixed radial prefactor K > 0.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
ReaxFF reactive force field · Example 1
Local harmonic bond-energy example
Problem & parameters. Near a stable isolated bond minimum, use the local quadratic energy with curvature k > 0.
Solution. The energy gradient vanishes at equilibrium. Retaining the second Taylor derivative gives ΔU = k(Δr)²/2.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local Taylor benchmark, not the full force field or a trained potential prediction; reactive changes and other coordinates are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
AMBER force-field family · Example 1
Local harmonic bond-energy example
Problem & parameters. Near a stable isolated bond minimum, use the local quadratic energy with curvature k > 0.
Solution. The energy gradient vanishes at equilibrium. Retaining the second Taylor derivative gives ΔU = k(Δr)²/2.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local Taylor benchmark, not the full force field or a trained potential prediction; reactive changes and other coordinates are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
CHARMM force-field family · Example 1
Local harmonic bond-energy example
Problem & parameters. Near a stable isolated bond minimum, use the local quadratic energy with curvature k > 0.
Solution. The energy gradient vanishes at equilibrium. Retaining the second Taylor derivative gives ΔU = k(Δr)²/2.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local Taylor benchmark, not the full force field or a trained potential prediction; reactive changes and other coordinates are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Machine-learned interatomic potential · Example 1
Local harmonic bond-energy example
Problem & parameters. Near a stable isolated bond minimum, use the local quadratic energy with curvature k > 0.
Solution. The energy gradient vanishes at equilibrium. Retaining the second Taylor derivative gives ΔU = k(Δr)²/2.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Local Taylor benchmark, not the full force field or a trained potential prediction; reactive changes and other coordinates are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
OPLS force-field family · Example 1
One torsional Fourier term
Problem & parameters. Retain only the first OPLS torsion coefficient V₁.
Solution. Set the other Fourier coefficients to zero and evaluate the remaining cosine term.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Single torsional energy contribution, not the full molecular force field.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Drude polarizable model · Example 1
Induced dipole in a uniform field
Problem & parameters. A charged Drude oscillator has harmonic stiffness k and charge q. Find its static induced dipole.
Solution. Balance kx = qE. Then p = qx = (q²/k)E, so α = q²/k.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Coarse-grained molecular model · Example 1
Gaussian coarse-coordinate free energy
Problem & parameters. Let a coarse variable have Gaussian probability proportional to exp(−q²/2).
Solution. Apply F = −kBT ln P, and remove the additive normalization constant.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Exactly solvable Gaussian coarse-graining example; it does not assert that arbitrary coarse models are harmonic.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Dissipative particle dynamics (DPD) · Example 1
Mean relative velocity under fixed pair drag
Problem & parameters. Hold pair distance and weight fixed; the mean relative velocity obeys dy/dτ = −y. Random force has zero mean.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Mean of a linear frozen-geometry pair reduction. DPD sample trajectories fluctuate and require a stochastic integrator.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Brownian dynamics · Example 1
One-dimensional mean-square displacement
Problem & parameters. For free Brownian motion in one dimension take D = 1 m²/s and initial position zero.
Solution. Integrate dx = √(2D)dW. Since the variance of W(t) is t, the mean-square displacement is 2Dt.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 5 on the vertical axis. Values are rounded for display.
Scope. Ensemble expectation, not a single random trajectory; illustrative diffusivity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Langevin dynamics · Example 1
Mean velocity after an impulse
Problem & parameters. A free Langevin particle has linear drag γ, mass m, mean initial speed v₀, and zero-mean thermal noise. Use τ = γt/m.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Ensemble mean velocity; the plotted smooth decay is not an individual noisy trajectory.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Kinetic Monte Carlo · Example 1
Probability of a first event
Problem & parameters. A kinetic Monte Carlo process has one constant total escape rate λ. Find the probability that its first event has occurred.
Solution. The survival probability solves S′ = −λS with S(0) = 1. Subtract S from one.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Waiting-time distribution for a fixed state and rate, not the entire evolving event network.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Cahn–Hilliard model · Example 1
A linear conserved-composition mode
Problem & parameters. Use dimensionless Cahn–Hilliard dynamics with M = a = κ = 1, quadratic free energy ac²/2, periodic boundaries, and initial perturbation cos x. Plot t = 1.
Solution. For wave number one, the amplitude satisfies A′ = −M(a+κ)A = −2A. Thus A(1) = exp(−2).
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -0.13534 on the vertical axis. Values are rounded for display.
Scope. Exact quadratic-free-energy special case, not nonlinear phase separation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Allen–Cahn model · Example 1
A relaxing Allen–Cahn mode
Problem & parameters. Take mobility, positive quadratic free-energy curvature, and gradient coefficient all equal to one, with initial cos x.
Solution. The local and gradient terms each contribute −A to the amplitude equation. Integrate A′ = −2A.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -0.13534 on the vertical axis. Values are rounded for display.
Scope. Linear quadratic-free-energy special case; domain walls of a double-well model are not represented.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Phase-field crystal model · Example 1
Linearized phase-field-crystal mode
Problem & parameters. Linearize ∂tψ = ∇²[(r+(1+∇²)²)ψ+ψ³] about ψ = 0 with r = 1; initial amplitude A₀ = 0.01 and wave number one.
Solution. The operator (1+∂xx) annihilates cos x. The remaining linear amplitude equation is A′ = −A.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -0.0036788 on the vertical axis. Values are rounded for display.
Scope. Linearized small-perturbation solution; the cubic term is omitted.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Potts grain-growth model · Example 1
Two-site Potts equilibrium alignment
Problem & parameters. For a three-state two-site Potts pair with energy −J when the states agree, compute the equilibrium agreement probability.
Solution. There are q agreeing states with Boltzmann weight exp(J/kBT), and q(q−1) disagreeing states with weight one. Normalize their sums.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.85898 on the vertical axis. Values are rounded for display.
Scope. Finite equilibrium toy problem, not a simulated grain-growth history.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Discrete dislocation dynamics · Example 1
Straight dislocation with constant mobility
Problem & parameters. Take one straight segment, constant force per length f = 1 N/m and mobility M = 1 m²/(N·s), starting at x = 0.
Solution. The overdamped mobility law gives constant velocity Mf; integrate with the initial position.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Illustrative coefficients; interactions, pinning, and changing segment geometry are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Population balance model · Example 1
Translated size distribution
Problem & parameters. For ∂tn+∂sn = 0 use n(s,0) = exp[−(s−2)²], constant growth G = 1, and compatible boundary inflow. Plot t = 1.
Solution. Along characteristics s−t is constant. Therefore n(s,t) = n₀(s−t).
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Ideal gas equation of state · Example 1
An ideal-gas isotherm
Problem & parameters. Hold temperature and amount of ideal gas fixed while varying its volume.
Solution. Solve pV = nRT for pressure and divide by the reference pressure nRT/V*.
Worked evaluation. Substitute the marked horizontal coordinate, 2.75, into the displayed formula to obtain 0.36364 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Van der Waals equation of state · Example 1
A supercritical van der Waals isotherm
Problem & parameters. Use the reduced van der Waals equation at T/Tc = 1.2.
Solution. Insert the critical scalings Vc = 3b, pc = a/(27b²), and Tc = 8a/(27Rb), then evaluate the reduced expression.
Worked evaluation. Substitute the marked horizontal coordinate, 2.3, into the displayed formula to obtain 1.06 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Peng–Robinson equation of state · Example 1
Peng–Robinson fixed-temperature curve
Problem & parameters. At fixed temperature choose aα/(RTb) = 2 and evaluate the Peng–Robinson pressure.
Solution. Divide its repulsive and attractive terms by RT/b and substitute v = Vₘ/b.
Worked evaluation. Substitute the marked horizontal coordinate, 3.75, into the displayed formula to obtain 0.26637 on the vertical axis. Values are rounded for display.
Scope. Illustrative EOS parameters; not a fitted fluid or a phase-equilibrium calculation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Soave–Redlich–Kwong equation of state · Example 1
Soave–Redlich–Kwong isotherm
Problem & parameters. At fixed temperature choose aα/(RTb) = 2 for the SRK equation.
Solution. Divide the EOS by RT/b and evaluate both terms using the reduced molar volume.
Worked evaluation. Substitute the marked horizontal coordinate, 3.75, into the displayed formula to obtain 0.25136 on the vertical axis. Values are rounded for display.
Scope. Illustrative parameters; the temperature dependence of α is fixed for this isotherm.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Virial equation of state · Example 1
A truncated virial compressibility
Problem & parameters. Use scaled second and third virial coefficients 0.2 and 0.05 over a dilute density interval.
Solution. Substitute the reduced density into Z = 1+Bρ+Cρ². At zero density it recovers the ideal-gas limit Z = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.1125 on the vertical axis. Values are rounded for display.
Scope. Truncated low-density illustrative expansion, not an extrapolation to dense fluids.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Gibbs-energy minimization · Example 1
Ideal binary mixing free energy
Problem & parameters. Take an ideal binary solution with equal pure-component reference energies. Find the composition dependence of its mixing free energy.
Solution. Sum the two ideal mixing contributions. Differentiation gives ln[x/(1−x)] = 0, so the minimum is at x = 1/2.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain -0.69315 on the vertical axis. Values are rounded for display.
Scope. Ideal-solution Gibbs term; real CALPHAD databases include additional phase and interaction terms. Conserved bulk composition constrains accessible equilibria.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
CALPHAD model · Example 1
Ideal binary mixing free energy
Problem & parameters. Take an ideal binary solution with equal pure-component reference energies. Find the composition dependence of its mixing free energy.
Solution. Sum the two ideal mixing contributions. Differentiation gives ln[x/(1−x)] = 0, so the minimum is at x = 1/2.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain -0.69315 on the vertical axis. Values are rounded for display.
Scope. Ideal-solution Gibbs term; real CALPHAD databases include additional phase and interaction terms. Conserved bulk composition constrains accessible equilibria.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
NRTL activity model · Example 1
Ideal-mixture activity limit
Problem & parameters. Set NRTL interaction parameters to zero; for UNIQUAC also take identical molecular sizes and shapes with zero interaction energies.
Solution. Under these restrictions the excess contribution vanishes and γ₁ = 1; activity is γ₁x₁.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Ideal-mixture limiting case only; unequal molecular sizes in UNIQUAC can retain a combinatorial contribution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
UNIQUAC activity model · Example 1
Ideal-mixture activity limit
Problem & parameters. Set NRTL interaction parameters to zero; for UNIQUAC also take identical molecular sizes and shapes with zero interaction energies.
Solution. Under these restrictions the excess contribution vanishes and γ₁ = 1; activity is γ₁x₁.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Ideal-mixture limiting case only; unequal molecular sizes in UNIQUAC can retain a combinatorial contribution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Debye–Hückel model · Example 1
Dilute ionic activity correction
Problem & parameters. For a monovalent ion in water near 25 °C use the Debye–Hückel limiting-law coefficient A = 0.509 (mol/L)⁻¹ᐟ².
Solution. Set charge magnitude to one in log₁₀γ = −Az²√I. Evaluate only at dilute ionic strengths.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain -0.035992 on the vertical axis. Values are rounded for display.
Scope. Limiting-law illustration; specific ion interactions and concentrated solutions are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Mass-action reaction kinetics · Example 1
First-order reactant consumption
Problem & parameters. For a single irreversible first-order reaction A → products in a constant-volume batch, use τ = kt and y = cA/cA0.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact one-mode reduction with constant coefficients; additional coupled physics is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Batch reactor model · Example 1
First-order reactant consumption
Problem & parameters. For a single irreversible first-order reaction A → products in a constant-volume batch, use τ = kt and y = cA/cA0.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact one-mode reduction with constant coefficients; additional coupled physics is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Arrhenius rate model · Example 1
Temperature dependence of an activated rate
Problem & parameters. Hold activation energy Ea > 0 and prefactor A constant.
