of a surface|thumb of 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. 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; see Combinatorial Techniques for Hexahedral Mesh Generation. 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.
Algebraic methods The grid generation by algebraic methods is based on mathematical
interpolation function. It is done by using known functions in one, two or three
dimensions taking arbitrary shaped regions. The computational domain might not be rectangular, but for the sake of simplicity, the domain is taken to be rectangular. The main advantage of the methods is that they provide explicit control of physical grid shape and spacing. The simplest procedure that may be used to produce boundary fitted computational mesh is the normalization transformation. For a nozzle, with the describing function y = x^2 the grid can easily be generated using uniform division in
y-direction with equally spaced increments in
x-direction, which are described by : \xi=x \, : \eta = \frac{y}{y_\max} \, where y_\max denotes the y-coordinate of the nozzle wall. For given values of (\xi, \eta), the values of (x, y) can be easily recovered.
Differential equation methods Like algebraic methods,
differential equation methods are also used to generate grids. The advantage of using the
partial differential equations (PDEs) is that the solution of grid generating equations can be exploited to generate the mesh. Grid construction can be done using all three classes of
partial differential equations.
Elliptic schemes Elliptic PDEs generally have very smooth solutions leading to smooth contours. 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. After extensive work done by Crowley (1962) and Winslow (1966) on PDEs by transforming physical domain into computational plane while mapping using
Poisson's equation, Thompson et al. (1974) have worked extensively on elliptic
PDEs to generate grids. In Poisson grid generators, the mapping is accomplished by marking the desired grid points (x,y) on the boundary of the physical domain, with the interior point distribution determined through the solution of equations written below : \xi_{xx} + \xi_{yy} = P(\xi, \eta) : \eta_{xx} + \eta_{yy} = Q(\xi, \eta) where, (\xi,\eta) are the co-ordinates in the computational domain, while P and Q are responsible for point spacing within D. Transforming above equations in computational space yields a set of two
elliptic PDEs of the form, : \alpha x_{\xi\xi} -2\beta x_{\xi\eta} + \gamma x_{\eta\eta} = -I^2 (Px_\xi + Qx_\eta) : \alpha y_{\xi\xi} -2\beta y_{\xi\eta} + \gamma y_{\eta\eta} = -I^2 (Py_\xi + Qy_\eta) where : \begin{align} \alpha & = x^2_\eta + y^2_\eta \\ \beta & = x_\eta x_\xi + y_\xi y_\eta \\ \gamma & = x^2_\xi + y^2_\xi \\ I & = \frac{\delta(x, y)}{\delta(\xi, \eta)} = y_\eta x_\xi - y_\xi x_\eta \end{align} These systems of equations are solved in the computational plane on uniformly spaced grid which provides us with the (x,y) co-ordinates of each point in physical space. The advantage of using
elliptic PDEs is the solution linked to them is smooth and the resulting grid is smooth. But, specification of P and Q becomes a difficult task thus adding it to its disadvantages. Moreover, the grid has to be computed after each time step which adds up to computational time.
Hyperbolic schemes This grid generation scheme is generally applicable to problems with open domains consistent with the type of
PDE describing the physical problem. The advantage associated with
hyperbolic PDEs is that the governing equations need to be solved only once for generating grid. 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) proposed a volume orthogonality method that uses Hyperbolic PDEs for mesh generation. For a 2-D problem, Considering computational space to be given by \Delta\xi = \Delta\eta = 1 , the inverse of the
Jacobian is given by, : x_\xi y_\eta - x_\eta y_\xi = I where I represents the area in physical space for a given area in computational space. The second equation links the orthogonality of grid lines at the boundary in physical space which can be written as : d\xi = 0 = \xi_x \, dx + \xi_y \, dy. For \xi and \eta surfaces to be perpendicular the equation becomes : x_\xi x_\eta + y_\xi y_\eta = 0. The problem associated with such system of equations is the specification of I. Poor selection of I may lead to shock and discontinuous propagation of this information throughout the mesh. While mesh being orthogonal is generated very rapidly which comes out as an advantage with this method.
