Feature-based multiblock finite element mesh generation
Computer-Aided Design
SIAM Journal on Scientific Computing
Shape-Up: Shaping Discrete Geometry with Projections
Computer Graphics Forum
MeshMed'12 Proceedings of the 2012 international conference on Mesh Processing in Medical Image Analysis
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Meshes containing elements with bad quality can result in poorly conditioned systems of equations that must be solved when using a discretization method, such as the finite-element method, for solving a partial differential equation. Moreover, such meshes can lead to poor accuracy in the approximate solution computed. In this paper, we present a nonlinear fractional program that relocates the vertex coordinates of a given mesh to optimize the average element shape quality as measured by the inverse mean-ratio metric. To solve the resulting large-scale optimization problems, we apply an efficient implementation of an inexact Newton algorithm that uses the conjugate gradient method with a block Jacobi preconditioner to compute the direction. We show that the block Jacobi preconditioner is positive definite by proving a general theorem concerning the convexity of fractional functions, applying this result to components of the inverse mean-ratio metric, and showing that each block in the preconditioner is invertible. Numerical results obtained with this special-purpose code on several test meshes are presented and used to quantify the impact on solution time and memory requirements of using a modeling language and general-purpose algorithm to solve these problems.