Solution. Insert the scaled temperature into k = A exp(−Ea/RT).
Worked evaluation. Substitute the marked horizontal coordinate, 0.55, into the displayed formula to obtain 0.16232 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Transition-state theory · Example 1
Transition-state rate at fixed activation free energy
Problem & parameters. Take transmission coefficient one and treat the molar activation free energy as constant over the displayed interval.
Solution. Divide the Eyring expression by its kBT/h prefactor and substitute the scaled temperature.
Worked evaluation. Substitute the marked horizontal coordinate, 0.55, into the displayed formula to obtain 0.16232 on the vertical axis. Values are rounded for display.
Scope. Illustrative fixed-barrier curve; real activation free energy can vary with temperature.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Michaelis–Menten kinetics · Example 1
A saturating occupancy or rate
Problem & parameters. For Michaelis–Menten set x = substrate/Km and y = v/Vmax. For Langmuir adsorption set x = KP and y = occupied-site fraction.
Solution. Solve the binding or adsorption balance to give occupied fraction x/(1+x). The half-saturation point is x = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 4, into the displayed formula to obtain 0.8 on the vertical axis. Values are rounded for display.
Scope. Single-substrate steady enzyme law or single-species equilibrium adsorption, as appropriate to the entry.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Langmuir adsorption isotherm · Example 1
A saturating occupancy or rate
Problem & parameters. For Michaelis–Menten set x = substrate/Km and y = v/Vmax. For Langmuir adsorption set x = KP and y = occupied-site fraction.
Solution. Solve the binding or adsorption balance to give occupied fraction x/(1+x). The half-saturation point is x = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 4, into the displayed formula to obtain 0.8 on the vertical axis. Values are rounded for display.
Scope. Single-substrate steady enzyme law or single-species equilibrium adsorption, as appropriate to the entry.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Langmuir–Hinshelwood kinetics · Example 1
Competing adsorption and surface reaction
Problem & parameters. Use the illustrative Langmuir–Hinshelwood rate r/r* = x/(1+x)², with other factors held constant.
Solution. Differentiate: the slope is (1−x)/(1+x)³. Thus the rate peaks at x = 1 and decreases under strong site blocking.
Worked evaluation. Substitute the marked horizontal coordinate, 4, into the displayed formula to obtain 0.16 on the vertical axis. Values are rounded for display.
Scope. One specified adsorption-limited rate law; the family contains many different mechanisms.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Fickian diffusion · Example 1
Binary concentration relaxation
Problem & parameters. Solve ∂τu = ∂ξξu with u(0,τ)=u(1,τ)=0 and initial sin(πξ), then plot τ = 0.1.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Fickian constant-diffusivity slab. Maxwell–Stefan reduces to this form for an ideal binary mixture with constant total concentration and diffusivity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Maxwell–Stefan diffusion · Example 1
Binary concentration relaxation
Problem & parameters. Solve ∂τu = ∂ξξu with u(0,τ)=u(1,τ)=0 and initial sin(πξ), then plot τ = 0.1.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Fickian constant-diffusivity slab. Maxwell–Stefan reduces to this form for an ideal binary mixture with constant total concentration and diffusivity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Advection–diffusion–reaction model · Example 1
Advected, diffused, reacting Gaussian
Problem & parameters. On the infinite line solve ut+ux = 0.1uxx−0.2u with u(x,0)=exp(−x²). Plot t = 1.
Solution. Advection translates the center by t. Diffusion increases the Gaussian width from 1 to 1+0.4t; first-order loss multiplies its conserved-mass diffusion solution by exp(−0.2t).
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 0.69195 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Continuous stirred-tank reactor (CSTR) · Example 1
CSTR outlet versus residence time
Problem & parameters. At steady state a well-mixed reactor consumes A by a first-order reaction at rate kcA.
Solution. Balance Qcin−Qcout−kVcout = 0 and solve for cout.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.28571 on the vertical axis. Values are rounded for display.
Scope. Constant-volume, isothermal, constant-flow reactor.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Plug-flow reactor (PFR) · Example 1
First-order plug-flow conversion
Problem & parameters. For an isothermal PFR with constant velocity u and first-order consumption k, use τ = kz/u and y = c/cin.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact axial concentration profile in ideal plug flow; the horizontal coordinate is residence time kz/u, not laboratory time.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Stokes creeping-flow model · Example 1
Pressure-driven laminar flow profile
Problem & parameters. Take steady, fully developed incompressible flow with constant viscosity between fixed parallel plates. For Hagen–Poiseuille use the equivalent diameter cut through a round pipe.
Solution. The axial momentum equation becomes a constant second derivative. Integrate twice and impose no slip at both walls to obtain a parabola.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact laminar benchmark. Plate and pipe pressure-to-maximum-speed factors differ; the plotted normalized profile is identical. DNS here resolves this simple laminar case.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Lubrication approximation · Example 1
Pressure-driven laminar flow profile
Problem & parameters. Take steady, fully developed incompressible flow with constant viscosity between fixed parallel plates. For Hagen–Poiseuille use the equivalent diameter cut through a round pipe.
Solution. The axial momentum equation becomes a constant second derivative. Integrate twice and impose no slip at both walls to obtain a parabola.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact laminar benchmark. Plate and pipe pressure-to-maximum-speed factors differ; the plotted normalized profile is identical. DNS here resolves this simple laminar case.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hagen–Poiseuille model · Example 1
Pressure-driven laminar flow profile
Problem & parameters. Take steady, fully developed incompressible flow with constant viscosity between fixed parallel plates. For Hagen–Poiseuille use the equivalent diameter cut through a round pipe.
Solution. The axial momentum equation becomes a constant second derivative. Integrate twice and impose no slip at both walls to obtain a parabola.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact laminar benchmark. Plate and pipe pressure-to-maximum-speed factors differ; the plotted normalized profile is identical. DNS here resolves this simple laminar case.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Direct numerical simulation (DNS) · Example 1
Pressure-driven laminar flow profile
Problem & parameters. Take steady, fully developed incompressible flow with constant viscosity between fixed parallel plates. For Hagen–Poiseuille use the equivalent diameter cut through a round pipe.
Solution. The axial momentum equation becomes a constant second derivative. Integrate twice and impose no slip at both walls to obtain a parabola.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact laminar benchmark. Plate and pipe pressure-to-maximum-speed factors differ; the plotted normalized profile is identical. DNS here resolves this simple laminar case.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Euler flow model · Example 1
A small-amplitude sound wave
Problem & parameters. Linearize inviscid Euler flow about a uniform rest state and use a sinusoidal pressure perturbation.
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. Linear acoustic limit of Euler flow, not a finite-amplitude compressible flow solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Potential-flow model · Example 1
Cylinder surface pressure
Problem & parameters. Find surface pressure for incompressible, inviscid, irrotational uniform flow around a circular cylinder without circulation.
Solution. Potential flow gives surface speed 2U∞ sin θ. Bernoulli’s equation then gives Cp = 1−(u/U∞)².
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. No viscosity or separation; this ideal model does not predict real cylinder drag.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Boundary-layer model · Example 1
A suddenly moving flat wall
Problem & parameters. A flat wall suddenly moves at speed U beneath an initially stationary semi-infinite viscous fluid.
Solution. With no streamwise variation, momentum reduces to diffusion. Similarity substitution and the wall/far-field conditions give the complementary error function.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 0.033895 on the vertical axis. Values are rounded for display.
Scope. Stokes’ first problem, an unsteady boundary-layer benchmark; not the Blasius spatially developing solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Darcy–Weisbach model · Example 1
Pipe pressure loss versus speed
Problem & parameters. Hold the Darcy friction factor f, pipe geometry, and density fixed.
Solution. Insert the mean speed into Darcy–Weisbach Δp = f(L/D)ρU²/2.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 2.25 on the vertical axis. Values are rounded for display.
Scope. Fixed-friction-factor illustration; f usually varies with Reynolds number and roughness.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Non-Newtonian power-law fluid · Example 1
Shear-thinning constitutive curve
Problem & parameters. Choose positive shear rates and power-law exponent n = 1/2, with reference stress K√(reference rate).
Solution. Substitute n = 1/2 into τ = Kγ̇ⁿ.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 1.4142 on the vertical axis. Values are rounded for display.
Scope. Steady shear constitutive evaluation; no low- or high-shear viscosity plateau is included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Bingham plastic model · Example 1
Bingham imposed-stress response
Problem & parameters. Increase a nonnegative applied shear stress on an ideal Bingham material.
Solution. Below yield, the shear rate is zero. Above yield, solve τ = τy+μpγ̇ for the rate.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Herschel–Bulkley model · Example 1
Herschel–Bulkley stress curve
Problem & parameters. Use exponent n = 1/2 and define g so that Kγ̇ⁿ/τy = √g. Evaluate the yielded branch.
Solution. Insert the chosen exponent into τ = τy+Kγ̇ⁿ.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 2.4142 on the vertical axis. Values are rounded for display.
Scope. Positive yielded branch only; at zero rate the unyielded model allows a range of stresses.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Oldroyd-B model · Example 1
Polymer stress relaxation at rest
Problem & parameters. After a small deformation, hold the fluid motionless. A homogeneous Oldroyd-B polymer shear stress obeys λdτp/dt+τp=0.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Zero-velocity, homogeneous stress-relaxation subproblem; convected terms vanish and the solvent stress is zero.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Large-eddy simulation (LES) · Example 1
Laminar-limit flow verification
Problem & parameters. Verify the molecular-viscosity momentum equation using fully developed plane Poiseuille flow with turbulent or subgrid stresses disabled.
Solution. A constant pressure gradient gives μu″ = dp/dx. Apply no slip at the two walls and normalize by the center speed.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Laminar-limit verification only; it neither models turbulence nor validates a RANS, LES, or DES closure.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Detached-eddy simulation (DES) · Example 1
Laminar-limit flow verification
Problem & parameters. Verify the molecular-viscosity momentum equation using fully developed plane Poiseuille flow with turbulent or subgrid stresses disabled.
Solution. A constant pressure gradient gives μu″ = dp/dx. Apply no slip at the two walls and normalize by the center speed.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Laminar-limit verification only; it neither models turbulence nor validates a RANS, LES, or DES closure.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Spalart–Allmaras model · Example 1
Spalart–Allmaras viscosity mapping
Problem & parameters. For nonnegative working variable χ evaluate the standard SA eddy-viscosity mapping with cv1 = 7.1.
Solution. Compute the damping function fv1 = χ³/(χ³+cv1³), then multiply by χ.
Worked evaluation. Substitute the marked horizontal coordinate, 10, into the displayed formula to obtain 7.3643 on the vertical axis. Values are rounded for display.
Scope. Algebraic closure contribution only, not a solution of the SA transport equation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
k–epsilon model · Example 1
k–epsilon eddy-viscosity closure
Problem & parameters. Hold dissipation ε = ε* fixed and use Cμ = 0.09.
Solution. Substitute k into νt = Cμk²/ε.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 0.36 on the vertical axis. Values are rounded for display.
Scope. Closure evaluation, not a prediction of k or ε from their coupled transport equations.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
k–omega model · Example 1
k–omega viscosity closure
Problem & parameters. Hold specific dissipation ω = ω* > 0 and use the basic νt = k/ω relation.
