Adaptive mesh refinement (AMR) in COMSOL Multiphysics is a powerful technique for improving solution accuracy while keeping computational costs under control. However, expression errors that arise during refinement steps can interrupt simulations, compromise results, and create significant frustration for analysts and engineers. These errors often appear unexpectedly when derived variables, boundary conditions, or material properties are re-evaluated on updated meshes. Correcting them requires methodical diagnostics, a deep understanding of COMSOL’s expression engine, and careful model configuration.
TL;DR: Expression errors in COMSOL adaptive mesh refinement simulations typically stem from invalid variable definitions, domain inconsistencies, or solver sequencing problems. Start by reviewing variable scopes, checking domain selections, and validating derived expressions under mesh updates. Use the Equation View and Probe tools to pinpoint failing terms. Systematically simplifying expressions and refining solver settings will usually resolve persistent issues.
To effectively fix expression errors, it is essential to first understand why they occur during adaptive remeshing.
Why Expression Errors Occur During Adaptive Mesh Refinement
During adaptive refinement, COMSOL modifies the mesh based on error estimates from previous solutions. Newly created elements require all expressions—material properties, source terms, coupling variables, and boundary conditions—to be re-evaluated. If any of those expressions depend on:
- Undefined variables in certain domains
- Invalid geometry selections
- Improper solver sequencing
- Non-converged intermediate results
- Interpolation functions outside their domain range
the remeshing process can trigger expression evaluation failures.
Image not found in postmetaMany users assume the mesh refinement itself is faulty. In reality, expression errors are almost always linked to model definitions, not the mesh algorithm.
Step 1: Read the Full Error Message Carefully
COMSOL provides detailed error logs in the Messages window. Do not stop at the first line. Expand the message completely.
Pay attention to:
- Variable name causing the issue
- Geometric entity number (domain, boundary, edge)
- Expression context (material, physics, weak contribution)
- Evaluation step (initialization, remeshing, nonlinear solver)
For example:
Failed to evaluate variable. Variable: comp1.T, Domain: 4
This indicates a domain-specific definition mismatch. The variable may not be defined in Domain 4 under certain refinement states.
Step 2: Verify Variable Scope and Domain Selections
One of the most common causes of expression errors during AMR is inconsistent domain selection.
Consider the following scenario:
- A variable is defined globally but relies on a domain-specific quantity.
- A boundary condition depends on a variable defined only in selected domains.
- A material property is assigned to some, but not all, newly refined regions.
When the mesh changes, domain numbering may not change, but the solver re-evaluates domain participation. If your definitions are conditional or restricted improperly, evaluation may fail.
Best Practices
- Use logical expressions (e.g., if statements) to guarantee valid evaluation everywhere.
- Avoid domain-limited variables unless absolutely necessary.
- Prefer physics-based definitions over manually specified expressions tied to geometry IDs.
Example correction:
Instead of:
k = k0*(T > 300)
Use:
k = if(T > 300, k0, k_ref)
This ensures the expression always returns a value.
Step 3: Check Derived Values and Probe Dependencies
Adaptive mesh refinement requires recomputation of error estimators. If these rely on derived values or probes that evaluate improperly on new mesh elements, simulations can halt.
To diagnose:
- Temporarily disable derived values.
- Remove probes linked to solution variables.
- Re-run without AMR to confirm baseline stability.
If the model runs normally without adaptive refinement, focus on expressions used in:
- Error estimation features
- Goal-oriented evaluation outputs
- User-defined indicators
Often, these expressions include gradients or higher-order derivatives that may not be well-defined immediately after mesh updates.
Step 4: Use Equation View for Transparent Diagnostics
The Equation View is an advanced but invaluable tool for resolving persistent expression errors.
It allows you to:
- Inspect how COMSOL assembles weak form contributions
- Identify hidden variable transformations
- Understand solver-generated auxiliary expressions
Enable Equation View in the Model Builder and check whether expression errors originate from:
- Automatic scaling factors
- Stabilization terms
- Constraint enforcement equations
This is particularly useful in multiphysics couplings where dependent variables interact across physics interfaces.
