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Sparse matrix computations. Dave O’Hallaron Intel Research Pittsburgh Sept 2007. element. node. t. mesh. simulation results. Galerkin discretization. FEM solver. Animation. Visualization Model. Mathematical Model. Computer Model. Numerical Model. Physical Model. Physical

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sparse matrix computations

Sparse matrix computations

Dave O’Hallaron

Intel Research Pittsburgh

Sept 2007

slide2

element

node

t

mesh

simulation results

Galerkin discretization

FEM solver

Animation

Visualization

Model

Mathematical

Model

Computer

Model

Numerical

Model

Physical

Model

Physical

System

Wave propagation equation

Material property model

Earthquake ground motion

Scientific Computing Process

slide3

t

Scientific Computing Workflow

Physical model

Mesh

Simulation

results

Solver

Mesh

generation

Visuali-

zation

Mesh

slide4

David O’Hallaron (CMU CS and ECE)

Jacobo Bielak (CMU CivE)

1994 northridge quake simulation
1994 Northridge Quake Simulation

20 seconds of an aftershock from the Jan 17, 1994 Northridge quake in San Fernando Valley of Southern California.

san fernando valley

lat. 34.38 long. -118.16

epicenter lat. 34.32 long. -118.48

x

San Fernando Valley

lat. 34.08 long. -118.75

San Fernando Valley
san fernando valley top view
San Fernando Valley (Top View)

Hard rock

epicenter

x

Soft soil

quake solver code

The disp vector records the

displacements of each mesh node during last three timesteps

Quake solver code

NODEVECTOR3 disp[3], M, C, M23;

MATRIX3 K;

/* matrix and vector assembly */

FORELEM(i) {

...

}

/* time integration loop */

for (iter = 1; iter <= timesteps; iter++) {

SMVP(K, disp[dispt], disp[disptplus]);

disp[disptplus] *= - IP.dt * IP.dt;

disp[disptplus] += 2.0 * M * disp[dispt] -

(M - IP.dt / 2.0 * C) *

disp[disptminus] - ...);

disp[disptplus] = disp[disptplus] / (M + IP.dt / 2.0 * C);

i = disptminus;

disptminus = dispt;

dispt = disptplus;

disptplus = i;

}

K is the adjacency matrix of the mesh, a labeled undirected graph

90% of time spent in sparse matrix vector product (SMVP) kernel