Journal of Engineering Research and Studies E-ISS 0976-79 Research Article SPECTRAL SOLUTIO OF STEADY STATE CODUCTIO I ARBITRARY QUADRILATERAL DOMAIS Alavani Chitra R 1*, Joshi Pallavi A 1, S Pavitran 2, Bedekar Smita S 1 Address for Correspondence 1 Interdisciplinary School of Scientific Computing, University of Pune, Pune 11007 2 Department of Mechanical Engineering,Vishwakarma Institute of Technology, Pune 11037 E Mail chitra.alavani@gmail.com ABSTRACT The Chebyshev spectral method is applied to steady state conduction problems with source terms. Arbitrary quadrilateral domains are considered. Both curved and straight boundaries are looked into. The Chebyshev derivative matrix is obtained in the physical space and utilised in obtaining the solution using the collocation method. The algorithm is tested for quadrilaterals with straight as well as curved boundaries. A general algorithm is also presented for an arbitrary curved quadrilateral. It can be seen that the algorithm yields spectral accuracy in all cases. In most cases, eight Chebyshev modes in each direction is sufficient to yield a high accuracy. The method is proposed with the idea of extending it to the equations of fluid flow. KEYWORDS: Chebyshev collocation, conduction, curved quadrilaterals. 1. ITRODUCTIO Spectral methods belong to the general class of weighted residual methods [2,3,]. Being global, the method is highly accurate. Spectral methods are in general computationally intensive as they result in dense matrices. The main advantage of the standard spectral methods relies on the exponential convergence property. The main drawback is their inability to handle complex geometries. Although there have been attempts to use the spectral method in irregular domains [10], these approaches usually involve either incorporating finite-element preconditioning or using the spectral element method. Heinrich in his work on spectral collocation method on a unit disc [6], mapped the unit square directly onto the unit disc by means of interpolation techniques avoiding the singularity of polar coordinates. To apply spectral methods on a complex geometry, one needs to divide the domain into triangular (tetrahedral in 3D) or quadrilateral elements. Results are available for spectral methods on triangular domains [5, 7]. Alfonso et al [1] have proposed a numerical method to approximate the solution of partial differential equations in irregular domains with no-flux boundary conditions. The main advantage of the method is its capability to deal with domains of arbitrary shape and its easy implementation through FFT routines. Kong and Wu [9] have implemented a Chebyshev tau method for irregular domains by embedding it in a larger regular one. The important aspect of the present work is that the Chebyshev collocation method is applied in the physical space and derivatives are determined in the same. The governing equation is solved in the actual domain and is not transformed. Thus, the algorithm is more general in the sense that any elliptic system can be solved, as the derivative matrices are only domain dependant. The paper is organized as follows. In the next section, the algorithm for the solving conduction equation is discussed. These are linear elliptic partial differential equations (PDE) with Dirichlet boundary condition on quadrilateral domains. umerical results and their discussion are presented in Section 3. The conclusions are presented in Section. 2. ALGORITHM An arbitrary quadrilateral in the domain is considered. In order to solve the problem of conduction, the standard Chebyshev (collocation) derivative matrices and have to be obtained. An one-one mapping is defined between the quadrilateral in and a square points in in the Gauss-Lobotto points in plane. This yields and. The collocation get mapped to the Chebyshev. The gradient and the mapping are elobarated in the subsequent subsections. 2.1 mapping In order to determine the gradient, one needs the mapping function as,.
Journal of Engineering Research and Studies E-ISS 0976-79 Figure (1), shows the mapping from domain to domain. In order to determine the gradient, one needs the mapping function as,. The mapping for straight sided and curved quadrilaterals is presented in the following sections. 2.1.1 Straight sided quadrilateral For quadrilaterals with straight sides, the mapping as outlined by [11] can be written as The derivatives and are obtained from the Chebyshev collocation derivative matrix (cf. [3]). The partial derivatives such as are expressed as (6) where J is the determinant of the Jacobian matrix of with respect to defined as (1) (7) where Here are the vertices of the quadrilateral. (2) 2.1.2 quadrilateral with curved sides For an arbitrary quadrilateral, collocation points on the boundary of the domain are obtained. This is done using the arc length. The mapping is obtained by solving the Laplacian equation for by and, given (3) The Jacobian is obtained directly from the mapping function. Thus, the differentiation matrices and are formed. The second order derivative matrices and are obtained by matrix multiplication. The elliptic (linear) PDE is then reduced to a linear algebraic system []. The matrix is dense and ill conditioned. Hence, iterative methods such as conjugate gradient are not suitable. A direct method such as LU decomposition is used to solve the linear system. 3. RESULTS AD DISCUSSIO The results for conduction with source terms in quadrilaterals with straight and curved sides are presented. The mapping is defined initially. The results are compared against the analytical (exact) solution. The boundary conditions are prescribed according to the exact solution. The norm is defined as error () With the boundary values of and corresponding to the boundary conditions. 2.2 gradient calculation and solution methodology The gradient is defined as (5) Where is the spectral solution, the exact one and is the number of Chebyshev modes in directions. The number of modes is taken to be the same for both. 3.1 mapping for quadrilateral with straight sides Consider a quadrilateral element with four nodes numbered and in an anti-clockwise direction, as shown in Figure (1).
