20071031 dijkstra

Information about 20071031 dijkstra

Published on January 11, 2008

Author: Sebastiana

Source: authorstream.com

Content

The Boundary Element Method: failure is an option:  The Boundary Element Method: failure is an option Willem Dijkstra Bob Mattheij October 31st, 2007 Problem description:  Problem description Problem A Problem B Method X solves Problem A Method X solves Problem B ? Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Boundary Element Method:  Boundary Element Method Green’s identity: Laplace problem: Boundary Element Method:  Boundary Element Method Boundary integral equation for Laplace problem: Constant approximations at each element: Matrix – vector notation: Slide7:  Boundary Element Method Slide8:  Boundary Element Method Advantages: Small matrices Tracking boundary only Disadvantages: Dense matrices Homogeneous media only Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide10:  Laplace problem Example: Laplace equation on a circle with radius R Plot cond(A) as function of R Not solvable when R=1 Slide11:  Laplace problem Example: mixed problem at ellipse Unknowns: Not solvable when a=4/3 Slide12:  N=24 Laplace problem Slide13:  Laplace problem Consequence: may expect (almost) singular linear systems Slide14:  Laplace problem Boundary integral equation to solve: Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide16:  Logarithmic capacity BIE not solvable on Γ logarithmic capacity of Γ is one Definition logarithmic capacity C(Γ): with Slide17:  Logarithmic capacity Properties: Slide18:  Recall: mixed problem at ellipse Logarithmic capacity Singular when a=4/3 Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies 5. Stokes problem Application Conclusions Slide20:  Rescale domain such that: Adjust fundamental solution: Add extra equation: Remedies Solve with: G1 Slide21:  Remedies Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide23:  Stokes problem Can also be solved with BEM fundamental solution Stokes equations Green’s identity for Stokes equations lead to Slide24:  Stokes problem Example: Dirichlet problem on a circle v is given, t unknown Not solvable at Slide25:  Stokes problem Example: Dirichlet problem on an ellipse v is given, t unknown Slide26:  Stokes problem Consequence: may expect (almost) singular linear systems Slide27:  Stokes problem No equivalent to logarithmic capacity No a priori information about critical domains Solve with: Remedy: add equations: G1 Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide29:  Application Blowing problem: Glass flow governed by Stokes equations Mixed boundary conditions BEM solves velocity at boundary Slide30:  Application Shape cond(A) Slide31:  Application Outline:  Outline Boundary Element Method Laplace problem Logarithmic capacity Remedies Stokes problem Application Conclusions Slide33:  Conclusions BEM does not always work for: Laplace problem (Dirichlet / mixed b.c.) Stokes problem (Dirichlet / mixed b.c.) Adding equations always works Unanswered questions: Critical domains for the Stokes problem Extension to general BVP Slide34:  W. D., R. M. M. Mattheij, A relation between the logarithmic capacity and the condition number of the BEM-matrices, Comm. Numer. Meth. Engrg. 23, 665-680, 2007 W. D., R. M. M. Mattheij Condition number of the BEM matrix arising from the Stokes equations in 2D, Eng. Anal. Bound. Elem., submitted E. Hille, Analytic Function Theory, volume II, Ginn, London, 1959 V. Dominguez, F.J. Sayas, A BEM-FEM overlapping algorithm for the Stokes equation, Appl. Math. Comput. 182, 691-710, 2006 References

Related presentations


Other presentations created by Sebastiana

2007 1 9 physical hazards
05. 03. 2008
0 views

2007 1 9 physical hazards

ElectricalSafety
22. 01. 2008
0 views

ElectricalSafety

Sea Power and Maritime
09. 01. 2008
0 views

Sea Power and Maritime

27831
15. 01. 2008
0 views

27831

endangered species
15. 01. 2008
0 views

endangered species

P078 00005
17. 01. 2008
0 views

P078 00005

film distributor
21. 01. 2008
0 views

film distributor

l2
21. 01. 2008
0 views

l2

203 Lec 21 St Lucia Conservation
22. 01. 2008
0 views

203 Lec 21 St Lucia Conservation

jane eyre
04. 02. 2008
0 views

jane eyre

The ministry of prayer
05. 02. 2008
0 views

The ministry of prayer

Adelante HHM 2007
12. 02. 2008
0 views

Adelante HHM 2007

ppt07
24. 01. 2008
0 views

ppt07

Prune
08. 01. 2008
0 views

Prune

Class1
09. 01. 2008
0 views

Class1

IRSL Q2 FY2006 Analyst Meet 17th
25. 01. 2008
0 views

IRSL Q2 FY2006 Analyst Meet 17th

returns2004
07. 02. 2008
0 views

returns2004

BackupAirways
14. 02. 2008
0 views

BackupAirways

VisualBasic
20. 02. 2008
0 views

VisualBasic

Question of rivers
11. 03. 2008
0 views

Question of rivers

pomeroy
15. 03. 2008
0 views

pomeroy

lockmart
21. 03. 2008
0 views

lockmart

case study cement plants
11. 02. 2008
0 views

case study cement plants

APESlec ppt ch13
03. 04. 2008
0 views

APESlec ppt ch13

Optimized PSTN Oct 05
08. 04. 2008
0 views

Optimized PSTN Oct 05

l5 mike
25. 02. 2008
0 views

l5 mike

3 1 11
16. 04. 2008
0 views

3 1 11

global2
24. 04. 2008
0 views

global2

030826wto
08. 05. 2008
0 views

030826wto

OlympicGames
02. 05. 2008
0 views

OlympicGames

customerforum 2007
30. 01. 2008
0 views

customerforum 2007

36188 Textilindustri
22. 01. 2008
0 views

36188 Textilindustri

Add Some Spice
26. 02. 2008
0 views

Add Some Spice

BMcCracken Presentation
04. 02. 2008
0 views

BMcCracken Presentation

TOP training pt1
13. 01. 2008
0 views

TOP training pt1

5 1 peter danielsson
23. 01. 2008
0 views

5 1 peter danielsson