36th AIAA Thermophysics Conference Hilton Walt Disney World. Orlando, Florida 23 - 26 June 2003

ITERATIVE DOMAIN DECOMPOSITION FOR PARALLEL COMPUTING OF LARGE-SCALE BEM HEAT TRANSFER MODELS

Eduardo Divo Department of Engineering Technology

University of Central Florida, Orlando, FL, 328 16-2450

Alain Kassab Department of Mechanical, Materials and Aerospace Engineering

University of Central Florida, Orlando, FL, 328 16-2450 kassab@mail.ucf.edu

Franklin Rodriguez Department of Mechanical, Materials and Aerospace Engineering

University of Central Florida, Orlando, FL, 328 16-2450

Abstract The solution of heat conduction problems using the boundary element method (BEM) only requires a surface mesh resulting in a fully-populated matrix for the final algebraic set of equations. This poses a serious challenge for large-scale problems due to storage requirements and computational times required by iterative solution of large set of non-symmetric equations. Approaches generally adopted to resolve this problem are: (1) artificial subsectioning of the geometry into a multi-region model in conjunction with algebraic block-solvers reminiscent of FEM frontal solvers, and (2) multipole methods in conjunction with non- symmetric iterative solvers such as GMRES. We propose to adopt a region-by-region iterative approach and is herein shown that the process converges efficiently, offers significant savings in memory, and does not require complex data-structure preparation. Several examples are presented and a 3-D film-cooled turbine blade geometry is considered.

Copyright 0 1997 by Eduardo Divo, Alain J. Kassab, and Franklin Rodriguez. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Introduction The boundary element method (BEM) is ideally suited for the solution of linear and non-linear heat conduction problems and is a particularly advantageous numerical method due to its boundary-only feature [ 1,2]. The BEM only requires a surface mesh to solve most heat conduction problems. However, the coefficient matrix of the resulting system of algebraic equations is fully-populated. For large-scale problems resulting in 3- D modeling of complex structures such as turbomachinery components, this poses very serious numerical challenges due both to large and often prohibitive storage requirements and to large computational times required by iterative solution of large sets of non-sparse equations. This problem has been approached in the BEM community two distinct methodologies: (1) the artificial subsectioning of the 3- D model, or domain decomposition, into a multi-region model in conjunction with block-solvers reminiscent of the FEM frontal solvers [3-61, and (2) the adoption of fast multipole methods in conjunction with non- symmetric iterative solvers such as GMRES [2,7-81. The first approach of domain decomposition produces a sparse block coefficient matrix that is efficiently stored and has been successfully implemented in commercial codes such as BETTI and GPBEST in the context of continuous boundary elements. However, the method requires generation of complex data-structures identifying connecting regions, star points, and

1

36th AIAA Thermophysics Conference23-26 June 2003, Orlando, Florida

AIAA 2003-4208

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

interfaces prior to analysis and lends itself with some difficulty to parallel implementation. The second approach offers an efficient alternative and lends itself to parallelization, however, it requires complete re- writing of the BEM code to adopt multipole formulation. Recently, a novel technique using wavelet decomposition has been proposed to reduce matrix storage requirements without need for a major alteration of traditional BEM codes [9 ] .

We propose to adopt the artificial subsectioning approach with the specific goal of tailoring the algorithm to parallelization, for which we employ a region-by-region iterative solver enforcing continuity of temperatures and heat fluxes at domain interfaces rather than a block solver. Each region is kept under 6,000 degrees of freedom to avoid delays due to hard drive paging and the heat conduction problem on each individual region is solved using the BEM with a non- symmetric pre-conditioned GMRES solver. This approach avoids generation of complex data-structures and lends itself ideally to parallel computation. Although it was reported in the literature that this process sometimes has difficulty converging for nonlinear problems [3,4], it is shown that when properly implemented using a dual temperature/flux averaging at the interfaces, the process converges very efficiently and can offer very substantial savings in memory. The approach is somewhat transparent to the user and offers a significant advantage in coupling to other field solvers such a Navier-Stokes finite volume solver, an approach taken by the authors to solve conjugate heat transfer (CHT) problems in large-scale turbomachinery problems[ IO]. Several numerical examples are presented to illustrate the technique.

q = h4 (T- Tm4)

Figure 1. BEM problem domain, boundary conditions, and single region discretization.

direct memory allocation also proportional to N 2 . With the aid of the boundary condition distribution, the system is re-arranged as

(1)

where (x} contains nodal unknowns T or q, whichever is not specified in the boundary conditions. The solution to the algebraic system for the boundary unknowns can be performed using a direct solution method such as LU decomposition requiring floating point operations proportional to N 3 or iterative methods such as Bi- Conjugate Gradient or General Minimization of Residuals (GMRES) that, in general, require floating

[HI{T) = [Gl{q) * [Al{zI = @)

point operations proportional to N 2 to achieve convergence.

