LOCAL DIFFERENTIAL QUADRATURE METHOD FOR ELLIPTIC EQUATIONS IN IRREGULAR DOMAINS

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1 LOCAL DIFFERENTIAL QUADRATURE METHOD FOR ELLIPTIC EQUATIONS IN IRREGULAR DOMAINS L.H. Shen and D.L. Young * Department of Civil Engineering and Hydrotech Research Institute National Taiwan University Taipei, TAIWAN * dlyoung@ntu.edu.tw ABSTRACT The differential quadrature (DQ) method is a numerical technique with high accuracy, but its application is sensitive to grid distribution and contracted by a problem limited to ill-conditioning matrix throughout. The restriction of this method to deal with the boundary is also limited to the problems with regular domain. In this article, the concept of localization and the technique in dealing with the irregular boundaries has been employed and proposed. The derivatives at a grid point are approximated using a weighted sum of the points in its neighborhood rather than at all grid points. With this concept, the computational cost can be efficiently reduced such that the global matrix will be improved to become a sparse and band matrix. The proposed method is applied to the potential flow problems and extended to the harbor resonance problem which is governed by the Helmholtz equations in the two-dimensional irregular domains. Numerical examples show that the present method produces very accurate results while maintaining good stability. The local DQ method enables us to solve more complicated problems and enhance the flexibility of DQ method significantly. INTRODUCTION The DQ method was introduced by Bellman et al. in the 1970s following the concept of integral quadrature (Bellman and Casti, 1971). The accuracy of this method can be very precise by means of a light mesh system and a few calculations. It widely applied to structural problems such as the plate vibration analysis (Bert and Malik, 1996) (Gutierrez and Laura, and1995 ) and composite structure (Bert and Malik, 1997). Shu et al. extended the DQ method to evaluate the eigenvalue analysis (Shu and Chew, 1997, 1998 and 1999) and simulate the flow and heat transfer problems (Shu et al., 1990, 199, 1994 and 1996). The fundamental theory of the DQ method is that any derivative at a point can be approximated by a weighted linear sum of all functional values along a straight line. Shu (000) developed the weighting coefficient as a high order polynomial that can be computed by the Lagrange polynomial interpolation with a recurrence relationship. This means that the numerical discretization of derivatives by the DQ method is also along a straight line. Due to this feature, the DQ method cannot be directly applied to irregular domain problems. Furthermore, it is not only sensitive to grid distribution but also requires that the number of grid points cannot be too large. This property is very apt to cause the ill-condition of global matrix. These defects greatly restrict the wider applications of the conventional DQ method. 135

2 This decade, a lot of have studied and done the discussions to the localize concept. This notion leads us to propose to localize differential quadrature to a small neighborhood so as to keep the balance of accuracy and stability. Zong and Lam (00) introduced a concept of localize into the DQ method. The damage of ill-conditioned matrix was eased off apparently. However, the difficulties of dealing with irregular domains still exist. In this study, an effective solution is proposed to overcome this difficult barrier. The local differential quadrature (LDQ) method was recently proposed by us. The method is based on the conventional differential quadrature (DQ) method. With the computed weighting coefficients, the method works in a very similar model as conventional finite difference schemes. In this paper, two two-dimensional cases are tested, and they are the steady heat problem in irregular domains. An eigen-value analysis for the Helmholtz equations with a two-dimensional Cassini-Oval shapes domain is the last application. Excellent numerical results are obtained. The success of these numerical simulations indicates the flexibility and good performance of the method in simulating linear problems with geometrical and dynamic complexity. LOCAL DIFFERENTIAL QUADRATURE METHOD The concept of conventional DQ method is able to be illustrated in Figure 1. Following the theory of integral quadrature, the DQ method assumes that any derivative at all discretization points can be approximated by a linear summation of all the functional values in the whole computational domain. This concept can be illustrated as follows, N ( x ) = a f( x ), i = 1,,..., N (1) i ij j x j = 1 N ( x ) ( ), 1,,..., i = bij f xj i = N () x j = 1 where N is the number of grid points, the subscript i, j are the global node index and local index in support of node i. The parameters a ij and b ij are the weighting coefficients as well as ( x i ) and x ( x ) i represent the first and second-order derivatives of the function f ( x ) at x point i. The complete form for the DQ method can be illustrated in equation (3). m N m f ( x ) = w f( x ), 1,,..., ij j i = N (3) m i x j = 1 The quantity of the polynomials is depending on the dimension orders of the computational domain. The weighting coefficient of each term of the approaching polynomials is decided by the regulation of Lagrange interpolating polynomial as above. 136

