Math gifts

- Art Gallery -

Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI , depending on the complexity of the domain and the type of mesh desired. The goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine (have small elements) in areas that are important for the subsequent calculations.

Meshes are used for rendering to a computer screen and for physical simulation such as finite element analysis or computational fluid dynamics. Meshes are composed of simple cells like triangles because, e.g., we know how to perform operations such as finite element calculations (engineering) or ray tracing (computer graphics) on triangles, but we do not know how to perform these operations directly on complicated spaces and shapes such as a roadway bridge. We can simulate the strength of the bridge, or draw it on a computer screen, by performing calculations on each triangle and calculating the interactions between triangles.

A major distinction is between structured and unstructured meshing. In structured meshing the mesh is a regular lattice, such as an array, with implied connectivity between elements. In unstructured meshing, elements may be connected to each other in irregular patterns, and more complicated domains can be captured. This page is primarily about unstructured meshes. While a mesh may be a triangulation, the process of meshing is distinguished from point set triangulation in that meshing includes the freedom to add vertices not present in the input. "Facetting" (triangulating) CAD models for drafting has the same freedom to add vertices, but the goal is to represent the shape accurately using as few triangles as possible and the shape of individual triangles is not important. Computer graphics renderings of textures and realistic lighting conditions use meshes instead.

Many mesh generation software is coupled to a CAD system defining its input, and simulation software for taking its output. The input can vary greatly but common forms are Solid modeling, Geometric modeling, NURBS, B-rep, STL or a point cloud.
Terminology

The terms "mesh generation," "grid generation," "meshing," " and "gridding," are often used interchangeably, although strictly speaking the latter two are broader and encompass mesh improvement: changing the mesh with the goal of increasing the speed or accuracy of the numerical calculations that will be performed over it. In computer graphics rendering, and mathematics, a mesh is sometimes referred to as a tessellation.

Mesh faces (cells, entities) have different names depending on their dimension and the context in which the mesh will be used. In finite elements, the highest-dimensional mesh entities are called "elements," "edges" are 1D and "nodes" are 0D. If the elements are 3D, then the 2D entities are "faces." In computational geometry, the 0D points are called vertices. Tetrahedra are often abbreviated as "tets"; triangles are "tris", quadrilaterals are "quads" and hexahedra (topological cubes) are "hexes."
Techniques
A mesh generated from an implicit surface

Many meshing techniques are built on the principles of the Delaunay triangulation, together with rules for adding vertices, such as Ruppert's algorithm. A distinguishing feature is that an initial coarse mesh of the entire space is formed, then vertices and triangles are added. In contrast, advancing front algorithms start from the domain boundary, and add elements incrementally filling up the interior. Hybrid techniques do both. A special class of advancing front techniques creates thin boundary layers of elements for fluid flow. In structured mesh generation the entire mesh is a lattice graph, such as a regular grid of squares. Structured mesh generation for regular grids is an entire field itself, with mathematical techniques applied to ensure high-polynomial-order grid lines follow the solution space smoothly and accurately. In block-structured meshing, the domain is divided into large subregions, each of which is a structured mesh. Some direct methods start with a block-structured mesh and then move the mesh to conform to the input; see Automatic Hex-Mesh Generation based on polycube. Another direct method is to cut the structured cells by the domain boundary; see sculpt based on Marching cubes.

Some types of meshes are much more difficult to create than others. Simplicial meshes tend to be easier than cubical meshes. An important category is generating a hex mesh conforming to a fixed quad surface mesh; a research subarea is studying the existence and generation of meshes of specific small configurations, such as the tetragonal trapezohedron. Because of the difficulty of this problem, the existence of combinatorial hex meshes has been studied apart from the problem of generating good geometric realizations. While known algorithms generate simplicial meshes with guaranteed minimum quality, such guarantees are rare for cubical meshes, and many popular implementations generate inverted (inside-out) hexes from some inputs.

Meshes are often created in serial on workstations, even when subsequent calculations over the mesh will be done in parallel on super-computers. This is both because of the limitation that most mesh generators are interactive, and because mesh generation runtime is typically insignificant compared to solver time. However, if the mesh is too large to fit in the memory of a single serial machine, or the mesh must be changed (adapted) during the simulation, meshing is done in parallel.
Types of meshes

See also Types of mesh.
Cell topology

Usually the cells are polygonal or polyhedral and form a mesh that partitions the domain. Important classes of two-dimensional elements include triangles (simplices) and quadrilaterals (topological squares). In three-dimensions the most-common cells are tetrahedra (simplices) and hexahedra (topological cubes). Simplicial meshes may be of any dimension and include triangles (2D) and tetrahedra (3D) as important instances. Cubical meshes is the pan-dimensional category that includes quads (2D) and hexes (3D). In 3D, 4-sided pyramids and 3-sided prisms appear in conformal meshes of mixed cell type.
Cell dimension

