Quantified against established engineering benchmarks.

Overview

Validated against Ghia, Ghia and Shin (1982).

Reference: Ghia, U., Ghia, K.N., Shin, C.T. (1982), J. Comput. Phys. 48(3), 387-411

Headline comparison — Centerline extrema: 0.8-9.4% relative deviation.

Geometry and simulation setup

1.0 m square cavity with a 0.1 m quasi-2D depth.

Re = 100 and 400; lid velocity U = 1 m/s; P2/P1 incompressible formulation; approximately 30 cells per side.

Parameter values

Results

v_max_x/v_min_x are reported but not gated on tight numeric agreement -- near-wall peak location is far more resolution-sensitive than bulk magnitude on this single, non-wall-refined mesh (~30 elements/side vs. Ghia's 129x129 grid); gated instead on qualitative circulation direction (v_max on the side nearer x=0, v_min nearer x=L), independently re-derived from first principles (lid drags +x at top -> turns down at right wall -> left along bottom -> up at left wall -> clockwise circulation).

Runs

Assessment

Centerline velocity extrema, signs, and circulation direction are compared with the published reference. The acceptance criterion covers bulk extrema and circulation topology; near-wall peak locations are reported separately because the mesh is not wall-refined.

Overview

Validated against Roshko and Williamson Strouhal range.

Reference: St = f*D/U at Re=100 ~ 0.16-0.17 (Roshko 1954; Williamson 1996 review)

Headline comparison — Strouhal number St = 0.1603 (reference interval 0.16-0.17).

Geometry and simulation setup

Circular cylinder in a widened quasi-2D cross-flow channel.

Re = 100; transient incompressible flow; force history sampled after shedding reaches a stationary amplitude.

Parameter values

Results

Assessment

The shedding frequency is extracted from the statistically stationary lift history. A controlled domain-width study quantifies blockage sensitivity before comparison with the reference interval.

Overview

A water column collapses into a dry tank and the resulting surge-front speed is compared with the Lobovsky et al. experiment.

The thin spanwise extrusion uses symmetry boundaries to model the published quasi-two-dimensional flow without artificial depth-wall drag.

Numerical setup

Results

Assessment

The refined active-coupling result meets its stated 15% experimental-comparison gate and preserves phase volume to near machine precision. Refining the active-coupling mesh improved the deviation from 16.10% on 9,600 cells to 13.10% on 38,400 cells.

Overview

Turbulent flow separates at a backward-facing step and reattaches downstream. The reattachment distance is the validation quantity: it is set by how fast the separated shear layer spreads, which makes it a demanding test of the turbulence closure rather than of the discretisation alone.

The same experiment is published as a validation case by NASA's Turbulence Modeling Resource and by SimScale, so the result below is compared against measurement and against both reference simulations.

Geometry

Fluid and flow conditions

Boundary conditions

Numerical setup

Result comparison

Reattachment length normalised by step height. SimScale's published figures are 6.84477 cm and 7.74 cm against a 0.0127 m step, i.e. 5.39 H computed against a 6.09 H experimental value.

SimScale solve this case with a finite-volume method (OpenFOAM) on 550,000 cells; the result below is finite element on 8,832 cells, so the two differ in discretisation as well as resolution.

Assessment

The computed reattachment sits 0.49% from NASA's published SST result, confirming the closure is implemented correctly, and is closer to the measured value than either reference simulation despite using far fewer cells.

All SST results lie above the measured 6.26 H except SimScale's, which lies well below it; the spread across implementations is larger than the experimental uncertainty, so the model rather than the solver sets the achievable accuracy for this quantity.

Overview

A plain three-dimensional airfoil mesh is compared with NASA’s two-dimensional NACA 0012 SSTm validation case. The full three-component velocity field is solved with spanwise symmetry boundaries.

The published result converged in 64 pseudo-iterations. Its lift, drag and peak suction pressure each satisfy the gallery’s 5% criterion.

Geometry and flow conditions

Boundary conditions

Numerical setup

Reference comparison

Pressure resolution and convergence

Leading-edge RMS errors use x/c < 0.05; trailing-edge RMS errors use x/c > 0.9. These are absolute Cp differences, not pointwise percentage errors.

Assessment and reproduction

All three headline quantities pass the 5% gate. NASA uses a compressible formulation at M = 0.15; TorsoCAE uses incompressible flow here, without a compressibility correction. This case does not establish grid independence.

