Sweeping moving objects has become one of the basic geometric operations used in engineering design, analysis and physical simulation. Despite their relevance, computing the boundary of the set swept by a non-polyhedral moving object is largely an open problem due to well known theoretical and computational difficulties of the envelopes. We have recently introduced a generic point membership cl...

We present a new Fast Marching algorithm for a non-convex eikonal equation modeling front evolutions in the normal direction. The algorithm is an extension of the Fast Marching Method since the new scheme can deal with a time-dependent velocity without any restriction on its sign. We analyze the properties of the algorithm and we prove its convergence in the class of discontinuous viscosity sol...

The level set method [8] has become a widely used numerical method for moving interfaces, e.g. see the many examples in [11, 7]. For many applications, the velocity of the interface is known only on the interface, while the level set method requires information about the interface speed at least in a neighborhood of grid points near the interface. To address this issue, velocity extensions are ...

This paper presents the development of a 2D high-order solver with spectral difference method for unsteady incompressible Navier–Stokes equations accelerated by a p-multigrid method. This solver is designed for unstructured quadrilateral elements. Time-marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic. An artificial compressib...

The existing Fast Marching methods which are used to solve the Eikonal equation use a locally continuous model to estimate the accumulated cost, but a discontinuous (discretized) model for the traveling cost around each grid point. Because the accumulated cost and the traveling (local) cost are treated differently, the estimate of the accumulated cost at any point will vary based on the directi...

We present a method to synthesize procedural models with global structures, such as growth plants, on existing surfaces at interactive time. More generally, our approach extends shape grammars to enable context-sensitive procedural generation on the GPU. Central to our framework is the unified representation of external contexts as texture maps. These generic contexts can be spatially varying p...

This article extends the theory of classical finite-difference summation-by-parts (FD-SBP) timemarching methods to the generalized summation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods are shown to retain: A and L-stability, as well as superconvergence of integral functionals when integrated with the quadrature associated with the discretization. This also implies that...

For a photonic crystal (PhC) of finite size, it is important to calculate its transmission and reflection spectra. For two-dimensional (2-D) PhCs composed of a square lattice of circular cylinders, the problem can be solved by an efficient method based on the Dirichletto-Neumann (DtN) map of the unit cell and a marching scheme using a pair of operators. In this paper, the DtN operator marching ...