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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 31, 1055-1068 (1991)

ACCURATE INTEGRATION OF SINGULAR KERNELS IN BOUNDARY INTEGRAL FORMULATIONS FOR

HELMHOLTZ EQUATION

P. S. RAMESH AND M. H. LEAN

Xerox Corporation, Design Research Institute, Engineering and Theory Center, ithaca, N Y 14823, U.S.A. and Xerox Corporation, Webster Research Center, Mechanical Engineering Sciences Laboratory, North Tarrytown, N Y 10291, U.S.A.

SUMMARY Boundary integral formulations for the 2D Helmholtz equation involve kernels in the form of modified Bessel functions. Accurate schemes for evaluating integrals of the kernels and their derivatives are presented. Special attention is paid to integrals involving singular and near singular kernels. Both boundary and domain integrals are considered. It is shown that, with the use of series expansion functions for the modified Bessel functions, the boundary integrals can be evaluated analytically in the neighbourhood of the singularity. For domain integrals, the behaviour of the kernels in the vicinity of the singularity is used to construct accurate numerical quadrature schemes. A transient heat conduction problem is formulated as a Helmholtz equation, solved, and compared against analytic solution to demonstrate the effectiveness of these schemes in relation to traditional methods. References are made to previous work to advocate the utility of the boundary integral method for non-linear and time-transient problems.

INTRODUCTION

The overall accuracy of boundary integral equation methods is largely dependent on the precision with which the various integrals are evaluated. The integrals involving singular and near singular kernels are often the most important. For Laplace and Poisson equations, the static Green functions are kernels that vary as ln(r) and l / r in two- and three-dimensions, respectively. A substantial amount of literature is available on accurate integration procedures for these kernels.’-4 A discussion of the numerical quadrature schemes for the evaluation of singular and nearly singular integrals involving kernels of the type l/P is available in Hayami and Brebbia.3 These schemes essentially involve co-ordinate transformations, with the quadrature sampling positions uniformly distributed in the transformed domains such that the distribution of quadrature points in the physical domain adequately represents the singular kernel. Another classical technique is to subtract out the singularity,” resulting in a simpler singular integral which can be evaluated accurately. An extension of this technique has been used in the acoustic wave scattering p r ~ b l e r n , ~ where the kernel is in the form of a Hankel function. A Taylor series expansion of the Hankel function is used to subtract out the singularity of ln(r) type and badly behaved terms of rln(r) and r21n(r) types. The largest wavenumber treated is 10.

For problems governed by the Helmholtz equation, such as eddy current induction heating: transient heat conduction using Laplace transforms’ and vorticity transport in Navier-Stokes flow,** the two-dimensional time-harmonic Green functions are kernels in the form of modified Bessel functions. These functions are characterized by a very rapidly varying region in the vicinity of the singularity, and a slowly changing asymptotic region beyond. For small arguments, the

0029-598 1/9 1/061055-14$07.00 0 1991 by John Wiley & Sons, Ltd.

Received 27 November 1989 Revised 6 July 1990

1056 P. S. RAMESH A N D M. H. LEAN

form of the singularity is ln(kr), where k is the wavenumber, and may be treated using any of the conventional methods noted above. Of particular interest are the large wavenumber cases, where the kernels decay very rapidly for small increases in the spatial distance from the source point. When this spatial distance is less than an element length, the accuracy of standard integration schemes deteriorates because of the inability to accurately sample the transition. Large wavenumbers are often encountered in the boundary integral solution of high speed Navier-Stokes flows and in electromagnetic skin effect problems, where their accurate treatment is crucial to ensuring precision in the definition of the boundary layer. The present study focusses on accurate integration schemes for the kernels of both boundary and domain integrals, involving modified Bessel functions, valid over a wide range of wavenumbers. For the boundary integral, the procedure employs a series expansion of the modified Bessel function in the vicinity of the singularity, and proceeds to integrate it analytically. It is shown that, with the use of simple recursion formulae, the series can be truncated appropriately and integrated exactly to give the desired accuracy. The region beyond this radius of influence is integrated numerically using standard Gauss-Legendre quadrature. For domain integrals, the singular region is sectioned into triangular subregions which are further partitioned about a critical kr to separate the rapidly and slowly varying regions. The singular region is integrated using an extension of the numerical scheme by Lean and Wexler.' The regular region is integrated by standard Gauss-Legendre quadrature.

