Augmented Thin Plate Spline Approximation in DRM - CiteSeerX

Augmented Thin Plate Spline Approximation in DRM

Sriganesh R. Karur and P.A. Ramachandran Department of Chemical Engineering, Washington University 1 Brookings Drive, Campus Box 1198, St. Louis, MO 63130 Tel:314-935-6531/Fax:314-935-7211/e-mail:[email protected]

Boundary Elements Communications, 6, 55-58 (1995)

Abstract

The dual reciprocity method (DRM) is a popular mathematical technique to solve non-homogeneous Poisson type equations. The method involves the approximation of the non-homogeneous term by a set of radial basis functions (RBF) and transferring the resultant domain integral to an equivalent boundary integral. In this work, the augmented thin plate spline (ATPS) is shown to be superior to the frequently used linear RBF for DRM approximation in two dimensions. Comparison of the DRM implementation with augmented and unaugmented thin plate spline is also provided.

Keywords:

Dual reciprocity method, Boundary element method, Radial basis functions, Augmented radial basis functions, Thin plate spline

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1 Introduction

The dual reciprocity method (DRM) is a class of boundary element methods (BEM) to solve non-homogeneous partial dierential equations (PDE), often referred to as Poisson type equations. In this method, the domain integral resulting from the non-homogeneous term, denoted as b, in a Poisson type equation is transferred to an equivalent boundary integral by approximating b at a certain number of boundary and internal nodes using a set of interpolating functions. The radial basis functions (RBF), which are functions of a single variable, rk , are widely used for this purpose. The variable rk is dened in 2-D as, p (1) rk = (x ? xk )2 + (y ? yk )2 where, (x; y) are the spatial variables and (xk ; yk ) are the positional coordinates of the kth node. Partridge et al1, have demonstrated the power of DRM in solving dierent kinds of non-homogeneous and non-linear PDE, using the linear RBF, 1 + rk as the set of interpolating functions. The linear function is being used for solving practical engineering problems for over a decade now2;3;4, although without any strong reason for its choice. However the theory of RBF has been well documented in the context of multivariate interpolation5. Recently Yamada and Wrobel6, Golberg and Chen7 have tried to link the theory of RBF with DRM approximation, to provide the much needed mathematical support for DRM. The rich space of RBF oers several choice in selecting suitable interpolating functions to be used in DRM, for example, the linear function, rk , the thin plate spline (TPS), rk2 ln rk , the Gaussian, exp(?rk2 = 2 ), the multiquadrics, 2 + rk2 , etc., where is a constant parameter. The thin plate spline acquires its name because it minimizes the thin plate functional Z @2f 2 @2f 2 @2f 2! 2 + 2 + 2 dx dy (2) @x @x@y @y R2 in a certain Hilbert space8 . In this paper, the augmented thin plate spline (ATPS) dened in Section (3), is used as the DRM interpolating function in 2-D. Golberg and Chen7 claim that the ATPS provides a natural basis for interpolation in 2-D. However, they have not shown the numerical implementation of this function in DRM. The purpose of this note is to show the increased accuracy in solution when ATPS is used in DRM as compared to the TPS and the frequently used linear function. In particular, when the non-homogeneous term in the Poisson type equation is a constant or a linear function in x or y, this interpolation scheme becomes exact and hence the domain discretization (in the form of internal nodes) is not necessary. At the same time, the method is general enough to handle a variety of both linear and non-linear b functions.