Journal of Engineering Research and Studies E-ISS 0976-79 The conduction equation with source terms is written as is solved with vertices at (1, 1), (5, 2), (, ) and (2, ). The exact solution is given by The results of the (9). (10), for different is presented in the Table of Figure (2). From that, it can be seen that convergence is obtained for. The equation (11) is solved with vertices at (0, 3), (3, 0), (0, 2) and ( 2, 0). The exact solution is given by The results of the () for different is presented in the Table of Figure (3). It can be observed that convergence is obtained for =. The reason for spectral convergence being obtained at a higher, compared to the previous case, is that the exact solution being an exponential one needs more terms of Chebyshev series for convergence. The asymptotic slope of the error versus curve on the log-log scale for equation (9) is -26 and for equation (11) is -25, indicating spectral accuracy. Figure 1: Mapping from natural to physical coordinate systems. 0.13 0.05 0.0002 9.117e-007 5.191e-015 Figure 2: for different order of on quadrilateral with points (1, 1), (5, 2), (, ) and (2, )
Journal of Engineering Research and Studies E-ISS 0976-79 36.0372 2.1031 0.0256 0.0192 3.0336e-0 Figure 3: for different order of on quadrilateral with points (0, 3), (3, 0), (0, 2) and ( 2, 0) 3.2 Quadrilateral with curved sides This section discusses results of PDEs solved on quadrilaterals with curved boundaries shown in Figure (). The mapping can be defined trivially as 3.3 arbitrary curved quadrilateral Here we will discuss the results of conduction equation solved on arbitrary curved domain bounded by four different curves as shown in Figure (5). The equation of the curve between point A and B is For equation 9, the results for different are presented in Table 1. As can be seen in that, one can say that convergence is obtained for. For equation (11), the error norm results for different is presented in Table 2. As can be seen in Table 2, it can be seen that convergence is obtained for. The asymptotic slope of the error versus curve on the log-log scale for equations (9) and (11) is -21, indicating spectral accuracy. Table 1: Variation of (for equation 9) with 0.011 5.636e-005 1.1953e-00 1.220e-011 9.1953e-015 Table 2: Variation of (for equation 11) with 1.915e-00 2.393e-007 1.193e-010 5.79e-01 5.27e-015 that between point B and C is between point C and D is and between point D and A is Collocation points on the boundary of the domain are obtained using the arc lengths of each curve. Then solving the Laplacian one gets all the collocation points in the interior of (x, y) domain. The mesh generated by using the mapping for curved quadrilaterals is given in Figure (6). The error norm results for equation (9) are given in Table 3. As one can observe, spectral convergence is obtained for. The results are presented for Equation 11. The variation of error norm for different is presented in Table. As can be seen in Table, one can say that convergence is obtained for. The asymptotic slope of the error versus curve on the log-log scale for equation (9) is -6 and for equation (11) is -5. This implies a high order accuracy, lower than the spectral one obtained in previous cases. This may be due to the mapping functions used and hence needs further investigation.
Journal of Engineering Research and Studies E-ISS 0976-79 Figure : Domain bounded by exponential curve and Figure 5: Domain bounded by four different curves Figure 6: Generated mesh in x, y domain
Journal of Engineering Research and Studies E-ISS 0976-79 Table 3: Table : Variation with 0.0066 3.001e-005 3.26e-006 6.63e-007 2.655e-00 for different order of 0.0039 1.963e-005 1.21e-006 5.033e-007 1.559e-00 [] Karniadakis, G. E., and Sherwin, S. J. Spectral/hp Element Methods for Computational Fluid Dynamics, second ed. Oxford Science Publications, 2005. [9] Kong, W., and Wu, X. Chebyshev tau matrix method for poisson type equations in irregular domain. Journal of Computational and Applied Mathematics 22 (200), 15 7. [10] Orszag, S. A. Spectral methods for problems in complex geometries. Journal of Computational Physics 37 (190), 70 92. [11] Pozrikidis, C. Introduction to Finite and Spectral Element Methods using MATLAB. Chapman and Hall/CRC, 2005. COCLUSIOS It can be seen that the algorithm yields spectral accuracy for conduction (elliptic systems). In most of the cases, spectral accuracy is obtained for. Thus, even though the matrices are dense, the relative size of the matrix is small, leading to a low computational cost. In the case of results shown in the Table of Figure 3, the error reduction is slower due to the exponential behaviour of the solution. For the case of arbitrary quadrilateral, the algorithm is of a higher lower than spectral. The algorithm can be extended to higher dimensions, transient cases and to fluid flow equations. REFERECES [1] Alfonso, B., V ıctor, M. P., and F., F. H. Spectral methods for partial differential equations in irregular domains: The spectral smoothed boundary method. SIAM Journal of Scientific Computing 2 (2006), 6 900. [2] Boyd, J. P. Chebyshev and Fourier Spectral Methods, second(revised) ed. Dover Publications, IC, 2001. [3] Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. Spectral Methods in Fluid Dynamics. Springer, 19. [] Canuto, C., Hussaini, M. Y., Quarteroni, A., and Zang, T. Spectral Methods Fundamentals in Single Domains. Springer, 2006. [5] Heinrichs, W. Spectral collocation on triangular elements. Journal of Computational Physics 15 (June 199), 73 757. [6] Heinrichs, W. Spectral collocation schemes on the unit disc. Journal of Computational Physics 199 (February 200), 66 6. [7] Heinrichs, W., and Loch, B. I. Spectral schemes on triangular elements. Journal of Computational Physics 173 (June 2001), 20 301.