Localized bi-Laplacian Solver on a Triangle Mesh and Its Applications

Partial differential equations(PDE) defined over a surface are used in various graphics applications, such as mesh fairing, smoothing, surface editing, and simulation. Often these applications involve PDEs with Laplacian or bi-Laplacian terms. We propose a new approach to a finite element method for solving these PDEs that works directly on the triangle mesh connectivity graph that has more connectivity information than the sparse matrix. Thanks to these extra information in the triangle mesh, the solver can be restricted to operate on a sub-domain, which is a portion of the surface defined by user or automatically self-adjusting. Our formulation permits us to solve high order terms such as bi-Laplacian by using a simple linear triangle element. We demonstrate the benefits of our approach on two applications: scattered data interpolation over a triangle mesh(painting), and haptic interaction with a deformable surface.