2 Dual Reciprocity Method

The Poisson type equation in general can be expressed as, r2u = b in

(3) where u is the dependent variable, is the domain of the object and b is the forcing function which in general is given by   @u ; @u b = f x; y; u; @x (4) @y

The boundary conditions are of the Dirichlet type (u = u0 on ?D ) or Neumann type (@u=@n = q0 on ?N ) or Robin type (a combination of both), where ?D + ?N = ?, the boundary enclosing . The integral formulation of Equation (3) is given by, Z  @u @G  Z (5) G @n ? u @n d? ? diui = G b d

?

where, di are coecients depending on the location of the source point i 9 and the weighting function, G, (also called as the generalized Green's function) is given by, G = ? 21 ln r 1

(6)

where r is the distance between the source point and a eld point. Equation (5) contains a domain integral corresponding to the non-homogeneous term, b. In DRM, this domain integral is transferred to an equivalent boundary integral by approximating the forcing function, b, by a set of interpolating functions at a certain number of boundary (N) and internal (L) nodes. The interpolating scheme is given by, b=

X

N +L k=1

k k

(7)

where, k are the RBF and k , the corresponding coecients. The essential part in DRM is to express k as the Laplacian of another function k such that,

r2 k = k

(8)

With this approximation for b, the domain integral in Equation (5) is converted to an equivalent boundary integral by applying the reciprocity theorem. Thus the integral formulation of Equation (3) is given by,

  Z NX Z  @u @G  +L  d ? ? di ik k G @@nk ? k @G G @n ? u @n d? ? di ui = @n ? k=1 ?

where

ik

is the value of

k

(9)

at the source point i.

3 Interpolating Functions

The accuracy of DRM largely depends on the properties of the interpolating functions, k . Some possible choices of k are listed below1;6;10;11, 8 rk > < (? rk22 ) (10) k = > exp : (r k2 2ln+(rrk2)) n2 n = 1; 2; 3


k

In the context of multivariate interpolation, while testing several RBF, Franke8 found that the thin plate spline along with the multiquadrics yield the best result for the interpolation of scattered data. In the present work, the former alone is considered because it does not have any user dependent empirical constant. The functions shown in Equation (10) are called unaugmented RBF which can be augmented by adding low order polynomial terms. Augmentation of the thin plate spline yields the ATPS and is given by,

X k

rk2 ln (rk ) + 1 + 2 x + 3 y

(11)

where i , i = 1 to 3 are constants. This function was originally discovered by Harder and Desmarais12, later theoretically investigated by Duchon13, Meinguet14 and recently analyzed by Powell5. The ATPS can be viewed as the 2-D analog of 1-D splines. Using this RBF at N boundary and L internal nodes, the DRM interpolation scheme becomes, b=

X? 2

N +L k=1




rk ln (rk ) k + 1 + 2 x + 3 y

(12)

The constants i are determined by the equilibrium constraints7;12,

X

N +L k=1

k =

X

N +L k=1

k x k = 2

X

N +L k=1

k yk = 0

(13)

This completes the interpolation scheme and now the interpolation coecients ~k , k = 1 to N + L along with ~i , i = 1 to 3 can be obtained from,

 ~  P~

P~T ~0

!   ~  ~ = b ~

(14)

~0

where ~ represents the matrix formed by the values of the RBF, k [= rk2 ln (rk )] at the interpolation nodes i and is given by, 0  1


1;N +L 1;1 CA .. ~ = B (15) @ ... . N +L;1


N +L;N +L P~ represents the matrix of low order polynomial terms as shown below,

0 1 1


1 1 P~ = @ x1 x2


xN +L A

(16)

y1 y2


yN +L

and ~b represents the values of b at the interpolating nodes. The advantages of using the ATPS as the interpolation function are: (i) the interpolation matrix is always non-singular5;15 (ii) under uniform mesh renement the convergence rate is of the order of (h4 ), h being the mesh spacing16 (iii) the interpolation is exact for a constant or a linear function of the position variables. Using the augmented RBF in DRM approximation, the RHS of Equation (9) is also augmented as follows,   Z NX +L  @G @ k (17) G @n ? k @n d? ? di ik k + I1 + I2 + I3 RHS = ? k=1

where I1 , I2 and I3 are given by,

  Z  @    @G 1 I1 = G @n ? 1 @n d? ? di i1 1 ?   Z  @  @G G @n2 ?

