Meshing in FEA: planning, refinement, convergence and meshing rules

The mesh turns a continuous structure into a finite set of equations a computer can solve, and its quality decides whether your results can be trusted. This Book Byte covers how to plan the meshing, why and how to refine a mesh, how to run a convergence study, the main 2D and 3D meshing rules, and the two rules to follow for mesh transitions.

Why do we mesh?

Reality: a continuous system subjected to continuous phenomena.

• Infinite number of points
• 3 translations and 3 rotations per point
• Infinite number of equations
• Governed by partial differential equations: cannot be solved by a computer

Model: a discrete system.

• Finite number of nodes
• 3 translations and 3 rotations per node
• Finite number of equations
• Governed by matrix equations: can be solved by a computer

Plan the meshing

1. Study the geometry in detail.
2. Clean up the geometry.
3. Select the element types.
4. Select the meshing base.
5. Select the joint modeling techniques.

Then mesh. Each step is detailed in Chapter 8 of the book.

Mesh refinement

Advantages and disadvantages of mesh refinement

A smaller mesh density gives less accuracy, less complexity and less computing time. A greater mesh density gives greater accuracy, more complexity and more computing time: the computing time is proportional to the square of the number of degrees of freedom.

It does not make sense to increase computing time by a factor of 2 or 3 to improve the accuracy by 1%. The key is to know your problem well, to achieve accuracy where it is needed.

Mesh refinement techniques

• Global element size reduction
• Local element size reduction
• Increasing the order of the elements
• Manual mesh adjustment
• Global adaptive mesh refinement
• Local adaptive mesh refinement
• Mesh refinement by splitting a coarse mesh

Each method is detailed in Chapter 8 of the book and in the FEA Academy courses.

Convergence study methodology: why?

• From a numerical accuracy standpoint, a converged mesh is one that produces no significant differences in the result when mesh refinement is introduced.
• Mesh convergence is a concern when producing a model to capture peak stress or strain.
• A convergence study must be conducted for the regions of peak stress, to ensure that the mesh size is fine enough to correctly capture the critical stress.

Convergence study guidelines

• Your organization should maintain a database on mesh convergence applied to its domain of activity.
• A coarse mesh produces inaccurate results. Estimate the deformed shape of your structure for a given loading before running your model.
• Plan the convergence study.
• Demonstrate mesh convergence by refining the mesh between successive runs.
• Do not use the results of a convergence study from another model, except in the case of similar stress gradients.
• Identify the critical regions with a coarse mesh.
• Select the appropriate element types.
• Identify the smallest details that must be captured.

2D meshing rules

Why mesh in 2D instead of 3D?

• If a high-quality mesh is required, a shell mesh is easier to create than a solid one.
• A 2D model uses fewer resources (less RAM, disk space and CPU time), especially for nonlinear analysis and larger models.
• Rather than modeling the thickness with additional nodes and elements, 2D models capture the wall thickness as a mathematical value, resulting in fewer equations to solve.
• The thickness is easier to change, since it is a property of the shell element.
• Post-processing 2D elements is easier and faster than post-processing solid elements.

The four most common 2D meshing errors

• Geometrical deviation (does not generate a fatal error)
• Wrong shell normal orientation (does not generate a fatal error)
• Free edges (can generate a fatal error)
• Duplicate elements (does not generate a fatal error)

3D meshing rules

Two methods of meshing a solid with tetrahedral elements

Hexahedral meshing techniques

• The hexahedral element (also called brick or HEX element) has been used intensively by many organizations to mesh solid parts.
• Over the years, however, the algorithms of TET elements, especially quadratic elements (TET10), have improved significantly; the same accuracy can now be achieved with TET10 or HEX8 elements.
• There are still advantages in using brick elements. The main one is that a brick mesh generates far fewer elements and nodes than a tetra mesh (in the order of ½ for elements and 1/50 for nodes), which gives an enormous reduction in computing time and easier handling of the model in pre- and post-processing.
• The great disadvantage of brick elements is that meshing a complex solid with them is time-consuming and requires experience and patience, because brick meshing relies on manual commands: automatic hexahedral meshing is generally not available in pre-processing software.

How to make a mesh transition

Mesh transitions may be used to:

• Connect regions with different mesh densities
• Connect different element types, for example a beam element to a solid element
• Make the transitions required to model the geometry of the structure

To ensure your FEA produces good results, follow these two basic rules when a mesh transition is required:

1. Never create a mesh transition in a region with a stress gradient.
2. Never create a mesh transition in a region of interest.

A mesh transition is not a wrong modeling technique, but it must not occur near a region of interest.

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