To illustrate the effectiveness of the present integration procedure, the transient heat conduction problem is considered. By taking finite-differences of the time derivative, the equation can be converted to a modified Helmholtz equation, where the wavenumber depends on the time step. It is shown that, for large wavenumbers, the numerical solution deviates considerably from the analytic solution where standard integration procedures are used. Solutions generated using the present integration strategy, however, compare extremely well with analytic solutions.

In recent years, the classical boundary integral methods have been extended successfully to treatment of non-linear problems.*- l o In many cases, the non-linear terms in the equations are manifested as domain integrals which contribute to the forcing function of the linear system of equations. The solution procedure is to assume step-wise linearity and iteratively refine on the interior field values evaluated at each iteration level till convergence is attained. Depending on problem physics, constitutive relationships may be used to ensure more accurate estimates as in the problem of ferromagnetic saturation" where volume magnetization is physically related to material characteristics at the media interface. Another example is vorticity transport in Navier-Stokes where the vorticity and fluid velocities in the interior are required to evaluate the non-linear convection transport terms. Both problems result in singular domain integrals arising from the evaluation of dependent variables and their derivatives. Therefore, the popular belief that the boundary integral method is not suited for non-linear problems requiring the evaluation of domain integrals may not be well justified. We would like to point out several factors that will contradict this belief. First, for fixed or invariant geometries, all the spatial integrals can be evaluated a priori and stored as influence coefficient matrices to be reused during iteration. The overhead incurred in the precomputation simplifies the iteration process to algebraic manipulation of products and sums of matrices. As a result, the non-linear solution phase is extremely rapid. Second, since integration is a smoothing process the domain grid is not required to be as fine as those used in domain methods requiring accurate treatment of nearest neighbour interactions. Third, the rank of the system matrix is dependent only on the discretiz- ation of the boundaries on which the governing equations are enforced, and therefore retains the inherent advantage of the boundary method in reduction of problem dimensionality. Finally, access to vectorization and parallel computing methods enables the possibility of addressing even

ACCURATE INTEGRATION OF SINGULAR KERNELS 1057

larger problem sets. A traditional bottleneck, the generation of matrix elements by evaluation of spatial integrals, has been recognized not to have data dependencies and therefore lends itself naturally to parallelization.

BOUNDARY INTEGRAL FORMULATION

The boundary integral formulation for the inhomogeneous Helmholtz equation

(1) 2 V 4 - k 2 4 = - k 2 4 0

defined in domain V with boundary S can be written as

where L = - 1/2n Ko(kr ) is the free space Green function, and k is the wavenumber. y is the Cauchy principal value of the boundary integral in equation (2). K O is the modified Bessel function of the second kind and order 0 and r is the distance between the source point and the field point. L behaves as ln(kr) as kr tends to zero and decays as e-k'/Jkr as kr tends to infinity. For small wavenumbers (k < l), L varies slowly with r and the integrals can be evaluated accurately using standard numerical quadrature schemes. Conversely for large wavenumbers (k 9 l), L varies rapidly over distances much smaller than the element lengths, and the singular and near singular integrals require special treatment. In the following sections the evaluation of the boundary and domain integrals in equation (2) is addressed.

EVALUATION OF BOUNDARY INTEGRALS

We now consider the boundary integrals in equation (2) evaluated over a straight line segment, as shown in Figure 1. A local co-ordinate system (5, q ) may be defined with origin at the source point, and 5 in the direction of the tangent and q in the direction of the outward normal to the boundary element (see Figure 1). Therefore,

r = Jt2 + q2

where K , is the modified Bessel function of the second kind and order 1. The notations Ko(k , 5, q ) and K , ( k , 5, q ) are used to represent Ko(kJC2 + q 2 ) and K l ( k J 1 2 + q2) , respectively. For linear shape functions, the boundary integrals in equation (2) are linear combinations of integrals of the

boundary element S;

Figure 1. Local co-ordinate system ( 5 , q ) for integration along the boundary element Sj with the source point at i

P. S. RAMESH AND M. H. LEAN 1058

following types:

Following Hall," the leading order singularity can be subtracted from the integrals in equations (4) and evaluated separately. For K O and K , the leading order behaviour of the singularity is ln(kr) and l/r, respectively. The advantage of this approach is that accurate schemes for evaluating logarithmic and l/r type singularities are available in the literature.' For large wavenumbers, the leading order behaviour of the singularity alone is not sufficient to resolve the integrals accurately and additional terms in the expansions of the modified Bessel functions need to be considered. K O and K can be expressed as the sums of infinite series:

e - k r m

Jkr i = l Ko(kr) = - 1 Ci

where Ai , Bi , Ci, Di, Ei and Fi are known functions of k (see Appendix). We define the singular region as r ES: klr - rsl < 2, where r, is the source point." Note that the singular region shrinks as k increases, which is consistent with the variation of the kernels for large k. The integrals are evaluated exactly in the singular regions by integrating the series analytically. The series can be truncated appropriately to attain the desired accuracy. In the examples presented here, the series are truncated after seven terms, and the residual error is 0(10-'). Numerical integration is used outside the singular region. Evaluating the integrals in equations (4) in the singular region, we obtain

m

I' = - r = l (+l,i - BiP2 , i )

I , = - ,x 2 P3,i - ?P4,i) r = 1 (". 4

i = 1


P l , i , P2 , i , P 3 , i and P4,i are given by the following formulae:

= - 1 arctan (5>; i = 0 tt

+ arctan (t)); i = - 1

P3, i = [ln(;t)ti-'dt

=f(ln(:t)-i); i i > 1

P4,i = ti-'dt s ti i'

- -* - i > , l

= Int; i = O

t ' 1. .

I = - 1

where I3 = arctan(t/q) and t = t2 + qZ. In a related study involving Hankel functions, Hall and Robertson' isolated the first three

terms of the type In(r), r In@), r2 In(r) from the Taylor series expansion of the singular kernel and treated them analytically. However, there are some important differences between the present scheme and Hall and Robertson's. Firstly, they treat only the singular element, whereas the

- -- -


singular region defined in the present scheme may extend beyond the singular element to neighbouring elements as well. Secondly and more importantly, the wavenumber dependence of the singular kernel is exploited by the present scheme and not by Hall and Robertson. As a result, we expect the present scheme to be applicable over a wide range of wavenumbers.

Figures 2 and 3 show the results of the integration of I, over a straight line segment of unit length with the source point at the centroid, i.e. q = 0 and 5 = [ - 0*5,05], plotted against the order of quadrature (NG). In Figure 2, k = 10 and in Figure 3, k = 100. In the curve indicated by legend Scheme 1, the integrals are evaluated by isolating the leading order singularity. In the curve indicated by legend Scheme 2, the integrals are evaluated analytically using equation (6) in the singular region (i.e. ( = [ - 0.2, 0.23 for k = 10 and 5 = [ - 002, 0.02) for k = 100). For k = 10, the accuracy of the integration can be greatly enhanced by choosing either Scheme 1 or Scheme 2 over the standard Gauss-Legendre quadrature scheme. However, for k = 100, both Scheme 1 and the standard Gauss-Legendre quadrature scheme perform poorly when compared

0.4 I \ -0- Scheme I

0.0

NG

Figure 2. Integral I , evaluated for q = 0, [ = [ - 0.5, (351 and k = 10, versus order of quadrature (NG)

-0- Scheme I

0.0

0 ,a I. I* 40 40 3s 8.

NG

Figure 3. Integral I , evaluated for q = 0, = [ - 0 5 , 0 5 J and k = 100, versus order of quadrature (NG)


with Scheme 2. This clearly demonstrates that, for large wavenumber applications, Scheme 2 is preferred. For both of these examples I , , I , and I, are uniformly zero.

DERIVATIVES OF THE DEPENDENT VARIABLE

In the boundary integral formulation of vorticity transport in Navier-Stokes f l o ~ s , ~ , ~ derivatives of equation (2) with respect to x and y are needed in order to compute the fluid velocities in V. Since derivatives of L and aL/an have a higher order of singularity, additional care must be taken while evaluating the integrals. Computed velocities are more accurate than can be obtained from traditional finite differencing. We can write

a a all a a t aY all aY at 8Y

a a aq a ay ax allax a y a x

+-- - ----

+-- - ----

Also,

The additional types of boundary integrals encountered when taking derivatives of equation (2) are

r


Evaluating the integrals in equations (7) in the singular region, we obtain

where P l , i , Pz,i, P 3 , i and P4,i have been defined earlier. Note that the J integrals are hypersingu- lar on the boundary, and equations (7) and (8) can be used to evaluate these integrals accurately close to the boundary. Accurate evaluation of the J integrals is very important in high speed Navier-Stokes Aows (large k) in order to pick up the boundary layer effects.

