Two derivative-free optimization algorithms for mesh quality improvement

High-quality meshes are essential in the solution of partial differential equations (PDEs), which arise in numerous science and engineering applications, as the mesh quality affects the solution accuracy, the solver execution time, and the problem conditioning. Mesh quality improvement is necessary when the mesh is of less than desirable quality (either from mesh generation or deformation). Nondifferentiable objective functions arise when the goal of the mesh optimization is to improve the worst quality element in the mesh. We propose two derivative-free methods for mesh optimization, namely the pattern search (PS) and multidirectional search (MDS) mesh quality improvement methods, to be used with nondifferentiable objective functions representing the overall mesh quality. Experimental results show that these two methods are successful in improving the worst quality mesh elements. The PS method yielded higher quality 2D meshes than did the MDS method; however, its execution time was longer. In the 3D case, most of the meshes converged to meshes of approximately the same quality because the initial meshes were fairly close to optimal. In 3D, the PS method required longer to execute than did the MDS method.