Mesh generation

Nozzle geometry

Computational mesh in physical space

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.[1]
For a nozzle, with the describing function ${\displaystyle y=x^{2}}$ the grid can easily be generated using uniform division in y-direction with equally spaced increments in x-direction, which are described by

${\displaystyle \xi =x\,}$

${\displaystyle \eta ={\frac {y}{y_{\max }}}\,}$

where ${\displaystyle y_{\max }}$ denotes the y-coordinate of the nozzle wall. For given values of (${\displaystyle \xi }$, ${\displaystyle \eta }$), the values of (${\displaystyle x}$, ${\displaystyle y}$) can be easily recovered.

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 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)[2] on PDEs by transforming physical domain into computational plane while mapping using Poisson's equation, Thompson et al. (1974)[3] have worked extensively on elliptic PDEs to generate grids. In Poisson grid generators, the mapping is accomplished by marking the desired grid points ${\displaystyle (x,y)}$ on the boundary of the physical domain, with the interior point distribution determined through the solution of equations written below

${\displaystyle \xi _{xx}+\xi _{yy}=P(\xi ,\eta )}$

${\displaystyle \eta _{xx}+\eta _{yy}=Q(\xi ,\eta )}$

where, ${\displaystyle (\xi ,\eta )}$ 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,

${\displaystyle \alpha x_{\xi \xi }-2\beta x_{\xi \eta }+\gamma x_{\eta \eta }=-I^{2}(Px_{\xi }+Qx_{\eta })}$

${\displaystyle \alpha y_{\xi \xi }-2\beta y_{\xi \eta }+\gamma y_{\eta \eta }=-I^{2}(Py_{\xi }+Qy_{\eta })}$

where

${\displaystyle {\begin{aligned}\alpha &=x_{\eta }^{2}+y_{\eta }^{2}\\\beta &=x_{\eta }x_{\xi }+y_{\xi }y_{\eta }\\\gamma &=x_{\xi }^{2}+y_{\xi }^{2}\\I&={\frac {\delta (x,y)}{\delta (\xi ,\eta )}}=y_{\eta }x_{\xi }-y_{\xi }x_{\eta }\end{aligned}}}$

These systems of equations are solved in the computational plane on uniformly spaced grid which provides us with the ${\displaystyle (x,y)}$ 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.[4]

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)[5] proposed a volume orthogonality method that uses Hyperbolic PDEs for mesh generation. For a 2-D problem, Considering computational space to be given by ${\displaystyle \Delta \xi =\Delta \eta =1}$, the inverse of the Jacobian is given by,

${\displaystyle x_{\xi }y_{\eta }-x_{\eta }y_{\xi }=I}$

where ${\displaystyle I}$ 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

${\displaystyle d\xi =0=\xi _{x}\,dx+\xi _{y}\,dy.}$

For ${\displaystyle \xi }$ and ${\displaystyle \eta }$ surfaces to be perpendicular the equation becomes

${\displaystyle x_{\xi }x_{\eta }+y_{\xi }y_{\eta }=0.}$

The problem associated with such system of equations is the specification of ${\displaystyle I}$. Poor selection of ${\displaystyle I}$ 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.

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 ${\displaystyle \eta =0}$ and the advancing the solutions to the outer surface of the object satisfying the boundary conditions along ${\displaystyle \xi }$ 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.

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.

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.[6]

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.[7]

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.

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.[8]

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.

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.

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, 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 whose primary topic is meshing are in bold.)

• 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 Symposium on Computational Geometry SoCG
• Numerical Geometry, Grid Generation and Scientific Computing (NUMGRID) (proceedings in Lecture Notes in Computational Science and Engineering)
• SIAM International Meshing Roundtable (SIAM IMR). An independent annual conference from 1992-2021, and a SIAM workshop concurrent with SIAM PP or SIAM CS&E since 2022. Refereed proceedings.
• SIGGRAPH (proceedings in ACM Transactions on Graphics)
• Symposium on Geometry Processing SGP (Eurographics) (proceedings in Computer Graphics Forum)
• World Congress on Engineering

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