All CFD analyses begin with a geometry model. It comes from a variety of sources in a multitude of representations. Mesh generation programs should properly read and process the geometry for users to create meshes successfully. Access to the geometry model is mostly by the mesh generation program.

A. CAD Interoperability

Concurrent engineering and outsourcing have elevated the importance of efficient product data exchange. Mechanical CAD (MCAD) models are built to define geometry from a manufacturing perspective. They contain much more information than is necessary for most CFD applications. MCAD and its issues with representation, finite tolerances, and translation are well understood. Moreover, when driven by the MCAD model data, mesh generation is made less robust, thus requiring additional user involvement and less repeatability, leading to increased design study uncertainty.

A separation of CAD topology is necessary, which is constructed to minimize geometric error, and mesh topology, which aims to minimize solution error. A promising approach would be the combination of so-called Pre-CAD, which includes a conceptual design parametric space, and MCAD data. In this approach, the conceptual design parametric space would be used to construct the mesh topology in a robust and repeatable way, while the mesh points and faces would lie on the geometrically accurate MCAD model.

B. Access to Geometry

Access to geometry has always been a requirement in a mesh generation process. Many commercial mesh generation tools provide the ability to import geometry from native CAD files or common formats such as IGES and STEP. Often, the final mesh is output to a CFD file format, and the linkage to the geometry model is discarded.

If the goal is to perform mesh adaptation, high-order mesh generation, and design through shape optimization, the geometry must be persistent throughout the CFD simulation process. A geometry kernel is needed beyond the initial mesh creation phase to achieve this persistence. This kernel must be lightweight and provide optimal querying capabilities for mesh generation, mesh adaptation, and high-order mesh elevation phases.

Fidelity Pointwise is working to address this need for a geometry kernel by developing an Application Programming Interface (API) for a lightweight version of our geometry kernel, named Geode. The geometry kernel is thread-safe and can be used in parallel on High-Performance Computing (HPC) systems. It has full querying functionality and limited geometry creation and modification capability. In the HPC environment, partitioning of CAD entities is expected to reduce the memory requirements per core for the kernel and provide optimal efficiency.

Mesh adaptation and high-order mesh generation both require access to the geometry to ensure the mesh conforms to the true shape of the boundaries.

A. Mesh adaptation

For a fully automated CFD process, the mesh must adapt as the flow solution evolves to ensure that all salient flow features are accurately resolved. For decades, research has focused on mesh adaptation schemes. Some methods use truncation errors or estimate errors in the solution by computing gradient fields of easily computed scalar quantities. These feature-based methods then identify regions in the solution space where the gradients are high, and the local mesh spacing is large.

Mesh adaptation can be performed by node repositioning. In Figure 1, the mesh was “squeezed” near the shocks on the surface and in the interior of the mesh. The number of elements did not change, nor did the mesh connectivity. The key to performing mesh adaptation is in the definition of the spacing field. Once spacing requirements are known, the actual mesh refinement or repositioning can be performed.

Figure 1. Adapted mesh for an inviscid CFD solution of the ONERA M6 wing using node repositioning.

In an HPC context, the mesh modules for adaptation must be parallel and thread-safe. The partitioning requirements of the mesh may not coincide with the flow solver requirements. The transfer of data between the mesh modules and the flow solver needs to be compact for efficiency and must allow for repartitioning. When significant repartitioning occurs, the processors' geometry distribution may need to adapt to the changing mesh.

B. High-order mesh generation

Finite element methods (FEM) applied to aerodynamic simulations have been evolving rapidly in recent years. Existing meshes can now be analyzed with FEM flow solvers to achieve second-order spatial accuracy using linear meshes. These methods can achieve higher-order accuracy by inserting additional nodes on and inside the elements, i.e., high-order elements.

The approach followed by most FEM practitioners to create high-order meshes is to modify linear meshes. Elevating linear volume meshes to a higher polynomial degree is relatively straightforward for flat geometries or meshes that do not have viscous type clustering. Curved boundaries can complicate the curving process. The process includes three or four basic steps.

Figure 2. Surface meshes near the wing tip trailing edge of the ONERA M6 wing for polynomial degrees 1-4.

In step 1, each element will receive additional nodes along edges, on faces, and in the interior. Step 2 involves placing the newly created boundary nodes on the true surface (the geometry model). Step 3 is an optional step where the surface nodes are smoothed. If there is a curvature on the edges in the tangential directions, this can complicate the surface refinement process. Access to the geometry model and geometry edge associativity is essential during the mesh smoothing process. In the final step, the interior volume nodes are smoothed. This is necessary for tightly clustered meshes in the normal direction of curved boundaries.

