Mesh Convergence Studies for FEA

Understanding Mesh Convergence in Finite Element Analysis

Finite Element Analysis (FEA) has become an indispensable tool for engineering firms across Atlantic Canada, enabling the simulation of complex structural, thermal, and fluid dynamics problems before physical prototypes are ever built. However, the accuracy of any FEA simulation is fundamentally dependent on the quality and refinement of the computational mesh. This is where mesh convergence studies become critical—they provide the mathematical assurance that your simulation results are reliable and independent of arbitrary mesh choices.

At its core, a mesh convergence study systematically refines the computational mesh until the solution stabilises within acceptable tolerances. Without performing this essential verification step, engineers risk making design decisions based on results that may be significantly influenced by numerical errors rather than actual physical behaviour. For industries prevalent in Nova Scotia and the Maritime provinces—including offshore energy, shipbuilding, and manufacturing—such errors can have serious safety and economic implications.

The Fundamentals of Mesh Refinement

The finite element method works by discretising a continuous domain into smaller, simpler elements. These elements approximate the governing differential equations through interpolation functions, and the accuracy of this approximation improves as elements become smaller and more numerous. However, increasing mesh density comes at a computational cost, making it essential to find the optimal balance between accuracy and efficiency.

Element Types and Their Characteristics

Different element types exhibit varying convergence behaviours that engineers must understand:

• Linear tetrahedral elements (TET4): These first-order elements are computationally inexpensive but converge slowly, often requiring very fine meshes to achieve accurate stress results. They are prone to shear locking in bending-dominated problems.

• Quadratic tetrahedral elements (TET10): With mid-side nodes, these elements provide significantly better accuracy per element and are generally recommended for structural analysis despite their higher computational cost.

• Linear hexahedral elements (HEX8): These elements offer excellent accuracy-to-cost ratios when geometry permits their use, though they require more careful mesh generation.

• Quadratic hexahedral elements (HEX20): The gold standard for structural analysis, these elements provide superior convergence characteristics and are particularly effective for capturing stress concentrations.

Mesh Quality Metrics

Beyond element size, mesh quality significantly impacts convergence behaviour. Key metrics include:

• Aspect ratio: The ratio of the longest to shortest element dimension should ideally remain below 3:1 for accurate results, though values up to 10:1 may be acceptable in regions of lower stress gradients.

• Jacobian ratio: This measure of element distortion should remain above 0.6 for reliable results, with values below 0.3 indicating severely distorted elements that may produce erroneous solutions.

• Skewness: Elements should maintain angles close to their ideal values (90° for quadrilaterals, 60° for triangles), with deviations beyond 45° considered problematic.

Conducting a Systematic Convergence Study

A properly executed mesh convergence study follows a methodical approach that ensures reliable results while optimising computational resources. This process is particularly important for engineering projects in Atlantic Canada, where tight project timelines and budget constraints demand efficient simulation workflows.

Step-by-Step Methodology

The recommended procedure for conducting a mesh convergence study involves the following steps:

• Define convergence criteria: Establish acceptable tolerance levels for key output parameters. Typical targets include 2-5% variation in maximum stress or displacement between successive mesh refinements.

• Select monitoring locations: Identify critical points where results will be tracked throughout the study. These should include areas of expected stress concentration, boundary condition application points, and locations of design interest.

• Create initial coarse mesh: Begin with a relatively coarse mesh that captures the basic geometry and loading conditions. This serves as the baseline for refinement.

• Systematic refinement: Progressively refine the mesh, typically doubling the element count or halving the characteristic element size with each iteration.

• Document and analyse results: Record all relevant output parameters for each mesh iteration and plot convergence curves to visualise the stabilisation of results.

Practical Refinement Strategies

Rather than uniformly refining the entire model, experienced analysts employ targeted refinement strategies that concentrate computational resources where they matter most:

Global-to-local refinement: Start with global mesh parameters, then apply local refinement to critical regions. For a typical pressure vessel analysis, the shell regions might use 10mm elements while nozzle intersections require 2mm elements to capture stress concentrations accurately.

Adaptive mesh refinement (AMR): Many modern FEA packages offer automated refinement based on error estimates from initial solutions. While convenient, these automated approaches should be validated against manual convergence studies for critical applications.

Submodelling techniques: For large assemblies common in shipbuilding and offshore structures—industries vital to Nova Scotia's economy—submodelling allows detailed analysis of local regions using boundary conditions extracted from coarser global models.

Interpreting Convergence Results

Understanding how to interpret convergence data separates competent analysts from exceptional ones. The behaviour of different output quantities during mesh refinement provides crucial insights into simulation reliability.

Convergence Behaviour of Different Quantities

Not all output parameters converge at the same rate:

• Displacements: These primary solution variables typically converge quickly, often stabilising within 1% after only 2-3 refinement iterations. Displacement results from relatively coarse meshes are generally reliable.

• Stresses: As derived quantities involving spatial derivatives of displacements, stresses converge more slowly and require finer meshes. Stress results may require 4-6 refinement iterations to achieve comparable accuracy.

