Curvature-based energy for simulation and variational modeling

Curvature-based energy and forces are used in a broad variety of contexts, ranging from modeling of thin plates and shells to surface fairing and variational surface design. The approaches to discretization preferred in different areas often have little in common: engineering shell analysis is dominated by finite elements, while spring-particle models are often preferred for animation and qualitative simulation due to their simplicity and low computational cost. Both types of approaches have found applications in geometric modeling. While there is a well-established theory for finite element methods, alternative discretizations are less well understood: many questions about mesh dependence, convergence and accuracy remain unanswered. We discuss the general principles for defining curvature-based energy on discrete surfaces based on geometric invariance and convergence considerations. We show how these principles can be used to understand the behavior of some commonly used discretizations, to establish relations between some well-known discrete geometry and finite element formulations and to derive new simple and efficient discretizations.