Basic Principles and General Methods of Finite Element Meshing

Basic Principles and General Methods of Finite Element Meshing

Basic Principles and General Methods of Finite Element Meshing

This article first analyzes the basic principles of finite element meshing, then scientifically classifies current typical meshing methods. It systematically examines the mechanisms, characteristics, and application scopes of various methods through examples, including mapping methods, grid-based methods, node-connecting element methods, topological decomposition methods, geometric decomposition methods, and sweeping methods. Finally, it elaborates on current research hotspots, summarizes hexahedral and surface meshing technologies, and prospects the development trends of finite element meshing.

Introduction

Finite element meshing is a crucial step in finite element numerical simulation analysis, directly affecting the accuracy of subsequent numerical calculation results. Meshing involves the shape and topological type of elements, element types, selection of mesh generators, mesh density, element numbering, and geometric primitives. In finite element numerical solutions, equivalent nodal forces, stiffness matrices, and mass matrices of elements are generated using numerical integration. Continuous elements, as well as in-plane components of shell, plate, and beam elements, use Gauss integration, while the thickness direction of shell, plate, and beam elements uses Simpson integration.

Basic Principles of Finite Element Meshing

The core idea of the finite element method is to discretize a structure—approximating a continuous body with simplified geometric elements—and solve it comprehensively based on deformation compatibility conditions. Therefore, finite element meshing must consider both accurate description of the geometric shape of each object and precise representation of deformation gradients. To establish a correct and reasonable finite element model, the following basic principles for meshing should be considered:

1. Mesh Quantity
   Mesh quantity directly affects calculation accuracy and time consumption. Increasing mesh quantity improves accuracy but also increases computation time. When the number of meshes is small, adding more meshes significantly enhances accuracy without a notable increase in time. However, once the mesh quantity reaches a certain level, further increases yield minimal accuracy improvements while drastically increasing computation time. Thus, a balance between these two factors is necessary when determining mesh quantity.

2. Mesh Density
   To adapt to the distribution characteristics of calculated data (such as stress), different parts of a structure require meshes of varying sizes. For example, areas near holes with stress concentration need refined meshes, while surrounding areas with smaller stress gradients can use coarser meshes. This reflects the principle of varying mesh density: dense meshes are needed in regions with large gradients of calculated data to better capture changes, while sparse meshes are used in regions with small gradients to reduce model size.

3. Element order
   Element order is closely related to finite element calculation accuracy. Elements are generally linear, quadratic, or cubic, with quadratic and cubic elements referred to as high-order elements. High-order elements have curved or curved-surface boundaries that better approximate the structure’s curved boundaries, and their high-degree interpolation functions can more accurately approximate complex field functions. However, higher-order elements increase the number of nodes, leading to larger model sizes for the same mesh quantity. Thus, a trade-off between accuracy and computation time is required.

4. Element Shape
   The quality of element shapes significantly impacts calculation accuracy; poorly shaped elements may even terminate computations. Key indicators for evaluating element shape include:

   • Aspect ratio (side length ratio, area ratio, or volume ratio), with reference to regular triangles, regular tetrahedrons, and regular hexahedrons.
   • Distortion, including in-plane torsion and out-of-plane warpage.
   • Node numbering, which affects the bandwidth and wavefront factor of the global stiffness matrix during solution, thereby influencing computation time and memory usage.

5. Element Compatibility
   Element compatibility means that forces and moments on an element can be transmitted to adjacent elements through nodes. To ensure compatibility:

   • Nodes of one element must also be nodes of adjacent elements (not internal or boundary-only nodes).
   • Common nodes of adjacent elements must have the same degree of freedom (DOF) properties. Note that meshes with the same DOFs are not necessarily compatible.

General Methods of Mesh Generation

Finite element meshing methods are difficult to categorize precisely and can be classified by element type, dimension, and automation level.

1. Mapping Method
   The basic idea is bidirectional mapping between the actual shape and a standard shape, involving three steps:

   • Map the physical domain to be meshed to a parameter space using appropriate mapping functions based on boundary parameter equations, forming a regular parameter domain.
   • Mesh the parameter domain.
   • Reverse-map the mesh from the parameter space to Euclidean space to generate the actual mesh.