@n d? ? di ?  Z  @  @G 3  I3 = G @n ? 3 @n d? ? di ? where 1 , 2 and 3 are determined by, I2 =

r2 1 = 1



2

r2 2 = x

(18)

2

(19)

3

(20)

r2 3 = y

(21)



i2 

i3



This completes the integral formulation and the required boundary integrals are computed by any standard technique such as the Gaussian quadrature. The functions k , k = 1 to N + L and j , j = 1 to 3 can be easily derived and are not presented here for brevity. After applying the boundary conditions the resulting system of linear algebraic equations is solved for the unknown variables along the boundary.

4 Illustration

In this section, two test cases are considered in this section to illustrate the accuracy of ATPS approximation in DRM. Case 1: Heat transfer with constant generation

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0.4

T=0

T=0

T=0

0.6 T=0

Figure 1: Geometry and boundary conditions for the illustration; b = constant; `' denotes the point used for comparing the numerical results. The problem of heat transfer with internal heat generation is solved to demonstrate the power of ATPS approximation in handling constant forcing functions. The BEM Benchmark problem # 3 is selected for this purpose. The problem specications are as follows: The temperature along the boundary of a 0:6m x 0:4m rectangular slab, as shown in Figure (1), is maintained at 00 C. There is a constant internal heat source of 106 W=m2 . If the thermal conductivity of the medium is constant and equals a numerical value of 52 W=m0 C then the center point temperature and the integral of the normal gradient of the temperature along the perimeter can be found out to be 310:10C and 4615:39 W/m, analytically. The governing dierential equation is given by, r2 T = ?106=52 (22) The above equation is rst solved by discretizing the boundary alone by equal quadratic elements along each of the axes without any internal nodes (L = 0). Table (1) shows that as the boundary elements are Table 1: Performance of ATPS versus Linear RBF and TPS without any internal nodes; BE = Number of Quadratic boundary elements. Quantity Integral of Gradient Center Point Temperature

RBF BE = 8 ATPS -4611.86 Linear -4543.42 TPS -4492.5 ATPS 309.63 Linear 302.78 TPS 296.95

BE = 10 -4613.65 -4546.57 -4493.87 309.97 303.22 297.17

BE = 16 -4614.75 -4548.63 -4498.68 310.04 303.33 297.53

BE = 20 Exact Solution -4615.45 -4549.31 -4615.39 -4500.19 310.08 303.39 310.1 297.63

increased (hence the boundary nodes), the numerical solution approaches the exact solution without the need of any internal nodes in the case of ATPS. This is due to the presence of the constant term in the ATPS which makes the interpolation scheme exact with only the boundary nodes. Without any internal nodes, the linear RBF and TPS do not perform well, because of poor approximation of the internal region. To obtain the exact solution using the linear RBF or the TPS, 20 boundary elements and 55 internal nodes are needed as shown in Table (2). Thus the ATPS oers considerable advantage in terms of reduction of degrees of freedom when the forcing function in the Poisson type equation is a constant. Similarly, when the b in Equation (3) is a linear function in x or y, use of ATPS in DRM obviates the need to place internal nodes for domain discretization. 4

Table 2: Performance of Linear RBF and TPS with internal nodes; BE = Number of Quadratic boundary elements and L = Number of internal nodes Quantity

RBF

BE = 10 BE = 10 BE = 10 BE = 20 BE = 20 L=1 L=9 L = 15 L = 15 L = 55 Integral Linear -4575.50 -4604.49 -4608.15 -4609.53 -4615.07 of Gradient TPS -4568.84 -4606.30 -4608.53 -4609.56 -4614.89 Center Point Linear 307.15 309.34 309.76 309.86 310.09 Temperature TPS 306.93 309.69 309.89 309.92 310.08 Case 2: Heat transfer with variable conductivity

A case of heat conduction problem with no heat generation and with a linear variation of thermal conductivity is solved to highlight the eectiveness of ATPS approximation in handling non-linear forcing function. The geometry and boundary conditions are shown in Figure (2). This choice of boundary conditions provides an analytical solution for comparing the numerical solution.

