A log-barrier method for mesh quality improvement and untangling Authors Shankar P. Sastry The Pennsylvania State University Suzanne M. Shontz Mississippi State University Stephen A. Vavasis Original Article

First Online: 24 November 2012 Received: 30 March 2012 Accepted: 04 October 2012 DOI :
10.1007/s00366-012-0294-6

Cite this article as: Sastry, S.P., Shontz, S.M. & Vavasis, S.A. Engineering with Computers (2014) 30: 315. doi:10.1007/s00366-012-0294-6
Abstract The presence of a few inverted or poor-quality mesh elements can negatively affect the stability, convergence and efficiency of a finite element solver and the accuracy of the associated partial differential equation solution. We propose a mesh quality improvement and untangling method that untangles a mesh with inverted elements and improves its quality. Worst element mesh quality improvement and untangling can be formulated as a nonsmooth unconstrained optimization problem, which can be reformulated as a smooth constrained optimization problem. Our technique solves the latter problem using a log-barrier interior point method and uses the gradient of the objective function to efficiently converge to a stationary point. The method uses a logarithmic barrier function and performs global mesh quality improvement. We have also developed a smooth quality metric that takes both signed area and the shape of an element into account. This quality metric assigns a negative value to an inverted element. It is used with our algorithm to untangle a mesh by improving the quality of an inverted element to a positive value. Our method usually yields better quality meshes than existing methods for improvement of the worst quality elements, such as the active set, pattern search, and multidirectional search mesh quality improvement methods. Our method is faster and more robust than existing methods for mesh untangling, such as the iterative stiffening method.

Keywords Mesh quality improvement Mesh optimization Mesh untangling Interior point method Log-barrier A preliminary version of a portion of these results appeared in shortened form in the Proceedings of the 2011 International Meshing Roundtable.

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