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:

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:

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:

$h$ = 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 corners
Load application points: Where forces are applied
Material boundaries: Interface between different materials
Contact regions: Areas in contact
Expected high gradients: Based on engineering judgment

Results Checking

Sanity Checks

Always verify:

Equilibrium: Reaction forces = applied loads
Symmetry: Symmetric problems give symmetric results
Boundary conditions: Displacements match constraints
Sign convention: Tension/compression correct
Order of magnitude: Results physically reasonable

Stress 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 mesh
Benchmark problems verify code and methodology
Error sources: Discretization, modeling, numerical, human
Quality metrics: Aspect ratio, Jacobian, skewness
Sanity checks: Equilibrium, symmetry, order of magnitude
Document everything: Assumptions, limitations, verification steps

What'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.