8.5 Mesh generation

Mesh generation is a critical step in scientific computing, transforming continuous domains into discrete elements for numerical simulations. It impacts accuracy and computational cost in applications like finite element analysis and computational fluid dynamics.

Different mesh types, including structured, unstructured, and hybrid, offer varying levels of flexibility and efficiency. Techniques like Delaunay triangulation and advancing front method create meshes, while quality metrics ensure optimal element shape and distribution for accurate results.

Mesh generation overview

• Mesh generation is a crucial step in many scientific computing applications, involving the discretization of a continuous domain into a set of discrete elements
• The quality and efficiency of the mesh directly impact the accuracy and computational cost of numerical simulations
• Key characteristics of meshes include element type (triangles, quadrilaterals, tetrahedra, hexahedra), element size, and element distribution

Importance in scientific computing

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• Meshes provide a discrete representation of the problem domain, enabling the application of numerical methods such as finite element analysis (FEA) and computational fluid dynamics (CFD)
• The accuracy of the numerical solution depends on the quality of the mesh, with higher-quality meshes leading to more accurate results
• Efficient mesh generation techniques can significantly reduce the computational cost of simulations by optimizing the number and distribution of elements

Key characteristics of meshes

• Element type: Meshes can consist of different element types, such as triangles and quadrilaterals in 2D, or tetrahedra and hexahedra in 3D
• Element size: The size of the elements determines the resolution of the mesh and affects the accuracy and computational cost of the simulation
• Element distribution: The distribution of elements should be adapted to the problem, with smaller elements in regions of high gradients or complex geometry, and larger elements in regions of low gradients or simple geometry

Types of meshes

• Meshes can be classified into three main categories based on their structure and element connectivity: structured meshes, unstructured meshes, and hybrid meshes
• The choice of mesh type depends on the complexity of the geometry, the desired accuracy, and the computational resources available

Structured meshes

• Structured meshes consist of a regular grid of elements, with each element having a fixed number of neighbors
• The connectivity between elements is implicit, making structured meshes computationally efficient
• Structured meshes are well-suited for simple geometries and problems with regular domains (rectangular or cylindrical)
• Examples of structured meshes include Cartesian grids and curvilinear grids

Unstructured meshes

• Unstructured meshes consist of irregularly connected elements, allowing for greater flexibility in capturing complex geometries
• The connectivity between elements is explicit and must be stored, leading to higher memory requirements compared to structured meshes
• Unstructured meshes are well-suited for problems with irregular domains or complex boundary conditions
• Examples of unstructured meshes include triangular and tetrahedral meshes

Hybrid meshes

• Hybrid meshes combine structured and unstructured elements to balance the advantages of both approaches
• Structured elements are used in regions with simple geometry or regular flow, while unstructured elements are used in regions with complex geometry or irregular flow
• Hybrid meshes can provide a good compromise between accuracy, flexibility, and computational efficiency
• Examples of hybrid meshes include quadrilateral-dominant meshes and prismatic boundary layer meshes

Mesh generation techniques

• Various techniques have been developed for generating meshes, each with its own strengths and weaknesses
• The choice of mesh generation technique depends on the type of mesh desired, the complexity of the geometry, and the available computational resources

Delaunay triangulation

• Delaunay triangulation is a popular technique for generating unstructured triangular meshes in 2D
• The Delaunay criterion ensures that no point lies inside the circumcircle of any triangle, resulting in well-shaped elements
• Delaunay triangulation can be extended to 3D using tetrahedra, but the resulting meshes may contain poorly shaped elements (slivers)
• Examples of Delaunay triangulation algorithms include the Bowyer-Watson algorithm and the incremental insertion algorithm

Advancing front method

• The advancing front method generates unstructured meshes by progressively adding elements from the boundary of the domain towards the interior
• The method starts with an initial front (boundary edges in 2D or boundary faces in 3D) and creates new elements by connecting front nodes to form optimal triangles or tetrahedra
• The advancing front method can handle complex geometries and generate high-quality elements near boundaries
• The method requires careful front management and intersection checking to ensure a valid mesh