3 Figure 1. The mesh generation type and related points for the conventional DQ method By the specified schemes and illustration, the complete linear matrix of derivational equations can be composed of the weighting coefficients along each aspect. The main advantage of the DQ method is that any order derivative can be easily and accurately determined. However, the important drawbacks of the DQ method are the problem of ill-conditioned global matrix and the difficulty in treating the irregular boundaries. In the proposed method, the referred ambit is narrowed down to a specific distance or count of points (Fig. ). Obviously the concept of localize is employed here. The derivatives at a grid point are approximated by a weighted sum of the points in its neighbourhood rather than of all grid points. Therefore, the 1-D general form in local DQ can be written as the following equation. n m p m f ( x ) = w f( x ), 1,,..., ij j i = N (4) m i x j = 1 The parameter n p is the number of supported reference points, the subscript i, j are the global node index and local index in supporting of node i. This approach is able to extend to the multi-dimensional domain effortlessly by the orthogonal discretization. The damage of ill-conditioned matrix alleviated efficaciously in virtue of the non-zero elements in the global matrix is widely decreased on each row. The counts of reference points decided by the requirement of accuracy order along each axis in the coordinate system respectively. The additional advantage is adopters are able to adjust the count of reference points locally with the properties of physics in each problem arbitrarily. 137

4 Figure. The relationship between computing and reference points As Figure 3 shows, one special mesh type can be composed by a set of simple orthogonal grid and the physical boundary of the problem. The boundary nodes are determined by the intersection of the grid line and physical boundary. By this technique, the Dirichlet type conditions at irregular boundaries can be easily treated. Figure 3: The mesh type and reference points for Dirichlet type boundary condition 138

5 Some additional treatments must be adopted in order to let the technique of preceding paragraphs can be applied to the derivational boundary conditions. The main theory of this technology stems φ φ φ nx + ny = from the following -D formula which is widely applied to the boundary x y n type numerical methods. Where nx and n y are the component on each axis of the unit normal vector. Following this notion, some additional computational points placed at the projective positions on the original gird lines by the boundary points (Fig. 4). Through the concept of localization, the extra nodes do not necessary to extend to the entire domain or to the opposite boundary. It only needs to ensure that the intact relation between each axis at the boundary points to maintain the completeness of derivational boundary conditions. Figure 4: The mesh type and reference points for Neumann boundary condition NUMERICAL SIMULATIONS In order to understand the performance and stability of the local DQ method in irregular domain problems, four kinds of different shaped problems in here brief tests. The -D steady heat problem with Dirichlet boundary conditions The first case is a steady heat equation problem. The governing equation is K ( T ) = Q, the source term Q here simplifies to zero and the thermal conductivity is a invariable constant. The 139

6 exact solution is formed with the following formula ln( R/ R) T R = T T + T, 1 ( ) ( 1) 1 ln( R/ R1) where R = x + y. Figure 5 shows the computational domain and boundary conditions. It is shaped into a quarter of an annular region. The boundary conditions are Dirichlet types along the radial axis and Neumann types on the rotational edges. q = 0 x T = 0 y T 1 = 1 R 1 = 1 R = x q = 0 Figure 5. The sketch of computational domain and boundary conditions As mentioned, the Dirichlet type boundary can be treated by the type of computational mesh directly which is illustrated in Figure 6. It is not necessary to transform the Cartesian coordinate into polar coordinate system. In this case, the orthogonal grid mesh is able to deal with the Neumann type boundary naturally. The governing equation can be discretized into the following matrices computation by the local DQ method. y The [ bx ] and [ ] (( Kx) [ bx] ( Ky) [ by] ){ T} { Q} + = (5) px ( ) [ ] ij j = 1 py ( ) [ ] ik k = 1 { } K = K = 1 ; Q = 0 x n y bx = w, i = 1,,..., N n by = w, i = 1,,..., N ; n = n = 7 by are the weighting coefficients for the x- and y-directive second-order derivations. The parameters n px and n py are the numbers of supported reference points on x- px py 140