Iterative Multi-Region BEM Approach The solution algorithm for the multi-region BEM iteration process consists on the steps to be described in this section which will specifically address a two- dimensional problem for ease of presentation. First, the problem domain R is identified along with the corresponding conditions over the boundary r. A typical problem definition along with the corresponding boundary conditions and a sample single-region BEM discretization is depicted below.

If a multi-region BEM iteration solution process were to be adopted instead, the domain is decomposed into K subdomains and each one is independently discretized. It is worth mentioning that the BEM discretizations of neighboring subdomains does not have to be coincident, this is, at the connecting interface, boundary elements and nodes from the two adjoining subdomains are not required to be structured following a sequence or particular position, the only requirement at the connecting interface is that it forms a closed boundary with the same path on both sides. The information between neighboring subdomains separated by an interface can be passed efficiently and accurately through a locally-supported radial-basis interpolation.

In a standard BEM solution process, a system of influence coefficient matrices and boundary values of size N will be formed [ 1,2], where N is the number of boundary nodes used to discretize the problem. The number of floating point operations required to arrive at the algebraic system is proportional to N 2 as well as

2

J 22

t t

Figure 2. BEM problem domain, boundary conditions, and single region discretization.

Figure 2 above shows the same problem depicted in Fig. 1 with a multi-region BEM discretization of four ( K = 4) subdomains. The boundary value problem will now be solved independently over each subdomain where initially, a guessed boundary condition is imposed over the interfaces in order to ensure the well- posedness of each sub-problem. For instance, the boundary value problem of subdomain 0, is transformed into the algebraic analogue of corresponding influence coefficient matrices and nodal boundary values as

V 2 ~ n , ( w ) = 0 3 Pnll{Tnl) = [Gn,l{qn,) (2)

The composition of this algebraic system requires floating point operations proportional to the square of the number boundary nodes in the subdomain (n2) as well as for direct memory allocation (n'). This new proportionality number n is roughly equivalent to

2N n x - K - t l

as long as the discretization along the interfaces has the same level of resolution as the discretization along the boundaries. Direct memory allocation requirement for later algebraic manipulation is now reduced to a proportion of n2 as the influence coefficient matrices can easily be stored in ROM memory for later use after the boundary value problems on remaining subdomains have been effectively solved. For the example shown here where the number of subdomains is K = 4 the new proportionality value n is approximately equal to n x 2N/5. This simple multi-region example reduces the memory requirements to about

n 2 / N 2 = (4/25) = 16% of the standard BEM approach.

The algebraic system for subdomain 01 is re-arranged, with the aid of given and guessed boundary conditions, as

I Po1 1 = 1% 1{4n, 1 (4) =+ [An,I{mJ = {bnJ

The solution of the new algebraic system of subdomain 01 requires now a number of floating point operations proportional to n 3 / N 3 = (8/125) = 6.4% of the standard BEM approach if a direct algebraic solution method is employed, or a number of floating point operations proportional to n 2 / N 2 = (4/25) = 16% of the standard BEM approach if an indirect algebraic solution method is employed. For both, floating point operation count and direct memory requirement the reduction is dramatic. However, as the first set of solutions for the subdomains were obtained using guessed boundary conditions along the interfaces, the global solution needs to follow an iteration process and convergence criteria.

Globally, the floating point count for the formation of the algebraic setup for all K subdomains must be multiplied by K , therefore, the total operation count for the coefficient matrices computation is given by,

n2 4K N 2 (K + 1)2

K-%

or, for this particular case with K = 4, K n 2 / N 2 = 16/25 = 64% of the standard BEM approach. Moreover, the more significant reduction is revealed in the RAM memory requirements as only the memory needs for one of the subdomains must be allocated at a time. The rest of the coefficient matrices for the remaining subdomains are temporarily stored in ROM memory until access and manipulation is required. Therefore, for this case of K = 4, the true memory reduction is n 2 / N 2 = 4/25= 16% of the standard BEM approach.