D

T = 0.0

C

. (k T) = 0

No Flux

No Flux

1.0

0.0

A 0.0

1.0 B T = 1.0 Figure 2: Geometry and boundary conditions for Case 2.

The dierential equation is of the form,

r
(krT ) = 0

(23)

k =1+T

(24)

where k is the thermal conductivity given as,

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With this substitution for k in Equation (23) and by after mathematical manipulation, the governing dierential equation becomes, " 2  2# 1 @T 2 r T = ? 1 + T @T (25) + @x @y The RHS of equation which is a non-linear function in T is approximated by the linear RBF and the ATPS. Two uniform meshes, (i) N = 16; L = 9 and (ii) N = 20; L = 81 are used. The gradient along the sides AB and CD are evaluted using DRM and compared with the exact solution in Table (3). From the results it is clear that even with a coarse mesh conguration, ATPS oers superior results when compared to the linear RBF. Table 3: Comparison of the gradients along AB and CD for diusion with linear variation of thermal conducitivity. Gradient RBF BE = 8 BE = 20 Exact Solution along L = 9 L = 81 AB ATPS 0.7469 0.7496 0.7500 Linear 0.7542 0.7507 CD ATPS ?1:487 ?1:496 ?1:500 Linear ?1:410 ?1:467

5 Conclusion

In this paper we show that the ATPS is a better approximation function in DRM for solving Poisson type equations, in 2-D. Two types of forcing functions are tested using the often used linear RBF and the ATPS. Especially when the non-homogeneity is of a constant form or a linear function in x or y, then no interpolation error is introduced due to the DRM approximation when ATPS is used. For such problems, it is not necessary to place any internal nodes for transferring the domain integral to an equivalent boundary integral. When the forcing function is non-linear, it is shown that the ATPS oers superior solution with reduced number of degrees of freedom.

6 References

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7. Golberg, M.A. and C.S. Chen, The theory of radial basis functions applied to the BEM for inhomogeneous partial dierential equations, Boundary Elements Communications, 5 57-61 (1994). 8. Franke, R., Scattered data interpolation, Mathematics of Computation, 38 181-200 (1982). 9. Brebbia, C.A. and J. Dominguez, Boundary Elements - An introductory course, Computational Mechanics Publications, Southampton and McGraw Hill Book Company, New York (1989). 10. Karur, S.R. and P.A. Ramachandran, Radial basis function approximation in the dual reciprocity method, submitted to Mathematical and Computer Modelling (1994). 11. Zheng, R., C.J. Coleman and N. Phan-Thien, A boundary element approach for non-homogeneous potential problems, Computational Mechanics, 7 279-288 (1991). 12. Harder, R.L. and R.N. Desmarais, Interpolation using surface splines, Journal of Aircraft, 9 189-191 (1972). 13. Duchon, J., Spline minimizing rotation-invariant seminorms in Sobolev spaces, in Constructive theory of functions of several variables, Lecture notes in mathematics 571, (Edited by W. Schempp & K. Zeller), pp. 85-100, Springer-Verlag, Berlin, (1977). 14. Meinguet, J., An intrinsic approach to multivariate spline interpolation at arbitrary points, in Polynomial and spline approximation - Theory and applications, (Edited by B.N. Sahney), pp. 163-190, D. Reidel Publishing Company, Holland, (1979). 15. Micchelli, C.A., Interpolation of scattered data: Distance matrices and conditionally positive denite functions, Constructive Approximation, 2 11-22 (1986). 16. Buhmann, M.D., Multivariate cardinal interpolation with radial-basis functions, Constructive Approximation, 6 225-255 (1990).

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