Interpolation-free space-time remeshing for the Burgers Equation

Abstract

SUMMARYIn this paper we present a mesh management strategy for use with discontinuous-Galerkin space±time ®nite element formulations of¯ow problems. We propose a re®nement technique for simplex-type meshes which requires no interpolation between grids at slab interfaces. The strategy uses an a posteriori spatial error estimator to tag re®nement or coarsening. Orientation of element edges along¯ow characteristics is accomplished by nodal displacement, and by a new diagonal-swapping technique to correct for the eects of misalignment due to h-re®nement. The swapping procedure realigns the mesh to improve the eectiveness of the h-adaptive process. Results are presented for the Burgers Equation using large time steps on a problem which exhibits merging and steepening fronts. #

A hybrid extended finite element/level set method for modeling phase transformations

“…Importantly, this projection can be restricted to the set of elements with enriched nodes at time steps t n and t n+1 . We note that similar projections are often employed in space-time formulations [30].…”

Generalized Euclidean Least Square Approximation

“…The applications of least-squares include robotics, manipulating parts with complex shapes in unstructured dynamic environments, planning end-effect or motions around singularities, writing on a blackboard, and other motion planning problems involving kinematic constraints [9]. Other works involving the use of interpolation in Euclidean space can be found in [10,11,12,13,14,15,16]. [17] made an investigation of the effects of interpolation error, using two error methods.…”

The Eulerian–Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations

“…The numerical methods developed more than three decades seem to serve as a satisfactory alternative to solve the unsteady Burger's equations. Most of the existing numerical methods in previous studies were reported successfully to be able to solve the Burgers' equations, such as the finite difference method (FDM) [8][9][10], the finite element method (FEM) [11,12] and the boundary element method (BEM) [13,14]. For example, for the Burgers' equation Bahadir [8] proposed a fully implicit finite difference scheme and Radwan [10] used a fourth-order compact scheme and the fourth-order Du Fort Frankel algorithm.…”

“…For example, for the Burgers' equation Bahadir [8] proposed a fully implicit finite difference scheme and Radwan [10] used a fourth-order compact scheme and the fourth-order Du Fort Frankel algorithm. In addition, Froncioni et al [11] proposed the discontinuous-Galerkin space-time finite element formulation using the simplextype meshes. In the meantime, Kutluay et al [12] used the least-squares quadratic B-spline FEM to handle the unsteady Burgers' equations.…”