Sparse linear solvers applied Sparse linear solvers applied to to parallel simulations of parallel simulations of underground flow underground flow in porous and fractured media in porous and fractured media A. Beaudoin A. Beaudoin 1 , J.R. De Dreuzy , J.R. De Dreuzy 2 , J. Erhel , J. Erhel 1 and H. and H. Mustapha Mustapha 1 1 - IRISA / INRIA, Rennes, France 1 - IRISA / INRIA, Rennes, France 2 - Department of Geosciences, University of Rennes 2 - Department of Geosciences, University of Rennes , , France France Matrix Computations and Scientific Computing Seminar Berkeley, 26 October 2005
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A. Beaudoin 1 , J.R. De Dreuzy 2 , J. Erhel 1 and H. Mustapha 1
Sparse linear solvers applied to parallel simulations of underground flow in porous and fractured media. A. Beaudoin 1 , J.R. De Dreuzy 2 , J. Erhel 1 and H. Mustapha 1. 1 - IRISA / INRIA, Rennes, France 2 - Department of Geosciences, University of Rennes , France. - PowerPoint PPT Presentation
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Sparse linear solvers applied to Sparse linear solvers applied to parallel simulations of parallel simulations of
underground flow underground flow in porous and fractured mediain porous and fractured media
A. BeaudoinA. Beaudoin11, J.R. De Dreuzy, J.R. De Dreuzy22, J. Erhel, J. Erhel11 and H. and H. MustaphaMustapha11
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
2( ) expY YY
C
rr
91 Y
3D fracture network with impervious matrix3D fracture network with impervious matrix
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
length distribution has a great impact : power law n(l) = l-a
3 types of networks based on the moments of length distribution
mean variation third moment3 < a < 4
mean variation2 < a < 3
mean variation third momenta > 4
EquationsEquations
Q = - K*Q = - K*grad (hgrad (h) )
div (Q) = 0div (Q) = 0 BoundaryBoundary conditions conditions
Flow modelFlow model
Fixed head
Nul flux
3D fracture network3D fracture network
Fix
ed
head
Fix
ed
head
Nul flux
Nul flux
2D porous medium2D porous medium
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
Numerical method for 2D heterogeneous Numerical method for 2D heterogeneous porous mediumporous medium
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
Finite Volume Method with a regular mesh
Large sparse structured matrix with 5 entries per row
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
n=32 zoom
Sparse matrix for 2D heterogeneous porous Sparse matrix for 2D heterogeneous porous mediummedium
Conforming Conforming triangular triangular
meshmesh
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
Mixed Hybrid Finite Element Method with unstructured mesh
Large sparse unstructured matrix with about 5 entries per row
Numerical method for 3D Numerical method for 3D fracture networkfracture network
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
Sparse matrix for 3D fracture Sparse matrix for 3D fracture networknetwork
N = 8181
Intersections and 7 fractures
zoom
Memory requirements for matrices A and LMemory requirements for matrices A and L
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
Complexity analysis with PSPASESComplexity analysis with PSPASES
CPU time of matrix generation, linear solving and flow computationCPU time of matrix generation, linear solving and flow computationobtained with two processorsobtained with two processors
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
Complexity analysis with PSPASESComplexity analysis with PSPASES
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
2D porous medium : memory size and CPU time 2D porous medium : memory size and CPU time with PSPASESwith PSPASES
Theory : NZ(L) = O(N logN) Theory : Time = O(N1.5)
Slope about 1 Slope about 1.5
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
3D fracture network : memory size and CPU time 3D fracture network : memory size and CPU time with PSPASESwith PSPASES
NZ(L) = O(N) ? Time = O(N) ?
Theory to be done
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
2D porous medium : condition number estimated by 2D porous medium : condition number estimated by MUMPSMUMPS
To be ckecked : scaling or not
Parallel Simulations of Underground Flow in Porous Parallel Simulations of Underground Flow in Porous and Fractured Mediaand Fractured Media
2D porous medium : residuals with PSPASES 2D porous medium : residuals with PSPASES