7 and y-directions respectively. Moreover, an equal value is adopted to be these two parameters for the isotropic heat conductivity in this case. The 99 computing points were employed in this simulation and the numerical result is shown as Figure 7. Furthermore, the temperature distribution of the numerical result along the radial direction is compared with the analytical solution (Fig. 8). The result shows very good accuracy which required very few computing cost. The maximum absolute error between the numerical -4 results and exact values is about Figure 6: The distribution of computational nodes Figure 7: The simulated temperature contour by proposed method 141

8 Local DQ 0.7 Analytical Figure 8. The comparison of temperature distribution of the numerical result and analytical solution along the radial direction The -D Poisson equation with Neumann boundary conditions The second example is employed to test the computational stabilities with the quantities of global nodes and local reference points. The former is to confirm the computing precision depending on the density of grid mesh and the latter observe the relationship with the accuracy order and matrix condition number. A -D Poisson equation problem with a capsule-shaped domain is chosen to enforce this test. The boundary conditions mixed by the Dirichlet and Neumann types. The Neumann boundary conditions are provided along the bevel edges and the Dirichlet at the curvy ends. The governing equation is u = sinx siny and the exact solution is u = sin x sin y. The sketch of computational domain and boundary conditions are shown in Figure 9. u = sin x sin0.9 y 0.8 u cos y sin x+ cos x sin y = n = n u cos y sin x cos x sin y u = sin x sin y Figure 9. The sketch of computational domain and boundary conditions 14

9 Figure 10. The distribution of computational nodes Figure 10 presents the mesh type of second example. There are 47, 904 and 3451 three different densities of computational points with the same type of grid mesh were adopted. Obviously the results are having very good agreement with the exact values and authentic numerical stability. Another property worth probing is the relationship between the quantity of reference points and accuracy. Therefore, it has also made a simplified analysis here to this property in this test. Figure 11 shows the contour of the numerical results. The relationship with the global computational points, local reference points, matrix condition number and accuracy are shown in Figure 1 and Table 1. The root mean square error in Fable 1 is defined as the following form, RMS- error = ( Numerical data - Exact solution) Number of computational nodes Figure 11. The contour map of numerical results by the LDQ method 143

10 (RMS-error) (Condition number) (Reference points: N ) Figure 1. The comparison of the maximum error, condition number and count of reference points L The parameters N Lx and N Ly are the counts of reference points along x- and y-direction. The count of reference points N is the sum of all directions. L Figure 13. The reference points on each direction Table 1: Maximum and RMS-error with different counts of computational points Computational points Max-error RMS-error 47.77* * * * * *

11 The -D Poisson equation with Neumann boundary conditions In order to identify the proposed model is attempted to verifying more -D Poisson equation problem in irregular domains with derivative boundary condition. A C-shaped domain is employed to test the -D Pisson problem. The governing equation is u = e θ ( x + y ), where 1 y u θ = tan ( ). The boundary conditions at R 1 = and R = 4 are = 0. The other sides, x n which are the ends of this domain, are on conditioned of two Dirichlet type boundary conditions, /4 u = e π 5 /4 and u = e π. The exact solution of this case is u = e θ. u =0 n R = 4 u= R 1 = 5 /4 e π 90 u = 0 n u= e π /4 Figure 14. The sketch of computational domain and boundary conditions As Figure 15 shows, the technique which abided by the mentioned concept at last section is able to deal with the Neumann boundary conditions instinctively. Identically, the results of this 3-D simulation are very good in agreement with the exact values such as the -D case. The contour map is shown in Figure 16. Total, 38 sums of computational nodes are employed in the -4 simulation. The maximum of absolute error approximates to and the root mean -4 square error is about