With respect to the algebraic solution of the system of Eqn. (4), if a direct approach as the LU decomposition is employed for all subdomains, the LU factors of the coefficient matrices for all subdomains can be computed only once at the first iteration step and stored in ROM memory or on disc for later use during the iteration process for which only a forward and a backward substitution will be required to solve the system at hand. This feature allows a significant reduction in the operational count through the iteration

3

process until convergence is achieved, as only a number of floating point operations proportional to n as opposed to n3 is required at each iteration step. To this computation time is added the access to ROM memory at each iteration step which is usually larger than the access to RAM. Alternatively, if the overall convergence of the problem requires few iterations, iterative solvers such as GMRES offer an efficient alternative.

The iteration process follows the initial step of guessing the interface conditions. This is a crucial step as more physical information is incorporated into the initial guess, the less iterations will be required to reduce the error. The simplest choice is to assume adiabatic conditions at the artificial interfaces. Results from several numerical studies show this approach leads to initial temperature fields that are far from the final temperature field and which are slow to update iteratively. A rather more efficient initial guess can be made using a physically-based 1 -D heat conduction argument for every node on the external surface to every node on the interface. The following algebraic initial guess for any interfacial node can be readily derived as

j= 1

where NT, Nq, and Nh are the number of first, second, and third kind boundary conditions specified at the external (non-interfacial) surfaces and

(7)

S, =E- 4 3=1 1rv l

with N = NT + Nq + Nh , the thermal conductivity of the medium is k , the film coefficient at the j-th convective surface is h,, the magnitude of the position vector from the interfacial node i to the external surface node j is rz, = IT^,^, while the area of element j denote by A, is readily computed as

+

r r+ l r+l

Once the initial temperatures are imposed as boundary conditions at the interfaces, a resulting set of normal heat fluxes along the interfaces will be produced. These are then non-symmetrically averaged in an effort to match the heat flux fkom neighboring subdomains. Considering a two-domain substructure the averaging at the interface is explicitly given as,

and,

(9)

to ensure the continuity condition of the averaged fluxes as qAla = - 9n,; I Compactly supported radial basis interpolation can be employed for the flux average to account for unstructured grids along the interface from neighboring subdomains.

Using these fluxes the BEM equations are again solved leading to mismatched temperatures along the interfaces for neighboring subdomains. These temperatures are interpolated, if necessary, from one side of the interface to the other side using compactly supported radial basis functions to account for the possibility of interface mismatch between the adjoining substructure grids. Once this is accomplished, the temperature is averaged out at each interface. Illustrating this for a two domain substructure,

and,

In case a real or physical interface exists and a thermal contact resistance is present between the connecting subdomains, the temperatures are averaged out with R" as the value of the thermal contact resistance imposing a jump on the interface temperature values. These now matched temperatures along the interfaces are used as the next set of boundary conditions.

Finally, in order to provide an improved initial guess to the BEM solver, a two-level resolution approach is employed. The initial guess provided by the 1-D physically-based procedure in Eqn. ( 6 ) is used to solve the conduction problem in a constant element BEM model, see Fig. 3.

4

/ X J

(a> (b) Figure 3. Two-level resolution approach: (a) constant element model and (b) bilinear discontinuous element model.

This constant-element computation is carried out using the same discretization as the bilinear model, however, all nodes are collapsed to a single central node, and this requires 1/64th of the work required to solve the bilinear model. This process utilizes a direct LU solver for each BEM subregion. Upon convergence of the constant element model, a full bilinear solution is subsequently computed with the constant element model providing the initial guess. At this point, the initial guess is close to the final temperature field and, consequently, the number of iterations to convergence will be very few, making the iterative GMRES solver more efficient to use.

The iteration process is continued until a convergence criterion is satisfied. A measure of convergence may be defined as the L:, norm of mismatched temperatures along all interfaces as:

This norm measures the standard deviation of BEM computed interface temperatures T' and averaged-out interface temperatures T,' in all subregions. The iteration routine can be stopped once this standard deviation reaches a small fraction E of AT, where AT is the maximum temperature span of the global field. It is noted that we refer to an iteration as the process by which an iterative sweep is carried out to average both

the interfacial fluxes and temperatures such that the above norm may be computed.

The process outlined above is ideally suited to parallel computing. For this purpose, we have built a small Windows-XP cluster consisting of ten P4 nodes with speeds ranging between: 1.7 Ghz - 2Ghz. These P4's are connected on a local network using 100 base-T Ethernet cards and a full-duplex Siemens switch. A parallel version of the code is implemented using the MPI libraries compiled under the COMPAQ Visual FORTRAN environment. The parallel code collapses to serial computation if a single processor is assigned when launching the application. Attention is now given to the numerical implementation.