The mesh is embedded in a geometric space that is typically two or three dimensional, although sometimes the dimension is increased by one by adding the time-dimension. Higher dimensional meshes are used in niche contexts. One-dimensional meshes are useful as well. A significant category is surface meshes, which are 2D meshes embedded in 3D to represent a curved surface.
Duality

Dual graphs have several roles in meshing. One can make a polyhedral Voronoi diagram mesh by dualizing a Delaunay triangulation simplicial mesh. One can create a cubical mesh by generating an arrangement of surfaces and dualizing the intersection graph; see spatial twist continuum. Sometimes both the primal mesh and its dual mesh are used in the same simulation; see Hodge star operator. This arises from physics involving divergence and curl (mathematics) operators, such as flux & vorticity or electricity & magnetism, where one variable naturally lives on the primal faces and its counterpart on the dual faces.
Mesh type by use

Three-dimensional meshes created for finite element analysis need to consist of tetrahedra, pyramids, prisms or hexahedra. Those used for the finite volume method can consist of arbitrary polyhedra. Those used for finite difference methods consist of piecewise structured arrays of hexahedra known as multi-block structured meshes. 4-sided pyramids are useful to conformally connect hexes to tets. 3-sided prisms are used for boundary layers conforming to a tet mesh of the far-interior of the object.

Surface meshes are useful in computer graphics where the surfaces of objects reflect light (also subsurface scattering) and a full 3D mesh is not needed. Surface meshes are also used to model thin objects such as sheet metal in auto manufacturing and building exteriors in architecture. High (e.g., 17) dimensional cubical meshes are common in astrophysics and string theory.
Mathematical definition and variants

What is the precise definition of a mesh? There is not a universally-accepted mathematical description that applies in all contexts. However, some mathematical objects are clearly meshes: a simplicial complex is a mesh composed of simplices. Most polyhedral (e.g. cubical) meshes are conformal, meaning they have the cell structure of a CW complex, a generalization of a simplicial complex. A mesh need not be simplicial because an arbitrary subset of nodes of a cell is not necessarily a cell: e.g., three nodes of a quad does not define a cell. However, two cells intersect at cells: e.g. a quad does not have a node in its interior. The intersection of two cells may be several cells: e.g., two quads may share two edges. An intersection being more than one cell is sometimes forbidden and rarely desired; the goal of some mesh improvement techniques (e.g. pillowing) is to remove these configurations. In some contexts, a distinction is made between a topological mesh and a geometric mesh whose embedding satisfies certain quality criteria.

Important mesh variants that are not CW complexes include non-conformal meshes where cells do not meet strictly face-to-face, but the cells nonetheless partition the domain. An example of this is an octree, where an element face may be partitioned by the faces of adjacent elements. Such meshes are useful for flux-based simulations. In overset grids, there are multiple conformal meshes that overlap geometrically and do not partition the domain; see e.g., Overflow, the OVERset grid FLOW solver. So-called meshless or meshfree methods often make use of some mesh-like discretization of the domain, and have basis functions with overlapping support. Sometimes a local mesh is created near each simulation degree-of-freedom point, and these meshes may overlap and be non-conformal to one another.
High-order elements

Many meshes use linear elements, where the mapping from the abstract to realized element is linear, and mesh edges are straight segments. Higher order polynomial mappings are common, especially quadratic. A primary goal for higher-order elements is to more accurately represent the domain boundary, although they have accuracy benefits in the interior of the mesh as well. One of the motivations for cubical meshes is that linear cubical elements have some of the same numerical advantages as quadratic simplicial elements. In the isogeometric analysis simulation technique, the mesh cells containing the domain boundary use the CAD representation directly instead of a linear or polynomial approximation.
Mesh improvement

Improving a mesh involves changing its discrete connectivity, the continuous geometric position of its cells, or both. For discrete changes, for simplicial elements one swaps edges and inserts/removes nodes. The same kinds of operations are done for cubical (quad/hex) meshes, although there are fewer possible operations and local changes have global consequences. E.g., for a hexahedral mesh, merging two nodes creates cells that are not hexes, but if diagonally-opposite nodes on a quadrilateral are merged and this is propagated into collapsing an entire face-connected column of hexes, then all remaining cells will still be hexes. In adaptive mesh refinement, elements are split (h-refinement) in areas where the function being calculated has a high gradient. Meshes are also coarsened, removing elements for efficiency. The multigrid method does something similar to refinement and coarsening to speed up the numerical solve, but without actually changing the mesh.