Solution. Divide the closure by the reference viscosity k*/ω*.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. Basic algebraic closure with fixed ω; model variants may include limiters.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
SST k–omega model · Example 1
SST shear-stress limiter
Problem & parameters. Hold positive k and ω fixed. Evaluate νt = a1k/max(a1ω,SF2) with a1 = 0.31.
Solution. Divide denominator and numerator by ω to expose the limiter transition at s = a1.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 0.31 on the vertical axis. Values are rounded for display.
Scope. Algebraic SST limiter illustration; blending functions and transport equations are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Reynolds-stress transport model · Example 1
Idealized return to isotropy
Problem & parameters. For a homogeneous Reynolds-stress anisotropy component use the reduced closure db/dt = −b/T with constant T.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Isolated linear return-to-isotropy term; production, transport, and changing dissipation are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Smagorinsky subgrid model · Example 1
Smagorinsky viscosity versus strain
Problem & parameters. Use Cs = 0.1 and constant filter width Δ.
Solution. Evaluate νt = (CsΔ)²|S|.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.025 on the vertical axis. Values are rounded for display.
Scope. Constant-coefficient closure; no dynamic procedure or wall damping is included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Volume-of-fluid (VOF) representation · Example 1
A transported smooth volume fraction
Problem & parameters. Advect the initial smoothed interface α(x,0) = [1−tanh(5x)]/2 at unit velocity with no compression term.
Solution. Characteristics give α(x,t) = α₀(x−t); evaluate t = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Exact scalar-advection benchmark with a deliberately smooth interface; interface reconstruction and multiphase momentum are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Euler–Euler two-fluid model · Example 1
Two-phase slip relaxation
Problem & parameters. For two homogeneous phases coupled only by linear interphase drag, scale time by the combined drag relaxation time and slip by its initial value.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Subtract the two phase momentum balances to obtain a decaying relative velocity; spatial transport, pressure gradients, and phase change are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Lagrangian particle tracking · Example 1
Particle acceleration under Stokes drag
Problem & parameters. A particle starts at rest in a uniform fluid of constant speed U and experiences linear drag only.
Solution. Solve τp v′ + v = U with v(0) = 0 using an integrating factor.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Dilute isolated-particle Stokes-drag reduction; no gravity or feedback on the fluid.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Fourier heat conduction · Example 1
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Groundwater flow model · Example 1
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Finite element method (FEM / FEA) · Example 1
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Finite volume method (FVM) · Example 1
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Finite difference method (FDM) · Example 1
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Boundary element method (BEM) · Example 1
Steady one-dimensional diffusion benchmark
Problem & parameters. Solve u″ = 0 on 0 < ξ < 1 with u(0) = 1 and u(1) = 0, constant transport coefficient, and no source.
Solution. Integrate twice to obtain u = A+Bξ. The two endpoint values give A = 1 and B = −1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. For numerical-method entries this is the exact target to verify against, not a computed discretization or convergence claim.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Transient heat equation · Example 1
Decaying heat-mode reference
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Spectral method · Example 1
Decaying heat-mode reference
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Reduced basis model · Example 1
Decaying heat-mode reference
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Physics-informed neural network (PINN) · Example 1
Decaying heat-mode reference
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Neural operator · Example 1
Decaying heat-mode reference
Problem & parameters. Use uτ = uξξ on the unit interval, zero end values, and u(ξ,0) = sin(πξ). Plot τ = 0.1.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact PDE benchmark. For reduced bases, PINNs, and neural operators, this is a reference target, not a claimed trained or computed prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Lumped-capacitance thermal model · Example 1
Single thermal capacitance cooling
Problem & parameters. A thermal capacitance C connects through resistance R to fixed ambient temperature. Set τ = t/(RC) and y = (T−T∞)/(T0−T∞).
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-node constant-property cooling example; multizone and multi-node networks have additional modes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Thermal resistance-capacitance network · Example 1
Single thermal capacitance cooling
Problem & parameters. A thermal capacitance C connects through resistance R to fixed ambient temperature. Set τ = t/(RC) and y = (T−T∞)/(T0−T∞).
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-node constant-property cooling example; multizone and multi-node networks have additional modes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Newton cooling model · Example 1
Single thermal capacitance cooling
Problem & parameters. A thermal capacitance C connects through resistance R to fixed ambient temperature. Set τ = t/(RC) and y = (T−T∞)/(T0−T∞).
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-node constant-property cooling example; multizone and multi-node networks have additional modes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Building thermal-zone model · Example 1
Single thermal capacitance cooling
Problem & parameters. A thermal capacitance C connects through resistance R to fixed ambient temperature. Set τ = t/(RC) and y = (T−T∞)/(T0−T∞).
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-node constant-property cooling example; multizone and multi-node networks have additional modes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Radiative transfer equation · Example 1
Uncollided beam attenuation
Problem & parameters. A steady beam traverses a homogeneous purely absorbing medium. Set τ = Σx and y = intensity / incident intensity.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact absorption-only transport benchmark, without scattering or emission. For Monte Carlo transport this is the expected value, not a sampled realization.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Neutron transport model · Example 1
Uncollided beam attenuation
Problem & parameters. A steady beam traverses a homogeneous purely absorbing medium. Set τ = Σx and y = intensity / incident intensity.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact absorption-only transport benchmark, without scattering or emission. For Monte Carlo transport this is the expected value, not a sampled realization.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Monte Carlo transport · Example 1
Uncollided beam attenuation
Problem & parameters. A steady beam traverses a homogeneous purely absorbing medium. Set τ = Σx and y = intensity / incident intensity.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact absorption-only transport benchmark, without scattering or emission. For Monte Carlo transport this is the expected value, not a sampled realization.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Stefan–Boltzmann surface model · Example 1
Net radiation to a fixed surrounding
Problem & parameters. A gray surface sees a large isothermal surrounding at T*, with constant emissivity.
Solution. Subtract incoming εσT*⁴ from outgoing εσT⁴. Positive net flux is outward.
Worked evaluation. Substitute the marked horizontal coordinate, 1.25, into the displayed formula to obtain 1.4414 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Surface-to-surface radiosity model · Example 1
Two gray parallel surfaces
Problem & parameters. Two infinite parallel diffuse-gray plates have emissivities ε and 0.8 and fixed unequal temperatures.
Solution. Add the two surface radiation resistances and the unit view-factor space resistance; solve the radiosity balance for q.
Worked evaluation. Substitute the marked horizontal coordinate, 0.525, into the displayed formula to obtain 0.46409 on the vertical axis. Values are rounded for display.
Scope. Equal facing areas, view factor one, and a nonparticipating gap.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Stefan phase-change problem · Example 1
Similarity-law melt-front position
Problem & parameters. Take a one-phase Stefan problem whose Stefan number selects similarity constant λ = 0.5.
Solution. The diffusion similarity coordinate makes the interface position s = 2λ√(αt). For this λ, the Stefan-number relation is Ste = √π λ exp(λ²) erf(λ) ≈ 0.5923.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 1.4142 on the vertical axis. Values are rounded for display.
Scope. Semi-infinite, one-phase conduction limit with a fixed boundary temperature; λ must be consistent with the material and thermal data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Enthalpy–porosity model · Example 1
Prescribed mushy-range liquid fraction
Problem & parameters. Choose the linear liquid-fraction law between solidus Ts and liquidus Tl.
Solution. Use zero fraction below Ts, linear interpolation inside the mushy interval, and unit fraction above Tl.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Constitutive phase-fraction example only; momentum damping and the transient enthalpy equation are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Linear elasticity (Hooke model) · Example 1
Uniform axial extension
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Orthotropic elasticity · Example 1
Uniform axial extension
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Truss model · Example 1
Uniform axial extension
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Representative volume element (RVE) · Example 1
Uniform axial extension
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
FE² computational homogenization · Example 1
Uniform axial extension
Problem & parameters. Apply uniform uniaxial strain to a homogeneous small-strain elastic bar with traction-free lateral surfaces.
Solution. The one-dimensional constitutive law is σ = Eε; divide by E. For an orthotropic solid use its modulus along a principal material axis.
Worked evaluation. Substitute the marked horizontal coordinate, 0.005, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Homogeneous linear reference for truss, RVE, and FE² entries; this is not a heterogeneous microscale simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Neo-Hookean hyperelasticity · Example 1
Incompressible neo-Hookean tension
Problem & parameters. Stretch an incompressible neo-Hookean solid uniaxially with traction-free transverse faces.
Solution. Incompressibility gives transverse stretches λ^−1/2. Eliminate the pressure using zero transverse stress, yielding μ(λ²−λ^−1).
Worked evaluation. Substitute the marked horizontal coordinate, 1.3, into the displayed formula to obtain 0.92077 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Mooney–Rivlin hyperelasticity · Example 1
Mooney–Rivlin uniaxial tension
Problem & parameters. Use an incompressible two-parameter Mooney–Rivlin material with C10 = C01 and μ = 2(C10+C01).
Solution. Differentiate the strain energy and eliminate transverse pressure. The axial stress is 2(C10+C01/λ)(λ²−1/λ).
Worked evaluation. Substitute the marked horizontal coordinate, 1.3, into the displayed formula to obtain 0.81453 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Ogden hyperelasticity · Example 1
One-term Ogden tension
Problem & parameters. Choose W = (2μ/α²)(λ1^α+λ2^α+λ3^α−3), α = 4, and incompressible uniaxial tension.
Solution. Set transverse stretches to λ^−1/2 and impose zero transverse stress. Then σ = (2μ/α)(λ^α−λ^−α/2).
Worked evaluation. Substitute the marked horizontal coordinate, 1.3, into the displayed formula to obtain 1.1322 on the vertical axis. Values are rounded for display.
Scope. The energy convention is stated explicitly because Ogden coefficient conventions vary.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Euler–Bernoulli beam model · Example 1
End-loaded cantilever deflection
Problem & parameters. A prismatic Euler–Bernoulli cantilever of length L carries a transverse tip force P.
Solution. Use bending moment M = P(L−x). Integrate EIw″ = M and apply zero displacement and slope at the clamped end.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.10417 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Timoshenko beam model · Example 1
Cantilever bending plus shear
Problem & parameters. Take an end-loaded Timoshenko cantilever with EI/(κGA L²) = 0.1.
Solution. Add the bending displacement to the shear contribution Px/(κGA). Divide by PL³/EI.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.15417 on the vertical axis. Values are rounded for display.
Scope. Linear prismatic beam; κ is the shear correction factor.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Kirchhoff–Love plate model · Example 1
Sinusoidally loaded plate centerline
Problem & parameters. Apply a single sinusoidal load mode to a simply supported rectangular plate. Plot its normalized centerline deflection.
Solution. A separable sin(πx/a)sin(πy/b) mode satisfies the simply supported displacement conditions. At y=b/2 it reduces to the displayed sine.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Kirchhoff–Love and compatible Mindlin single-mode solutions share this normalized shape but have different bending/shear amplitude formulas.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Mindlin–Reissner plate model · Example 1
Sinusoidally loaded plate centerline
Problem & parameters. Apply a single sinusoidal load mode to a simply supported rectangular plate. Plot its normalized centerline deflection.
Solution. A separable sin(πx/a)sin(πy/b) mode satisfies the simply supported displacement conditions. At y=b/2 it reduces to the displayed sine.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Kirchhoff–Love and compatible Mindlin single-mode solutions share this normalized shape but have different bending/shear amplitude formulas.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Shell model · Example 1
Thin spherical shell membrane stress
Problem & parameters. A thin spherical shell of radius R and thickness t carries uniform internal pressure p.
Solution. Balance pressure on a hemisphere against the circumferential membrane force: pπR² = 2πRtσ.
