o ak r idge n ational l aboratory u.s. d epartment of e nergy 1 parallel solution of the 3-d laplace...

1

OAK RIDGE NATIONAL LABORATORYU.S. DEPARTMENT OF ENERGY

Parallel Solution of the 3-D Laplace Equation Using a Symmetric-Galerkin Boundary Integral

Approximation

Talisha HaywoodPhysics major w/ Emphasis in Computational Science

Wofford College, Spartanburg, SC 29303E-mail: [email protected]

Research Alliance for Minorities

Mentor: Leonard J. GrayComputer Science and Mathematics Division

E-mail: [email protected]


2

Problem

How do we develop a parallel solution

of the 3-D Laplace Equation using a Symmetric-Galerkin Boundary Integral

Approximation?


3

Background

Boundary element method: fundamental technique used for solving partial differential equations

Discretizes boundary integral equation to find system of equations from which the boundary values can be found

After a problem has been solved, normal flux and potential are known everywhere in domain


4

Steps

Investigate Laplace Equation Model and edit the single processor algorithm Investigate the code ScaLapack and its procedure Develop a parallel algorithm Test the algorithm


5

Laplace Partial Differential Equation

Integral equation on domain boundary (e.g. surface of a cube)

Either potential or flux is specified everywhere on the boundary


6

Single Processor Algorithm

Two Integral Equations Discretize equations then simplify to matrix

form: (H[potential]=G[flux])

Rearrange H and G columns Move known values to the right-hand side Move unknown values to the left-hand side

Simplify rearranged matrix to (Ax=b) form Call the routine that solves the finite system of

linear equations


7

Code Behavior

Based on a cube divided into triangular elements

Boundary value is given, 0 or 1 Input: number of elements, number of nodes Output: coordinates of elements and nodes Eight subroutines

- bem( ) -hyp( )

-bem_c( ) -hyp_c( )

-bem_ae( ) -hyp_ae

-bem_av( ) -hyp_av( )]


8

Breakdown of subroutine bem( ) [all non-touching elements]

H and G is a sum of integrals (EP and EQ)

Integrals are done for each pair of elements (EP,EQ)

EP integral generates 3 elements P1 P2 P3

EQ integral generates 3 elements Q1 Q2 Q3

After the loop a test should be made: Do I have to do this integral?


9

ScaLapack

Scalable Linear Algebra Package

Parallel libraries that support PVM and MPI

Designed to solve linear algebra problems on distributed memory parallel computers


10

ScaLapack Procedure

ScaLapack routines help in setting up number of processes

Each process constructs its own matrix Each processor receives all of the code Each processor has to check whether it has to

do certain integrals Sum up matrix elements and solve the linear

equation


11

Parallel Algorithm

Execute linear solve routine in parallel

Call PDGESV

4 basic steps to call ScaLapack routine1.Initialize process grid

2.Data distribution

3.Call the routine

4.Release process grid


12

Results

Techniques developed will work for boundary integral equations other than Laplace Equation

Applications Electrochemistry Thermal Analysis


13

Summary

Investigated the Laplace Equation

Modeled single processor algorithm

Edited single processor algorithm to develop a parallel algorithm

Applied ScaLapack


14

Acknowledgements

This research was performed under the Research Alliance for Minorities Program administered through the Computer Science and Mathematics Division, Oak ridge National Laboratory. This program is sponsored by the Mathematical, Information, and Computational Sciences Division; Office of Advanced Scientific Computing Research; U. S. Department of Energy. Oak Ridge National Laboratory is managed by UT-Battelle, LLC, for the U.S. Department of Energy under contract DE-AC05-00OR22725

I would like to thank my mentor Leonard J. Gray and Bill Shelton for introducing me to another level of mathematics and computer science

o ak r idge n ational l aboratory u.s. d epartment of e nergy 1 parallel solution of the 3-d laplace...

Documents