For continuous changes, nodes are moved, or the higher-dimensional faces are moved by changing the polynomial order of elements. Moving nodes to improve quality is called "smoothing" or "r-refinement" and increasing the order of elements is called "p-refinement." Nodes are also moved in simulations where the shape of objects change over time. This degrades the shape of the elements. If the object deforms enough, the entire object is remeshed and the current solution mapped from the old mesh to the new mesh.
Practitioners

The field is highly interdisciplinary, with contributions found in mathematics, computer science, and engineering. Meshing R&D is distinguished by an equal focus on discrete and continuous math and computation, as with computational geometry, but in contrast to graph theory (discrete) and numerical analysis (continuous). Mesh generation is deceptively difficult: it is easy for humans to see how to create a mesh of a given object, but difficult to program a computer to make good decisions for arbitrary input a priori. There is an infinite variety of geometry found in nature and man-made objects. Many mesh generation researchers were first users of meshes. Mesh generation continues to receive widespread attention, support and funding because the human-time to create a mesh dwarfs the time to set up and solve the calculation once the mesh is finished. This has always been the situation since numerical simulation and computer graphics were invented, because as computer hardware and simple equation-solving software have improved, people have been drawn to larger and more complex geometric models in a drive for greater fidelity, scientific insight, and artistic expression.
Community activities

Literature on Mesh Generation list website
Journals

Meshing research is published in a broad range of journals. This is in keeping with the interdisciplinary nature of the research required to make progress, and also the wide variety of applications that make use of meshes. About 150 meshing publications appear each year across 20 journals, with at most 20 publications appearing in any one journal. There is no journal whose primary topic is meshing. The journals that publish at least 10 meshing papers per year are in bold.

Advances in Engineering Software
American Institute of Aeronautics and Astronautics Journal (AIAAJ)
Algorithmica
Applied Computational Electromagnetics Society Journal
Applied Numerical Mathematics
Astronomy and Computing
Computational Geometry: Theory and Applications
Computer-Aided Design (CAD) often including a special issue devoted to extended papers from the IMR (see conferences below)
Computer Aided Geometric Design (CAGD)
Computer Graphics Forum (Eurographics)
Computer Methods in Applied Mechanics and Engineering
Discrete and Computational Geometry
Engineering with Computers
Finite Elements in Analysis and Design
International Journal for Numerical Methods in Engineering (IJNME)
International Journal for Numerical Methods in Fluids
International Journal for Numerical Methods in Biomedical Engineering
International Journal of Computational Geometry & Applications
Journal of Computational Physics (JCP)
Journal on Numerical Analysis
Journal on Scientific Computing (SISC)
Transactions on Graphics (ACM TOG)
Transactions on Mathematical Software (ACM TOMS)
Transactions on Visualization and Computer Graphics (IEEE TVCG)
Lecture Notes in Computational Science and Engineering (LNCSE)
Computational Mathematics and Mathematical Physics (CMMP)

Conferences

Conferences whose primary topic is meshing are in bold.

Conferences, Workshops, Summerschools list website

Aerospace Sciences Meeting AIAA (15 meshing talks/papers)
Canadian Conference on Computational Geometry CCCG
CompIMAGE: International Symposium Computational Modeling of Objects Represented in Images
Computational Fluid Dynamics Conference AIAA
Computational Fluid Dynamics Conference ECCOMAS
Computational Science & Engineering CS&E
Conference on Numerical Grid Generation ISGG
Eurographics Annual Conference (Eurographics) (proceedings in Computer Graphics Forum)
Geometric & Physical Modeling SIAM
International Conference on Isogeometric Analysis IGA
International Meshing Roundtable IMR
International Symposium on Computational Geometry SoCG
Numerical Geometry, Grid Generation and Scientific Computing (NUMGRID) (proceedings in Lecture Notes in Computational Science and Engineering)
SIGGRAPH (proceedings in ACM Transactions on Graphics)
Symposium on Geometry Processing SGP (Eurographics) (proceedings in Computer Graphics Forum)
World Congress on Engineering

Workshops

Workshops whose primary topic is meshing are in bold.

Conference on Geometry: Theory and Applications CGTA
European Workshop on Computational Geometry EuroCG
Fall Workshop on Computational Geometry
Finite Elements in Fluids FEF
MeshTrends Symposium (in WCCM or USNCCM alternate years)
Polytopal Element Methods in Mathematics and Engineering
Tetrahedron workshop

Resources
Mesh generators

Many commercial product descriptions emphasize simulation rather than the meshing technology that enables simulation.

Lists of mesh generators (external):
Free/open source mesh generators
Public domain and commercial mesh generators
ANSA Pre-processor
ANSYS
CD-adapco and Siemens DISW
Comet Solutions
CGAL Computational Geometry Algorithms Library
Mesh generation
2D Conforming Triangulations and Meshes
3D Mesh Generation
CUBIT CUBIT
Gmsh
Hextreme meshes
MeshLab
MSC Software
Omega_h Tri/Tet Adaptivity
Open FOAM Mesh generation and conversion
Salome Mesh module
TetGen
TetWild
TRIANGLE Mesh generation and Delaunay triangulation

Multi-domain partitioned mesh generators

These tools generate the partitioned meshes required for multi-material finite element modelling.