Quadtree/Octree decomposition

• Quadtree (2D) and octree (3D) decomposition methods recursively subdivide the domain into quadrants or octants based on a specified error criterion
• The resulting mesh consists of structured elements (squares or cubes) of different sizes, with smaller elements in regions of high error and larger elements in regions of low error
• Quadtree/Octree meshes are efficient for adaptive mesh refinement and multigrid methods
• The method can be combined with other techniques (triangulation or tetrahedralization) to generate unstructured meshes

Voronoi diagrams

• Voronoi diagrams partition the domain into regions (Voronoi cells) based on a set of discrete points (seeds)
• Each Voronoi cell contains all points that are closer to its seed than to any other seed
• The dual of a Voronoi diagram is a Delaunay triangulation, connecting seeds whose Voronoi cells share an edge
• Voronoi diagrams can be used to generate unstructured meshes by placing seeds strategically and triangulating the resulting Voronoi cells

Mesh quality metrics

• Mesh quality metrics quantify the shape, size, and orientation of elements in a mesh
• High-quality meshes have elements that are well-shaped, appropriately sized, and aligned with the flow or solution gradients
• Poor mesh quality can lead to numerical instabilities, inaccurate solutions, and slow convergence of iterative solvers

Aspect ratio

• The aspect ratio measures the ratio of the longest edge to the shortest edge of an element
• Ideal elements have an aspect ratio close to 1 (equilateral triangles or regular tetrahedra), while highly stretched elements have large aspect ratios
• High aspect ratio elements can cause numerical instabilities and interpolation errors, particularly in regions of high gradients

Skewness

• Skewness measures the deviation of an element from its ideal shape (equilateral triangle, square, or regular tetrahedron)
• Skewed elements have angles that are significantly smaller or larger than the ideal angle (60° for triangles, 90° for quadrilaterals, 60° for tetrahedra)
• Highly skewed elements can lead to inaccurate interpolation and gradient calculations

Orthogonality

• Orthogonality measures the angle between element edges and the lines connecting element centroids
• Ideal elements have orthogonal edges, with angles close to 90°
• Non-orthogonal elements can cause errors in gradient calculations and slow convergence of iterative solvers

Smoothness

• Smoothness measures the change in size and shape of adjacent elements
• A smooth mesh has gradual transitions in element size and shape, without abrupt changes between neighboring elements
• Abrupt changes in element size or shape can cause interpolation errors and numerical instabilities

Mesh adaptation and refinement

• Mesh adaptation and refinement techniques modify the mesh during the simulation to improve accuracy and efficiency
• The goal is to concentrate elements in regions of high solution gradients or complex geometry, while coarsening the mesh in regions of low gradients or simple geometry

Error estimation techniques

• Error estimation techniques quantify the local error in the numerical solution, providing guidance for mesh adaptation
• A posteriori error estimators use the computed solution to estimate the error, while a priori error estimators use information about the problem and the discretization scheme
• Examples of error estimators include gradient-based estimators, residual-based estimators, and goal-oriented estimators

h-adaptivity vs p-adaptivity

• h-adaptivity involves modifying the mesh by adding or removing elements (refining or coarsening) based on the estimated error
• p-adaptivity involves modifying the polynomial degree of the approximation within each element, without changing the mesh topology
• hp-adaptivity combines both approaches, adapting both the mesh size and the polynomial degree to optimize accuracy and efficiency

Adaptive mesh refinement (AMR)

• Adaptive mesh refinement (AMR) is a technique for dynamically refining the mesh in regions of high error or interest
• AMR methods typically use a hierarchy of nested meshes, with each level having a different resolution
• Elements are refined or coarsened based on error indicators, ensuring that the mesh resolution is appropriate for the local solution behavior
• Examples of AMR frameworks include block-structured AMR and tree-based AMR

Mesh generation software

• Mesh generation software tools automate the process of creating meshes from geometric models or data
• These tools offer various features and capabilities, ranging from basic mesh generation to advanced mesh optimization and adaptation