The 48.8-minute runtime covers this 64-iteration run from interpolated baseline fields. The case folder includes the exact initial state, frozen results, reference data and the complete validated project. It is not a cold-start timing or a demonstrated end-to-end speedup.

The downloadable journal expects initial_state.npz beside it, as supplied in the repository case folder, or an explicit TORSO_NACA_INITIAL_STATE path. Future validation runs write results.json; the gallery uses the separate frozen results_validated.json.

Overview

Validated against Exact parabolic channel-flow solution.

Reference: Standard exact solution of the 2D Stokes equations for pressure-driven channel flow (e.g. White, Viscous Fluid Flow); u(y)=U_max*(1-(2y/H)^2), dp/dx=-8*mu*U_max/H^2

Headline comparison — Velocity profile: 1.75%; pressure gradient: 1.74% relative deviation.

Geometry and simulation setup

Long rectangular channel represented as a quasi-2D three-dimensional domain.

Creeping flow; Taylor-Hood P2/P1 velocity-pressure spaces; pressure-driven inlet-to-outlet flow.

Parameter values

Results

Runs

Assessment

Velocity samples at three streamwise stations are compared with the exact parabola, and the recovered pressure gradient is checked independently.

Overview

Validated against Stokes law, F = 6 pi mu R U.

Reference: Stokes, G.G. (1851), exact unbounded-domain creeping-flow drag law

Headline comparison — Drag: 5.99% relative deviation at L/R = 30.

Geometry and simulation setup

Refined sphere inside successively larger outer flow domains.

Creeping flow; L/R = 6, 12, 20 and 30; fixed converged sphere resolution; 10% finite-domain acceptance threshold.

Parameter values

Results

Domain-size sweep at a fixed, independently-validated sphere refinement level (surface_sizes=0.05*R). Pass requires the largest-domain error vs. the cited UNBOUNDED formula to be under 10.0% AND monotonically decreasing across the sweep -- demonstrated convergence to the cited value, not an assumed one.

Levels

Assessment

The assessment requires monotonic convergence toward the unbounded Stokes-law drag as the domain expands. Lateral-force symmetry and cross-solver agreement provide independent consistency checks.

Validation scope and animation

The header animation shows a finer Torso CAD mesh (139,701 Tet4 cells, 4 mm filter). All numerical validation comparisons below use the available coarse reference data (14,027 Tet4 cells, 10 mm filter).

The original example publishes fine-mesh images but supplies coarse-mesh numerical files. The animation is a separate illustrative run; it is not evidence of quantitative agreement with the published fine-mesh design.

Geometry, constraints and loading

The reference coarse mesh is used unchanged for the numerical comparison. Its non-design bearing annulus and four bolt sleeves remain solid. The finer workflow imports STEP, removes the bearing insert, creates the cylindrical bore, partitions the protected regions and generates a new conforming tetrahedral mesh.

Nodes on the four bolt bore surfaces are fixed. A rigid bearing coupling at (101.6, −2.07, 0) mm applies three separate force cases: F1 = (281, 0, 0) N; F2 = (298, −298, 0) N; F3 = (0, −562, 0) N.

Optimization parameters

Coarse-mesh numerical comparison

The published retained state is file073.vtk, iteration 73. Torso stopped after 66 iterations with a two-cycle oscillation. This is reported as oscillation, not mathematical convergence.

Stress and mass reference values come from the published CalculiX coarse log. The compliance reference column is Torso’s reanalysis of the published retained design; no independent CalculiX compliance value is used.

Retained-material topology and cross-check

Volume-weighted intersection over union of retained design material is 94.51%. Protected cells are excluded from this overlap metric. The overlap is reported separately from the 5% stress, mass and compliance criteria.

As an independent finite-element cross-check, Torso reanalysis of the published coarse design gives 524.967311 MPa maximum element stress versus 524.967279 MPa in the CalculiX log. A fresh four-rank reanalysis reproduced the archived numerical metrics before publication.

Finer animation provenance

The animation contains 76 samples of stored retained-material surface frames, displayed with a fixed camera and no smoothing or shape editing. The faint outer shape provides the original design envelope.

This finer run completed 150 iterations and stopped at the iteration limit. Its final design mass fraction is 10.00474% and summed compliance is 0.197255635 J. These values are provided as run metadata, not as comparisons with the fine CalculiX image.

Limits and reproduction

The current Torso BESO stopping logic does not include the reference’s five-iteration mean failure-index test. The coarse result stopped on oscillation; the animation run stopped at its iteration cap. Neither stopping reason establishes optimization convergence.