MDM(Multiple Domain Meshing) generates unstructured tetrahedral and hexahedral meshes for a composite domain made up of heterogeneous materials, automatically and efficiently

QMDM (Quality Multi-Domain Meshing) produces a high quality, mutually consistent triangular surface meshes for multiple domains

QMDMNG, (Quality Multi-Domain Meshing with No Gap), produces a quality meshes with each one a two-dimensional manifold and no gap between two adjacent meshes.

SOFA_mesh_partitioning_tools generates partitioned tetrahedral meshes for multi-material FEM, based on CGAL.

Articles

Another Fine Mesh, MeshTrends Blog, Pointwise
Mesh Generation & Grid Generation on the Web
Mesh Generation group on LinkedIn

Research groups and people

Mesh Generation people on Google Scholar
David Bommes, Computer Graphics Group, University of Bern
David Eppstein's Geometry in Action, Mesh Generation
Jonathan Shewchuk's Meshing and Triangulation in Graphics, Engineering, and Modeling
Scott A. Mitchell
Robert Schneiders

Models and meshes

Useful models (inputs) and meshes (outputs) for comparing meshing algorithms and meshes.

HexaLab has models and meshes that have been published in research papers, reconstructed or from the original paper.
Princeton Shape Benchmark
Shape Retrieval Contest SHREC has different models each year, e.g.,
Shape Retrieval Contest of Non-rigid 3D Watertight Meshes 2011
Thingi10k meshed models from the Thingiverse

CAD models

Modeling engines linked with mesh generation software to represent the domain geometry.

ACIS by Spatial
Open Cascade

Mesh file formats

Common (output) file formats for describing meshes.

NetCDF
Genesis/Exodus
XDMF
VTK/VTU
MEDIT
MED/Salome
Gmsh
ANSYS mesh
OFF
Wavefront OBJ
PLY
STL

meshio can convert between all of the above formats.
Mesh visualizers

Blender
Mesh Viewer
Paraview

Books

Grid Generation Methods - Liseikin, Vladimir D.

Tutorials

Cubit tutorials

See also

Delaunay triangulation
Fortune's algorithm
Grid classification
Mesh parameterization
Meshfree methods
Parallel mesh generation
Periodic Table of the Finite Elements
Principles of grid generation
Polygon mesh
Regular grid
Ruppert's algorithm
Stretched grid method
Tessellation
Types of mesh
Unstructured grid

References
Edelsbrunner, Herbert (2001), "Geometry and Topology for Mesh Generation", Applied Mechanics Reviews, Cambridge University Press, 55 (1): B1–B2, Bibcode:2002ApMRv..55B...1E, doi:10.1115/1.1445302, ISBN 978-0-521-79309-4.
Frey, Pascal Jean; George, Paul-Louis (2000), Mesh Generation: Application to Finite Elements, Hermes Science, ISBN 978-1-903398-00-5.
P. Smith and S. S. Sritharan (1988), "Theory of Harmonic Grid Generation" (PDF), Complex Variables, 10 (4): 359–369, doi:10.1080/17476938808814314
S. S. Sritharan (1992), "Theory of Harmonic Grid Generation-II", Applicable Analysis, 44 (1): 127–149, doi:10.1080/00036819208840072
Thompson, J. F.; Warsi, Z. U. A.; Mastin, C. W. (1985), Numerical Grid Generation: Foundations and Applications, North-Holland, Elsevier.
CGAL The Computational Geometry Algorithms Library
Oden, J.Tinsley; Cho, J.R. (1996), "Adaptive hpq-Finite Element Methods of Hierarchical Models for Plate- and Shell-like Structures", Computer Methods in Applied Mechanics and Engineering, 136 (3): 317–345, Bibcode:1996CMAME.136..317O, doi:10.1016/0045-7825(95)00986-8
Steven J. Owen (1998), A Survey of Unstructured Mesh Generation Technology, pp. 239–267
Shimada, Kenji; Gossard, David C. (1995), Bubble Mesh: Automated Triangular Meshing of Non-Manifold Geometry by Sphere Packing, ACM, pp. 409-419, doi:10.1145/218013.218095, ISBN 0-89791-672-7

Undergraduate Texts in Mathematics

Graduate Texts in Mathematics

Graduate Studies in Mathematics

Mathematics Encyclopedia

World

Index

Hellenica World - Scientific Library

Retrieved from "http://en.wikipedia.org/"
All text is available under the terms of the GNU Free Documentation License