How do you deal with mesh-related issues such as boundary conditions, singularities, and discontinuities?

1 What are boundary conditions?

Boundary conditions are the constraints that define how the physical problem interacts with the surrounding environment. They can be specified as values or functions of the variables at the boundaries of the domain, or as relations between the variables and their derivatives. Boundary conditions are essential for ensuring the uniqueness and physical realism of the solution, but they can also introduce errors or difficulties if they are not properly implemented or matched with the mesh.

2 What are singularities?

Singularities are points or regions where the solution or its derivatives become infinite or undefined. They can arise from the geometry, the physics, or the numerical method of the problem. Singularities can cause numerical instability, convergence issues, or loss of accuracy if they are not handled correctly. Some possible ways to deal with singularities are to refine the mesh near the singularity, to use special elements or schemes that can capture the singularity, or to apply regularization techniques that smooth or remove the singularity.

3 What are discontinuities?

Discontinuities are points or regions where the solution or its derivatives change abruptly or discontinuously. They can result from the presence of shocks, interfaces, or material changes in the problem. Discontinuities can also affect the numerical stability, convergence, and accuracy of the solution if they are not resolved properly. Some possible ways to deal with discontinuities are to use adaptive mesh refinement, to use special elements or schemes that can handle discontinuities, or to apply artificial viscosity or diffusion terms that reduce the discontinuities.

4 How to choose a mesh type and size?

The mesh type and size are two important factors that influence the quality and performance of the numerical simulation. The mesh type refers to the shape and arrangement of the elements that form the mesh, such as triangles, quadrilaterals, hexahedra, or tetrahedra. The mesh size refers to the length or area of the elements that determine the resolution of the mesh. The choice of the mesh type and size depends on the geometry, the physics, and the numerical method of the problem, as well as on the computational resources and time available.

5 How to check and improve the mesh quality?

The mesh quality is a measure of how well the mesh conforms to the geometry and the physics of the problem. A poor mesh quality can lead to errors, instability, or slow convergence of the solution. Some common indicators of the mesh quality are the aspect ratio, the skewness, the orthogonality, and the smoothness of the elements. There are various tools and methods to check and improve the mesh quality, such as mesh generators, mesh optimizers, mesh quality metrics, and mesh quality plots.

6 How to verify and validate the mesh and the solution?

The verification and validation of the mesh and the solution are essential steps to ensure the reliability and credibility of the numerical simulation. The verification is the process of checking if the mesh and the solution are free of errors and consistent with the numerical method. The validation is the process of checking if the mesh and the solution are accurate and realistic compared to the physical problem. Some common methods to verify and validate the mesh and the solution are grid convergence studies, error estimation, sensitivity analysis, and comparison with experimental or analytical data.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?