Problem Statement

Given a bounded domain $\Omega\subset \mathbb{R} ^3$ the goal is to approximate the solution $\mbox{\boldmath {$E$ }}$ (electric field) to Maxwell's equations

$$\nabla\times(\nabla\times \mbox{\boldmath {$E$ }})-k^2\epsilon_r\mbox{\boldmath {$E$ }}=0$$

in $\Omega$ where k is the wave number of the radiation (the wave-length of the radiation is $\lambda=2\pi/k$) and $\epsilon_r$ is the (possibley complex and spatially varying) relative permittivity of the material in $\Omega$. In addition the solution must satisfy the generalized impedance boundary condition:

$$-(1+Q)\mbox{\boldmath {$E$ }}\times\mbox{\boldmath {$\nu$ }}+... ...$E$ }})\times\mbox{\boldmath {$\nu$ }}=\mbox{\boldmath {$g$ }}$$

on the boundary of $\Omega$ where $\mbox{\boldmath {$\nu$ }}$ is the unit outward normal to the boundary of $\Omega$. Using different combinations of Q and the data function $\mbox{\boldmath {$g$ }}$ we can simulate different types of boundaries. For example Q=0 and $\mbox{\boldmath {$g$ }}=0$ is a classical absorbing boundary condition, while Q=1 gives a perfect conductor. The UWVF shares many similarities with finite element codes. First the region of interest is subdivided into tetrahedra using an automatic mesh generator (this would be an interesting subject for parallelization, but far too difficult for this semester!). Note that the resulting mesh is unstructured and may have elements of very different sizes. A cut through a simple mesh is shown in Figure 1.

  

Figure 1: A cut through mesh for the domain between a sphere and a cube. The goal is to simulate scattering from the inner sphere using the outer sphere as an artificial boundary to terminate the mesh. This is a small test mesh with around 71,000 tetrahedra.

Once the mesh is available the UWVF code performs two operations

• The tetrahedra are processed and a large, sparse complex valued matrix is generated. In addition a right hand side vector is also computed.
• The matrix problem is solved by a simple iterative process requiring only matrix multiplies.

Even for small numbers of tetrahedra, this can be very time consuming and very memory intensive. For example, a very small test problem using 71,528 tetrahedra with 20 basis functions per tetrahedron currently requires 2.5GByte of memory and runs for 4,859 seconds (in serial mode) on a 195 MHz Silicon Graphics Origin-2000. Realistic problems of simulating the interaction of microwaves with the human head will require far more memory and computer time.

We note that the assembly phase is a small fraction of the overall computer time.

The UWVF is not a standard finite element method (it is closer to a discontinuous Galerkin finite element method in spirit), so many of the node based parallelization schemes in the literature need to be adapted in this case. In particular, at the assembly stage, when computing the matrices for one tetrahedron, it is necessary to use data from surrounding tetrahedra (unlike finite elements in which this process is entirely local). However the sparsity structure of the resulting matrix is quite simple.