EVALUATION O F DOMAIN INTEGRALS OVER THE SINGULAR ELEMENT

As mentioned earlier, singular domain integrals are encountered in boundary integral formulations of non-linear and time-transient phenomena. We now consider singular domain integrals involving modified Bessel functions, in particular the integration of the kernel L and its derivatives over a quadrilateral element containing the source point P. Following the sectioning scheme suggested by Lean and Wexler' for logarithmic and l/r type kernels, four triangular subelements can be constructed with P as the common vertex (see Figure qa)), and the volume integral can be evaluated as the sum of the integrals of the subelements. Each triangular subelement can be mapped into a simplex element (see Figure qb)) using the following transformation:

where vertex P is mapped onto t = 0. Quadrature data assembled on the simplex in (t, q ) can be mapped into each triangular subelement using equation (9). The quadrature points in (x, y) space will fan out from point P, thus automatically guaranteeing a larger density of sampling positions in the vicinity of P. Effectively, the non-linear transformation from a square to a triangle


Figure 4. Sectioning scheme for singular domain integrals

introduces a Jacobian that regularizes the l/r type singularity. The Jacobian of this transformation is

J = N V ( - x1 + X Z ) + X I - X P ) ( - Y l + Yz)

n n

11 J , = J J k ~ , ( k tfz) <L((l- q ) x t + rlxz - x,)dtdrl

1 r

The integrands in equations (12) are now regular functions because l K , ( k ( ) tends to 0 and k t K , ( k t ) tends to 1 as 5 tends to 0. However, a plot of k < K , ( k ( ) and k ( K , ( k ( ) versus k5 in Figure 5 suggests that the largest variation in the values of the integrands will occur when k( < 8, i.e. 5 < 8/k. This implies that for large wavenumbers ( k % i), sufficient quadrature sampling positions are required in the range 0 =G < < 8/k in order to resolve the integrals accurately. This can be easily achieved by further partitioning the integral over the simplex into two parts as

1064

1 .o

0.9

0.7

P. S. RAMESH A N D M. H. LEAN

0 0.5

0.4

0.3

0.2

0.1

0.0 I 0

I I I 15. 20. 8 lo'

5.

k<

Figure 5. ktK,(kl) and k l K , ( k t ) versus k t

follows: 1 s. c ( . . . . a ) d l d n t j ; i ) = 8 , k ( . . . . . )d[dq

and evaluating each separately using numerical quadrature. Figure 6 shows results of the integration of I,, over a square element of unit length with the

source point at the centroid and k = 100, plotted against the total number of quadrature sampling positions within this singular element (NVS). The standard Guass-Legendre quadrature converges very slowly, and is therefore highly undesirable. A direct implementation of the

. . . . . . , . - Gauss-Legendre -1 I 1- Sectioning w ~ o Partitioning 1

200 b 0 0 000 800 1000 DO

NVS

Figure 6. Integral I , , evaluated over a square element of unit length with the source point at the centroid and k = 100, versus total number of quadrature sampling positions within this singular element (NVS)


sectioning scheme suggested by Lean and Wexler' for the static Green functions requires a very large number of quadrature points. Excellent results are obtained when this sectioning scheme is combined with partitioning, wherein the integrals within each triangular subregion are further partitioned into 0 < t < 8 /k and 8 / k < t < 1 regions prior to integration.

APPLICATION TO TRANSIENT HEAT CONDUCTION PROBLEM

The superiority of the present integration schemes over standard numerical integration schemes for boundary integral formulations of the Helmholtz equation can be demonstrated by consider- ing the transient heat conduction problem:

Classical boundary integral formulations for this problem' involve domain integrals and kernels which vary in both time and space. An alternate formulation is possible by rewriting the time derivative in equation (13) using differences, i.e.

where n signifies the time level and A t is the time step. The truncation error involved in the approximation in equation (14) is O(At), implying that small At is required to follow the transients accurately. Equation (13) can now be rewritten as a modified Helmholtz equation:

v 2 4 n + l - k 2 4 " + l = - k24n (15) where k 2 = l /aAt. Note that evaluation of the V24 term at the n + 1 time level is consistent with a fully implicit formulation. The coefficient matrices obtained from the boundary and domain integrations do not change if At is constant. Consequently, the integrations need to be performed just once.