   Limitations of this method include:

   • Poor geometric feature orientation, making full automation difficult (especially for 3D regions).
   • Weak local mesh control.
   • Strong interdependence of mesh density between mapped blocks (adjusting density in one block requires adjustments in others).
   • Poor adaptability to complex shapes, requiring pre-division of the target domain into mappable sub-regions, which is tedious and requires extensive manual interaction.

2. Grid-Based Method (Space Decomposition Method)
   The algorithm flow is:

   • Cover the object with a set of non-intersecting grids, placing nodes either at regular grid points or randomly within grid cells.
   • Detect intersections between grids and the object, retaining grids fully or partially within the target region and deleting those entirely outside.
   • Adjust, clip, or re-decompose grids intersecting the object boundary to better approximate the target region.
   • Mesh internal and boundary grids to obtain the finite element mesh for the entire target region.

3. Node-Connecting Element Method
   This method generally involves two steps:
   - Place the nodes evenly on the boundary and within the valid region according to mesh density requirements.

   • Connect these nodes into triangular or tetrahedral meshes based on specific criteria.

4. Topological Decomposition Method
   Proposed by Wordenwaber from the University of Cambridge, UK, this method focuses on topological factors rather than specific element shapes. It assumes all mesh vertices are boundary vertices of the target, then uses a triangulation algorithm to cover the target with the minimum number of triangles.

5. Geometric Decomposition Method
   Its key feature is simultaneous generation of nodes and elements. This method heavily considers the geometric characteristics of the domain to be meshed, ensuring high-quality elements.

6. Sweeping Method
   This method generates high-dimensional meshes by rotating, sweeping, or extruding basic discrete elements. It is relatively simple to implement and is featured in most commercial CAD software and finite element preprocessing tools. However, it is only suitable for simple 3D shapes and relies heavily on human-machine interaction, resulting in low automation.

Research Hotspots

In recent years, finite element analysis has been widely applied in various engineering fields, and the theoretical basis of meshing technology has become increasingly mature. Research in finite element meshing has shifted from 2D planar problems to 3D solids, with focus moving from triangular (tetrahedral) meshes to quadrilateral (hexahedral) meshes, emphasizing fully automatic mesh generation and adaptive meshing.

1. Hexahedral Meshing
   Current algorithms for hexahedral element mesh generation include mapping element methods, element conversion methods, grid-based methods, multi-subregion methods, sweeping methods, and projection methods:

   • Mapping element method: First divide the 3D solid into several large 20-node hexahedral regions, then use shape function mapping to convert each region into small 8-node hexahedral elements.
   • Element conversion method: Convert other element types into hexahedral elements.
   • Grid-based method: Generate a hexahedral mesh template and overlay it onto the 3D solid to be meshed.
   • Multi-subregion method: Decompose the complex target domain into simple subregions, mesh each subregion with hexahedrons, then combine them into a global mesh.
   • Sweeping method: Generate hexahedral meshes by rotating, sweeping, or extruding 2D quadrilateral meshes.
   • Projection method: Use high-quality tetrahedral meshes as projection templates, controlling projection paths and scaling through correspondence between template nodes and key surface points of the target solid.

2. Surface Meshing
   Thin-shell structures commonly used in engineering are composed of free-form surfaces. 3D surfaces, a degenerate form of 3D solids, have wide applications in finite element meshing. Current surface meshing methods are roughly divided into direct methods and mapping methods:

   • Direct method: Meshes directly in the physical space of the surface, referencing local geometric features and adopting different strategies based on surface conditions.
   • Mapping method: First maps surface boundaries to a 2D parameter space, meshes the parameter space, then reverses the mapping to the physical space to form the surface mesh.

Prospect

Currently, finite element mesh generation technology is quite mature, enabling fully automatic 3D meshing. However, significant research space remains. Challenges such as meshing efficiency and element quality have not been fully resolved and require further improvement.

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