Parabolic schemes 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. Nakamura (1982) and Edwards (1985) developed the basic ideas for parabolic grid generation. The idea uses either of
Laplace or the
Poisson's equation and especially treating the parts which controls elliptic behavior. The initial values are given as the coordinates of the point along the surface \eta = 0 and the advancing the solutions to the outer surface of the object satisfying the boundary conditions along \xi edges. The control of the grid spacing has not been suggested until now. Nakamura and Edwards, grid control was accomplished using non uniform spacing. The parabolic grid generation shows an advantage over the hyperbolic grid generation that, no shocks or discontinuities occur and the grid is relatively smooth. 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.
Variational methods This method includes a technique that minimizes
grid smoothness,
orthogonality and volume variation. This method forms mathematical platform to solve grid generation problems. In this method an alternative grid is generated by a new
mesh after each iteration and computing the grid speed using
backward difference method. This technique is a powerful one with a disadvantage that effort is required to solve the equations related to grid. Further work needed to be done to minimize the
integrals that will reduce the CPU time.
Unstructured grid generation The main importance of this scheme is that it provides a method that will generate the grid automatically. Using this method, grids are segmented into blocks according to the surface of the element and a structure is provided to ensure appropriate connectivity. To interpret the data
flow solver is used. 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. The information storage in structured scheme is
cell to cell instead of grid to grid and hence the more memory space is needed. Due to random cell location, the solver
efficiency in unstructured is less as compared to the structured scheme. Some points are needed to be kept in mind at the time of grid
construction. The grid point with high resolution creates difficulty for both structured and unstructured. For example, in case of
boundary layer, structured scheme produces elongated grid in the direction of flow. On the other hand, unstructured grids require a higher cell
density in the boundary layer because the cell needs to be as
equilateral as possible to avoid errors. We must identify what information is required to identify the cell and all the neighbors of the cell in the
computational mesh. We can choose to locate the
arbitrary points anywhere we want for the unstructured grid. A point insertion scheme is used to insert the points independently and the cell connectivity is determined. This suggests that the point be identified as they are inserted.
Logic for establishing new connectivity is determined once the points are inserted. Data that form grid point that identifies grid cell are needed. As each cell is formed it is numbered and the points are sorted. In addition the neighbor cell information is needed.
Adaptive grid 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. So the grid may or may not be the best for the given problem. Adaptive methods are used to improve the
accuracy of the solutions. The adaptive method is referred to as ‘h’ method if mesh refinement is used, ‘r’ method if the number of grid point is fixed and not redistributed and ‘p’ if the order of solution scheme is increased in finite-element theory. The multi dimensional problems using the equidistribution scheme can be accomplished in several ways. The simplest to understand are the Poisson Grid Generators with control function based on the equidistribution of the weight function with the
diffusion set as a multiple of desired cell volume. The equidistribution scheme can also be applied to the unstructured problem. The problem is the connectivity hampers if mesh point movement is very large.
Steady flow and the time-accurate flow calculation can be solved through this adaptive method. The grid is refined and after a predetermined number of iteration in order to adapt it in a steady flow problem. The grid will stop adjusting to the changes once the solution converges. In time accurate case coupling of the
partial differential equations of the physical problem and those describing the grid movement is required.
Image-based meshing Machine learning in mesh generation Recent advances in artificial intelligence (AI) and machine learning (ML) have significantly impacted mesh generation, automating traditionally labor-intensive processes and improving accuracy in computational simulations. AI-driven techniques, such as neural networks and reinforcement learning, can predict optimal mesh configurations, adaptively refine meshes, and reduce manual intervention in finite element analysis (FEA) and computational fluid dynamics (CFD). Companies like NVIDIA, Ansys, and Siemens have integrated AI-based mesh generation tools into their simulation software, accelerating workflows in aerospace, automotive, and biomedical engineering. == Cell topology ==