C. H- and P- adaptation

The most efficient approach to mesh adaptation involves performing both h- and p-adaptation. The former involves inserting mesh points to reduce the local mesh edge length (h), while the latter involves elevating the polynomial degree (p) of an element. Certain regions of the flow where discontinuities exist, such as sharp edges and shock waves, can be modeled using high-order elements but should be handled through mesh refinement of linear elements. Other regions of the flow where the solution is smooth can be handled through mesh order elevation. The combination will permit optimal use of the degrees of freedom of the flow field solution.

To perform h-p adaptation properly, the flow solver and mesh generation program must work together. The forward part of communication (mesh to the solver) is well defined. This is typically done through files. The reverse part of the communication is when the flow solver instructs the mesh generator about were to refine and how (h- and/or p- adaptation). This can involve some means of specifying a size field. Once the mesh generator has modified the mesh and communicated it back to the flow solver, the solution must be transferred to the new mesh.

To approach the goal of automating mesh generation realistically requires using kernels or functions that are modularized and available in an HPC context. Some important mesh generation modules are smoothing, viscous mesh extrusion, isotropic volume mesh generation, and Delaunay reconnection.

A. Mesh smoothing

Mesh smoothing plays an important role in many meshing operations. However, many unstructured mesh smoothing techniques do not adequately control mesh quality. Some smoothing methods used for unstructured meshes are Spring Analogy, Linear-Elastic Smoothing, Winslow Elliptic Smoothing, and Weighted Condition Number Optimization-based Smoothing.

B. Normal extrusion module

The normal extrusion module is predominantly used to create clustered meshes in viscous regions. It starts with a defined surface mesh comprised of triangles and quadrilaterals. The prismatic and hexahedral mesh is grown in the normal direction following a preset growth rate. It terminates when the extruded elements reach an isotropic shape or when element quality constraints stop it to prevent collisions and inverted elements. Adaptation is very difficult using these meshes, and the prismatic mesh can introduce extremely poor-quality elements.

C. Hexahedral Dominant Volume Mesh Generation

An alternative to isotropic tetrahedral volume mesh generation is using hexahedral meshes. These meshes are rapidly generated and can accommodate solution adaptation requirements. They are most popular in non-viscous applications. These meshes can work with extruded viscous meshes. These methods are typically very fast in serial because the operation counts are low. They do not have good parallel performance. The main reason to go parallel with these methods is to enable larger mesh sizes.

D. Delaunay Mesh Generation

Probably the most important module is the triangle and tetrahedral mesh generation module. It is applicable in several areas, from generating the entire isotropic mesh, creating the stitch mesh in the hybrid scheme, and performing mesh topology optimization in an adaptation pass. Tetrahedral mesh generation schemes are predominantly serial in operation. Parallel operations are possible but usually require modification to the normal process, resulting in slightly different meshes from the serial result.

The NASA study describes the mesh generation process as a roadblock and a dominant cost in terms of human intervention. Progress must be made to automate the mesh generation phase of the simulation. Scripting is an integral component of automated mesh generation. Like many commercial tools, Fidelity Pointwise has a scripting language that users apply to repetitive operations.

Some scripts are so sophisticated that the entire mesh can be constructed at the click of a button. However, these tend to be cases where the configuration has minor variations from previous runs, and the same meshing strategy and topology are simply re-applied. Organizations invest significant effort in making robust scripts. Deviation from the assumed strategy and mesh topology is not permitted and will usually result in the script failing.

If one assumes the starting point for automated mesh generation is a water-tight geometry model, then fully automated mesh generation is possible today, although restrictions must be imposed concerning the mesh topology and spacing information.

The complete build recipe may not be necessary to achieve an outcome that satisfies the user’s intent. Each additional piece of information can be a clue to the automation process. The clues can take the form of specific quantities, such as edge spacing. They can also be more abstract, such as labels using keywords to indicate the type of boundary or curve. The current challenges are too great to be attempted in isolation. Collaborative efforts will accelerate the progress toward automated, intelligent mesh generation.

Reference:

1. Karman, Steve L., Wyman, Nick J., and Steinbrenner, John P., “Mesh Generation Challenges: A Commercial Software Perspective,” AIAA paper no. 2017-3790, June 2017.

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