This comparison does not certify connectivity of every protected bolt region, manufacturability or fatigue life.

Download journal provides the finer CAD workflow shown in the animation. It expects the original Bearing_Bracket_Assy_STEP214.STEP uploaded under cadex/. The repository case includes run_validation.py for frozen-evidence verification and README.md with the separate coarse MPI benchmark rerun command.

Reference: https://github.com/calculix/beso/wiki/Example-3:-airplane-bearing-bracket. The result data include the source archive checksum, reference iteration, numerical metrics and animation provenance.

Overview

Validated against Closed-form isotropic-hardening response.

Headline comparison — Stress-strain response: 1.50% maximum relative deviation.

Geometry and simulation setup

Prismatic bar loaded monotonically in uniaxial tension.

J2 yield surface with linear isotropic hardening; load sweep spans elastic response, first yield, and post-yield hardening.

Parameter values

Results

closed form derived from code's R(alpha)=H*alpha isotropic hardening law (plasticity_j2.py HardeningLaw); stress read from fixed-end reaction force / area (equilibrium-exact, end-effect free average axial stress)

Points

Assessment

The computed reaction stress is compared point-by-point with the closed-form elastic-plastic law.

Overview

Validated against Lame thick-cylinder solution.

Headline comparison — Elastic stress field: 0.59% maximum relative deviation.

Geometry and simulation setup

Quarter-sector of a thick-walled cylinder under internal pressure.

J2 isotropic hardening; two elastic pressure levels define the quantitative acceptance criterion; higher pressures exercise plastic onset.

Parameter values

Results

Assessment

Radial and hoop stresses are compared with Lame theory before yield. Plastic stress saturation is reported as a separate qualitative trend.

Overview

Validated against Hertz (1882).

Headline comparison — Peak contact pressure within 4.27% of closed-form Hertz theory (finest mesh); contact radius within 11.68%.

Geometry and simulation setup

20 mm steel sphere pressed onto a 200x200x100 mm steel box, small flat cap trimmed opposite the contact pole for load application.

E=210 GPa, nu=0.3 (both bodies); displacement-controlled cap loading; MFEM/Tribol native frictionless contact, interior-point solver.

Parameter values

Results

Displacement-controlled cap (removes the frictionless rigid-body null space); F_meas is the solver-reported contact force (Lagrange multiplier of the gap constraint, mapped to nodal force via J^T*l), not a preset target. PRIMARY: peak contact pressure p0_fem vs. Hertz p0(F_meas), converged via mesh refinement study (see convergence.json / plots/mesh_convergence.png). SECONDARY: force-weighted RMS contact radius vs. Hertz a(F_meas). Indentation depth is not used: the cap is 1.7R from the pole, so prescribed cap displacement includes bulk sphere compression, not just Hertzian approach (confirmed directly, not assumed -- see run_validation.py docstring).

Mesh convergence runs

Assessment

Cap is displacement-controlled (not pressure-controlled) to remove the rigid-body null space that pure normal contact leaves in a freely-loaded sphere. The resulting contact force is solver-reported (Lagrange multiplier of the gap constraint, mapped to nodal force via J^T*l), not a preset target. Peak contact pressure and force-weighted RMS contact radius are compared against Hertz theory at that measured force; indentation depth is not used since displacement-controlled cap motion includes bulk sphere compression, not just Hertzian approach. A two-level mesh refinement study confirms convergence: both errors roughly halve (p0 9.93%->4.27%, a 21.53%->11.68%) between the two resolutions.

Overview

Validated against Boussinesq (1885); Sneddon (1965).

Reference: Boussinesq (1885); Sneddon, I.N. (1965), Int. J. Engng Sci. 3; see also Johnson, K.L., Contact Mechanics (1985), Ch. 3.4

Headline comparison — Contact-face indentation within 0.18% of rigid-flat-punch theory at the solver-measured contact force, at a domain depth 120x the punch radius; both near-contact and far-field mesh resolution independently convergence-tested..

Geometry and simulation setup

20 mm diameter cylindrical punch, 1000x stiffer than the substrate to approximate a rigid indenter, pressed into a 400x400x1200 mm elastic block.

E1=210 TPa (punch), E2=210 GPa, nu=0.3 (block); displacement-controlled punch; MFEM/Tribol native frictionless contact, interior-point solver.