Numerical Results A series of numerical examples are now presented to illustrate the approach. Results are provided for serial as well as parallel implementations. The first three examples are carried out serially and are used to illustrate the implementation of the algorithm for progressively larger models and the final example is carried out in parallel and illustrates the possibility of computing a large scale problem of 85,000 degrees of freedom on a PC cluster. In all cases, unless otherwise stated, we set E = lop3.

Conduction Model of a Rectangular Bar A 3-D 5x1 rectangular bar is initially discretized in a single-region using 550 equally spaced bilinear isoparametric discontinuous boundary elements with a total of 2,200 nodes around the boundary. Figure 4(a) shows the BEM discretization for this problem along with the converged isotherm distribution in Figure 4(b) The boundary conditions are distributed as first kind on both end faces with a temperature T = O"C, adiabatic on the bottom face, and convective on the remaining three surfaces with T, = 100°C and h = 1 W / m 2 K . The thermal conductivity was imposed as k = 1 WImK.

Next, the 3-D slab is divided into 5 subdomains each discretized with 1 50 bilinear isoparametric discontinuous boundary elements with 600 nodes. Figure 5(a) shows the BEM discretization for this problem. A converged solution is found after just four iterations and the isotherm distribution is plotted in Figure 5(b). Adiabatic conditions are used as an initial guess at the subsection interface nodes. The plot of the L2 norm progression is shown in Fig. 6 for the first 10 iterations, however, the level of convergence of E . AT,,, was achieved after only 4 iterations. Table 1 reveals the memory requirement proportions for each

5

case and the computation time for the algebraic setup and solution after the 4 iterations for which Lz = 0.012.

T 0 10 20 30 40 50 60 70 80 90 I C 0

(b) Figure 4. Rectangular bar example: (a) single region BEM discretization and (b) Isotherm distribution of single-region 3D rectangular slab.

Table 1. Memory and time comparison for 3D slab

I

Time for Solution I 374s I 86s

2

(b) Figure 5. Five subsection Model for a rectangular bar: (a) BEM discretization and (b) isotherm distribution of 5-subsection 3D rectangular slab.

Figure 6. Lz norm progression for the 5-region 3D slab.

Conduction Model in a Thrust Vector Control Vane A 3-D thrust vector-control (TVC) vane is initially discretized in a single-region using 6 10 equally spaced bilinear isoparametric discontinuous boundary elements

6

with a total of 2,440 nodes around the boundary. Figure 7 shows the BEM discretization for this problem along with the isotherm distribution. The boundary conditions were imposed as insulated on the bottom surface, convective on the back surface with T, = 200°C and h = 100 W / m 2 K , and convective with T, = 4000°C and h(z ) = 1000(~/z,,,)2 W/m2Kfor a maximum of h = 1000 W / m 2 K on the leading edge. A thermal conductivity of IC = 14.9 W/mKwas used.

2

k

2

k:

T 4Wo

3ow

3800

3700

s w 3500

3400

33w

(b) Figure 7. (a) BEM discretization and (b) converged isotherm distribution of single-region 3D Vane.

Next, the 3-D TVC vane is divided into 3 subdomains with the larger discretized using about 300 bilinear isoparametric discontinuous boundary elements with 1,200 nodes. Figure 8(a) shows the BEM discretization for this problem and the isotherm distribution is plotted in Figure 8(b) after the 1st iteration using the adiabatic initial guess. The isotherm distribution is shown in Figure 9(a) after 43 iterations and in Figure 9(b) at convergence after 8 1 iterations.

2 c:

2

h:

T 4My)

39w

3800

37w

3m

35w

34w

3303

(b) Figure 8. (a) BEM discretization and (b) Isotherm distribution of 3-region 3D vane after 1 iteration.

The plot of the Lz norm progression is shown in Figure 10 comparing the iterations for the cases of adiabatic initial guess at the subsection interfacial nodes and the 1-D physically-based initial guess in Eqn. (6) . When an adiabatic initial guess is made at the subregion interfaces, 81 iterations where required to reach the level of convergence of E . AT, while the physically- based initial guess provided an initial error norm of 0.045 (vs. 0.32 with the adiabatic guess) which lead to less than 45 iterations to achieve convergence. This clearly demonstrates the effectiveness of the proposed physically-based initial guess in reducing the computational load of the iterative process in this case. Table 3 reveals the memory requirement proportions for each case and the computation time for the algebraic setup and solution. Although final time to solution is comparable, larger problems could not be tackled by the 1 region approach due to memory requirements and round-off error and prohibitive storage requirements.

