Validation & Verification
Getting a solution from FEA software is easy. Getting a correct solution requires care. This lesson covers the critical practices of Verification (solving the equations right) and Validation (solving the right equations).
V&V: Two Different Questions
Verification
"Are we solving the equations correctly?"
Verification checks that the mathematical model is solved accurately:
Is the mesh fine enough?
Are the elements behaving correctly?
Is the solver converging?
Compares: FEA results vs. analytical solutions or refined meshes
Validation
"Are we solving the right equations?"
Validation checks that the physical model represents reality:
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Are boundary conditions realistic?
Is the material model appropriate?
Are we capturing the right physics?
Compares: FEA results vs. experimental data or real-world behavior
The Verification Process
1. Code Verification
Ensure the FEA software itself is correct:
Patch tests: Simple problems where elements must give exact answers
Constant stress state
Rigid body motion
Linear displacement field
If elements fail patch tests, the formulation is flawed.
2. Mesh Convergence Study
The most important verification step:
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Watch how stress results converge as mesh is refined. The exact solution is known for this benchmark problem. Process:
Start with a coarse mesh
Refine the mesh (halve element size)
Compare key results (stress, displacement, etc.)
Repeat until results stabilize
Convergence criteria:
Results change < 5% between refinements
Or asymptotically approach a limit
3. Convergence Rate
For h-refinement (smaller elements):
$$\text{Error} \propto h^p$$
Where:
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$h$ = element size
$p$ = convergence rate (depends on element order)
Element Type Expected Rate Linear (CST, Q4) $p = 1$ for stress Quadratic (LST, Q8) $p = 2$ for stress
Richardson extrapolation: Use convergence rate to estimate exact solution:
$$u_{exact} \approx u_h + \frac{u_h - u_{2h}}{2^p - 1}$$
4. Energy Norm Convergence
A more robust convergence measure:
$$\|e\|_E = \sqrt{\int_\Omega (\sigma - \sigma_h)^T [D]^{-1} (\sigma - \sigma_h) \, dV}$$
Monitors the error in strain energy — captures global accuracy.
Benchmark Problems
Always verify against known solutions:
Patch Test Problems
Test What It Checks Constant stress Basic element formulation Rigid body motion No spurious strains Linear displacement Completeness
Classical Benchmarks
Problem Analytical Solution Key Output Cantilever beam Euler-Bernoulli Tip deflection Plate with hole Kirsch solution Stress concentration Thick cylinder Lamé solution Hoop stress Hertz contact Hertz theory Contact pressure
NAFEMS Benchmarks
Standardized test cases with published reference solutions:
LE1: Elliptic membrane
LE10: Thick plate
T1-T4: Thermal problems
Error Sources
1. Discretization Error
Cause: Finite elements can't represent exact solution
Symptoms:
Results change with mesh refinement
Stress discontinuities between elements
Fix: Refine mesh, use higher-order elements
2. Modeling Error
Cause: Simplified geometry, loads, or physics
Examples:
2D approximation of 3D problem
Ignoring nonlinearities
Simplified boundary conditions
Fix: Better physical modeling
3. Numerical Error
Cause: Floating-point arithmetic, solver tolerance
Symptoms:
Different results on different computers
Sensitivity to units
Fix: Use double precision, tight solver tolerances
4. Human Error
Cause: Mistakes in setup
Common errors:
Wrong units
Incorrect material properties
Missing or wrong boundary conditions
Inverted elements
Fix: Systematic checking, peer review
Mesh Quality Checks
Before solving, verify mesh quality:
Element Quality Metrics
Metric Ideal Acceptable Poor Aspect ratio 1 < 5 > 10 Jacobian ratio 1 > 0.5 < 0.3 Skewness 0° < 45° > 60° Warpage (3D) 0° < 15° > 30°
Where to Refine
Stress concentrations: Holes, notches, sharp cornersLoad application points: Where forces are appliedMaterial boundaries: Interface between different materialsContact regions: Areas in contactExpected high gradients: Based on engineering judgmentResults Checking
Sanity Checks
Always verify:
Equilibrium: Reaction forces = applied loadsSymmetry: Symmetric problems give symmetric resultsBoundary conditions: Displacements match constraintsSign convention: Tension/compression correctOrder of magnitude: Results physically reasonableStress Continuity
At element boundaries:
Displacement: Should be continuous (satisfied by definition)Stress: May be discontinuous (normal for FEA)Large stress jumps indicate:
Mesh too coarse
Poor element quality
Singularity nearby
Error Estimation
Many FEA codes provide error indicators:
ZZ error estimator: Based on stress recovery
$$\eta = \frac{\|\sigma^ - \sigma_h\|}{\|\sigma^ \|}$$
Where $\sigma^*$ is smoothed (recovered) stress.
$\eta < 5\%$: Excellent
$\eta < 10\%$: Good
$\eta > 20\%$: Refine mesh
The Validation Process
Comparison with Experiments
Comparison What It Validates Strain gauge data Local strain accuracy Displacement measurement Global stiffness Modal analysis Natural frequencies Fatigue testing Life prediction
Sources of Discrepancy
Experimental uncertainty:
Measurement error
Specimen variability
Boundary condition approximations
Modeling limitations:
Material property uncertainty
Geometric simplifications
Physics not captured
Acceptable Agreement
Depends on application:
Research: < 5% error
General engineering: < 10%
Preliminary design: < 20%
Best Practices Checklist
Before Analysis
[ ] Understand the physics
[ ] Choose appropriate element types
[ ] Define realistic boundary conditions
[ ] Verify material properties
[ ] Plan mesh refinement strategy
During Analysis
[ ] Check mesh quality metrics
[ ] Monitor solver convergence
[ ] Watch for warnings/errors
[ ] Verify boundary condition application
After Analysis
[ ] Check equilibrium
[ ] Perform convergence study
[ ] Compare with benchmarks if available
[ ] Review stress discontinuities
[ ] Sanity check all results
[ ] Document assumptions and limitations
Common Mistakes
1. Trusting Default Meshes
Problem: Auto-generated mesh may be too coarse
Solution: Always do convergence study
2. Ignoring Singularities
Problem: Stresses at sharp corners → infinity
Solution:
Use stress at distance from corner
Apply fillet radius
Use fracture mechanics approach
3. Over-Constraining
Problem: Too many boundary conditions → artificial stress
Solution: Apply minimum constraints needed
4. Unit Errors
Problem: Mixing units (mm vs m, MPa vs Pa)
Solution: Check unit consistency before solving
5. Blind Faith in Results
Problem: Accepting results without verification
Solution: Always question, always verify
Key Takeaways
Verification: Are we solving the math correctly? (Convergence studies)Validation: Are we modeling the physics correctly? (Experiments)Mesh convergence is essential — never trust a single meshBenchmark problems verify code and methodologyError sources: Discretization, modeling, numerical, humanQuality metrics: Aspect ratio, Jacobian, skewnessSanity checks: Equilibrium, symmetry, order of magnitudeDocument everything: Assumptions, limitations, verification stepsWhat's Next
With verification and validation understood, the final lesson brings everything together with Practical FEA — real-world workflow, tips from industry, and a complete example problem.
