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Abstract: In the numerical solution of partial differential equations (PDEs), high-quality meshes are crucial for the stability, accuracy, and convergence of the associated PDE solver. Mesh quality improvement is often performed to improve the quality of meshes before use in numerical solution of the PDE. Mesh smoothing (performed via optimization) is one popular technique for improving the mesh quality; it does so by making adjustments to the vertex locations. When an inefficient mesh quality metric is used to design the optimization problem, and hence also to measure the mesh quality within the optimization procedure, convergence of the optimization method can be much slower than desired. However, for many applications, the choice of mesh quality metric and the optimization problem should be considered fixed. In this paper, we propose a simple mesh quality metric alternation scheme for use in the mesh optimization process. The idea is to alternate the use of the original inefficient mesh quality metric with a more efficient mesh quality metric on alternate iterations of the mesh optimization procedure in order to reduce the time to convergence, while solving the original mesh quality improvement problem. Typical results of using our application scheme to solve mesh quality improvement problems yield approximately 40-55% improvement in comparison to the original mesh optimization procedure. More frequent use of the efficient metric results in greater speed-ups. [Copyright &y& Elsevier]

Abstract: 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. [ABSTRACT FROM AUTHOR]

We propose a hybrid mesh deformation algorithm which uses the direction of the boundary deformation to determine the positions of the interior mesh vertices in the deformed mesh. Our goal is to produce meshes on deformed domains which maintain mesh ‘similar’ element shape and possess no inverted elements. The hybrid mesh deformation algorithm consists of two steps, anisotropic finite element-based mesh warping (FEMWARP) followed by multiobjective mesh optimization. The first step estimates the interior vertex positions on the deformed mesh using the boundary deformation to choose appropriate partial differential equation (PDE) coefficients in the anisotropic FEMWARP method. As a second step, we find the local optimal mesh with no inverted elements on the deformed domain by employing multiobjective mesh optimization with one term controlling element shape and a second term designed to untangle inverted elements. Numerical results show that our hybrid algorithm outperforms existing mesh deformation algorithms in terms of mesh quality and number of inverted elements and is able to preserve ‘similar’ element shape on the deformed domain while eliminating inverted elements on the deformed domain. [ABSTRACT FROM AUTHOR]