High-Order Continuous Geometrical Validity

Abstract

We propose a conservative algorithm to test the geometrical validity of simplicial (triangles, tetrahedra), tensor product (quadrilaterals, hexahedra), and mixed (prisms) elements of arbitrary polynomial order as they deform linearly within a time interval. Our algorithm uses a combination of adaptive Bézier refinement and bisection search to determine if, when, and where the Jacobian determinant of an element’s polynomial geometric map becomes negative in the transition from one configuration to another. In elastodynamic simulation, our algorithm guarantees that the system remains physically valid during the entire trajectory, not only at discrete time steps. Unlike previous approaches, physical validity is preserved even when our method is implemented using floating point arithmetic. Hence, our algorithm is only slightly slower than existing non-conservative methods while providing guarantees and while being an easy drop-in replacement for current validity tests. To prove the practical effectiveness of our algorithm, we demonstrate its use in a high-order Incremental Potential Contact (IPC) elastodynamic simulator and experimentally show that it prevents invalid, simulation-breaking configurations that would otherwise occur using non-conservative methods.