Open-source tools

• Open-source mesh generation tools are freely available and often developed by the academic community
• Examples of open-source tools include Gmsh, Netgen, and TetGen
• Open-source tools provide flexibility and customization options, but may have limited documentation and support

Commercial packages

• Commercial mesh generation packages are developed and maintained by software companies, often as part of larger simulation software suites
• Examples of commercial packages include ANSYS Meshing, Pointwise, and Simcenter STAR-CCM+
• Commercial packages often offer advanced features, user-friendly interfaces, and professional support, but can be expensive

Comparison of features and capabilities

• When choosing a mesh generation tool, it is important to consider factors such as:
  • Supported element types and mesh formats
  • Geometry import and repair capabilities
  • Mesh quality control and optimization options
  • Scripting and automation features
  • Integration with simulation software
  • Performance and scalability for large models
  • Documentation, tutorials, and user support

Challenges in mesh generation

• Despite advances in mesh generation techniques and software, several challenges remain in creating high-quality meshes for complex problems

Complex geometries

• Generating meshes for complex geometries, such as those with small features, sharp corners, or narrow gaps, can be difficult
• Geometry cleanup and repair operations may be necessary to ensure a valid and conforming mesh
• Capturing geometric details while maintaining a reasonable mesh size and quality can be challenging

High aspect ratio elements

• Some problems, such as boundary layer flows or thin structures, require elements with high aspect ratios to accurately capture the solution behavior
• Generating high aspect ratio elements with good quality is difficult, as they are prone to numerical instabilities and interpolation errors
• Special techniques, such as anisotropic meshing or boundary layer meshing, may be needed to handle high aspect ratio elements

Mesh size and computational cost

• The mesh size directly affects the computational cost of the simulation, with larger meshes requiring more memory and longer solution times
• Finding the optimal balance between mesh resolution and computational cost is challenging, particularly for problems with a wide range of scales
• Adaptive mesh refinement and parallel computing can help manage mesh size and computational cost, but require careful implementation and load balancing

Ensuring mesh quality

• Generating a mesh with consistently high quality elements is difficult, particularly for complex geometries or problems with strong solution gradients
• Poor mesh quality can lead to numerical instabilities, inaccurate solutions, and slow convergence of iterative solvers
• Mesh quality metrics and optimization techniques can help improve mesh quality, but may require manual intervention and iterative refinement

Applications of mesh generation

• Mesh generation is a fundamental step in many scientific computing applications, enabling the discretization of complex problems for numerical simulation

Finite element analysis (FEA)

• Finite element analysis (FEA) is a numerical method for solving partial differential equations (PDEs) in solid mechanics, heat transfer, and other engineering fields
• FEA relies on a mesh to discretize the problem domain into elements, over which the solution is approximated using polynomial basis functions
• The accuracy and efficiency of FEA simulations depend on the quality and resolution of the mesh

Computational fluid dynamics (CFD)

• Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical simulation to analyze fluid flow, heat transfer, and related phenomena
• CFD simulations require a mesh to discretize the fluid domain and solve the governing equations (Navier-Stokes equations) over the elements
• Mesh generation for CFD often involves creating boundary layer elements to capture near-wall flow behavior and adapting the mesh to resolve flow features (shocks, vortices, etc.)

Electromagnetics and wave propagation

• Electromagnetic simulations involve solving Maxwell's equations to analyze the propagation of electromagnetic waves in complex structures (antennas, waveguides, etc.)
• Mesh generation for electromagnetics requires creating a conformal mesh that accurately represents the geometry and material interfaces
• The mesh must also be fine enough to resolve the wavelength of the electromagnetic waves, which can lead to large meshes and high computational costs

Multiphysics simulations

• Multiphysics simulations involve coupling multiple physical phenomena, such as fluid-structure interaction, electro-mechanical coupling, or thermo-mechanical coupling
• Mesh generation for multiphysics problems must account for the different length scales and solution behavior of each physics, often requiring specialized techniques (interface capturing, moving meshes, etc.)
• Ensuring compatibility between the meshes for different physics and maintaining mesh quality during the simulation are key challenges in multiphysics mesh generation