Parameter values

Results

Displacement-controlled punch top; F_meas is the solver-reported contact force, not a preset target. PRIMARY metric: mean uz on the punch's bottom/contact face (not the top face, which includes the punch's own bulk axial compression under load) vs. the rigid-flat-punch closed form delta(F) = F/(2*E_star*a) evaluated at F_meas, where E_star = E2/(1-nu2^2) is the single half-space modulus the closed form assumes.

delta_contact_face within 0.18% of the rigid-flat-punch closed form at the solver-measured contact force, at box depth h/a=120 with far-field mesh convergence-tested (0.64% -> 0.18% on size_max refinement, same sign both times, confirmed against the raw displacement field); near-contact mesh converged separately (0.2 percentage points between refinement levels); measured stiffness independent of load level (4x range) and of contact damping (10x range).

Near contact mesh convergence

Box depth sensitivity

Far field mesh convergence

Load level sensitivity

Damping sensitivity

Modulus ratio sensitivity

Assessment

Punch top face is displacement-controlled; the contact force is solver-reported, not a preset target. Primary metric is mean indentation on the punch contact face compared to the classical rigid-flat-punch closed-form solution evaluated at the measured force. Two independent mesh studies confirm convergence: refining the contact-zone mesh changes the result by only 0.2 percentage points, and refining the far-field mesh at the production domain depth moves the error 0.64% -> 0.18%, the same direction both times (the only physically admissible one -- a finite domain can only be stiffer than the idealized infinite half-space, never softer). A separate depth study isolates domain depth, not lateral extent, as the dominant geometric factor: deepening the box alone reduces the error from 12.5% to 0.6% before far-field refinement.

Overview

Validated against Nonlinear pendulum period from the elliptic integral.

Headline comparison — Oscillation period: 0.20% relative deviation.

Geometry and simulation setup

Flexible solid link attached to a reference frame by an RBE2 surface coupling and a revolute joint.

Gravity-driven generalized-alpha dynamics; 800 steps at dt = 0.01 s; finite-rotation joint kinematics.

Parameter values

Results

Assessment

The free-end trajectory supplies the oscillation period, which is compared with the amplitude-dependent nonlinear pendulum solution.

Overview

Validated against Independent rigid-body double-pendulum ODE.

Headline comparison — Angular trajectory: 1.31 degree RMS deviation over 1.0 s.

Geometry and simulation setup

Two flexible solid links connected through reference frames, RBE2 attachments, and revolute joints.

Gravity-driven generalized-alpha dynamics; 1200 steps at dt = 0.0025 s; comparison interval limited to the first 1.0 s, before sensitivity-driven trajectory divergence dominates.

Parameter values

Results

Rms angle error by window

Assessment

Angular trajectories are compared with an independently integrated rigid-body ODE over the defined comparison interval. Energy evolution is retained as a secondary diagnostic rather than the acceptance quantity.

Overview

Validated against Constant-moment rigid rectangular crank.

Headline comparison — Angular trajectory: 0.668% acceleration and 0.564% final-angle deviation.

Geometry and simulation setup

Flexible rectangular crank coupled to a ground revolute joint through an RBE2 root attachment.

Tangential follower traction; Newmark average-acceleration integration; 200 steps at dt = 0.005 s; one complete nominal revolution.

Parameter values

Results

Assessment

The unwrapped free-end trajectory is compared with the independently calculated constant-moment rigid-body solution. Pivot drift and radius consistency separately verify the finite-rotation constraints.

Overview

Validated against Exact rigid slider-crank closure relation.

Headline comparison — Piston trajectory: 0.000145% RMS error relative to stroke.

Geometry and simulation setup

Flexible crank and connecting rod coupled to a guided flexible piston through three revolute joints and an RBE2 frame network.

12 N m follower moment; Newmark average-acceleration integration; 185 steps at dt = 0.005 s; 378.52 degrees of continuous crank travel.

Parameter values

Results

Assessment

The computed piston trajectory is compared throughout the motion with the exact rigid slider-crank closure relation. Rod-length consistency, piston-guide error, crank-root drift, and the full-rank 17-row joint Jacobian provide independent constraint checks.

Overview

Validated against Bisshopp and Drucker (1945) elastica.

Reference: Bisshopp & Drucker, 'Large Deflection of Cantilever Beams', Q. Appl. Math. 3(3), 272-275 (1945)

Headline comparison — Resultant tip displacement: 0.95% relative deviation.

Geometry and simulation setup

2.0 x 0.1 x 0.1 m cantilever with quadratic hexahedral elements.

Large transverse end traction; 20 load steps; finite-strain geometric nonlinear formulation.

