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NASA NTRS1990

Concurrent Finite-Element Analysis On Hypercube Computers

Improved approach to execution of finite-element codes on hypercube and similar concurrent data processors increases efficiency of computation for many different types of problems. Based on flexible general model of computation on, and communication among, large-node parallel processors. Hybrid combining direct methods within subdomains and preconditioned-conjugate-gradient (PCG) iteration on remaining boundary system to obtain method of solution both robust and efficient. Preserves general structure and function of conventional sequential finite-element-method software. Generalized to both distributed- and shared-memory multicomputers, eliminating degree of machine specificity restricting general usefulness.

Lyzenga, G. A.↗

NASA NTRS1987

Solving Finite-Element Problems on a Concurrent Processor

By use of "conjugate gradients" technique, concurrent efficiency greater than 90 percent. Algorithm applies method of conjugate gradients to iterative solution of finite-element problems on concurrent processor. With algorithm, iteration rates nearly proportional to number of processors. For sufficiently large problems, fraction of proportional speedup achieved, called concurrent efficiency, exceeds 90 percent. Results indicate future application of this and related algorithms to large finite-element problems depend primarily upon applicability of iteratiave techniques, not upon issues of concurrency or efficiency.

Lyzenga, G. A.↗

NASA NTRS1987

Solving finite element equations on concurrent computers

This paper discusses the development of a concurrent algorithm for the solution of systems of equations arising in finite element applications. The approach is based on a hybrid of direct elimination method and preconditioned conjugate iteration. Two different preconditioners are used; diagonal scaling and a concurrent implementation of incomplete LU factorization. First, an automatic procedure is used to partition the finite element mesh into sub-structures. The particular mesh partition is chosen to minimize an estimate of the cost for evaluating the solution using this algorithm on a concurrent computer. These procedures are implemented in a finite element program on the JPL/CalTech MARK III hypercube computer. An overview of the structure of this program is presented. The performance of the solution method is demonstrated with the aid of a number of numerical test runs, and its advantages for concurrent implementations are discussed. Efficiency and speed-up factors over sequential machines for the numerical examples are highlighted.

Nour-Omid, B.↗

NASA NTRS1985

On the penetration of a hot diapir through a strongly temperature-dependent viscosity medium

The ascent of a hot spherical body through a fluid with a strongly temperature-dependent viscosity has been studied using an axisymmetric finite element method. Numerical solutions range over Peclet numbers of 0.1 - 1000 from constant viscosity up to viscosity variations of 100,000. Both rigid and stress-free boundary conditions were applied at the surface of the sphere. The dependence of drag on viscosity variation was shown to have no dependence on the stress boundary condition except for a Stokes flow scaling factor. A Nusselt number parameterization based on the stress-free constant viscosity functional dependence on the Peclet number scaled by a parameter depending on the viscosity structure fits both stress-free and rigid boundary condition data above viscosity variations of 100. The temperature scale height was determined as a function of sphere radius. For the simple physical model studied in this paper pre-heating is required to reduce the ambient viscosity of the country rock to less than 10 to the 22nd sq cm/s in order for a 10 km diapir to penetrate a distance of several radii.

Daly, S. F.↗

NASA NTRS1985

A critical assessment of viscous models of trench topography and corner flow

Stresses for Newtonian viscous flow in a simple geometry (e.g., corner flow, bending flow) are obtained in order to study the effect of imposed velocity boundary conditions. Stress for a delta function velocity boundary condition decays as 1/R(2); for a step function velocity, stress goes as 1/R; for a discontinuity in curvature, the stress singularity is logarithmic. For corner flow, which has a discontinuity of velocity at a certain point, the corresponding stress has a 1/R singularity. However, for a more realistic circular-slab model, the stress singularity becomes logarithmic. Thus the stress distribution is very sensitive to the boundary conditions, and in evaluating the applicability of viscous models of trench topography it is essential to use realistic geometries. Topography and seismicity data from northern Hoshu, Japan, were used to construct a finite element model, with flow assumed tangent to the top of the grid, for both Newtonian and non-Newtonian flow (power law 3 rheology). Normal stresses at the top of the grid are compared to the observed trench topography and gravity anomalies. There is poor agreement. Purely viscous models of subducting slables with specified velocity boundary conditions do not predict normal stress patterns compatible with observed topography and gravity. Elasticity and plasticity appear to be important for the subduction process.

Zhang, J.↗

NASA NTRS1985

Finite elements and the method of conjugate gradients on a concurrent processor

An algorithm for the iterative solution of finite element problems on a concurrent processor is presented. The method of conjugate gradients is used to solve the system of matrix equations, which is distributed among the processors of a MIMD computer according to an element-based spatial decomposition. This algorithm is implemented in a two-dimensional elastostatics program on the Caltech Hypercube concurrent processor. The results of tests on up to 32 processors show nearly linear concurrent speedup, with efficiencies over 90 percent for sufficiently large problems.

Lyzenga, G. A.↗

NASA NTRS1984

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

Three-dimensional finite element modeling techniques are used to synthesize geodetic and seismological results for 1979 Imperial Valley earthquake. The strategy pursued consists of two principal steps. In the first step, the seismologically-derived coseismic fault slip is taken as a function of position in the fault plane and is applied directly to the three-dimensional dislocation model. In the second step, a physical model of stresses and constitutive parameters is perturbed so as to reproduce the observed fault slip. Hence, the principal features of the coseismic slip pattern are explained by a stress-driven fault model in which: (1) a spatially unresolved asperity is found equivalent to a stress drop of 18 MPa averaged over an area of 15 sq km, and (2) driving stress is essentially absent on the fault segment overlapping the 1940 earthquake rupture zone.

