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?

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?

The Verification Process

1. Code Verification

Ensure the FEA software itself is correct:

• 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:

• Start with a coarse mesh
• Refine the mesh (halve element size)
• Compare key results (stress, displacement, etc.)
• Repeat until results stabilize

• 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)

$$u_{exact} \approx u_h + \frac{u_h - u_{2h}}{2^p - 1}$$

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

Classical Benchmarks

NAFEMS Benchmarks

Standardized test cases with published reference solutions:

• LE1: Elliptic membrane
• LE10: Thick plate
• T1-T4: Thermal problems

Error Sources

1. Discretization Error

• Results change with mesh refinement
• Stress discontinuities between elements

2. Modeling Error

• 2D approximation of 3D problem
• Ignoring nonlinearities
• Simplified boundary conditions

3. Numerical Error

• Different results on different computers
• Sensitivity to units

4. Human Error

• Wrong units
• Incorrect material properties
• Missing or wrong boundary conditions
• Inverted elements

Mesh Quality Checks

Before solving, verify mesh quality:

Element Quality Metrics

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:

$$\eta = \frac{\|\sigma^ - \sigma_h\|}{\|\sigma^\|}$$

Where $\sigma^*$ is smoothed (recovered) stress.

• $\eta < 5\%$: Excellent
• $\eta < 10\%$: Good
• $\eta > 20\%$: Refine mesh

The Validation Process

Comparison with Experiments

Sources of Discrepancy

• Measurement error
• Specimen variability
• Boundary condition approximations

• 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

2. Ignoring Singularities

• Use stress at distance from corner
• Apply fillet radius
• Use fracture mechanics approach

3. Over-Constraining

4. Unit Errors

5. Blind Faith in Results

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.