A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry Processing

Abstract

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in ${\mathbb{R}}^2$ or ${\mathbb{R}}^3$, such as curves or surfaces. Such manifold PDEs often involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold's interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of the closest point method (CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM's finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver. We demonstrate our method's convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations.

26 pages

References in corpus (12)

• Complex Patterns in a Simple System
• Solving eigenvalue problems on curved surfaces using the Closest Point Method
• An RBF-FD closest point method for solving PDEs on surfaces
• Calculus on Surfaces with General Closest Point Functions
• A least-squares implicit RBF-FD closest point method and applications to PDEs on moving surfaces
• A Practical Walk-on-Boundary Method for Boundary Value Problems
• PDEs on moving surfaces via the closest point method and a modified grid based particle method
• Cahn-Hilliard on Surfaces: A Numerical Study
• Sum-of-Squares Geometry Processing
• Efficient 2D Simulation on Moving 3D Surfaces
• Solving variational problems and partial differential equations that map between manifolds via the closest point method
• A closest point method library for PDEs on surfaces with parallel domain decomposition solvers and preconditioners