Simple and effective variational optimization of surface and volume triangulations

Optimizing surface and volume triangulations is critical for advanced numerical simulations. We present a simple and effective variational approach for optimizing triangulated surface and volume meshes. Our method minimizes the differences between the actual elements and ideal reference elements by minimizing two energy functions based on conformal and isometric mappings. We derive simple, closed-form formulas for the values, gradients, and Hessians of these energy functions, which reveal important connections of our method with some well-known concepts and methods in mesh generation and surface parameterization. We then introduce a simple and efficient iterative algorithm for minimizing the energy functions, including a novel asynchronous step-size control scheme. We demonstrate the effectiveness of our method experimentally and compare it against Laplacian smoothing and other mesh optimization techniques.