Worked evaluation. Substitute the marked horizontal coordinate, 0.01, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Thin-shell membrane approximation, away from supports and local bending disturbances.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Cable and membrane models · Example 1
Hanging cable shape
Problem & parameters. An ideal flexible cable supports its own uniform weight per arc length; choose a = horizontal tension / weight per length.
Solution. Force balance gives y″ = √(1+y′²)/a. Symmetry at the lowest point integrates to the catenary.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Self-weight catenary, not the parabolic approximation for uniform load per horizontal span.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
von Mises J2 plasticity · Example 1
Ideal uniaxial elastic-perfectly-plastic response
Problem & parameters. Load monotonically in uniaxial tension from an unstressed state, with no hardening.
Solution. Use Hooke’s law until σ = σy. Further strain is plastic while stress stays at σy.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Uniaxial case where J2 and Tresca coincide; multiaxial yield surfaces differ.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Tresca yield model · Example 1
Ideal uniaxial elastic-perfectly-plastic response
Problem & parameters. Load monotonically in uniaxial tension from an unstressed state, with no hardening.
Solution. Use Hooke’s law until σ = σy. Further strain is plastic while stress stays at σy.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Uniaxial case where J2 and Tresca coincide; multiaxial yield surfaces differ.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Drucker–Prager plasticity · Example 1
Pressure-dependent Drucker–Prager strength
Problem & parameters. Define the illustrative yield line q−0.5p−c=0 with compression-positive pressure p.
Solution. Solve the stated yield function for q.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. A specified pressure/deviatoric convention and slope; different parameter mappings to friction angle exist.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Mohr–Coulomb model · Example 1
Mohr–Coulomb shear strength
Problem & parameters. Use cohesion c > 0 and friction angle 30 degrees.
Solution. Insert the normal stress into τf = c+σn tan φ with compression positive.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 2.1547 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Johnson–Cook model · Example 1
Johnson–Cook strain-hardening factor
Problem & parameters. Set B/A = 0.5, n = 0.5, strain rate equal to its reference value, and homologous temperature zero.
Solution. The rate and thermal factors become one. Evaluate the remaining A+Bεpⁿ hardening term.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.3536 on the vertical axis. Values are rounded for display.
Scope. Illustrative constants, not a calibrated metal response.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Crystal plasticity · Example 1
One slip-system power law
Problem & parameters. For positive resolved shear choose rate sensitivity m = 0.2 and fixed slip resistance g.
Solution. Evaluate γ̇ = γ̇0(τ/g)^(1/m).
Worked evaluation. Substitute the marked horizontal coordinate, 0.75, into the displayed formula to obtain 0.2373 on the vertical axis. Values are rounded for display.
Scope. Single-system constitutive evaluation; lattice rotation and hardening are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Maxwell viscoelastic model · Example 1
Maxwell stress relaxation
Problem & parameters. Apply a step strain ε₀ to a Maxwell spring–dashpot series element and hold it fixed. Scale stress by Eε₀ and time by η/E.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Exact one-mode reduction with constant coefficients; additional coupled physics is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Kelvin–Voigt model · Example 1
Kelvin–Voigt creep after a stress step
Problem & parameters. Apply constant stress σ₀ at t = 0 to an initially undeformed parallel spring and dashpot.
Solution. Solve ηε′+Eε = σ₀ with zero initial strain.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Standard linear solid · Example 1
Standard-linear-solid relaxation
Problem & parameters. Apply a fixed strain step to a standard linear solid with relaxed modulus E∞ = 0.4E0.
Solution. Its relaxation modulus is E∞+(E0−E∞)exp(−t/τ). Multiply by the imposed strain.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.44925 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Norton creep law · Example 1
Norton creep-rate sensitivity
Problem & parameters. At fixed temperature use Norton exponent n = 3 and reference rate Aσ*³.
Solution. Substitute the stress into ε̇ = Aσ³.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Steady creep constitutive law at fixed material state and temperature.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Linear elastic fracture mechanics (LEFM) · Example 1
Crack-tip opening stress on the forward ray
Problem & parameters. Use the leading mode-I elastic crack-tip field on θ = 0.
Solution. The angular factor equals one on the forward ray. Evaluate KI/√(2πr).
Worked evaluation. Substitute the marked horizontal coordinate, 1.025, into the displayed formula to obtain 0.98773 on the vertical axis. Values are rounded for display.
Scope. Near-tip linear-elastic asymptotic field, outside the process zone; the singular tip itself is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Cohesive-zone model · Example 1
Triangular cohesive traction law
Problem & parameters. Choose peak traction at half the complete-separation opening and linear loading/softening branches.
Solution. Connect (0,0), (δc/2,tmax), and (δc,0). The work of separation is the triangle area tmaxδc/2.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Monotonic prescribed cohesive law; unloading and mixed-mode effects are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Phase-field fracture model · Example 1
One-dimensional AT2 crack profile
Problem & parameters. Minimize the isolated AT2 crack-surface functional with d(0)=1 and d→0 far from the crack, without mechanical driving away from x=0.
Solution. The Euler equation is d−ℓ²d″=0 on each half-line. Select decaying exponentials and enforce symmetry.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Stationary isolated crack-profile benchmark, not a coupled fracture-growth solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Paris fatigue crack-growth law · Example 1
Paris-law crack growth for exponent two
Problem & parameters. Use da/dN=C(ΔK)² and ΔK=Δσ√(πa) with constant stress range and geometry factor one.
Solution. Substitute ΔK to obtain da/dN = CΔσ²πa. Separate variables and apply a(0)=a₀.
Worked evaluation. Substitute the marked horizontal coordinate, 0.75, into the displayed formula to obtain 2.117 on the vertical axis. Values are rounded for display.
Scope. Only within the Paris regime; threshold, instability, and changing geometry are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Miner cumulative damage rule · Example 1
Single-amplitude fatigue damage
Problem & parameters. Apply constant-amplitude cycles with a fixed fatigue life Nf.
Solution. Miner’s sum has one term n/Nf; the conventional failure threshold is D=1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Linear accumulation hypothesis, not a physical guarantee of failure at exactly D=1.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Archard wear model · Example 1
Wear volume versus sliding distance
Problem & parameters. Hold wear coefficient k, normal force W, and hardness H constant.
Solution. Integrate dV/ds = kW/H from zero initial wear.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Steady Archard wear regime with no changes in contact, debris, or material properties.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Newton–Euler rigid-body model · Example 1
Constant-force translation
Problem & parameters. A rigid body starts at rest with constant net force-to-mass ratio 1 m/s² along one axis and zero net torque.
Solution. Newton’s law gives constant acceleration. Integrate twice with zero initial position and velocity.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 3.125 on the vertical axis. Values are rounded for display.
Scope. Single translational degree of freedom; the remaining forces, torques, and rotational motion are set to zero.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Six-degree-of-freedom flight model · Example 1
Constant-force translation
Problem & parameters. A rigid body starts at rest with constant net force-to-mass ratio 1 m/s² along one axis and zero net torque.
Solution. Newton’s law gives constant acceleration. Integrate twice with zero initial position and velocity.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 3.125 on the vertical axis. Values are rounded for display.
Scope. Single translational degree of freedom; the remaining forces, torques, and rotational motion are set to zero.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Lagrangian mechanics · Example 1
One conservative vibration mode
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hamiltonian mechanics · Example 1
One conservative vibration mode
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Mass–spring–damper model · Example 1
One conservative vibration mode
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Modal superposition model · Example 1
One conservative vibration mode
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Multibody dynamics · Example 1
One conservative vibration mode
Problem & parameters. Choose a single unconstrained linear mode with zero damping, initial displacement A, and zero velocity.
Solution. Either force balance or the quadratic energy gives q″+ω²q=0. Apply the initial conditions to select the cosine.
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Exact single harmonic mode; multibody constraints and other modal couplings are absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Duffing oscillator · Example 1
Duffing equilibrium force curve
Problem & parameters. For positive linear and cubic stiffness choose ℓ=√(k/β). Find the force needed to hold a static displacement.
Solution. Set velocity and acceleration to zero in the Duffing equation. Normalize F=kx+βx³.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Static hardening equilibrium curve, not a forced nonlinear transient or resonance calculation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Linear acoustic wave model · Example 1
Linear traveling-wave snapshot
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Transmission-line acoustic model · Example 1
Linear traveling-wave snapshot
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Maxwell electromagnetic model · Example 1
Linear traveling-wave snapshot
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Transmission-line electrical model · Example 1
Linear traveling-wave snapshot
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Elastic seismic-wave model · Example 1
Linear traveling-wave snapshot
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Magnetohydrodynamics (MHD) · Example 1
Linear traveling-wave snapshot
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Finite-difference time-domain (FDTD) · Example 1
Linear traveling-wave snapshot
Problem & parameters. Use a one-dimensional sinusoidal wave in a uniform, lossless linear medium. Plot the normalized field at time zero.
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. An acoustic, electromagnetic, elastic, or linear Alfvén-wave reference as appropriate. For MHD this is the small transverse perturbation of a uniform magnetized equilibrium; for FDTD it is an exact target, not a discretized result.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Helmholtz acoustic model · Example 1
One-dimensional standing acoustic mode
Problem & parameters. Solve p″+k²p=0 with pressure-release endpoints and choose the first nonzero eigenmode.
Solution. Both endpoint conditions select kL=π. Normalize the remaining arbitrary amplitude by its maximum.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Electrostatic Poisson model · Example 1
Uniform-charge potential between grounded planes
Problem & parameters. Solve φ″ = −ρ/ε for constant charge density between φ(0)=φ(L)=0.
Solution. Integrate the constant second derivative twice. The grounded endpoints fix both integration constants.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.125 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Magnetostatic model · Example 1
Magnetic field around a straight wire
Problem & parameters. Consider the exterior of a long straight wire of radius a carrying steady current I in vacuum.
Solution. Ampère’s law around a circle gives 2πrB=μ0I.
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 0.33333 on the vertical axis. Values are rounded for display.
Scope. Exterior field of an ideal long wire; end effects are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Eddy-current model · Example 1
AC skin-depth amplitude
Problem & parameters. A sinusoidal magnetic field penetrates a homogeneous conducting half-space with skin depth δ.
Solution. The diffusion equation at angular frequency ω has a decaying complex solution exp[−(1+i)x/δ]. Its amplitude is exp(−x/δ).
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Linear conductor with constant conductivity and permeability; displacement current neglected.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Magnetic-circuit model · Example 1
Linear magnetic circuit
Problem & parameters. Use a single magnetic circuit of fixed reluctance ℛ with no leakage.
Solution. Solve NI=ℛΦ for flux.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1.5 on the vertical axis. Values are rounded for display.
Scope. Linear unsaturated material and fixed geometry.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Jiles–Atherton hysteresis model · Example 1
Anhysteretic magnetization curve
Problem & parameters. Evaluate the Langevin-form anhysteretic component of a Jiles–Atherton model, using its zero-field limit M=0.
Solution. Insert h=He/a into Ms[coth(h)−1/h]. The apparent singularity is removable; the small-field slope is 1/3.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Anhysteretic reference only, not the history-dependent hysteresis loop.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Geometrical optics · Example 1
Refraction from air into glass
Problem & parameters. A ray crosses a plane interface from index 1 into index 1.5.
Solution. Use Snell’s law n1 sin θ1=n2 sin θ2 and solve for the refracted angle.
Worked evaluation. Substitute the marked horizontal coordinate, 40, into the displayed formula to obtain 25.374 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Scalar diffraction model · Example 1
Single-slit far-field diffraction
Problem & parameters. Illuminate a slit of width a uniformly with monochromatic coherent light and observe the Fraunhofer pattern.
