Know Your Mesh – Basic Methodologies

Consider the simulation of flow field past a solid obstruction. Generating a grid to solve a discretization scheme shall proceed in one of few routes. First a surface grid covering the boundaries of the solid body may be generated. In subsequent step, If a finite-difference scheme is employed, then a curvilinear coordinate system aligned with the grid lines shall be created and the differential form of the governing equations shall be transformed to meet the coordinate system. Since the grid was created to conform with the surface of the body, the equations may be discretized in a computational domain easily. This form of grid generation is known as structured method grid generation.

The relation between physical and computational grid in structured method grid generation

The subsequent step may be approached instead with finite-volume or finite-element techniques by which the integral form of the governing equations shall be discretized and the information concerning the geometrical form of the grid shall be incorporated directly into the discretization, so grid transformation is not resorted to. This form of grid generation is known as unstructured method grid generation.

Another strategy for grid generation is to employ a non-body conformal Cartesian grid, while still representing the surface of the body through some kind of surface grid, such that the Cartesian volume grid shall be generated with no regard to this surface grid. The solid boundary will now “cut” through the Cartesian volume grid and because the grid does not conform with the solid boundary, the generation of boundary conditions would require a modification for the equations in the vicinity of the boundary through the introduction of a forcing function that reproduces the effect of the boundary. In what follows, the equations may be discretized using finite-difference, finite-volume, or a finite element technique without resorting to transformation of coordinates or complicated discretization operations. This form of grid generation is known as Immersed Boundary Method (IBM).

Structured Method Grid Generation

In a structured mesh, the connectivity of adjacent node points is identical everywhere in the interior of the grid. There are several methods to perform structured grid generation: Algebraic interpolation methods and methods based on solving partial differential equations.
As in the first paragraph, the generation of a boundary-conforming structured coordinate system is accomplished by determination of the values of the curvilinear coordinates in the interior of a physical region from specified values on the boundary of the region. The equivalent problem in the transformed region is the determination of values of the physical coordinates in the interior of the transformed region from specified values on the boundary of this region. Since the solution of such a boundary-value problem however, is a classic problem of partial differential equations (PDE), it is straightforward to take the coordinates to be solutions of a system of partial differential equations.

A structured grid may be generated according to the solution of  hyperbolic or parabolic PDE by numerically solving the partial differential equations and marching in the direction of one curvilinear coordinate between two boundary surfaces in three dimensions, each of these PDE methods shall allow efficient grid generation for certain cases such as grid between the two boundaries of a double-connected region for which each of the boundaries is specified (parabolic PDE) and unbounded regions where the exact location of a computational outer boundary is not as important (hyperbolic PDE), but none of these kinds of PDE will allow the entire boundaries of a general region be specified. If the coordinate points are specified on the entire closed boundary of the physical region, the equations must be elliptic.
Another form of structured grid generation is by algebraic interpolation. Since the problem of generating a curvilinear coordinate system is actually a problem of generating values of the Cartesian coordinates in the interior of a rectangular transformed region from values on the boundaries, it can be constructed directly by interpolation (Lagrange, Hermite or other polynomial interpolation scheme) from the boundaries.

Multiblock structured mesh

There are several advantages and shortcomings upon invoking a structured method grid generation approach and I shall refer to some.
One advantage of structured method grid generation is that its solvers typically require a lower amount of memory and execute faster for a given mesh size because they are optimized for the structured layout of the grid. Another advantage of structured method grid generation is that you can simplify some operations on the grid because you know the index or indices of a cell or point. Yet another important advantage is the “alignment with the flow” feature. This feature is extremely valuable considering problems such as filter and derivatives commutation in LES and provides a solid ground for mesh convergence procedures for example.
The main shortcomings of the structured grid generation method relies in its ability (or lack of one) to generate a grid when complex geometries are involved. For a partial achievement of such a task one would require a multiblock approach (decomposing the domain to many blocks such that each is structured method grid generated) and possibly defeaturing many geometry oriented singularities (shallow cones, wedges, etc…) or if they seem important to the flow pattern creating a highly refined block, all of which may prove quite prohibitive considering the computational cost. The second drawback of the method is the time, expertise and experience required to lay out an optimal block structure for an entire model. In many cases it is the practitioner past experience which guides him how to seed the boundary. Commonly known structured method grid generation softwares include GridPro and Pointwise with the later being highly comfortable due to its technology for decomposition of the domain to multiblocks.

