A Spectral-Element Approach for the Eikonal Equation

Abstract

A novel spectral-element approach for the eikonal equation is presented based on the work of Tucker et al. This approach is readily implemented in standard finiteelement flow solvers developed for the Navier-Stokes equations. Results are presented for distance function (level set) for three planar configurations: circular cylinder, NACA 0012 airfoil, and T106 airfoil. Results demonstrate that the approach is capable of arbitrary order of accuracy. For configurations with strong convex curvature, additional curvature-based dissipation is added to the eikonal equation to generate a weak solution consistent with the entropy condition.

Arbitrary Order Solutions for the Eikonal Equation using a Discontinuous Galerkin Method

“…As a consequence the solution is limited to one side of the boundary, i.e., we limit ourselves to computing positive or negative values for the distance s. Now we introduce the gradient vector q = [u, v, w] T = ∇s. As the original equation was found to be very difficult to converge to a steady state solution using a high-order solver [14,20], we decouple the solution of the gradient field and the scalar distance function. To get a set of partial differential equations for the gradients, we take the derivative of equation ( 2) with respect to Cartesian coordinate directions x = [x, y, z] T :…”

“…the form used in [14]. However, for cases where the solution is not smooth, leading to shocks forming in the gradients equation, the solution becomes multi-valued and depends on the initial solution.…”

“…As in [14], we find using a global eigenvalue γ = 1/f to improve convergence. For the viscous flux we use the second method of Bassi and Rebay [3].…”

“…Following the philosophy of Tucker [28], a discontinuous Galerkin solver for the eikonal equation was developed in [14] for body-fitted, general unstructured meshes achieving high-order accuracy. The motivation for this approach is that, while some algorithmic efficiency is lost, the eikonal equation is solved in parallel, leveraging the same data structures and algorithms as the physics simulation for which the solution is required.…”

“…As in [14], this work first focuses on computing an approximate signed distance function s, resulting as f = 1 and s = 0 on boundaries Γ. We seek to leverage our work on discontinuous-Galerkin spectral-element methods (DGSEM) for the Navier-Stokes equations [7,8,6] to develop an eikonal solver.…”