Using a multifrontal sparse solver in a high performance, finite element code

We consider the performance of the finite element method on a vector supercomputer. The computationally intensive parts of the finite element method are typically the individual element forms and the solution of the global stiffness matrix both of which are vectorized in high performance codes. To further increase throughput, new algorithms are needed. We compare a multifrontal sparse solver to a traditional skyline solver in a finite element code on a vector supercomputer. The multifrontal solver uses the Multiple-Minimum Degree reordering heuristic to reduce the number of operations required to factor a sparse matrix and full matrix computational kernels (e.g., BLAS3) to enhance vector performance. The net result in an order-of-magnitude reduction in run time for a finite element application on one processor of a Cray X-MP

Modeling of the surface static displacements and fault plane slip for the 1979 Imperial Valley earthquake

Synthesis of geodetic and seismological results for the 1979 Imperial Valley earthquake is approached using three-dimensional finite element modeling techniques. The displacements and stresses are calculated elastically throughout the modeled region. The vertical elastic structure in the model is derived from compressional and shear wave velocities as used in the seismic data analysis (Fuis et al., 1981) combined with a sediment density profile. Two strategies for applying initial conditions are followed in this modeling. In the first strategy, a sample seismological estimate for fault plane slip is used to predict the resultant surface motions. We show that the geodetic strain results over distances of tens of kilometer from the fault (Snay et al., 1982) are basically consistent with the model seismic fault displacements. Geodetic results from within a few kilometers of the fault trace (Mason et al., 1981) seem to require more slip at shallow depths than appears at seismic time scales. This is consistent with the occurrence of aftercreep at shallow depths in less well-consolidated material, which would bring surface displacements into line with maximum slip at depth, but not greatly affect the net moment. In the second strategy, we consider stresses on the fault plane, rather than displacements, as model variables. To constrain this part of our numerical modeling, we assume that the fault driving stress is governed by ambient tectonic stress and an opposing Coulomb friction derived from experiment. The coseismic stress drop from point to point on the failed fault is given by the difference between the tectonic shear stress and the frictional stress. After arriving at such a uniform model which adequately represents the Snay et al. results, we further modify a small region near the seismic “asperity” to make the fault plane motions qualitatively and quantitatively resemble the model of coseismic motions used in the first strategy. The observed offset on the fault trace (Sharp et al., 1982) is approximated in this final stress-driven model by removing the driving stress on the southern third of the fault. Thus, the principal features of the coseismic slip pattern are explained by a stress-driven fault model in which: (a) a spatially unresolved asperity is found equivalent to a stress drop of 18 MPa averaged over an area of 15 km^2, and (b) driving stress is essentially absent on the fault segment overlapping the 1940 earthquake rupture zone

Subduction, back-arc spreading and global mantle flow

The shapes and orientations of Benioff zones beneath island arcs, interpreted as marking the location of subducted lithosphere, provide the best presently available constraints on the global convective flow pattern associated with plate motions. This global flow influences the dynamics of subduction. Subduction zone phenomena therefore provide powerful tests for models of mantle flow. We compute global flow models which, while simple, include those features which are best constrained, namely the observed plate velocities, applied as boundary conditions, and the density contrasts given by thermal models of the lithosphere and subducted slabs. Two viscosity structures are used; for one, flow is confined to the upper mantle, while for the other, flow extends throughout the mantle. Instantaneous flow velocity vectors match observed Benioff zone dips and shapes for the model which allows mantle-wide flow but not for the upper mantle model, which has a highly contorted flow pattern. The effect of trench migration on particle trajectories is calculated; it is not important if subduction velocities are greater than migration rates. Two-dimensional finite element models show that including a coherent high viscosity slab does not change these conclusions. A coherent high viscosity slab extending deep into the upper mantle would significantly slow subduction if flow were confined to the upper mantle. The maximum earthquake magnitude, M_w, for island arcs correlates well with the age of the subducted slab and pressure gradient between the trench and back-arc region for the whole mantle, but not the upper mantle, flow model. The correlations with orientations of Benioff zones and seismic coupling strongly suggest that the global return flow associated with plate motions extends below 700 km. For both models, regions of back-arc spreading have asthenospheric shear pulling the back-arc toward the trench; regions without back-arc spreading have the opposite sense of shear, suggesting global flow strongly influences back-arc spreading

Subduction, back-arc spreading and global mantle flow

The shapes and orientations of Benioff zones beneath island arcs, interpreted as marking the location of subducted lithosphere, provide the best presently available constraints on the global convective flow pattern associated with plate motions. This global flow influences the dynamics of subduction. Subduction zone phenomena therefore provide powerful tests for models of mantle flow. We compute global flow models which, while simple, include those features which are best constrained, namely the observed plate velocities, applied as boundary conditions, and the density contrasts given by thermal models of the lithosphere and subducted slabs. Two viscosity structures are used; for one, flow is confined to the upper mantle, while for the other, flow extends throughout the mantle. Instantaneous flow velocity vectors match observed Benioff zone dips and shapes for the model which allows mantle-wide flow but not for the upper mantle model, which has a highly contorted flow pattern. The effect of trench migration on particle trajectories is calculated; it is not important if subduction velocities are greater than migration rates. Two-dimensional finite element models show that including a coherent high viscosity slab does not change these conclusions. A coherent high viscosity slab extending deep into the upper mantle would significantly slow subduction if flow were confined to the upper mantle. The maximum earthquake magnitude, M_w, for island arcs correlates well with the age of the subducted slab and pressure gradient between the trench and back-arc region for the whole mantle, but not the upper mantle, flow model. The correlations with orientations of Benioff zones and seismic coupling strongly suggest that the global return flow associated with plate motions extends below 700 km. For both models, regions of back-arc spreading have asthenospheric shear pulling the back-arc toward the trench; regions without back-arc spreading have the opposite sense of shear, suggesting global flow strongly influences back-arc spreading

Finite Element Solution of Thermal Convection On A Hypercube Concurrent Computer

Numerical solutions to thermal convection flow problems are vital to many scientific and engineering problems. One fundamental geophysical problem is the thermal convection responsible for continental drift and sea floor spreading. The earth's interior undergoes slow creeping flow (~cm/yr) in response to the buoyancy forces generated by temperature variations caused by the decay of radioactive elements and secular cooling. Convection in the earth's mantle, the 3000 km thick solid layer between the crust and core, is difficult to model for three reasons: (1) Complex rheology -- the effective viscosity depends exponentially on temperature, on pressure (or depth) and on the deviatoric stress; (2) the buoyancy forces driving the flow occur in boundary layers thin in comparison to the total depth; and (3) spherical geometry -- the flow in the interior is fully three dimensional. Because of these many difficulties, accurate and realistic simulations of this process easily overwhelm current computer speed and memory (including the Cray XMP and Cray 2) and only simplified problems have been attempted [e.g. Christensen and Yuen, 1984; Gurnis, 1988; Jarvis and Peltier, 1982]. As a start in overcoming these difficulties, a number of finite element formulations have been explored on hypercube concurrent computers. Although two coupled equations are required to solve this problem (the momentum or Stokes equation and the energy or advection-diffusion equation), we will concentrate our efforts on the solution to the latter equation in this paper. Solution of the former equation is discussed elsewhere [Lyzenga, et al, 1988]. We will demonstrate that linear speedups and efficiencies of 99 percent are achieved for sufficiently large problems