A FAST ITERATIVE METHOD FOR SOLVING THE EIKONAL EQUATION ON TETRAHEDRAL DOMAINS

Abstract

Generating numerical solutions to the eikonal equation and its many variations has a broad range of applications in both the natural and computational sciences. Efficient solvers on cutting-edge, parallel architectures require new algorithms that may not be theoretically optimal, but that are designed to allow asynchronous solution updates and have limited memory access patterns. This paper presents a parallel algorithm for solving the eikonal equation on fully unstructured tetrahedral meshes. The method is appropriate for the type of fine-grained parallelism found on modern massively-SIMD architectures such as graphics processors and takes into account the particular constraints and capabilities of these computing platforms. This work builds on previous work for solving these equations on triangle meshes; in this paper we adapt and extend previous two-dimensional strategies to accommodate three-dimensional, unstructured, tetrahedralized domains. These new developments include a local update strategy with data compaction for tetrahedral meshes that provides solutions on both serial and parallel architectures, with a generalization to inhomogeneous, anisotropic speed functions. We also propose two new update schemes, specialized to mitigate the natural data increase observed when moving to three dimensions, and the data structures necessary for efficiently mapping data to parallel SIMD processors in a way that maintains computational density. Finally, we present descriptions of the implementations for a single CPU, as well as multicore CPUs with shared memory and SIMD architectures, with comparative results against state-of-the-art eikonal solvers.

Keywords: Hamilton–Jacobi equation; eikonal equation; graphics processing unit; parallel algorithm; shared memory multiple-processor computer system; tetrahedral mesh.

Fig. 1.1

Examples of body-fitting meshes used for numerical simulation. On the left, we present the surface of a heart model mesh used for bioelectric computation. On the right, we present a cross section of a lens model used for the simulation of geometric optics (the green region denotes air while red denotes the location of the lens).

Fig. 2.1

Diagram denoting components of the local solver. We compute the value of the approximation at the vertex v₄ from the values at vertices v₁, v₂ and v₃. The vector n denotes the wave propagation direction that intersects with the triangle Δ_1,2,3 at v₅.

Fig. 2.2

Diagram denoting the strategy used to deal with obtuse tetrahedra. We split the obtuse angle ω₄ to create three virtual tetrahedra used within the local solver.

Fig. 3.1

2D representation of the outer surface of vertex v formed by the one-ring tetrahedra: the polygon formed by the bold line segments is analogous to the outer triangular surface in a tetrahedral mesh.

Fig. 3.2

One-ring-strip data structure: in this figure, T_i is a tetrahedron, x_i, y_i and z_i represent the coordinates of the ith vertex, f_i is the inverse of speed on a vertex. Φ_i denotes the value of the solution at the ith vertex. I_i in STRIP represents the data structure for the one-ring strip of the ith vertex each of which has q indices pointing (shown as arrows) to the value array.

Fig. 4.1

Blobs mesh and its cross section. The different colors in the cross section represent different materials indices of refraction (speed functions).

Fig. 4.2

Color maps and level curves of the solutions on the cube and heart meshes. Left: the ellipse speed function (Speed 2). Right: the isotropic constant function (Speed 1).

Fig. 4.3

Color maps and level curves of the solutions on the lens model with boundary as given by the figure on the left.

Fig. 4.4

Color maps and level curves of the solutions on the blobs model with boundary as given by the figure on the left.