Step 5: Confirm Interpolation and External Functions
Interpolation functions are frequent sources of AMR-related errors.
During adaptive remeshing:
- New spatial points may fall outside interpolation ranges.
- Temporal solvers may query unexpected parameter values.
- Material property evaluations may extrapolate inconsistently.
Always configure interpolation functions to:
- Allow extrapolation when physically justified
- Extend defined ranges slightly beyond expected limits
- Use smooth transitions rather than sharp cutoffs
If using imported datasets (e.g., temperature-dependent material curves), verify they fully cover the simulation domain.
Step 6: Review Solver Sequencing and Initial Values
Expression errors can occur because solution-dependent variables are evaluated before the solver has properly updated them.
This is common in:
- Nonlinear stationary solvers
- Time-dependent simulations with remeshing events
- Segregated multiphysics approaches
Solutions include:
- Switch to a fully coupled solver temporarily for testing
- Add consistent initial values for all dependent variables
- Increase nonlinear iteration limits
- Apply continuation methods (parameter ramping)
Adaptive remeshing introduces additional nonlinear complexity. Ensuring solver robustness often eliminates expression failures indirectly.
Step 7: Simplify Expressions Systematically
Complex nested expressions increase the likelihood of evaluation breakdowns during refinement.
For example:
sigma = f(T)*exp(-Ea/(R*T))*sqrt(abs(grad(T)))
If any term becomes undefined (e.g., sqrt of negative value due to numerical noise), the entire solve halts.
A reliable debugging approach:
- Replace the full expression with a constant.
- Verify adaptive refinement runs.
- Reintroduce terms incrementally.
- Identify which sub-expression triggers the fault.
This controlled isolation process quickly identifies unstable mathematical constructs.
Step 8: Validate Mesh Quality Metrics
Although expression errors are usually model-related, mesh quality still matters.
Poor-quality elements can produce:
- Artificially high gradients
- Division by near-zero Jacobians
- Numerically unstable derivative evaluations
Inspect:
- Element quality distribution
- Skewness and aspect ratios
- Minimum element size thresholds
If element quality deteriorates significantly during refinement, consider:
- Adjusting refinement thresholds
- Limiting maximum refinement levels
- Adding local mesh controls
Step 9: Ensure Consistent Physics Coupling Across Interfaces
Multiphysics simulations exacerbate expression errors. Variables may be transferred between physics interfaces using:
- Multiphysics nodes
- General extrusion operators
- Boundary pair couplings
If a source variable is undefined on newly refined geometric entities, expression evaluation fails.
Check that:
- Couplings apply to all relevant domains
- Source and destination meshes remain compatible
- Mapped operators use robust interpolation settings
This is especially critical in fluid-structure interaction, heat transfer with phase change, and electrochemical models.
Preventative Strategies for Stable AMR Simulations
Rather than solving expression errors reactively, implement preventative modeling habits:
- Define variables globally whenever possible
- Use conditional guarding for sensitive operations (division, logarithms, square roots)
- Avoid referencing geometric entity numbers in expressions
- Test models thoroughly without AMR before enabling it
- Implement continuation strategies prior to adaptive refinement
Well-structured models rarely encounter persistent expression breakdowns during remeshing.
Conclusion
Expression errors in COMSOL adaptive mesh refinement simulations are serious but manageable issues. They almost always signal inconsistencies in variable definition, domain applicability, solver timing, or interpolation behavior rather than flaws in the adaptive meshing algorithm itself. By systematically analyzing error messages, verifying variable scopes, simplifying complex expressions, and ensuring solver robustness, engineers can restore simulation stability with confidence.
Adaptive mesh refinement remains one of the most powerful tools for achieving high-fidelity simulation results. With disciplined modeling practices and rigorous debugging techniques, expression errors become solvable technical challenges rather than disruptive obstacles.
Careful diagnostics, structured simplification, and disciplined solver configuration are the keys to long-term AMR reliability.