We consider the transient heat conduction problem in a square domain, 0 < x, y < 1, with the boundaries at y = 0 , l insulated, and the boundaries at x = 0,l at I#I = 0 and 4 = 1, respectively. a is taken to be 1 and the initial condition is 4(x, y, t = 0) = 0. In Figure 7, the error between the analytic and numerical solution a t x = 0-5, y = 0.5 is plotted as a function of time, for k2 = 10 (corresponding to At = 0.1) and k 2 = 100 (corresponding to At = 0.01). For At = 0.1, both the present integration schemes and standard numerical integration scheme using NG = 4 and NVS = 16 perform similarly. The dominant error in these solutions is primarily due to the truncation error term in the difference approximation. However, for At = 0.01, the accuracy of the standard numerical integration scheme deteriorates, leading to accumulation of errors during the transients.

In Figure 8, the errors between the analytic and numerical solutions are plotted along the centreline (y = 0.5) as a function of time. Again the standard numerical quadrature using NG = 4 and NVS = 16 is compared against the present scheme. It is evident that the accuracy of the present integration schemes does not deteriorate at large wavenumbers (small At), whereas the standard numerical quadrature scheme is inadequate for these cases. In particular, for k2 = lo00 (At = 0-001), the standard numerical integration solution departs from the exact analytical solution by as much as 70 per cent.

For the above example, 40 equally spaced boundary elements and a uniform 10 x 10 volume mesh were used for k 2 = 10, 100. For k2 = 1O00, a finer mesh was used (80 boundary elements

1066 P. S. RAMESH AND M. H. LEAN

0.07

0.06

0.05

Id 0.04 e w 0.03

0.02

0.01

0.00

-,Ol ' I I I I I I I I I 1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.0 1.0

Time

Figure 7. Error between numerical and analytic solution at (x, y) = (0.5, 0.5), plotted against time for: (a) k2 = 10 (At = 01); (b) k2 = 100 (At = 0.01)

and 20 x 20 volume mesh). In general, finer grids are required for larger wavenumbers owing to the rapidly decaying kernel. As a rule of thumb, the grid spacing should be roughly one-fourth of the characteristic wavelength.'

CONCLUDING REMARKS

Accurate integration schemes for modified Bessel functions encountered in the solution of Helmholtz equation in two-dimensional space are outlined. Both boundary and domain integrals of the kernel and its derivatives are considered. The analytical behaviour of the singular and near singular regions is used to develop accurate integration procedures. Special treatment is required for large wavenumber applications, where the singular kernels depart considerably from logarithmic and l/r type behaviours. A simple transient heat conduction problem is formulated as a Helmholtz equation, solved, and compared against analytic solution to illustrate the efficacy of the proposed schemes in relation to standard methods. Although the present study considers only linear shape functions and straight line elements, it is possible to extend the schemes to include general polynomial shape functions and curved elements.

ACKNOWLEDGEMENTS

The computational support for this work was provided by the Cornell National Supercomputer Facility, a resource of the Center for Theory and Simulation in Science and Engineering (Cornell Theory Center), which receives major funding from the National Science Foundation and IBM Corporation, in addition to support from New York State and members of the Corporate Research Institute.

ACCURATE INTEGRATION OF SINGULAR KERNELS

Standard Numerical Intepration Present lntegratiort s,-},-

1067

Error

0 0

\

Figure 8. Error distributions between numerical and analytic solution along y = 0 5 plotted against time for: (a) k2 = 10 (At = 0.1); (b) kZ = 100 (At = 0.01); (c)'kZ = loo0 (At = 0001)

APPENDIX

For completeness, we present here the coefficients of the first seven terms of the modified Bessel function expansions in equation (9, as given in Abramovitz and Stegun." The residual error incurred in truncating the series is of order 0(10-7) to O(10-9).

P. S. RAMESH A N D M. H. LEAN 1068

i 1 2 3 4 5 6 7

A; 1-0000000 - 35 156229 3.0899424 1.2067492 0,2659732 0.0360768 Oa0458 13

B; Ci ,057721566 1.25331414 042278420 - 0.07832358 023069756 0.02189568

0.00262698 040587872

0-00000740 0.00053208

0-03488590 - 001062446

0-00010750 -0.00251540

0; 050000000 0.87890594 0.51498869 0.15084934 0.02658733 0-00301532 0*00032411

E ; 1 *00000000 0.1 5443 144

-0.67278579 -0.18156897 - 001 9 19402 -000110404 - 0.00004686

Fi 1.25331 414 0.23498619

0.0 1504268

040325614

- 0.03655620

-0.00780353

- 0.00068245

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