Solution. Integrate the phase factor across the slit to obtain sinc amplitude; square its magnitude. Use the continuous limit I/I0=1 at u=0.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Gaussian beam model · Example 1
Gaussian beam transverse intensity
Problem & parameters. At a fixed axial plane, take a fundamental paraxial Gaussian beam with 1/e² intensity radius w.
Solution. Square the Gaussian field amplitude exp(−r²/w²) to obtain its intensity.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. One transverse cut at a fixed plane; w changes with axial distance.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Drude–Lorentz optical model · Example 1
Lossless Drude dielectric response
Problem & parameters. Take the free-electron Drude limit with zero collision rate, no Lorentz resonances, and background permittivity one.
Solution. Solve the harmonic free-electron displacement equation and insert the induced polarization into D=ε0E+P.
Worked evaluation. Substitute the marked horizontal coordinate, 1.75, into the displayed formula to obtain 0.67347 on the vertical axis. Values are rounded for display.
Scope. Lossless frequency-domain special case; the zero-frequency singular point is excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Lumped RLC circuit model · Example 1
RC charging limit of an RLC circuit
Problem & parameters. Set inductance to zero and apply a voltage step Vs to a series resistor and initially uncharged capacitor.
Solution. Kirchhoff’s law gives RCV′+V=Vs. Solve the first-order initial-value problem.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. RC limiting circuit, not a general second-order RLC transient.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Shockley diode model · Example 1
Ideal diode current
Problem & parameters. Evaluate the Shockley diode law without series resistance or reverse breakdown.
Solution. Substitute the thermal-voltage-scaled bias into the exponential current law.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Ebers–Moll transistor model · Example 1
Forward-active transistor collector current
Problem & parameters. Use the forward-active Ebers–Moll branch and neglect the reverse junction contribution.
Solution. Keep the αFIES[exp(VBE/VT)−1] term and divide by its prefactor.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 6.3891 on the vertical axis. Values are rounded for display.
Scope. Forward-active approximation; no saturation, Early effect, or breakdown.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
MOSFET square-law model · Example 1
Long-channel MOSFET saturation
Problem & parameters. Use a long-channel MOSFET in strong-inversion saturation with constant mobility and no channel-length modulation.
Solution. Set VDS at or above overdrive and integrate the gradual-channel charge relation to obtain ID=β(VGS−Vth)²/2.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 2.25 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
BSIM compact-model family · Example 1
Weak-inversion current benchmark
Problem & parameters. Use an ideal weak-inversion exponential trend at fixed drain bias as a compact-model check.
Solution. A Boltzmann subthreshold charge law gives current proportional to exp(VGS/nVT); normalize at VGS=V*.
Worked evaluation. Substitute the marked horizontal coordinate, -1.5, into the displayed formula to obtain 0.22313 on the vertical axis. Values are rounded for display.
Scope. Asymptotic benchmark only; not the complete BSIM equations or a result from a foundry model card.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Drift–diffusion semiconductor model · Example 1
Uniform-carrier drift current
Problem & parameters. Take uniform electron density n, fixed mobility μ, and a low-field steady state. The density gradient is zero.
Solution. The diffusion contribution vanishes. Evaluate the conventional drift-current magnitude law J=qnμE.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Low-field isothermal drift limit; carrier heating and higher hydrodynamic moments are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hydrodynamic carrier model · Example 1
Uniform-carrier drift current
Problem & parameters. Take uniform electron density n, fixed mobility μ, and a low-field steady state. The density gradient is zero.
Solution. The diffusion contribution vanishes. Evaluate the conventional drift-current magnitude law J=qnμE.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Low-field isothermal drift limit; carrier heating and higher hydrodynamic moments are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Nernst equilibrium potential · Example 1
Equilibrium potential versus activity ratio
Problem & parameters. For Ox+ne− ⇌ Red use ideal specified activities and fixed temperature.
Solution. Set the reaction electrochemical free-energy change to zero and rearrange the Nernst relation.
Worked evaluation. Substitute the marked horizontal coordinate, 5.05, into the displayed formula to obtain 1.6194 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Butler–Volmer kinetics · Example 1
Symmetric Butler–Volmer polarization
Problem & parameters. Set anodic and cathodic transfer coefficients to one half, with one-electron charge convention.
Solution. Subtract the two exponentials in Butler–Volmer to obtain twice the hyperbolic sine.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Tafel approximation · Example 1
Anodic Tafel relation
Problem & parameters. Use the anodic high-overpotential regime where the cathodic exponential is negligible.
Solution. From j≈j0 exp(αFη/RT), take logarithms and solve for overpotential.
Worked evaluation. Substitute the marked horizontal coordinate, 55, into the displayed formula to obtain 4.0073 on the vertical axis. Values are rounded for display.
Scope. Asymptotic approximation, plotted well above j/j0=1; not valid near equilibrium.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Poisson–Nernst–Planck model · Example 1
Screened potential in a dilute electrolyte
Problem & parameters. At zero ionic flux, linearize a symmetric dilute electrolyte near equilibrium next to a planar wall.
Solution. Boltzmann ionic populations linearize Poisson’s equation to φ″=φ/λD². Select the decaying solution and impose the wall potential.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Debye–Hückel equilibrium limit of PNP, requiring |zFφ|≪RT; no driven ionic transport.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Doyle–Fuller–Newman (DFN/P2D) model · Example 1
Spherical-particle average concentration balance
Problem & parameters. Start with a spherical active particle of radius R and mean concentration c*. Impose constant outward molar flux jout.
Solution. Integrate spherical diffusion over particle volume: d(c̄)/dt=−(surface/volume)jout=−3jout/R. Apply the initial average.
Worked evaluation. Substitute the marked horizontal coordinate, 0.125, into the displayed formula to obtain 0.625 on the vertical axis. Values are rounded for display.
Scope. Exact particle mass balance shared by DFN, SPM, and SPMe. It does not give the radial profile, terminal voltage, electrolyte dynamics, or a usable-capacity prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Single-particle battery model (SPM) · Example 1
Spherical-particle average concentration balance
Problem & parameters. Start with a spherical active particle of radius R and mean concentration c*. Impose constant outward molar flux jout.
Solution. Integrate spherical diffusion over particle volume: d(c̄)/dt=−(surface/volume)jout=−3jout/R. Apply the initial average.
Worked evaluation. Substitute the marked horizontal coordinate, 0.125, into the displayed formula to obtain 0.625 on the vertical axis. Values are rounded for display.
Scope. Exact particle mass balance shared by DFN, SPM, and SPMe. It does not give the radial profile, terminal voltage, electrolyte dynamics, or a usable-capacity prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Single-particle model with electrolyte (SPMe) · Example 1
Spherical-particle average concentration balance
Problem & parameters. Start with a spherical active particle of radius R and mean concentration c*. Impose constant outward molar flux jout.
Solution. Integrate spherical diffusion over particle volume: d(c̄)/dt=−(surface/volume)jout=−3jout/R. Apply the initial average.
Worked evaluation. Substitute the marked horizontal coordinate, 0.125, into the displayed formula to obtain 0.625 on the vertical axis. Values are rounded for display.
Scope. Exact particle mass balance shared by DFN, SPM, and SPMe. It does not give the radial profile, terminal voltage, electrolyte dynamics, or a usable-capacity prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Equivalent-circuit battery model · Example 1
Battery polarization under a current step
Problem & parameters. Apply a constant current I to an initially relaxed single-RC battery polarization branch.
Solution. Solve CpVp′+Vp/Rp=I. The terminal-voltage drop also includes any separate series ohmic resistance.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. One branch with fixed parameters; state of charge and open-circuit voltage are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Darcy porous-flow model · Example 1
Darcy flux versus pressure gradient
Problem & parameters. Let G=−dp/dx be positive, and hold permeability k and viscosity μ constant.
Solution. Solve μu/k = G.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 1.5 on the vertical axis. Values are rounded for display.
Scope. Single-phase creeping flow in a homogeneous porous medium.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Brinkman porous-flow model · Example 1
Brinkman flow between porous walls
Problem & parameters. Solve μe u″−μu/k+G=0 between no-slip walls ±H. Choose screening length ℓ=√(μe k/μ) and H/ℓ=2.
Solution. Add a constant particular solution kG/μ to the symmetric cosh homogeneous solution, then enforce the wall values.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.7342 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Forchheimer model · Example 1
Forchheimer inertial pressure loss
Problem & parameters. Choose velocity and gradient scales so that the linear and quadratic drag coefficients are both one.
Solution. Substitute positive velocity into G=av+bv|v| and apply the chosen scaling.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 3.75 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Richards equation · Example 1
Linearized unsaturated-head relaxation
Problem & parameters. Linearize moisture capacity and hydraulic conductivity about a uniform reference state, neglect gravity, and solve the resulting diffusion equation on a slab.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Constant-coefficient linearization of Richards’ equation. The nonlinear retention and conductivity changes are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
van Genuchten retention model · Example 1
van Genuchten water retention
Problem & parameters. Choose n=2 and m=1−1/n=1/2 for a drying retention curve.
Solution. Insert the chosen parameters into Se=[1+(α|h|)^n]^−m.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.37139 on the vertical axis. Values are rounded for display.
Scope. Retention relation only; hysteresis and conductivity are not evaluated.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Biot poroelasticity · Example 1
A single consolidation pressure mode
Problem & parameters. Use one-dimensional linear consolidation with drained ends and an initial excess pore-pressure mode sin(πx/L). Plot cvt/L²=0.1.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact single-mode Terzaghi solution and a compatible one-dimensional poroelastic reduction; not an arbitrary initial loading history.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Terzaghi consolidation model · Example 1
A single consolidation pressure mode
Problem & parameters. Use one-dimensional linear consolidation with drained ends and an initial excess pore-pressure mode sin(πx/L). Plot cvt/L²=0.1.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Exact single-mode Terzaghi solution and a compatible one-dimensional poroelastic reduction; not an arbitrary initial loading history.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Modified Cam-Clay model · Example 1
Modified Cam-Clay yield ellipse
Problem & parameters. Hold preconsolidation pressure pc and critical-state slope M fixed. Plot the compression-positive yield locus.
Solution. Solve q²+M²p(p−pc)=0 for the nonnegative q branch.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. Yield-surface geometry only; hardening and stress-path evolution are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Saint-Venant shallow-water model · Example 1
Linear shallow-water surface wave
Problem & parameters. Linearize shallow-water dynamics about rest at constant depth H; the wave speed is √(gH).
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. Small free-surface perturbation in a constant-depth channel, without friction or dispersion.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Kinematic-wave routing · Example 1
Constant-speed routing pulse
Problem & parameters. Use the linear routing equation ht+hx=0 with initial Gaussian pulse exp(−x²).
Solution. The pulse is constant along characteristics x−t, hence it translates without changing shape.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Constant-celerity reduction; nonlinear depth-dependent routing can distort or steepen the pulse.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Rainfall–runoff model · Example 1
Linear-reservoir recession
Problem & parameters. After rainfall stops, let storage S obey S′=−S/K and outflow Q=S/K. Normalize either by its initial value.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-reservoir rainfall–runoff component; no new rain, infiltration, or additional routing stores.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Advection–dispersion groundwater model · Example 1
Dispersing tracer plume
Problem & parameters. On an infinite line take velocity one, dispersion coefficient 0.1, and initial concentration exp(−x²).
Solution. Translate the Gaussian by vt and broaden its squared width to 1+4Dt, adjusting amplitude to conserve mass.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 0.84515 on the vertical axis. Values are rounded for display.
Scope. Homogeneous advection–dispersion with no reactions or sorption.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Numerical weather prediction · Example 1
Isothermal hydrostatic atmosphere
Problem & parameters. Use an ideal gas at constant temperature and constant gravity, with density ρ0 at height zero.
