Elsevier

Computers & Chemical Engineering

Volume 32, Issue 12, 22 December 2008, Pages 2891-2896
Computers & Chemical Engineering

The finite differences method for solving systems on irregular shapes

https://doi.org/10.1016/j.compchemeng.2008.02.005Get rights and content

Abstract

A relatively simple and efficient symbolic-numerical procedure based on the finite differences method for solving partial differential equations on systems of irregular shapes is presented. The new concept is based on the spline parameterization of the irregular domain. The curvilinear domain of the real system is transformed to the rectangular domain by spline functions where the finite differences method is used to solve the transformed system of depended variables. The numerical results are then transported back to the original irregular shape of the system. In order to present the symbolic-numerical technique effectively, the Laplace's equation of heat transfer with the Dirichlet and the Neumann boundary conditions in different 2D curvilinear domains is considered. The proposed technique is applied for the non-steady-state heat transfer by conduction as well. Numerical experiments were performed to justify the proposed method.

Introduction

An important element of the numerical solution of partial differential equations by the finite-difference method (FDM) or the finite-element method (FEM) on general regions is a grid which represents the physical domain in a discrete form (Lisejkin, 1999). The FEM is generally considered more suitable for the treatment of irregular boundaries due to its flexibility in dividing the problem domain into elements of various sizes and shapes. While the FEM can be applied to uniform and non-uniform meshes, the FDM requires a rectangular grid, which is impossible in the physical plane of practically all real-life problems. Therefore, it is necessary to use a coordinate transformation and map the irregular region into a regular one in the computational plane (Cebeci, Shao, Kafyeke, & Laurendeau, 2005). According to the literature, three major classes of techniques corresponding to algebraic methods, differential equation methods, also called the body-fitted (or boundary-fitted) coordinate (BFC) methods, and conformal mapping methods have mainly been used so far to overcome these difficulties. Although algebraic methods have a major advantage of rapid computation, they are generally less preferred to partial differential equation (PDE) methods due to the lack of grid smoothness (Koo & Leap, 1998). PDE methods are based on solving a system of PDEs, the most widely used of which are elliptic equations. Since the early 1980s the use of BFC in the numerical solution of physical problems involving arbitrary geometries has become one of the primary numerical approaches. Thompson, Warsi, and Mastin (1982), Thompson, Warsi, and Mastin (1995) and Knupp and Steinberg (1993) gave a comprehensive review of numerical grid generation methods. Recently, the overall approach to the numerical solution of a physical problem in boundary-fitted curvilinear coordinates is more or less a well-established procedure (Beji & Nadaoka, 2004). It appears from the literature that conformal mapping and the solution of elliptic partial differential equations are the most widely used principles for the generation of a curvilinear body-fitted coordinate system (He, Fan, & Cen, 2000; Jie, Jie, Mao, & Li, 2004; Koo & Leap, 1998; Lee & Leap, 1994; Tsay & Hsu, 1997). The inherent smoothness and boundary slope discontinuities which do not propagate into the field are the advantages of the system of elliptic partial deferential equations as a means of coordinate system generation (Tsay & Hsu, 1997). Ramanathan and Kumar (1988) compared the BFC method with FEM for solving transient heat conduction problems in complex geometries. They showed that the BFC method is more accurate and economical in terms of both computational time and storage requirement than the FEM. However, the generation of the grid by solving three-dimensional systems of partial differential equations in domains with complex geometry may be the most time-consuming part of the calculations (Lisejkin, 1999). According to the same author, the meshes still limit the efficiency of the numerical methods for the solution of partial differential equations. Recently, the numerical instability in finite-difference approximations of partial differential equations on a skewed mesh, and its effect on the solution quality was presented by You, Rajat, Wang, and Moin (2006). The authors performed the truncation error and modified the wave number analysis to reveal various effects of mesh skewness and non-uniformity. A comparison of the spine and the elliptic mesh generation methods in case of dynamics of sessile droplet evaporation was presented by Widjaja et al. (2007).

The present paper explains the methodology and the symbolic-numerical solution procedure based on the finite differences method for solving partial differential equations on systems of irregular shapes. The new concept is based on the parameterization of the curvilinear region of the physical domain by inherent smooth splines. The mapping of the rectangular domain to the irregular domain is used to build the corresponding PDE. The procedure is automated by the developed mathematical code based on the repeated chain rule. The classical finite differences method was then applied to solve the system on the rectangular domain. The transport of numerical results back to the original irregular shape of the system is automated as well by the developed mathematical code written in the Mathematica computational tool (Wolfram, 2006). In the present work Laplace's equation with prescribed and mixed boundary conditions on different 2D curvilinear domains is considered and the solutions were numerically verified. The proposed technique is demonstrated for time dependent problems as well and it can be applied to the higher dimensions settings of the second-order PDE with appropriate boundary conditions.

Section snippets

Splines

Let us consider a series of points T1, T2, …, Tn. Let M is equal (Bartels, Beatty, & Barsky, 1987):M=161331363030301410andBt(T1,T2,T3,T4)=[t3,t2,t,1]MTwhere M is the B-spline basis matrix, T is the B-spline geometry vector for the curve segment defined by four successive points, and [t3, t2, t, 1] is defined as the row vector (Foley et al., 1996).

Now we take the first four points T1, T2, T3, T4 of an polygonal line and assign them cubic polynomial mapping Bt(T1, T2, T3, T4), so that the T

Results and discussion

Besides the horseshoe-shaped 2D domain (Fig. 4), an arbitrary geometrical shape is chosen to graphically demonstrate the application of the finite differences method for solving systems on irregular shapes by the proposed symbolic-numerical procedure. The domain of a selected shape formed by spline patches is presented in Fig. 5.

As example, the numerical results of the steady-state heat transfer by conduction for selected boundaries conditions on the domains of the irregular shapes are

Conclusions

The numerical solutions based on finite differences method of Laplace's equation and time dependent problem of heat transfer by conduction on different 2D curvilinear domains were performed and theoretically described. The methodology of the proposed symbolic-numerical solution procedure for solving the partial differential equations on the systems of irregular shapes is based on the parameterization of the curvilinear region of physical domain by inherent smooth splines, mapping of the

Acknowledgments

This work was supported by Grant P2-0191 Chemical Engineering, provided by the Ministry of Education, Science and Sport of the Republic of Slovenia.

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