Slade, M. A.↗

NASA NTRS1984

A critical assessment of viscous models of trench topography and corner flow

Stresses for Newtonian viscous flow in a simple geometry (e.g., corner flow, bending flow) are obtained in order to study the effect of imposed velocity boundary conditions. Stress for a delta function velocity boundary condition decays as 1/R(2); for a step function velocity, stress goes as 1/R; for a discontinuity in curvature, the stress singularity is logarithmic. For corner flow, which has a discontinuity of velocity at a certain point, the corresponding stress has a 1/R singularity. However, for a more realistic circular-slab model, the stress singularity becomes logarithmic. Thus the stress distribution is very sensitive to the boundary conditions, and in evaluating the applicability of viscous models of trench topography it is essential to use realistic geometries. Topography and seismicity data from northern Hoshu, Japan, were used to construct a finite element model, with flow assumed tangent to the top of the grid, for both Newtonian and non-Newtonian flow (power law 3 rheology). Normal stresses at the top of the grid are compared to the observed trench topography and gravity anomalies. There is poor agreement. Purely viscous models of subducting slables with specified velocity boundary conditions do not predict normal stress patterns compatible with observed topography and gravity. Elasticity and plasticity appear to be important for the subduction process.

Zhang, J.↗

NASA NTRS1984

The distribution of earthquakes with depth and stress in subducting slabs

The global variation of Benioff zone seismicity with depth and the orientation of stress axes of deep and intermediate earthquakes is explained using numerical models of subducting slabs. Models that match the seismicity and stress require a barrier to flow at the 670 km seismic discontinuity. The barrier may be a viscosity increase of at least an order of magnitude or a chemical discontinuity. Instantaneous flow is subparallel to the slabs for models with a viscosity increase but contorted for models with a chemical barrier. Log N (number of earthquakes) decreases linearly to 250-300 km depth and increases thereafter. Stress magnitude in the models shows the same pattern, in accord with experiments showing N proportional to e(k-sigma), with k a constant and sigma the stress magnitude. The models predict downdip compression in the slabs at depths below 300-400 km, as observed for earthquake stress axes.

Vassiliou, M. S.↗

NASA NTRS1984

Finite elements and the method of conjugate gradients on a concurrent processor

An algorithm for the iterative solution of finite element problems on a concurrent processor is presented. The method of conjugate gradients is used to solve the system of matrix equations, which is distributed among the processors of a MIMD computer according to an element-based spatial decomposition. This algorithm is implemented in a two-dimensional elastostatics program on the Caltech Hypercube concurrent processor. The results of tests on up to 32 processors show nearly linear concurrent speedup, with efficiencies over 90% for sufficiently large problems.

Lyzenga, G. A.↗

NASA NTRS1983

Two-dimensional Finite Element Modeling for Modeling Tectonic Stress and Strain

Techniques of finite element analysis in two dimensional plane strain were applied to problems of geophysics and tectonics. More specifically, the flexibility of the finite element method was employed to address problems involving geological complexity and fault interactions. The modeling of effective anisotropy in material elastic properties proved useful in describing the deformation of faulted crustal blocks. The applications of this modeling work to problems of actual tectonics in southern California was explored. Preliminary models show encouraging agreement with measured tectonic strain in this region, and modeling work was done to gain an understanding of the stress state in a locked fault region with future seismic potential.

Lyzenga, G. A.↗

NASA NTRS1983

Anelastic response of the earth to a dip slip earthquake

The deformation induced by a vertical dip slip earthquake is examined using a variety of rheologic models. In this way the complications of dipping faults are avoided, and the phenomenon of transient peripheral warping is clearly revealed. A thrust fault dipping at 30 deg is investigated, and the important effects of dip and the existence of a slab on the asymmetry of strain pulses propagated into the overthrust and subducted lithosphere are demonstrated. One of the signal results of the study is the essential similarity of the strain patterns for Newtonian and non-Newtonian flow laws: the two rheologies give nearly identical strain field geometries. The principal difference between the two, which is readily observable, is in their time evolution. Relaxation in non-Newtonian rheologies tends to be initially fast, then slow at times that are late in comparison with relaxation in a Newtonian rheology. The possibility of simply recalling the time dependence of a Newtonian solution to obtain an approximate solution to a non-Newtonian problem is demonstrated.

Melosh, H. J.↗

NASA NTRS1983

Subduction, back-arc spreading and global mantle flow

It is pointed out that the subducted lithosphere associated with Benioff zones provides the only direct evidence about the flow in the earth's interior associated with plate motions. It is the primary objective of the present investigation to study the relation between the orientation of subducting lithosphere and the flow patterns (both local and global) near subduction zones. Most of the calculations conducted are based on simple flow models for radially symmetric, Newtonian viscous spheres. The investigation is concerned with the possibility that a simple model of global mantle flow could account for some features of subduction zones. It is found that such a model can account for the orientation of the seismic zones, and, in addition, also for features related to back-arc spreading and perhaps the maximum earthquake size.

Hager, B. H.↗