Parameter values

Results

Mesh convergence

Assessment

The final tip displacement and direction are compared with a numerically evaluated elliptic-integral reference at 44.8 degrees tip rotation.

Overview

Validated against the stable antisymmetric second-mode elastica derived by Bigoni (2015).

At normalized end shortening |u1(L)|/L = 1.2, the TorsoCAE centerline differs from the independent analytical solution by 0.0785% L maximum and 0.0557% L RMS.

The reference boundary-value problem converged to a maximum RMS residual below 1.0e-9.

Geometry and simulation setup

The 490 x 25 x 1.5 mm strip uses the experimental dimensions reported by Bigoni.

A structured 2,415-node, 1,280-element Hex8 mesh is solved in 96 load increments on two MPI ranks.

Parameter values

Results

Assessment scope

The small antisymmetric imperfection selects the second branch; axial shortening alone is not claimed to discover the secondary bifurcation.

This case validates the equilibrium shape. Automatic snap-through and reaction magnitude are outside its acceptance criterion and are not presented as validated results.

Overview

Validated against the finite-rotation objectivity sequence of Glaser and Armero (1997).

The S-shaped deformation is held fixed while a rigid rotation is superposed from 0 to 90 degrees.

The maximum relative co-rotated reaction drift is 1.95e-11 and the maximum relative von Mises drift is 5.39e-11.

Geometry and simulation setup

The beam has length 1.0, height 0.2, width 1.0, and a prescribed right-end transverse shift of 0.2.

A structured 28-node, 6-element Hex8 mesh is solved on two MPI ranks.

Parameter values

Results

Assessment scope

The constitutive model differs from the enhanced-strain element in the reference, so response magnitudes are not compared.

The persisted result currently exposes reaction forces rather than reaction moments; this case validates axial/shear reaction and stress objectivity without claiming the reference moment curve.

Overview

Validated against Mesh self-convergence for the nonlinear 3D setup.

Reference: Cook, R.D. (1974); geometry per the widely used Cook 1974 tapered-panel benchmark

Headline comparison — 0.12% change across the finest resolutions.

Geometry and simulation setup

Tapered Cook membrane extruded into a three-dimensional solid.

500 Pa end load; four mesh resolutions from 247 to 1888 displacement DOFs; load-bisected nonlinear solve.

Parameter values

Results

Mesh convergence

Assessment

The assessment quantifies asymptotic mesh stability for this nonlinear three-dimensional parameterization. No external displacement reference is asserted for this specific loading configuration.

Overview

Validated against Independent rigid compound-pendulum ODE.

Headline comparison — Angular trajectory: 0.47 degree RMS deviation over the first half-swing.

Geometry and simulation setup

Flexible plate suspended from a finite-rotation pivot attachment.

Gravity-driven geometric nonlinear dynamics; mass properties for the reference are computed from geometry rather than fitted.

Parameter values

Results

Root/tip markers (x=0 and x=1 face centroids) tracked over time from the SCVZ result-frame history; rotation angle recovered via atan2 of their relative vector (any two rigid-body material points suffice -- the pivot itself, tag 7, need not be one of the markers). Compared against an independent nonlinear rigid compound-pendulum ODE (scipy solve_ivp DOP853, rtol=1e-12) using the plate's own mass/I_pivot/pivot-to-COM distance, computed directly from its geometry (not fit), released from the same rest-horizontal initial condition.

Rms angle error by window

Assessment

The trajectory is compared with an independent compound-pendulum ODE whose mass properties are derived directly from the modeled geometry. This reference is used because the relevant published beam response is available only graphically and does not support a precise numerical comparison.

Overview

Validated against Treloar (1944) real experimental data, 3-term Ogden fit.

Reference: Treloar, L.R.G. (1944), Trans. Faraday Soc. 40; data digitized via github.com/OVGU-CoMe/treloar_fit (Data_Treloar_UT.mat)

Headline comparison — R-squared = 0.970 against real Treloar stress data.

Geometry and simulation setup

4.0 x 2.0 x 2.0 homogeneous hyperelastic block.

3-term Ogden material fit simultaneously to real Treloar uniaxial, equibiaxial, and pure-shear data; real measured stretch applied as a displacement boundary condition; 20 nonlinear load steps.

Parameter values

Results

Real Treloar-measured STRETCH applied as an axial-only displacement BC (lateral faces free); FEM nominal stress (from Cauchy stress at the known homogeneous J) compared directly to Treloar's actual measured stress.