Structured mesh for a 2D airfoil

Unstructured Method Grid Generation

Unstructured grid methods utilize an arbitrary set of elements to fill the domain. Because the arrangement of elements has genarally no prerequisite for the pattern, the grid is called unstructured. In contrast to structured method grid generation, where breaking up the domain to multiblocks when complex geometry describing boundaries are involved and strict alignment of the elements might be a prerequisite of the analysis code or necessary to capture the physics, in unstructured method grid generation the node valence requirement is absent, allowing any number of elements to meet a single grid point. By that other forms of 3D entities such as tetrahedral or prism may be formed.

Unstructured tetrahedron meshing for external flow in automotive application (by Pointwise)

Most techniques for unstructured method grid generation employ either Octree,Delaunay or advancing front methodologies.
The most popular tetrahedral techniques employ the Delaunay criterion according to which a node must not be contained within the sphere passing through all four vertices of the tetrahedron within the mesh. As this is only a criteria, there are methods for defining how exactly to locate these interior nodes such as Point Insertion and Boundary-Constrained Triangulation.

CFD case study of the DrivAer model with ANSA and OpenFOAM (by BETA CAE Systems)

With Octree, cubes containing the geometric model are recursively subdivided until a desired resolution threshold is achieved. In Octree grid generation methodology irregular cells created using cubes intersecting the surface often require a significant amount of surface intersection calculation. Codes such as that of FloEFD by Mentor Graphics use the Octree methodology for grid generation.

In advancing Front methodologies the tetrahedral are built progressively inward and the front is maintained where new elements are formed. Some operations are included to ensure opposing fronts do not overlap. The Advancing Front mesh generation is available as part of ANSYS meshing.

As tetrahedral mesh is the most popular meshing methodology for unstructured method grid generation, when the geometry and the domain shall allow it, hexahedral mapped meshing generated by sweeping, generally incorporating decomposition of the geometry directly, will produce the most desirable results. Such a grid generation methodology is applied in the CUBIT code. There are also direct and indirect hexahedral generation methodologies which may be applied when mapped meshing by sweeping is not applicable. Indirect hexahedral mesh is achieved by applying a tetrahedron mesh and subdividing each tetrahedron into four hexahedral (which may produce quite a poor element quality) Strategies for direct hexahedra mesh generation include: whisker weaving (based on constructing a Spatial Twist Continuum (STC) and fitting hexahedral elements aided by the STC), Plastering (placing elements at the boundary and advancing to the center of the volume) and Hex-Dominant (subdividing the domain to sub-domains of mapped hexhedral and completing with tetrahedral mesh), the last being the most robust and applicable in the ANSYS code.

The main advantage of unstructured method grid generation is its ease to apply for even the most complex geometries. Another advantage lies in its level of automation which therefore, requires little user time or effort by the practitioner.

Its main drawback is the lack of user control when laying out the mesh. Typically any user involvement is limited to the boundaries of the mesh with the mesh generator automatically filling the interior. Another major drawback is the unstructured flow solvers which typically require more memory with longer execution times than structured grid solvers on a similar mesh and may produce larger errors due the reliance on CAD imported geometry.

Immersed Boundary Method (IBM):

Although found some reemergence in the past few years, the general concept was actually developed by Peskin in 1972 for the simulation of blood flow and cardiac mechanics coupling.
In IBM a non-body conformal Cartesian grid is employed. The immersed boundary would still be represented by a surface grid, but the Cartesian volume grid is generated without regarding the immersed boundary surface grid resulting in the solid boundary “cutting” through this Cartesian volume grid. Because of the non-conformity of the solid boundary and the Cartesian grid, applying boundary conditions requires modifying the equations in the vicinity of the solid boundary by means of a forcing function that reproduces the effect of the boundary.
The methodologies for generating a forcing function are the Continuous Forcing approach and the Discrete Forcing Approach.
In the continuous forcing approach the forcing function is included in the continuous momentum and continuity set of equations for the entire domain to be subsequently discretized on a Cartesian grid leading to a system of forced discrete equations.
In contrast, in the discrete forcing approach the equations are first discretized on a Cartesian grid disregarding the immersed boundary and subsequently the discretization in the cells near the immersed boundary is adjusted to account for its presence.
The main advantage of the method is in its ability to simplify tremendously the task of grid generation. In contrast to structured method grid generation no additional operations are required to account for the terms associated with grid transformation. As to unstructured methods for grid generation, the main advantage of IBM is its ability to efficiently invoke geometric multigrid methods and line-iterative techniques which leads to a per-grid reduction in operations, reducing the computational cost.
Another application of which IBM stands above others is when moving boundaries are present. approaching such a case study with unstructured method grid generation would require the generation of new mesh to conform with the moving surface of the body as the calculation progresses. In IBM on the other hand, the non-body conforming mesh would alleviate such a problematic feature.

A snapshot of Large Eddy Simulation of a 5-bladed rotor wake in hover with a novel multiblock IBM
(by Technion CFD Lab)

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