7

z

h:

0 IS-

0 3 - '

T 4wo

i 39m

3803

37w

36W

35M

3100

33w

___ AdlabsIIL ~nil i l~! ronhlion

PhwEallY b rad 8rId eucm __-

i: T

4wo

3903

38W

3700

3 m

35w

34w

33w

Table 2. Memory and time comparison for 3D TVC vane problem. For the 3-region: (a) adiabatic guess and

(c) physically-based initial guess. 1 1-Region 1 3-R. (a) I 3-R. (b) 1

I I I I I

Conduction Model of a Plenum-Cooled Blade Here, a 3-D blade with a lOcm chord and 14cm in the spanwise direction cooled by two plena. The blade is discretized using GridProTM [ 1 11 into 6 subdomains with a surface grid of a total of nearly 6,000 bilinear elements or nearly 24,000 degrees of freedom, see Fig. 1 1. Each subdomain is kept at a discretization level near

Y

E z

Figure 10. Progression of the LZ norm normalized by AT,,,- 700K for the 3-region 3D vane comparing the iteration for the cases of adiabatic initial guess at the

guess. subsection interfacial nodes and physically-based initial (b)

Figure 11. BEM grid for 3D cooled blade and its substructure.

8

to 1,000 bilinear boundary elements. Adiabatic conditions are imposed on the top and bottom surfaces of the blade. Convective boundary conditions are imposed on all other surfaces. The film coefficient on the outer surface of the blade is taken as h = 1000 W/m2K with the reference temperature taken as 1000K, while the cooling plena are both imposed with film coefficients h = 500 W / m 2 K with the reference temperature taken as linearly varying from 300Kto 400K in the increasing z-direction of the cooling plenum closest to the leading edge, while linearly varying from 500Kto 400K in the decreasing z- direction of the cooling plenum closest to the trailing edge.

The physically-based initial guess at the subsection interfaces using Eqn. ( 6 ) provided an excellent starting point for the iteration which converged in 16 steps to

Y

T 650 700 750 800 850 900 950 1000

Y

0 -2500 -2000 -1500 -1000 -500 0 500 1000 1500 2000 2500

provide an L2 iterative norm, defined in Eqn. (13), of L2 = 0.00O11698ATm,,. In this case, the constant element BEM model was employed to provide an initial guess to the bilinear BEM model. The constant element model converged in four iterations with a total computational time of 89 seconds while the bilinear element model required only one iteration to achieve convergence after a total computational time of 3,951 seconds. The resulting temperature plots illustrated in Figure 12(a) reveal a very smooth distribution across all blocks. The resulting surface heat fluxes are presented in Fig. O(b) revealing a very smooth distribution from a minimum of - 180,000 W/m2 to a maximum of 230,000 W/m2. As noted the domain decomposition method illustrated so far in terms of serial computation is ideally suited for parallel implementation, and the next example demonstrates this point.

Parallel BEM Conduction Model for a Film-Cooled Turbine Blade This example addresses the implementation of the iterative domain decomposition under a parallel environment. The configuration under consideration is a Honeywell film-cooled blade [ 101 which is discretized using the algebraic grid generator GridProrM [ 1 11. The discretization corresponds to 21,306 bi-linear elements corresponding to 85,224 degrees of freedom, see Fig. 13. A typical subregion is shown in Fig. 13(c), while the full domain decomposition is shown in Fig. 13(d). Each subdomain is kept to under 1,600 elements and there are a total of 20 subdomains.

The thermal conductivity of the blade is taken as that of Inconel IC = 1.34 Btu/hr . in . R. Adiabatic boundary conditions are imposed at the upper and lower spanwise surfaces. The remaining are imposed with convective conditions. The heat transfer coefficient is taken as a constant, h = 10 Btulhr. in2 . R, and the reference temperature is taken to vary quadratically along the chord with a maximum value of 2,500 R at the leading and trailing edges and a minimum at the mid-chord with a value of 1,500 R. The convective condition is imposed on all non-adiabatic surfaces, including cooling holes and plenum.

Figure 12. Converged surface solution: (a) temperature distribution and, (b) heat flux distribution.

9

Figure 14. Composite of domain decomposition of Honeywell film-cooled blade.