Points

Assessment

The primary acceptance criterion is direct agreement between the finite-element nominal stress and Treloar's real measured stress at the same real measured stretch. Finite-element agreement with the material model's own closed-form solution is reported separately as a numerical solver check.

Overview

Validated against Treloar (1944) real experimental data, 3-term Ogden fit.

Reference: Treloar, L.R.G. (1944), Trans. Faraday Soc. 40; data digitized via github.com/OVGU-CoMe/treloar_fit (Data_Treloar_ET.mat)

Headline comparison — R-squared = 0.833 against real Treloar stress data.

Geometry and simulation setup

4.0 x 4.0 x 0.5 homogeneous hyperelastic sheet.

Same 3-term Ogden material as the uniaxial case, fit simultaneously to all three real Treloar loading modes; real measured equibiaxial stretch applied as displacement boundary conditions on both in-plane faces.

Parameter values

Results

Real Treloar-measured STRETCH applied as axial-only displacement BCs on both in-plane faces (thickness free); FEM nominal stress (from Cauchy stress at the known homogeneous J) compared directly to Treloar's actual measured stress.

Points

Assessment

The primary acceptance criterion is direct agreement between the finite-element nominal stress and Treloar's real measured equibiaxial stress at the same real measured stretch. Finite-element agreement with the closed-form solution is reported separately as a numerical solver check.

Overview

Validated against Treloar (1944) real experimental data, 3-term Ogden fit.

Reference: Treloar, L.R.G. (1944), Trans. Faraday Soc. 40; data digitized via github.com/OVGU-CoMe/treloar_fit (Data_Treloar_PS.mat)

Headline comparison — R-squared = 0.739 against real Treloar stress data.

Geometry and simulation setup

4.0 x 8.0 x 0.5 homogeneous hyperelastic strip, gripped along both long edges.

Same 3-term Ogden material as the uniaxial and equibiaxial cases; real measured pure-shear stretch applied as an axial displacement boundary condition with both long edges held to enforce the plane-strain state.

Parameter values

Results

Real Treloar-measured STRETCH applied as an axial-only displacement BC; width rigidly gripped (uy=0) on both long edges to enforce the plane-strain pure-shear state; FEM nominal stress (from Cauchy stress at the known homogeneous J) compared directly to Treloar's actual measured stress.

Points

Assessment

The primary acceptance criterion is direct agreement between the finite-element nominal stress and Treloar's real measured pure-shear stress at the same real measured stretch. Finite-element agreement with the closed-form solution is reported separately as a numerical solver check.

Overview

Validated against Euler-Bernoulli beam solution.

Headline comparison — Tip deflection: 1.02% relative deviation.

Geometry and simulation setup

2.0 x 0.1 x 0.1 m solid cantilever with quadratic hexahedral elements.

Steel elastic properties; 1000 N resultant end traction; fixed root; mesh-convergence sweep.

Parameter values

Results

Mesh convergence

Assessment

The transverse tip displacement is compared with beam theory while mesh refinement checks convergence of the three-dimensional solid model.

Overview

Validated against Kirsch (1898), stress concentration Kt = 3.

Headline comparison — Stress concentration factor: 0.17% relative deviation.

Geometry and simulation setup

Symmetry-reduced plate sector with a 0.01 m circular hole and finite thickness.

1 MPa remote tension; quadratic tetrahedral mesh; symmetry constraints on the cut planes.

Parameter values

Results

Mesh convergence

Assessment

The maximum principal stress at the hole is normalized by the far-field stress and compared with the Kirsch concentration factor.

Overview

Validated against Ko, Lee and Bathe (2017), reference displacement -0.3024.

Reference: Ko, Y., Lee, P.-S., and Bathe, K.-J. (2017), Performance of the MITC3+ and MITC4+ shell elements in widely-used benchmark problems, Computers & Structures 193, 187-206, doi:10.1016/j.compstruc.2017.08.003

Headline comparison — Free-edge displacement: 0.623% relative deviation.

Geometry and simulation setup

Quarter-symmetry cylindrical roof with radius 25, full length 50, 40 degree half-angle and thickness 0.25.

E = 4.32e8; nu = 0; surface self-weight = 90; regular MITC4+ quadrilateral meshes; two-rank MPI.

Parameter values

Results

Runs

Assessment

The vertical displacement at the midpoint of the free edge is tracked through four mesh levels and compared with both the accepted -0.3024 reference and the published MITC4+ convergence sequence.

Overview

Airborne sound transmission between two rooms through a panel, from meshed CAD bodies.