Solution. Combine dp/dz=−ρg with p=ρRsT; integrate dρ/dz=−ρ/H, H=RsT/g.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Hydrostatic column benchmark only. For stellar structure this approximates a thin isothermal layer, not an entire star; radiation, convection, and dynamics are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
General circulation model (GCM) · Example 1
Isothermal hydrostatic atmosphere
Problem & parameters. Use an ideal gas at constant temperature and constant gravity, with density ρ0 at height zero.
Solution. Combine dp/dz=−ρg with p=ρRsT; integrate dρ/dz=−ρ/H, H=RsT/g.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Hydrostatic column benchmark only. For stellar structure this approximates a thin isothermal layer, not an entire star; radiation, convection, and dynamics are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Stellar structure model · Example 1
Isothermal hydrostatic atmosphere
Problem & parameters. Use an ideal gas at constant temperature and constant gravity, with density ρ0 at height zero.
Solution. Combine dp/dz=−ρg with p=ρRsT; integrate dρ/dz=−ρ/H, H=RsT/g.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Hydrostatic column benchmark only. For stellar structure this approximates a thin isothermal layer, not an entire star; radiation, convection, and dynamics are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Earth system model (ESM) · Example 1
One-box climate response to a forcing step
Problem & parameters. For a constant radiative-forcing step F, use CΔT′=F−λΔT with positive linear feedback parameter λ and initially zero anomaly.
Solution. Apply an integrating factor to the one-box energy balance; the equilibrium anomaly is F/λ.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Reduced global-mean energy balance. For ESM this is an illustrative diagnostic reduction, not a full Earth-system forecast.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Energy-balance climate model · Example 1
One-box climate response to a forcing step
Problem & parameters. For a constant radiative-forcing step F, use CΔT′=F−λΔT with positive linear feedback parameter λ and initially zero anomaly.
Solution. Apply an integrating factor to the one-box energy balance; the equilibrium anomaly is F/λ.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Reduced global-mean energy balance. For ESM this is an illustrative diagnostic reduction, not a full Earth-system forecast.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Ocean circulation model · Example 1
Linear barotropic ocean-wave reference
Problem & parameters. Use a constant-depth, nonrotating, inviscid shallow-water reduction of ocean circulation.
Solution. A sinusoidal traveling-wave solution is u = sin[2π(ξ−τ)]. Set τ = 0 to obtain the plotted snapshot.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. Single linear barotropic mode; rotation, stratification, mixing, and realistic boundaries are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Sea-ice thermodynamic-dynamic model · Example 1
Conduction-limited ice growth
Problem & parameters. Assume zero initial thickness, fixed surface-to-freezing temperature difference ΔT, and conductive flux kΔT/h through the ice.
Solution. Balance latent heat: ρLh′=kΔT/h. Integrate h²=2kΔTt/(ρL) and choose t*=ρLℓ²/(2kΔT).
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 1.4142 on the vertical axis. Values are rounded for display.
Scope. Stefan growth limit with no ocean heat flux, snow insulation, or ice dynamics.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Lifting-line model · Example 1
Finite-wing lift slope
Problem & parameters. Use lifting-line theory for an ideal elliptically loaded wing of aspect ratio eight and two-dimensional slope 2π per radian.
Solution. The induced angle reduces the effective angle. Solve CL=a0[α−CL/(πeAR)] for CL.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Small-angle attached-flow approximation; no stall prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Blade-element momentum model · Example 1
Ideal actuator-disk power
Problem & parameters. Use the ideal nonrotating actuator-disk limit underlying axial momentum theory.
Solution. Mass, momentum, and energy balances give the displayed coefficient. Differentiating yields a maximum 16/27 at a=1/3.
Worked evaluation. Substitute the marked horizontal coordinate, 0.25, into the displayed formula to obtain 0.5625 on the vertical axis. Values are rounded for display.
Scope. Momentum-theory benchmark for BEM; blade geometry, swirl, drag, tip losses, and high-induction corrections are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Bicycle vehicle model · Example 1
Steady kinematic bicycle turning
Problem & parameters. Assume low-speed rolling without tire slip for a vehicle of wheelbase L.
Solution. The front-wheel geometry gives turn radius R=L/tanδ; curvature is 1/R.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Kinematic limit, not a high-speed dynamic tire-force model.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Quarter-car suspension model · Example 1
Wheel-hop-free suspension mode
Problem & parameters. Hold the unsprung mass fixed, set damping to zero, and release the sprung mass from displacement A.
Solution. The reduced quarter-car equation is ms z″+ks z=0; ω=√(ks/ms).
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Single-mode constrained reduction, not the full two-degree-of-freedom road response.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Pacejka tire model · Example 1
Illustrative Magic Formula tire force
Problem & parameters. Choose B=10, C=1.3, E=0, zero offsets, and fixed load in the basic Pacejka Magic Formula.
Solution. With E=0 the curvature correction drops out. Evaluate the sine of the scaled arctangent.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Illustrative coefficients, not a calibrated tire or a combined-slip model.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
AC power-flow model · Example 1
Lossless power-angle relation
Problem & parameters. Use two fixed voltage magnitudes connected by a purely reactive line; for a generator use the analogous fixed internal-voltage coupling.
Solution. The lossless AC circuit gives P=(V1V2/X)sinδ.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Steady electrical-power term; the swing-equation rotor transient and voltage dynamics are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Swing-equation generator model · Example 1
Lossless power-angle relation
Problem & parameters. Use two fixed voltage magnitudes connected by a purely reactive line; for a generator use the analogous fixed internal-voltage coupling.
Solution. The lossless AC circuit gives P=(V1V2/X)sinδ.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. Steady electrical-power term; the swing-equation rotor transient and voltage dynamics are not solved.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
DC power-flow approximation · Example 1
Small-angle DC power flow
Problem & parameters. Use nearly equal fixed bus voltage magnitudes, negligible resistance, and small angle difference.
Solution. Linearize sinδ≈δ in the lossless AC transfer formula.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. DC power-flow approximation; it does not calculate reactive power or voltage magnitudes.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
State-space model · Example 1
First-order unit-step response
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Transfer-function model · Example 1
First-order unit-step response
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Bond-graph model · Example 1
First-order unit-step response
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Electrical analog model · Example 1
First-order unit-step response
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hardware-in-the-loop model · Example 1
First-order unit-step response
Problem & parameters. Use the scalar state equation y′+y=1 with y(0)=0, or transfer function 1/(s+1).
Solution. The homogeneous response is Ce^−τ and the constant particular response is one. The initial state gives C=−1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Exact linear plant reference. For bond graphs/electrical analogs use a single storage-and-resistance element; for HIL this is a reference trajectory, not measured hardware data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hybrid dynamical model · Example 1
Bouncing-ball flight and one impact
Problem & parameters. Drop a ball from 1 m with g=9.81 m/s². At first ground contact reverse velocity and multiply its magnitude by restitution e=0.8. Plot before the second impact.
Solution. The first impact occurs at ti=√(2/g). Integrate constant gravity before and after the velocity reset with continuous height.
Worked evaluation. Substitute the marked horizontal coordinate, 0.55, into the displayed formula to obtain 0.30139 on the vertical axis. Values are rounded for display.
Scope. Ideal instantaneous first bounce; air resistance and contact deformation are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
System-dynamics stock-flow model · Example 1
Stock with constant inflow and linear outflow
Problem & parameters. An initially empty stock receives constant inflow q and drains at rate kS.
Solution. Solve S′=q−kS with S(0)=0.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Single stock, constant coefficients, and no delays or saturation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Discrete-event simulation · Example 1
Deterministic event accumulation
Problem & parameters. Identical events occur at Δt,2Δt,… with zero events completed at t=0.
Solution. Count the positive integer multiples of Δt not exceeding t. The floor function gives the exact event count.
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 3 on the vertical axis. Values are rounded for display.
Scope. Simple scheduled-event benchmark; a discrete-event model need not have periodic arrivals.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Agent-based physical-system model · Example 1
Mean position of independent moving agents
Problem & parameters. Agents start at mean position zero, have constant mean velocity 1 m/s, and do not interact.
Solution. Each agent has x=x0+vt. Average this relation over agents; the mean is linear in time.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Noninteracting kinematic benchmark; not an emergent many-agent simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Markov state model · Example 1
Two-state continuous-time occupation
Problem & parameters. Two states exchange population at equal rate k. Initially all probability is in state one.
Solution. Use P2=1−P1 in P1′=−kP1+kP2. Solve the resulting first-order equation.
Worked evaluation. Substitute the marked horizontal coordinate, 2, into the displayed formula to obtain 0.50916 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Kalman state estimator · Example 1
Kalman gain versus measurement noise
Problem & parameters. For one scalar measurement with observation coefficient one, hold the positive prior variance fixed.
Solution. Insert H=1 into K=P−H/(H²P−+R).
Worked evaluation. Substitute the marked horizontal coordinate, 5, into the displayed formula to obtain 0.16667 on the vertical axis. Values are rounded for display.
Scope. Single measurement update; not a full dynamic filter trajectory.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Model predictive control · Example 1
One-step unconstrained predictive control
Problem & parameters. Let xnext=x+u and minimize (x+u)²+ρu² with ρ=1 and no constraints.
Solution. Differentiate the quadratic cost with respect to u, set 2(x+u)+2ρu=0, and solve.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain -0 on the vertical axis. Values are rounded for display.
Scope. Analytical horizon-one MPC example; longer horizons and constraints change the feedback law.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hodgkin–Huxley membrane model · Example 1
Passive membrane voltage relaxation
Problem & parameters. Set sodium and potassium conductances to zero, hold leak reversal potential EL fixed, and normalize V−EL by its initial value. Use τ=gLt/Cm.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Passive leak-only reduction of Hodgkin–Huxley; action potentials and voltage-dependent gates are deliberately excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
FitzHugh–Nagumo model · Example 1
FitzHugh–Nagumo voltage nullcline
Problem & parameters. For v′=v−v³/3−w+I set I=0 and find the zero-fast-derivative curve.
Solution. Set v′=0 and solve for w.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0 on the vertical axis. Values are rounded for display.
Scope. A phase-plane nullcline, not a trajectory or the complete system equilibrium; equilibria also lie on the recovery nullcline.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hill muscle model · Example 1
Hill force–velocity curve
Problem & parameters. Use (F+a)(v+b)=(F0+a)b, with a/F0=0.25 and vmax=bF0/a.
Solution. Solve the hyperbolic force–velocity equation for F and substitute the normalized speed.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.16667 on the vertical axis. Values are rounded for display.
Scope. Steady concentric shortening only; activation and length effects are held fixed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Windkessel circulation model · Example 1
Windkessel diastolic pressure decay
Problem & parameters. With zero inflow, a two-element Windkessel discharges through resistance R from compliance C. Use τ=t/(RC) and normalize pressure above venous pressure.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Constant-compliance diastolic interval, not a full pulsatile cardiac cycle.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Pennes bioheat model · Example 1
Uniform tissue heating with perfusion
Problem & parameters. Take spatially uniform tissue with constant heat source Q, blood heat-exchange coefficient W>0, and initial tissue temperature equal to arterial temperature Ta.
Solution. The Pennes balance reduces to ρc T′=Q−W(T−Ta). Solve the linear initial-value problem.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.91792 on the vertical axis. Values are rounded for display.
Scope. Uniform-temperature reduction; no spatial conduction, temperature-dependent perfusion, or safety prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Reaction–diffusion morphogenesis model · Example 1
One linear reaction–diffusion mode
Problem & parameters. Solve ut=uxx−u with zero ends and initial sin(πx), then plot t=1.