12 Figure 15. The distribution of computational nodes Figure 16. The contour of u The eigen-analysis for the two dimensional Helmholtz equation. Problems of eigen-analysis are very classical test for a developing numerical model. In order to validate the proposed scheme, the associated eigenvalue problem with a Oval of Cassini shapes domain Ω of, r (θ ) = cos θ sin θ 0 θ π, y = r (θ ) sin θ and x = r (θ ) cos θ 146

13 for homogeneous Dirichlet boundary conditions was solved by the local DQ method. Thus the Helmholtz equations φ + λ φ = 0 governed this analysis, the boundary conditions φ = 0 and the computational domain is illustrated with Figure 17. In this application, the Singular Value Decomposition (SVD) (Press et al., 199) is adopted to analyze the non-trivial solutions. The eigen-analysis with SVD technique is very common for the field of time-harmonic wave propagation. The boundary type numerical methods are widely used to solve this equation, such as the boundary element method (BEM) (Chen et al., 003) and the method of fundamental solutions (MFS) (Young et al., 006). There are 35 computing nodes adopted to compose the computational mesh (Fig. 18). The first singular value of the weighting coefficient matrix is plotted in Figure 19. The eigenvalues, which are computed by the local DQ method, is presented and compared with the results by MFS (Young et al., 006) in Table. The eigen-mode through low to high frequency (first to twelfth) have a rational trend (shown in Fig. 0). These results illustrate that the proposed method is reliable for analysis of the electromagnetic waveguide problems with irregular-shaped domains Figure 17. The sketch of computational domain Figure 18. The distribution of computational nodes 147

14 Figure 19. The minimum singular value plot for Oval of Cassini domain Table. Comparison of the numerical solutions for Oval of Cassini shaped case Mode MFS 80 nodes (Young et al. 006) LDQ 35 nodes

15 Figure 0. The 1 st to 1 th eigenvalues and -modes of a peanut-shaped problem 149

16 CONCLUSIONS The traditional DQ method has high accuracy but unsuitable to apply to the extensive calculation amount and question with the irregular boundary. A localized differential quadrature method is developed in this paper to handle more complicated and irregular domain problems. It is characterized by approximating the derivatives at a grid point using the weighted sum of the points in its neighbourhood rather than all the grid points. Some uncomplicated treatments at boundaries make irregular problems of form can be simulated too. In doing so, stability is enhanced, and accuracy is guaranteed by using any large number of grid points. The results obtained from the present method are compared with analytical results, both having very good agreements. The examples show the improved capability of the present method over traditional DQ method. Moreover, compared with traditional DQ approximations, the present method requires less CPU time on permutation and localized DQ computations by the banded computing matrix. Additionally, the proposed method also has high potential in replacing conventional high order FDM in virtue of the numerical accuracy order of local DQ can be varied very flexibly. ACKNOWLEDGEMENT The National Science Council of Taiwan is gratefully appreciated for providing the financial support for this research work under the Grant No. NSC E and NSC E REFERENCES Bellman R.E. and J. Casti Differential quadrature and long-term integration, J. Math Anal Appl, Vol 34, pp Bellman R.E., B.G. Kashef, and J. Casti Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations, Journal of Computational Physics. Vol 10, pp Bert C.W. and M. Malik Free vibration analysis of tapered rectangular plates by differential quadrature method: a semi-analytical approach. J Sound Vib, Vol 190, Iss 1, pp Bert C.W. and M. Malik Differential quadrature: a powerful new technique for analysis of composite structures. Compos Struct, Vol 39, Iss 3-4, pp Chen, J.T., L.W. Liu and H.K. Hong Spurious and true eigensolutions of Helmholtz BIEs and BEMs for a multiply connected problem, Proceedings of the Royal Society A, Vol 459, pp

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