Headline comparison - level difference within 0.51 dB of the mass law below coincidence and 0.31 dB of Cremer above it; power-balance residual 3.5e-16.

Results

Overview

Structure-borne sound through plate joints recovered from meshed CAD bodies.

Headline comparison - every bending coupling loss factor within 0.20% of the closed form.

Results

Overview

Validated against Euler-Bernoulli bending frequencies.

Headline comparison — Modes 1 and 2: 1.19% and 2.58% frequency deviation.

Geometry and simulation setup

2.0 x 0.1 x 0.0125 m slender solid cantilever.

Quadratic hexahedral discretization; fixed root; sparse generalized stiffness-mass eigenproblem.

Parameter values

Results

Mesh convergence

Assessment

The first two bending frequencies are tracked through mesh refinement and compared with the analytical beam frequencies.

Overview

Validated against Leissa plate-frequency solution.

Headline comparison — First three modes: 0.17%, 0.84% and 1.30% frequency deviation.

Geometry and simulation setup

1.0 x 0.7 x 0.01 m thin rectangular solid plate.

Quadratic hexahedral discretization; simply-supported edge constraints; sparse generalized eigenproblem.

Parameter values

Results

Mesh convergence

Assessment

Three identified bending modes are compared with analytical thin-plate frequencies across mesh refinement.

Overview

Validated against Closed-form undamped SDOF step response.

Headline comparison — Overshoot: 2.76%; period: 1.76% relative deviation.

Geometry and simulation setup

Slender steel cantilever reduced to its dominant first bending response.

Suddenly applied constant tip traction; 170 EDMC-1 time steps; zero algorithmic dissipation for the validation run.

Parameter values

Results

Assessment

Peak dynamic amplification and the measured oscillation period are compared independently with the equivalent SDOF solution.

Overview

Validated against Euler-Bernoulli first bending frequency.

Headline comparison — First bending frequency: 1.15% relative deviation.

Geometry and simulation setup

2.0 x 0.1 x 0.1 m solid cantilever.

Step-like end loading; 420 transient steps at dt = 7.9537e-4 s; frequency extracted from the displacement spectrum.

Parameter values

Results

Assessment

The dominant FFT peak of the transient response is compared with the analytical first bending frequency.

Overview

Validated against Exact linear Fourier-conduction solution.

Reference: Standard 1D steady-state Fourier conduction (textbook exact solution)

Headline comparison — Agreement within 7.5e-9% for temperature and 2.1e-7% for heat flux.

Geometry and simulation setup

Rectangular slab with one-dimensional heat flow between opposite faces.

Constant isotropic conductivity; prescribed hot and cold face temperatures; adiabatic remaining faces.

Parameter values

Results

FEM temperature field at every mesh node compared to the exact linear profile T(x) = T_hot + (T_cold-T_hot)*x/L; FEM heat flux magnitude (saved per-cell in the npz) compared to the exact uniform flux q = k*(T_hot-T_cold)/L.

Assessment

Temperature at every node and the recovered conductive heat flux are compared with the exact linear profile.

Overview

Validated against Exact logarithmic radial Fourier solution.

Reference: Standard 1D radial Fourier conduction in a hollow cylinder (textbook exact solution)

Headline comparison — Temperature: 0.0013%; median heat flux: 0.0078% relative deviation.

Geometry and simulation setup

90-degree hollow-cylinder wedge with radii 0.05 and 0.10 m and 0.02 m axial length.

k = 50 W/(m K); inner wall at 400 K; outer wall at 300 K; radial cut and axial faces adiabatic.

Parameter values

Results

FEM temperature at every mesh node compared to the exact logarithmic profile T(r) = T_in + (T_out-T_in)*ln(r/a)/ln(b/a); FEM heat flux magnitude (per-cell) compared to the exact q(r) = k*(T_in-T_out)/(r*ln(b/a)) (median relative error reported for flux, since DG0 cell-center flux samples very close to the curved boundaries are mesh-resolution-limited, not a solver defect -- see README).

Assessment

Nodal temperature and cell heat flux are compared with the exact logarithmic radial profiles.

Overview

Validated against sigma = -E alpha deltaT.

Reference: Standard restrained thermal expansion result, sigma = -E*alpha*delta_T (e.g. Boresi & Schmidt, Advanced Mechanics of Materials)

Headline comparison — Axial thermal stress: 1.20% relative deviation.

Geometry and simulation setup

1.0 m bar with a 0.05 x 0.05 m square cross-section.

