Control Cell Size Gradation for Desired CFD Solution Accuracy

Mesh generation in computational fluid dynamics (CFD) must achieve a balance between solution accuracy and simulation convergence time. The generated mesh should be fine enough to accurately resolve the flowfield and, at the same time, should be coarse enough to converge in a reasonable amount of time. Hence, it is essential for the mesh cell sizes to vary across the computational domain, with finer cells being used in boundary layer regions, no-slip walls, and other flow features that need a higher resolution; and larger cells being used elsewhere for computational efficiency. An additional requirement is that the cell size must blend smoothly from fine to coarse.

Design Factors for Cell Size Gradation in Fidelity Pointwise

The three design factors to be taken into consideration for local element size gradation in Fidelity Pointwise are - 

• Mesh Control: In a bottom-up mesh generator such as Fidelity Pointwise, the volume meshing is truly a boundary value problem - the volume mesh's cell sizes are driven by the surface element sizes and user-controllable blending functions. Any additional controls for cell size gradation must integrate smoothly into that bottom-up paradigm.
• Mesh Robustness: The cell size gradation method must work with meshes in which relevant length scales vary by six or more orders of magnitude as driven by both geometry and flow physics. Examples of such meshes include viscous simulations of fully appended aircraft and submarines.
• Mesh Quality: The element size, shape, and gradation must not negatively impact the flow solver's convergence or solution accuracy.

Different Approaches to Cell Size Gradation

• The simplest method of grading a mesh's cell size in a bottom-up method is through the mesh's topology; apply a cell size gradation to the mesh's boundaries and blend those gradations onto the interior. While this method is quite robust, its effects are very local. To make its effects more global, one must create and interact with the mesh's topology, which can be quite cumbersome.
• Octree methods have the proven ability to resolve both local and global effects. A minor limitation is that mesh gradation must follow level set rules.
• A network of radial basis functions (RBFs) can help achieve mesh gradation. RBFs can be formulated as a simple interpolation scheme that provides local control. By extending the method to a network of RBFs, global influence can be had on the mesh's gradation. A further benefit of RBFs for a prior mesh size gradation is the ability to implement them in a manner that is interactive and easily controlled by the user.

Sources and Shapes in Radial Basis Functions

Figure 1. Example of distributing user-specific element size along a line shape.

With the mathematics of the RBF defined, one needs only a method for defining the source. In this implementation, sources are defined by geometric primitives called shapes. Shapes can be 0-D (point), 1-D (line, curve, circle), 2-D (circle, quad, disk, polygon), or 3-D (sphere, box, frustum, cylinder, cone, swept polygon). These shapes are created in a sketch-based interface and can overlap with each other and the geometry of the object being meshed.

Implementing radial basis functions and source shapes allows users to apply a priori element size gradation to a tetrahedral mesh. Below are two examples of source shapes used for cell size refinement in the wake region.

In Figure 2, for wake refinement behind an automobile, a source shape is defined. Further, two cell sizes are specified on the source, with a smaller one immediately behind the automobile and a larger one downstream.

Figure 2. A box source in the wake region behind an automobile(top), the box source refines the mesh behind an automobile(bottom).

In Figure 3, the two-volume sources behind the aircraft not only overlap each other but also overlap the aircraft geometry.

Figure 3. Two overlapping sources are placed behind the aircraft – a flat box source is used in the wing's wake region, and a cone source is used behind the engine nacelle(top); the cone source refines the mesh behind the nacelle(bottom).

For more information on cell size gradation in Fidelity Pointwise, read Controlling Localized Element Size Gradation in an Unstructured Mesh by clicking the button below -