Boundary Collocation Methods: Review and Application to Composite Media P. A. Ramachandran Washington University St. Louis, MO Lecture Presented at UNLV

OUTLINE OF LECTURE  Introduction: Boundary Collocation  MFS: Diffusion-Reaction (D-R) Problem  Trefftz method: Laplace Equation  Trefftz: D-R Problem  MFS for Composite Regions  Effective Media Properties.  Concluding Remarks

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Boundary Collocation CONSIDER Lu=0 where L is a linear differential operator. Find or choose a set of basis function G i which satisfy PDE SOLUTION IS EXPRESSED AS: where b i are expansion coefficients. Fit the boundary conditions at selected points called the collocation points to find b i

PROTOTYPE EQUATIONS  Laplace Equation  Diffusion-Reaction Problems  Poisson equation and Non-linear and time dependent problems  These can be reduced to linear operator type by use of particular solutions; u = v + w;  MESH FREE METHODS

APPLICATION AREAS  LAPLACE EQUATON  HEAT TRANSFER  POTENTIAL FLOW  ELECTRIC POTENTIAL;  ELECTROCHEMICAL SYSTEMS  DIFFUSION WITH REACTION (Porous catalyst)  ACOUSTICS: (Helmholtz Equation)  STOKES FLOW: VECTOR POISSON EQUATION  LINEAR ELASTICITY: VECTOR EQUATION  MAGNETIC FIELD: VECTOR EQUATION.

MFS for Diffusion-Reaction Problem CONSIDER on  SOLUTION IS EXPRESSED AS: G i is a solution to PDE; Also known as fundamental solution or free space Greens function r i is the distance from source point i to any field point. Source point Collocation point 

Gradient at the boundary where Constants b i, determined using boundary collocation to satisfy boundary condition. Choose a set of ‘M’ Collocation Points are located on the boundary. Then for each point, j, we have: or Dirichlet or Neumann

Equations can be assembled in a matrix form (COE) is G ji or Q ji; (RHS) is either u j or p j Size of COE matrix is M (collocation Points) x N( source points). M=NDirect Collocation M>N Least Square Fitting MATLAB: b = COE \ RHS’ or b = pinv (COE) *RHS Estimate of Solution Accuracy (chi-square error)

Test Problem 1  =10; u=1 Geometry and Boundary Condition Dirichlet condition Source point Collocation point

Results for Test Problem 1 40 Collocation Points; 40 source points located on a circle; Circle radius 1 to 10. Center concentration = Average Flux = Chi-Square error 2e-06 to 2e-14. Condition number O(1.E+15) Convergence test was done with M = 160 and 20 to 160 source points.

Variations on the Theme: Method of Complete Solutions Regular Solutions: Chan and Tanaka Example: Circular domain with dirichlet Completeness of the basis functions? Method of complete solutions Annular region: Both functions are needed.

Comparison of K0 and I0 basis functions for test problem #1.

Test Problem 2 u =1

Results for Test Problem # 2

Singularity at the points where the boundary conditions change. Special functions may be needed to capture singularity. HEURISTIC RULE: Points close to the Neumann Boundary. SOURCE POINT LOCATION FOR MIXED BOUNDARY CONDITIONS

MFS for Laplace equation Basis Functions 1, ln r i, i = 1 to N r i = distance from the source point, i. N+1Coefficients to be found by either direct collocation or by the least square method

Laplace Equation: Example Test Problem (Golberg and Chen) Test Problem: Oval of Cassini ! Least Square error (n=39)

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Method of Regular Solutions: Trefftz Method For Laplace Equation on a simple domain the following basis functions hold. b 0 = 1 b1 = r cos  b 2 = r sin  b 3 = r 2 cos 2  b 4 = r 2 sin 2  etc. Obtained by Separation of Variables. NB = Number of basis function Solution for u is then r  r,  polar coordinate from some interior point, preferably the centroid of the area

Trefftz Method: Illustration Test Problem: Oval of Cassini (Golberg and Chen) Boundary Condition: u = exp(x)cos(y) Least Square fitting of B.C. was used. Treftz Series converges rapidly with the following coefficients: b 0 = 1b 1 = 0.5b 3 = b 5 = b 6 = b 7 = b9 = Coefficients are Fourier’s series of the analytic continuation of the boundary conditions on an unit circle. Advantage: Only a few terms are needed. No need to locate sources. No source points needed

Trefftz Functions: Derivations Separation of Variables in Polar Coordinates Trignometric Euler-Cauchy

T-Treftz Method: Multiple Domains Complete Set of Basis Functions are therefore as follows; Center (origin) 1, ln r r n cos n  r n sin n  n = 1,2,….etc.

T-Treftz Method: Diffusion-Reaction Problem Separation of Variables: u = R(r)T(  )  -Solutions: Trigonometric Functions R-Solutions: Modified Bessel Functions I 0 (  r),K 0 (  r) I n (  r)cos(n  ),K n (  r)sin(n  ) n = 1,2,3,….etc If r = 0 belongs to the domain, the K n functions are excluded

Diffusion-Reaction Problem T-Method Results for theTest Problem #1: Square geometry; Dirichlet on all sides. Fitted Coefficients 0 I 0 (  r) I n (  r)cos(n  ), = 0; n = 1, 2,3 I n (  r)cos(n  ), for n = 4 n = n = n = e+05 n = I n (  r) sin (n  ) = 0; NO K n

Concentration profiles along the y axis. The solution matches well with MFS. T-Method Results for theTest Problem #1: Square geometry; Dirichlet on all sides.

MFS for composite Regions: Problem Statement Motivation: Transport properties of a composite Material Shaded Region has different conductivity

Transport Properties of Composite Media Simple Models Series ModelParallel Model Variational Methods gives tighter bounds

Representation of Boundary Conditions A Simplified Two Region Case  E = External Boundary with prescribed boundary condition  C = Common Boundary for Region II Also  II for this case (  C =  II ) Matching Boundary Conditions on  C on  C EE CC n II nInI

Representation by MFS Region I Region II Solution Source point Collocation point

Matrix Representation of Solution Region I Region II EE CC Collocation points

Compact Matrix Representation Matching Conditions can be expressed as or and Equation can be compacted as Dirichlet boundary, Modify for Neumann points NOTE: due to matching condition Region I NOTE:

Test Problem I I AB CD Circular Inclusion in a Square geometry of unit length Flux along AB = Flux along CD k eff = Average Flux on AB/100 Porosity of Region II = ¶R 2, R 0.5 Porosity of Region I = 1 - ¶R 2

Trefftz method and Fundamental solution method provide comparable results; Trefftz method is more general (no source points needed). Method of complete solutions needs to be investigated; Initial results are promising; Optimal placement of source points? Statistical methods? MFS useful and easy to implement for composite regions. Need to explore this for other problems. Summary and Conclusions