Solution. The Laplacian and decay each multiply the mode by a negative constant; its amplitude solves A′=−(π²+1)A.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 1.9028e-05 on the vertical axis. Values are rounded for display.
Scope. One-species linear stable subproblem, not a two-species Turing pattern or nonlinear morphogenesis prediction.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Monod growth model · Example 1
Monod nutrient limitation
Problem & parameters. Evaluate growth rate at prescribed substrate concentration with fixed Monod parameters.
Solution. Normalize μ=μmax S/(Ks+S) by μmax and substitute S/Ks.
Worked evaluation. Substitute the marked horizontal coordinate, 4, into the displayed formula to obtain 0.8 on the vertical axis. Values are rounded for display.
Scope. Growth-rate relation only; substrate depletion and biomass evolution are not integrated.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Physiologically based compartment model · Example 1
Single well-mixed compartment washout
Problem & parameters. After an initial dose, use one well-mixed compartment with first-order elimination, no further input, and τ=kt.
Solution. Separate dy/dτ = −y, integrate ln(y) = −τ + C, and apply y(0) = 1.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. One-compartment limiting case; interorgan exchange, binding, and nonlinear metabolism are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Boltzmann kinetic equation · Example 1
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Vlasov–Poisson model · Example 1
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Vlasov–Maxwell model · Example 1
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Direct simulation Monte Carlo (DSMC) · Example 1
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Particle-in-cell (PIC) · Example 1
Homogeneous Maxwellian velocity marginal
Problem & parameters. Take a spatially uniform equilibrium with zero drift and the normalized Gaussian velocity marginal. For collisionless plasma use zero fields and a neutralizing background.
Solution. The homogeneous force-free streaming terms vanish. Maxwellian collisions balance for Boltzmann equilibrium; integrating the Gaussian fixes its normalization.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Equilibrium distribution or exact kinetic benchmark. DSMC and PIC would estimate it using particles; this plot is not a finite-particle sample.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Neutron diffusion approximation · Example 1
Subcritical neutron-density diffusion mode
Problem & parameters. On a slab solve nτ=nξξ with zero extrapolated-end values and initial sin(πξ), ignoring reactions in this illustrative diffusion subproblem.
Solution. The sine satisfies both zero end values. Its second derivative is −π² times itself; the amplitude solves a′ = −π²a.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.37271 on the vertical axis. Values are rounded for display.
Scope. Diffusion-only benchmark; absorption and fission terms would modify the mode growth/decay rate.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Point reactor kinetics · Example 1
Prompt-only subcritical neutron decay
Problem & parameters. Set delayed-neutron fraction and external source to zero, take constant negative reactivity ρ, and prompt generation time Λ.
Solution. Point kinetics reduces to n′=(ρ/Λ)n. Integrate with n(0)=n0.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Prompt-only idealization, not a realistic startup, shutdown, or reactor-safety calculation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Bateman decay-chain model · Example 1
Daughter buildup in a two-step decay chain
Problem & parameters. Initially N1=N10 and N2=0. Let the parent decay to the daughter with λ2=2λ1 and unit branching fraction.
Solution. Solve N1=N10exp(−λ1t). Insert into N2′+λ2N2=λ1N1 and integrate using an integrating factor.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.075347 on the vertical axis. Values are rounded for display.
Scope. Analytical solution or constitutive evaluation for the stated special case. It is not a general solution of the full model family.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Newtonian gravitational N-body model · Example 1
Circular two-body orbital coordinate
Problem & parameters. Reduce an isolated gravitational system to two point masses with total mass M in a circular relative orbit of radius a.
Solution. Balance relative centripetal acceleration n²a against GM/a². The Cartesian x coordinate then follows a cosine.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Exact two-body circular orbit, not a general many-body solution; a one-coordinate time trace is shown.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
General relativity model · Example 1
Gravitational time dilation outside a sphere
Problem & parameters. For a stationary observer outside a nonrotating spherical mass, compare proper time with Schwarzschild coordinate time at infinity.
Solution. Set spatial coordinate increments to zero in the Schwarzschild line element and take the square root of its time coefficient.
Worked evaluation. Substitute the marked horizontal coordinate, 4.525, into the displayed formula to obtain 0.88261 on the vertical axis. Values are rounded for display.
Scope. Exterior vacuum Schwarzschild solution, r>rs. A static observer cannot remain at the horizon.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
FLRW cosmological model · Example 1
Matter-dominated cosmic expansion
Problem & parameters. Take a spatially flat FLRW universe with pressureless matter only and zero cosmological constant.
Solution. Mass conservation gives ρ∝a^−3. Friedmann’s equation then gives ȧ∝a^−1/2; integrate from the big-bang branch.
Worked evaluation. Substitute the marked horizontal coordinate, 1.51, into the displayed formula to obtain 1.3162 on the vertical axis. Values are rounded for display.
Scope. Matter-only special case, not a fit to the present universe.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Smoothed particle hydrodynamics (SPH) · Example 1
Normalized SPH kernel section
Problem & parameters. Evaluate the standard one-dimensional cubic-spline smoothing kernel of support radius 2h.
Solution. Use q=|x|/h, apply the inner and outer polynomial branches, and normalize their integral to one.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 0.66667 on the vertical axis. Values are rounded for display.
Scope. Kernel evaluation, not a complete SPH flow or solid simulation.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Discrete element method (DEM) · Example 1
Elastic DEM contact
Problem & parameters. Choose a linear frictionless normal-contact spring with stiffness k, no damping, and positive overlap.
Solution. The prescribed contact law is F=kδ. Before contact, F=0.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.5 on the vertical axis. Values are rounded for display.
Scope. One elastic contact contribution; many-particle dynamics and tangential friction are excluded.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Lattice Boltzmann method (LBM) · Example 1
Uniform LBM shear-mode decay target
Problem & parameters. Use a small-amplitude periodic transverse shear wave in the low-Mach hydrodynamic limit; plot νt/L²=0.02.
Solution. The continuum transverse velocity obeys diffusion. Its wave number 2π/L fixes the exponential decay rate.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 5.5604e-17 on the vertical axis. Values are rounded for display.
Scope. Exact continuum benchmark for LBM, not a finite-lattice prediction; compressibility and lattice errors must be checked separately.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Material point method (MPM) · Example 1
Uniform bar extension benchmark
Problem & parameters. Apply a uniform small axial strain of 0.01 to a homogeneous elastic bar.
Solution. Integrate du/dx=0.01 with u(0)=0.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.005 on the vertical axis. Values are rounded for display.
Scope. Exact continuum target for MPM; grid transfer and particle quadrature errors are not represented.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Homogenization · Example 1
Parallel-layer effective modulus
Problem & parameters. Two perfectly bonded parallel axial bars share the same strain, with modulus ratio E2/E1=4.
Solution. Average stress is [(1−f)E1+fE2] times the common strain. Divide by strain to obtain the effective axial modulus.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Exact iso-strain parallel-bar construction; generally an upper-bound estimate for other microstructures.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
QM/MM coupling · Example 1
Coupled-region harmonic reference
Problem & parameters. As a consistency check, choose a common harmonic coordinate whose total coupled-region energy is kq²/2 and whose effective mass is m.
Solution. The total force is −kq. Solve m q″+kq=0 with q(0)=A and q′(0)=0.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Prescribed harmonic reference only; no electronic calculation, interface force transfer, or adaptive region simulation is performed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Atomistic–continuum coupling · Example 1
Coupled-region harmonic reference
Problem & parameters. As a consistency check, choose a common harmonic coordinate whose total coupled-region energy is kq²/2 and whose effective mass is m.
Solution. The total force is −kq. Solve m q″+kq=0 with q(0)=A and q′(0)=0.
Worked evaluation. Substitute the marked horizontal coordinate, 3.1416, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Prescribed harmonic reference only; no electronic calculation, interface force transfer, or adaptive region simulation is performed.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Fluid–structure interaction (FSI) · Example 1
Added-mass structural oscillation
Problem & parameters. Approximate fluid loading as a constant added mass ma=m on an undamped spring-supported body.
Solution. Combine the masses: (m+ma)q″+kq=0. The frequency becomes √[k/(m+ma)].
Worked evaluation. Substitute the marked horizontal coordinate, 6.2832, into the displayed formula to obtain -0.26626 on the vertical axis. Values are rounded for display.
Scope. Linear added-mass reduction of FSI; no viscous drag, free-surface, or flow-field solution.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Thermomechanical coupling · Example 1
Fully restrained thermal expansion
Problem & parameters. A one-dimensional elastic bar is prevented from expanding while its temperature rises uniformly.
Solution. Total strain is σ/E+αΔT. Set it to zero and solve for stress.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain -1 on the vertical axis. Values are rounded for display.
Scope. Small-strain constant-property axial model, with tension positive; uniform heating produces compression.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Proper orthogonal decomposition (POD) · Example 1
Exactly rank-one snapshot reconstruction
Problem & parameters. All snapshots are scalar multiples of sin(πx). Reconstruct the snapshot at dimensionless time one using one POD mode.
Solution. The snapshot matrix has rank one. Its only nonzero spatial mode is proportional to sin(πx), with coefficient exp(−t).
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.36788 on the vertical axis. Values are rounded for display.
Scope. Exact rank-one constructed data set; real POD truncation can incur substantial error.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Gaussian-process surrogate · Example 1
One-observation Gaussian-process posterior mean
Problem & parameters. Use a zero-mean, unit-variance squared-exponential GP, unit length scale, and one noiseless observation y(0)=1.
Solution. The one-by-one training covariance is one. The conditional mean k(x,0)K^−1y equals exp(−x²/2).
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Analytical posterior mean under the stated kernel; it is not a physical law, and posterior uncertainty is not shown.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Digital twin framework · Example 1
Digital-twin reference cooling trajectory
Problem & parameters. Use a lumped thermal model with a known constant cooling time as an ideal reference for a thermal digital twin.
Solution. Solve the first-order heat balance analytically; compare actual sensor data with this reference in a real implementation.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 0.082085 on the vertical axis. Values are rounded for display.
Scope. Reference physics only. No sensors, online updates, or actual equipment measurements are included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Bayesian model calibration · Example 1
Gaussian conjugate posterior density
Problem & parameters. Use prior θ~Normal(0,1) and one measurement y=1 with independent Normal(0,1) measurement noise.
Solution. Add prior and data precisions to obtain variance 1/2; precision-weight the means to obtain posterior mean 1/2. Normalize the Gaussian.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.56419 on the vertical axis. Values are rounded for display.
Scope. Exact conjugate scalar calibration example; not a calibrated engineering system.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Polynomial chaos expansion · Example 1
First-order polynomial chaos response
Problem & parameters. Use a linear response to a uniform random input. Expand in the first two Legendre polynomials.
Solution. Since P0=1 and P1=ξ, coefficients are 2 and 0.5. The mean is 2 and variance is 0.25/3.
Worked evaluation. Substitute the marked horizontal coordinate, 0, into the displayed formula to obtain 2 on the vertical axis. Values are rounded for display.
Scope. Exact degree-one expansion for the chosen response, not a surrogate fitted to arbitrary simulation data.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Geometrically scaled physical model · Example 1
Geometric volume scaling
Problem & parameters. Scale all dimensions of a shape by the same positive length ratio.
Solution. Volume is the product of three lengths; multiply the three identical scale factors.
Worked evaluation. Substitute the marked horizontal coordinate, 0.5, into the displayed formula to obtain 0.125 on the vertical axis. Values are rounded for display.
