# Delaunay Refinement Mesh Generation

### Key Takeaways

• Delaunay refinement mesh generation is the process to smooth out the initial coarse mesh to capture the details of fluid-structure interactions in complex geometries.

• The process involves adding nodes to the existing mesh and combining fluid and structure meshes to precisely capture the deformation.

• CFD tools provide specialized simulation options such as the Arbitrary Lagrangian-Eulerian to account for the effect of fluid-structure interactions.

Example of Delaunay refinement mesh generation

The accuracy of flow model simulation in computational fluid dynamics (CFD) is highly dependent on the quality of mesh generation. The options for generating triangular meshes are efficient, considering their accuracy and meshing flexibility for a wide range of geometries. Delaunay triangulation is a helpful method for generating non-overlapping triangles for highly defined to irregular geometries.

Oftentimes, a certain set of algorithms has to be implemented to produce high-quality elements from Delaunay triangulation. This is the process of Delaunay refinement mesh generation. Let us discuss this concept in detail and explore its benefits in capturing fluid-structure interactions for a fluid flow.

## Delaunay Refinement Mesh Generation

The primary purpose of Delaunay refinement is to improve the quality of the mesh. Delaunay triangulation includes dividing the discrete set of points into a set of non-overlapping triangles that adhere to the Delaunay criterion. Two things to note:

• → No vertex should lie within the circumcircle of the mesh triangle.
• → It is preferable for triangles to be equiangular, although triangles of different sizes may be used.

Thus, generated mesh is more stable due to well-shaped elements, minimizing the numerical errors caused by overlapping.

Nevertheless, this initial mesh is coarse and needs refinement to smooth it out. The Delaunay refinement mesh generation process inserts additional points into the existing mesh and connects them using Delaunay triangulation to generate an even finer mesh. Such mesh is detailed, so computation is the most reliable and efficient way to accomplish this task.

Such a refinement process has several advantages:

1. Triangles are well-defined and have a good aspect ratio. This minimizes computational errors during simulation.

2. The mesh is graded, i.e., it improves the simulation of complex geometries and flow patterns by capturing small-scale features with appropriate mesh resolutions. Around curvatures or large stress gradient regions, the element density is higher.

3. When automated, the refinement process reduces the time and effort required for generating and optimizing the mesh, minimizes error, and improves the quality of the mesh.

The chart below provides a basic overview of the process involved in Delaunay refinement mesh generation.

## Capturing Fluid-Structure Interactions With Delaunay Refinement Mesh Generation

For complex geometry, Delaunay refinement mesh generation facilitates the capture of fluid-structure interactions (FSI) in CFD simulation. FSI analysis is important in fluid dynamics to understand complex fluid flows and their effect on the deformation of solid structures. The interaction of the fluid and solid domain can be generated and refined in the mesh using the following method:

1. Discretize the domain. Use Delaunay triangulation to generate a coarse mesh for the entire flow domain.

2. Perform mesh refinement. Following the Delaunay refinement mesh generation, add additional nodes to the mesh to optimize the resolution and improve the quality of the fluid mesh.

3. Generate mesh for the solid structure. This can be done using specialized methods like adaptive ALE* meshing or other methods that can capture the deformation of the solid structure.

4. Combine the fluid and solid structure meshes. Define the appropriate interface and boundary conditions to combine the two domains.

5. Perform the simulation. Use the appropriate FSI algorithm (such as ALE*, CEL*, or IBM*).

```*ALE - Arbitrary Lagrangian-Eulerian
*CEL - Coupled Eulerian-Lagrangian
*IBM - Immersed boundary method```

## Optimizing FSI Simulation in CFD With Delaunay Refinement Mesh Generation

Delaunay refinement is an effective method of capturing the complex details of fluid-surface interactions. CFD tools can generate the initial coarse mesh with Delaunay triangulation, which can be refined around the solid structure by adding nodes and elements to the mesh to achieve optimal mesh resolution. CFD tools typically offer specialized simulation options such as the ALE, which allows the mesh to move and deform to account for the solid structure deformation due to fluid forces.

This deformation can be visualized in the CFD tool to understand flow behavior. The degree of mesh motion can be calculated by solving the partial differential equation associated with the mesh. System designers prefer the use of Cadence’s CFD solvers to compute these governing equations for mesh refinement. With high-quality mesh generation, CFD tools such as Fidelity Pointwise support reliable simulation of fluid-structure interactions in a wide range of aerospace, automotive, or biomedical system design applications.