Mesh generation
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.
Tetrahedra are often abbreviated as "tets"; triangles are "tris", quadrilaterals are "quads" and hexahedra (topological cubes) are "hexes."
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.
A special class of advancing front techniques creates thin boundary layers of elements for fluid flow.
Another direct method is to cut the structured cells by the domain boundary; see sculpt based on Marching cubes.
the grid can easily be generated using uniform division in y-direction with equally spaced increments in x-direction, which are described by where
Using its smoothness as an advantage Laplace's equations can preferably be used because the Jacobian found out to be positive as a result of maximum principle for harmonic functions.
The initial point distribution along with the approximate boundary conditions forms the required input and the solution is the then marched outward.
Steger and Sorenson (1980)[5] proposed a volume orthogonality method that uses Hyperbolic PDEs for mesh generation.
The solving technique is similar to that of hyperbolic PDEs by advancing the solution away from the initial data surface satisfying the boundary conditions at the end.
The specifications of initial values and selection of step size to control the grid points is however time-consuming, but these techniques can be effective when familiarity and experience is gained.
When an unstructured scheme is employed, the main interest is to fulfill the demand of the user and a grid generator is used to accomplish this task.
Due to random cell location, the solver efficiency in unstructured is less as compared to the structured scheme.
A problem in solving partial differential equations using previous methods is that the grid is constructed and the points are distributed in the physical domain before details of the solution is known.
In time accurate case coupling of the partial differential equations of the physical problem and those describing the grid movement is required.
Important classes of two-dimensional elements include triangles (simplices) and quadrilaterals (topological squares).
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.
One can create a cubical mesh by generating an arrangement of surfaces and dualizing the intersection graph; see spatial twist continuum.
Three-dimensional meshes created for finite element analysis need to consist of tetrahedra, pyramids, prisms or hexahedra.
Surface meshes are also used to model thin objects such as sheet metal in auto manufacturing and building exteriors in architecture.
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.
So-called meshless or meshfree methods often make use of some mesh-like discretization of the domain, and have basis functions with overlapping support.
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.
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.
Improving a mesh involves changing its discrete connectivity, the continuous geometric position of its cells, or both.
In adaptive mesh refinement, elements are split (h-refinement) in areas where the function being calculated has a high gradient.
The multigrid method does something similar to refinement and coarsening to speed up the numerical solve, but without actually changing the mesh.
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.
