Keywords: mesh optimization, objective function, mesh quality, untangling, smoothing, steepest decent method.

In the past thirty years, much research work has been done on the issue of improving mesh quality. Mesh smoothing is the most commonly used technique, which reposition nodes to improve mesh quality without changing the topology. A very popular mesh smoothing technique is Laplacian smoothing [1,2], which relocates the interior nodes by solving a Laplacian equation. The method is simple to implement and computationally inexpensive, but it has a drawback that it does not guarantee the improvement of mesh quality. In recent years, a new type of smoothing technique based on the optimization of mesh quality measures [3,4,5,6,7,8] has been devloped and is proven to be robust and effective, however, most existing optimization based smoothing methods are restricted to valid meshes, and an additional untangling scheme is required for invalid meshes.

In this paper, we propose a new mesh optimization scheme, which makes mesh untangling and smoothing simultaneous. Firstly, we review the mesh quality measure for quadrilaterals. The mesh quality measure is constructed by decomposing a quadrilateral to four sub-triangles. Given a sub-mesh, a corresponding composite objective function for local sub-mesh optimization is formulated. Then, an additional term is considered to construct a new objective function, which ensures that the level sets of the objective function is convex for the entire domain. The optimization approach is performed in an iterative Guass-Seidel-like scheme by sweeping over all the adjustable nodes iteratively until convergence is achieved. In each step, only one node is adjustable while other nodes are fixed. The local optimization approach is to find the optimal position by maximizing the new objective function. A steepest descent method is used to solve the local optimization problem.

Lastly, several numerical examples using the current approach are presented to demonstrate that the current approach is robust and effective for both invalid and valid meshes.

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