• Reaction forces: These integral quantities converge rapidly and can serve as useful verification metrics alongside displacement results.

• Singular stress values: At geometric discontinuities like sharp corners, theoretical stress values approach infinity, meaning stresses will never truly converge. Engineers must apply appropriate interpretation techniques such as stress linearisation or hot-spot stress methods.

Recognising Problematic Convergence Patterns

Several convergence patterns indicate potential issues requiring investigation:

Oscillatory convergence: If results oscillate rather than monotonically approaching a stable value, this may indicate mesh quality issues, inappropriate element types, or problems with boundary condition application.

Divergent behaviour: Results that increase without bound as the mesh refines often indicate stress singularities or modelling errors that must be addressed before meaningful results can be obtained.

Premature convergence: Apparent stabilisation followed by significant changes with further refinement suggests the initial mesh was too coarse to capture important physical phenomena.

Application Examples from Maritime Industries

The principles of mesh convergence apply across diverse engineering applications common to Nova Scotia and Atlantic Canada. Understanding how these principles manifest in specific contexts helps engineers develop appropriate analysis strategies.

Offshore Platform Structural Analysis

For offshore structures operating in the demanding North Atlantic environment, mesh convergence studies are essential for fatigue-critical connections. A typical tubular joint analysis might begin with 25mm shell elements for global response, then refine to 3-5mm solid elements at the weld toe region where fatigue cracks initiate. Convergence criteria typically target hot-spot stress values within 3% variation, as these directly impact calculated fatigue life.

Marine Vessel Hull Structures

Hull structural analysis for vessels built in Maritime shipyards requires careful attention to mesh convergence in way of structural discontinuities. Frame-to-shell connections, hatch corners, and equipment foundations all require local mesh refinement. Industry standards such as those from classification societies typically specify minimum element sizes relative to plate thickness—commonly 50mm × 50mm elements for plate thicknesses up to 25mm, with proportional refinement for thicker structures.

Industrial Equipment and Pressure Vessels

Manufacturing facilities throughout Nova Scotia rely on pressure vessels and industrial equipment that must meet ASME or CSA standards. Convergence studies for nozzle-to-shell intersections typically require element sizes of t/4 to t/2 (where t is the local wall thickness) to accurately capture membrane and bending stress distributions for code compliance assessment.

Best Practices and Quality Assurance

Implementing robust mesh convergence practices requires organisational commitment to quality assurance processes that ensure consistent, reliable simulation results.

Documentation Requirements

Professional engineering practice demands thorough documentation of mesh convergence studies:

• Mesh parameters: Record global and local element sizes, element types, and mesh quality statistics for each iteration.

• Convergence plots: Include graphical representation of key parameters versus mesh density or element count.

• Final mesh justification: Document the rationale for selecting the production mesh, including convergence criteria achieved.

• Computational resources: Note solution times and hardware specifications to inform future project planning.

Verification and Validation Context

Mesh convergence studies form part of the broader verification and validation (V&V) framework essential for simulation credibility. Verification confirms that the computational model correctly solves the governing equations (solving the equations right), while validation confirms that those equations appropriately represent physical reality (solving the right equations). Mesh convergence specifically addresses discretisation error within the verification domain.

Common Pitfalls to Avoid

Experience with FEA projects across Atlantic Canada has identified several common mistakes that compromise mesh convergence studies:

• Insufficient refinement range: Testing only 2-3 mesh densities may not reveal true convergence behaviour. A minimum of 4-5 mesh iterations is recommended.

• Ignoring mesh quality degradation: Aggressive refinement can sometimes degrade element quality, offsetting accuracy gains from smaller elements.

• Inconsistent refinement approach: Changing element types or mesh topology between iterations can confound convergence interpretation.

• Overlooking boundary layer effects: In fluid dynamics or contact analyses, boundary layer mesh resolution requires separate convergence consideration.

Partner with Atlantic Canada's Trusted Engineering Experts

Mesh convergence studies represent just one aspect of the rigorous analytical approach required for reliable engineering simulation. The complexity of modern FEA projects demands expertise developed through years of practical experience across diverse industries and applications.

Sangster Engineering Ltd. brings decades of professional engineering experience to clients throughout Nova Scotia, Atlantic Canada, and beyond. Our team understands the unique challenges facing Maritime industries—from offshore energy development to marine transportation and industrial manufacturing. We combine advanced simulation capabilities with practical engineering judgement developed through countless successful projects.

Whether you require comprehensive FEA services, independent verification of existing analyses, or guidance in developing your organisation's simulation capabilities, our Amherst-based team is ready to assist. Contact Sangster Engineering Ltd. today to discuss how our expertise in finite element analysis and mesh convergence methodology can support your next project's success.

Partner with Sangster Engineering

At Sangster Engineering Ltd. in Amherst, Nova Scotia, we bring decades of engineering experience to every project. Serving clients across Atlantic Canada and beyond.

Contact us today to discuss your engineering needs.