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Parallel Solution of the 3-D Laplace Equation Using a Symmetric-Galerkin Boundary Integral Approximation Talisha Haywood Physics major w/ Emphasis in Computational Science Wofford College, Spartanburg, SC 29303 E-mail:

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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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

Talisha Haywood Physics major w/ Emphasis in Computational Science Wofford College, Spartanburg, SC 29303 E-mail: haywoodtalisha@hotmail.com Research Alliance for Minorities Mentor: Leonard J. Gray Computer Science and Mathematics Division E-mail: ljg@ornl.gov

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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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Problem How do we develop a parallel solution

  • f the 3-D Laplace Equation using a

Symmetric-Galerkin Boundary Integral Approximation?

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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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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

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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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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
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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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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

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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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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

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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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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( )]
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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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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?

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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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ScaLapack

  • Scalable Linear Algebra Package
  • Parallel libraries that support PVM and MPI
  • Designed to solve linear algebra problems on

distributed memory parallel computers

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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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ScaLapack Procedure

  • ScaLapack routines help in setting up number
  • f 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

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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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Parallel Algorithm

  • Execute linear solve routine in parallel
  • Call PDGESV
  • 4 basic steps to call ScaLapack routine
  • 1. Initialize process grid
  • 2. Data distribution
  • 3. Call the routine
  • 4. Release process grid
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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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Results

  • Techniques developed will work for boundary

integral equations other than Laplace Equation

  • Applications
  • Electrochemistry
  • Thermal Analysis
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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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Summary

  • Investigated the Laplace Equation
  • Modeled single processor algorithm
  • Edited single processor algorithm to develop a

parallel algorithm

  • Applied ScaLapack
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OAK RIDGE NATIONAL LABORATORY

U.S. DEPARTMENT OF ENERGY

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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