Scope. Geometric similarity alone does not ensure force, material, or dynamic similarity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Wind-tunnel model · Example 1
Dynamic pressure in a wind-tunnel test
Problem & parameters. Use fixed air density and reference dynamic pressure q*=ρU*²/2.
Solution. Evaluate q=ρU²/2 and divide by q*.
Worked evaluation. Substitute the marked horizontal coordinate, 1, into the displayed formula to obtain 1 on the vertical axis. Values are rounded for display.
Scope. Test-planning relation, not measured wind-tunnel data; Reynolds and Mach similarity require separate checks.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hydraulic flume model · Example 1
Froude-similar velocity scaling
Problem & parameters. Use the same gravitational acceleration and match Froude number U/√(gL) between a model and prototype.
Solution. Equate the two Froude numbers and solve for the velocity ratio.
Worked evaluation. Substitute the marked horizontal coordinate, 0.505, into the displayed formula to obtain 0.71063 on the vertical axis. Values are rounded for display.
Scope. Gravity-dominated similarity appropriate to free-surface flumes; Reynolds, Weber, and other dimensionless groups may not also match.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Dimensional-analysis similarity model · Example 1
Froude-similar velocity scaling
Problem & parameters. Use the same gravitational acceleration and match Froude number U/√(gL) between a model and prototype.
Solution. Equate the two Froude numbers and solve for the velocity ratio.
Worked evaluation. Substitute the marked horizontal coordinate, 0.505, into the displayed formula to obtain 0.71063 on the vertical axis. Values are rounded for display.
Scope. Gravity-dominated similarity appropriate to free-surface flumes; Reynolds, Weber, and other dimensionless groups may not also match.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Shake-table structural model · Example 1
Undamped shake-table reference transfer
Problem & parameters. For an undamped single-degree-of-freedom oscillator with sinusoidal base motion, calculate the steady absolute displacement below resonance.
Solution. Insert harmonic motions into mX″+k(X−Y)=0. Solve (k−mΩ²)X=kY for the amplitude ratio.
Worked evaluation. Substitute the marked horizontal coordinate, 0.4, into the displayed formula to obtain 1.1905 on the vertical axis. Values are rounded for display.
Scope. Ideal steady reference, not shake-table measurements; the undamped resonance singularity is outside the plotted range.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Photoelastic model · Example 1
Photoelastic fringe order
Problem & parameters. Use a transparent specimen of thickness t, stress-optic coefficient C, and monochromatic wavelength λ.
Solution. The principal refractive-index difference is CΔσ. Optical path retardation is CtΔσ; divide by wavelength to obtain fringe order.
Worked evaluation. Substitute the marked horizontal coordinate, 2.5, into the displayed formula to obtain 2.5 on the vertical axis. Values are rounded for display.
Scope. Uniform stress through thickness and linear stress-optic law; this is not a fringe photograph.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Ornstein-Zernike equation · Example 1
OZ structure factor with prescribed direct correlation
Problem & parameters. Assume rho times the Fourier-transformed direct correlation is −exp[−(kℓ)²]. Find S(k) from the OZ relation.
Solution. Fourier transformation gives h_hat=c_hat/(1−rho c_hat). Substitute into S=1+rho h_hat to obtain the displayed expression. At k=0, S=1/2; at large k, S tends to 1.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 0.90465 on the vertical axis. Values are rounded for display.
Scope. A prescribed-correlation algebraic benchmark, not a self-consistent closure solution or measured scattering spectrum.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Percus-Yevick closure · Example 1
Percus-Yevick hard-sphere contact value
Problem & parameters. Use the analytical three-dimensional, monodisperse hard-sphere PY solution to evaluate its contact pair distribution as packing fraction varies.
Solution. The PY hard-sphere solution gives virial-route Z=(1+2φ+3φ²)/(1−φ)². The hard-sphere contact theorem Z=1+4φg(σ+) then gives g(σ+)=(1+φ/2)/(1−φ)² after subtraction and cancellation. At φ=0 use the limit g=1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.225, into the displayed formula to obtain 1.8522 on the vertical axis. Values are rounded for display.
Scope. Contact-value evaluation of the PY approximation; the analytical OZ/PY solution is taken as the starting result. Thermodynamic routes are not identical.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Hypernetted-chain (HNC) closure · Example 1
Dilute HNC Gaussian-core pair distribution
Problem & parameters. Take the zero-density limit of an equilibrium soft Gaussian-core fluid with beta epsilon=1. Find its pair distribution.
Solution. As density tends to zero, OZ gives h=c and hence gamma=0. HNC reduces to the two-particle Boltzmann factor exp(−beta u); insert the specified Gaussian repulsion. At r=0, g=exp(−1).
Worked evaluation. Substitute the marked horizontal coordinate, 1.5, into the displayed formula to obtain 0.89997 on the vertical axis. Values are rounded for display.
Scope. Exact dilute two-particle limit for this specified potential; at finite liquid density, solve the coupled HNC/OZ equations instead.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Carnahan-Starling hard-sphere equation of state · Example 1
Hard-sphere pressure amplification
Problem & parameters. For a monodisperse hard-sphere fluid, compute pressure relative to ideal-gas pressure from packing fraction using Carnahan-Starling.
Solution. Insert φ into the numerator and denominator. For example, at φ=0.3 the numerator is 1.363 and denominator 0.343, giving Z=3.97376. The dilute limit is Z=1.
Worked evaluation. Substitute the marked horizontal coordinate, 0.225, into the displayed formula to obtain 2.716 on the vertical axis. Values are rounded for display.
Scope. Constitutive evaluation of the approximate fluid EOS; no attractive forces, mixture effects, or solid phase are included.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Stokes-Einstein diffusion relation · Example 1
Brownian sphere diffusion in a viscous solvent
Problem & parameters. Take T=298 K and solvent viscosity eta=0.001 Pa s. Estimate D for dilute no-slip spherical probes with radii between 10 and 200 nm.
Solution. Convert radius from nm to m and substitute into Stokes-Einstein. At R=100 nm, D=2.18273×10⁻¹² m²/s. Doubling radius halves diffusivity.
Worked evaluation. Substitute the marked horizontal coordinate, 105, into the displayed formula to obtain 2.0788e-12 on the vertical axis. Values are rounded for display.
Scope. Chosen constant solvent viscosity, not a measured water-property curve. Continuum, no-slip, dilute-sphere assumptions apply.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Green-Kubo viscosity relation · Example 1
Viscosity integral for an exponential stress correlation
Problem & parameters. Assume the equilibrium intensive shear-pressure autocorrelation C(t)=C0 exp(−t/τ), with C0>0. Calculate the running Green-Kubo viscosity integral.
Solution. Integrate C0 exp(−s/τ) from 0 to t to obtain C0τ[1−exp(−t/τ)]. Multiply by V/(kBT) and divide by its infinite-time limit. At t=3τ, 95.0213% of the assumed total is recovered.
Worked evaluation. Substitute the marked horizontal coordinate, 3, into the displayed formula to obtain 0.95021 on the vertical axis. Values are rounded for display.
Scope. Analytical exponential-correlation benchmark; real liquid stress correlations may oscillate or have long tails. This is not a molecular-dynamics measurement.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Einstein crystal heat-capacity model · Example 1
Einstein oscillator heat capacity
Problem & parameters. For 3N identical oscillators, calculate the normalized heat capacity versus temperature.
Solution. Each oscillator has thermal energy hbar omega/[exp(hbar omega/kBT)−1]. Differentiate and divide the total by 3NkB. At T=ThetaE, the result is e/(e−1)²=0.920674.
Worked evaluation. Substitute the marked horizontal coordinate, 1.05, into the displayed formula to obtain 0.92772 on the vertical axis. Values are rounded for display.
Scope. Exact evaluation within the single-frequency harmonic Einstein model; the acoustic low-temperature cubic law is absent.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Debye phonon model · Example 1
Debye low-temperature cubic law
Problem & parameters. In the regime T much smaller than ThetaD, estimate lattice heat capacity using the leading Debye asymptote.
Solution. Extend the Debye integral upper limit to infinity. Its value is 4pi⁴/15, so multiplying by 9(T/ThetaD)³ gives 12pi⁴(T/ThetaD)³/5. Doubling temperature multiplies this leading term by eight.
Worked evaluation. Substitute the marked horizontal coordinate, 0.0275, into the displayed formula to obtain 0.0048619 on the vertical axis. Values are rounded for display.
Scope. Low-temperature analytical asymptote only; the plotted range stops at T/ThetaD=0.05. Use the finite-cutoff integral outside this regime.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Sommerfeld free-electron model · Example 1
Sommerfeld electronic heat capacity
Problem & parameters. Find the leading electronic heat capacity of a three-dimensional free-electron gas at fixed electron number and low temperature.
Solution. The fixed-number Sommerfeld expansion gives U/N=(3/5)EF+(pi²/4)(kBT)²/EF to this order. Differentiate with respect to temperature, and use EF=kB TF. At T/TF=0.01, Ce/(NkB)=0.0493480.
Worked evaluation. Substitute the marked horizontal coordinate, 0.0255, into the displayed formula to obtain 0.12584 on the vertical axis. Values are rounded for display.
Scope. Leading low-temperature contribution of ideal electrons only; excludes lattice heat capacity, band corrections, interactions, and superconductivity.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Tight-binding electronic model · Example 1
Nearest-neighbor tight-binding band
Problem & parameters. Use a one-dimensional chain with one orbital per site and positive nearest-neighbor hopping t. Find its band over half the Brillouin zone.
Solution. Insert amplitudes c_n=exp(ikna) into E c_n=epsilon0 c_n−t(c_(n+1)+c_(n−1)). Divide by c_n and combine the two exponentials as 2cos(ka). The band runs from epsilon0−2t to epsilon0+2t.
Worked evaluation. Substitute the marked horizontal coordinate, 1.5708, into the displayed formula to obtain -1.2246e-16 on the vertical axis. Values are rounded for display.
Scope. One-orbital, orthonormal, noninteracting chain. The other half-zone follows by inversion symmetry; real semiconductor bands generally need multiple orbitals.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Nearly-free-electron model · Example 1
Nearly-free-electron avoided crossing
Problem & parameters. Let ER=hbar²(G/2)²/(2m) and a real lattice Fourier coupling VG=0.1 ER. Calculate the lower branch near k=G/2.
Solution. The two free plane-wave energies are ER(1+q)² and ER(q−1)². Diagonalizing their 2 by 2 matrix gives the displayed lower eigenvalue. At q=0 the energies are 0.9 ER and 1.1 ER, separated by 0.2 ER.
Worked evaluation. Substitute the marked horizontal coordinate, 0.15, into the displayed formula to obtain 0.70627 on the vertical axis. Values are rounded for display.
Scope. Exact two-state diagonalization, approximate nearly-free-electron physics. Only the lower branch on one side of the Bragg plane is plotted; remote plane waves are omitted.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.
Harmonic lattice dynamics · Example 1
Monatomic harmonic-chain dispersion
Problem & parameters. Take identical masses m separated by a, joined by nearest-neighbor springs K. Find the normal-mode dispersion over half the Brillouin zone.
Solution. Insert u_n=A exp[i(qna−omega t)] into m u_n″=K(u_(n+1)+u_(n−1)−2u_n). This gives omega²=(4K/m)sin²(qa/2). Select the nonnegative frequency. At small q the sound speed is a sqrt(K/m).
Worked evaluation. Substitute the marked horizontal coordinate, 1.5708, into the displayed formula to obtain 0.70711 on the vertical axis. Values are rounded for display.
Scope. One-dimensional harmonic monatomic chain; no optical branch, anharmonic scattering, or measured material parameters.
Use the model entry’s references and assumptions for the full formulation. This curve is an analytical illustration, not measured product data.