A load balancing program was written to distribute the computational tasks optimally. Once the number of processors has been chosen, a sample BEM problem is run on each processor to determine its relative computational capability. Subsequently, an optimization algorithm is invoked to distribute the appropriate subdomains to each processor by minimizing an objective function which contains information with regards to subdomain sizes and relative computational capability of each assigned processor.

The physically based initial guess is used to begin the iteration using a constant element model with the bilinear nodes collapsed to the center node. The initial L2 norm of was reduced to a value of 5 . 10-3ATm,, in 17 iterations on the constant element model. It took 415 seconds to set up the BEM equations and carry out the LU factors and 357 seconds to perform the iterations Upon convergence this solution was extrapolated to the nodes of the bilinear model element by element Using this initial guess, the iteration was carried out on the bilinear element model converging after just one iteration effectively reducing the La norm to a value of 5 . 10-3AT,,, on the bilinear element model. The BEM equations for the bilinear model were solved using a pre-conditioned GMRES for each subdomain. It took 967 seconds to setup the bilinear BEM equations and 1,483 seconds to perform the iterations. The total time to solution, accounting for load balancing, the initial guess, the constant element solution, and the bilinear element solution was 3,222 seconds or just under 54 minutes. The converged isotherm distribution is shown in Figure 15.

(b) trailing edge for the blade with slot cooling.

(c) a typical subdomain. Figure 13. Honeywell film-cooled blade, discretization

and typical domain decomposition.

10

Figure 15. Converged temperature profile.

Conclusions We adopt a domain decomposition method with a region-by-region iterative solution for large-scale BEM models of heat conduction problems. Several conduction examples are presented to illustrate the approach. Numerical tests reveal that the process converges very efficiently case and offer very substantial savings in memory. Moreover, the technique does not require complex data structure preparation or re-writing of traditional BEM codes. The approach is somewhat transparent to the user, a significant advantage in coupling BEM to other field solvers such a Navier-Stokes finite volume solver, an approach taken by the authors to solve conjugate heat transfer problems in large-scale turbomachinery models.

Acknowledgement This research is supported by a grant from NASA Glenn Research Center, NAG3-2691.

References Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C., Boundary Element Techniques, Springer-Verlag, Berlin, 1984. Wrobel, L., The Boundary Element Method - applications in therrno-fluids and acoustics, Yol. I , John Wiley and Sons, 2002. Azevedo, J.P.S. and Wrobel, L.C., Non-Linear Heat Conduction in Composite Bodies: A Boundary Element Formulation, Znt. J. Numer. Meth. Eng., Vol. 26, pp. 19-38, 1988.

4.

5 .

6.

7.

8.

9.

10.

11.

Bialecki, R. and Nhalik, R., Solving Nonlinear Steady State Potential Problems in Inhomogeneous Bodies Using the Boundary Element Method, Num. Heat Trans., Part B, Vol. 15, pp. 79-96, 1989. Bialecki, R.A., Merkel, M., Mews, H., and Kuhn, G., In-and Out-Of-Core BEM Equation Solver with Parallel and Nonlinear Options, Int. J . Num. Methods in Engineering, Vol. 39,

Kane, J.H., Kashava-Kumar, B.L., and Saigal, S., An Arbitrary Condensing, Non-Condensing Strategy for Large Scale, Multi-zone Boundary Element Analysis, Comp. Methods App. Mech. Eng., Vol. 79, pp. 219-244, 1990. Greengard, L. and Strain, J., A Fast Algorithm for the Evaluation of Heat Potentials, Comm. Pure Appl. Math., Vol. 43, pp. 949-963, 1990. Hackbush, W. and Nowak, Z. P., On the Fast Multiplication in the Boundary Element Method by Panel Clustering, Numerische Mathematik, Vol. 54,

Bucher, H. and Wrobel, L.C., A Novel Approach to Applying Wavelet Transforms in Boundary Element Method, Ahances in Boundary ElementTechniques, 11, Denda, M., Aliabadi, M.H., and Charafi, A., (eds.), Hogaar Press, Switzerland,

Kassab, A., Divo, E., Heidmann, J., Steinthorsson, E., and Rodriguez, F., "BEMRVM Conjugate Heat Transfer Analysis of a Three-Dimensional Film Cooled Turbine Blade," International Journal for Numerical Methods in Heat and Fluid Flow, (accepted for publication).

Program Development Corporation, GridProTM az3000-User's guide and Reference Manual, White Plains, New York, 1997.

pp. 4215-4242, 1996.

pp. 463-491, 1989.

pp. 3-11.

11