E = 210 GPa; nu = 0.3; alpha = 12e-6 1/K; uniform 100 K rise; both ends restrained axially.

Parameter values

Results

Temperature forced uniform (fixed_temp on all 6 faces at the same value); axial displacement restrained at both ends, lateral faces free; stress read directly from the interior (mid-bar) stress field, well away from either end.

Assessment

Interior axial stress is compared with the exact fully restrained thermal-expansion stress, away from end effects.

Overview

Validated against Timoshenko (1925), equal-modulus special case.

Reference: Timoshenko, S. (1925) 'Analysis of Bi-Metal Thermostats', J. Opt. Soc. Am. 11(3); equal-thickness equal-modulus (n=m=1) special case kappa=1.5*(alpha1-alpha2)*deltaT/h, derived here independently via composite-beam equilibrium (N=0, M=0) and cross-checked against that known special case.

Headline comparison — Thermally induced tip deflection: 1.28% relative deviation.

Geometry and simulation setup

0.5 x 0.02 x 0.02 m two-layer cantilever strip.

E = 150 GPa in both layers; alpha = 19e-6 and 12e-6 1/K; uniform 100 K rise; quadratic elements.

Parameter values

Results

Runs

Assessment

Computed thermally induced curvature and tip displacement are compared with the independently derived equal-thickness Timoshenko formula.

Overview

Validated against Turek and Hron (2006); FEATFLOW reference data.

Reference: Turek, S., Hron, J. (2006), Proposal for Numerical Benchmarking of FSI; official FSI2 time series from wwwold.mathematik.tu-dortmund.de/~featflow.

Headline comparison — Point-A response frequency within 0.6% and amplitude within 2.2% of the FEATFLOW reference.

Geometry and simulation setup

Turek-Hron channel: rigid cylinder (r=0.05m) with a thin elastic flag (0.35x0.02m) clamped to its downstream side, in a 2.5x0.41m quasi-2D channel.

rho_fluid=1000, mu=1.0 (Re=100); rho_solid=10000, E=1.4 MPa, nu=0.4; parabolic inlet ramped to full FSI2 profile over 2s; dt=0.002s; Bossak fluid+solid time integration, MVQN coupling, Dirichlet-Neumann interface.

Parameter values

Results

Point A response, settled window (t>9.7s, verified stable across two independent sub-windows to <0.1%).

Assessment

Frequency and amplitude of the self-excited flapping response agree with the published FEATFLOW FSI2 reference to within 0.6% and 2.2% respectively, well inside the 10% tolerance used across TorsoCAE's FSI benchmarks. The small amplitude offset is consistent with the measured quantity combining the flag tip's X and Y displacement (the reference's own Ux is genuinely nonzero, -2 to -28mm) rather than reporting Uy alone.

Reference and purpose

Ishihara et al., Computer Modeling in Engineering & Sciences 136 (2023), 1187–1206; DOI 10.32604/cmes.2023.024614. Table 6 provides the 3 mm-wide P-15 strong-coupling displacement.

This case checks a three-dimensional, two-way coupled electroelastic response against an independently published finite-element result.

Geometry and physical setup

20 x 3 x 0.5 mm bimorph with two 0.25 mm PZT layers, both poled through thickness. The interface electrode is held at 1 V; outer electrodes are grounded.

The left end follows Figure 5 Case A: axial displacement fixed on the end face, transverse displacement fixed along line AD, and vertical displacement fixed along line CD. Other faces are mechanically free.

Material and discretization

P-15 Table 1: C11=137, C12=88.1, C13=88.3, C33=120, C44=17.1 and C66=24.4 GPa; e31=-10.8, e33=8.7, e15=14.4 C/m²; epsilon11=1.90e-8 and epsilon33=1.24e-8 F/m.

Torso CAD partitioning retains the internal electrode. The Torso mesh has 960 Hex27 cells on 40 x 6 x 4 divisions; the published mesh has 960 Hex20 cells. Both solution fields are quadratic.

Measured result

The measurement point is (20, 1.5, 0) mm. The saved numerical field and its checksum are preserved with the validation case.

Scope and limitations

This is a static strongly coupled benchmark. It does not validate harmonic response or transient dynamics.

Table 2 of the paper is a separate 100 mm case; its tabulated value appears inconsistent with the paper’s own one-way formula and P-15 constants, so this gallery case uses Table 6 only.

The gallery image shows actual solved displacement colored in physical µm; geometry deformation is multiplied by 3500 only for visibility.