PAMM · Proc. Appl. Math. Mech. 7, 2020129–2020130 (2007) / DOI 10.1002/pamm.200700951
A singular function boundary integral method for elliptic problems with singularities Christos Xenophontos∗1 , Georgios Georgiou 1 , and Miltiades Elliotis 1
1
Department of Mathematics and Statistics, University of Cyprus, PO BOX 20537, Nicosia 1678, Cyprus.
In this work we present a singular function boundary integral method for elliptic problems with boundary singularities. In this method, the approximation is constructed from the truncated asymptotic expansion for the solution near the singular point and the Dirichlet boundary conditions are weakly enforced by means of Lagrange multiplier functions. The resulting discrete problem is posed and solved on the boundary of the domain, away from the point of singularity. We are able to show that the method approximates the generalized stress intensity factors, i.e. the coefficients in the asymptotic expansion, at an exponential rate. © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
1
Introduction
Planar elliptic boundary value problems with boundary singularities were greatly discussed in the last few decades. Many different methods have been proposed for the solution of such problems, ranging from special mesh-refinement schemes to sophisticated techniques that incorporate, directly or indirectly, the form of the local asymptotic expansion, which is known in many occasions. These methods aim to improve the accuracy and resolving the convergence difficulties that are known to appear in the neighborhood of such singular points. The local (asymptotic expansion for the) solution, centered at the singular point, in polar coordinates (r, θ) is of the general form: u(r, θ) =
∞
αi rμi fi (θ) ,
(1)
i=1
where μi are the eigenvalues and fi are the eigenfunctions of the problem, which are uniquely determined by the geometry and the boundary conditions along the boundaries sharing the singular point. The singular coefficients αi , also known as generalized stress intensity factors [6] or flux intensity factors [2], are determined by the boundary conditions in the remaining part of the boundary. Knowledge of the singular coefficients is of importance in many engineering applications, especially in fracture mechanics. In this work we present a Singular Function Boundary Integral Method (SFBIM) (see [7] and the references therein), in which the unknown singular coefficients are calculated directly, as opposed to via post-processing, which is the most commonly used technique (cf. [3]). In order to describe the method, we consider, for simplicity, the Laplacian problem described graphically in Figure 1 below.
Fig. 1 A two-dimensional Laplace equation problem with one boundary singularity.
∗
Corresponding author: e-mail:
[email protected], Phone: +357 22 892610, Fax: +357 22 892601
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
2020130
ICIAM07 Contributed Papers
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The Singular Function Boundary Integral Method
The SFBIM is based on the approximation of the solution by the leading terms of the local solution expansion Nα u ¯ = i=1 α ¯ i W i , where Nα is the number of singular functions used, which are defined by W i ≡ rμi fi (θ), and α ¯i are the approximate singular coefficients. (We note that this approximation is valid only if Ω is a subset of the convergence domain of the local solution (1).) We begin by multiplying the differential equation by W i , integrating over Ω, and using Green’s second identity twice along with the fact that the the singular functions are harmonic and exactly satisfy the boundary conditions along S1 and S2 , to obtain ∂W i ∂W i ∂u i i ∂u W −u −u W dS + dS = 0, ∀i ∈ N. (2) ∂n ∂n ∂n ∂n S4 S3 Note that the dimension of the problem is reduced by one, which leads to a considerable reduction of the computational cost. We approximate u by u¯ and impose the Neumann condition along S4 by simply substituting the normal derivative by the known function g. The Dirichlet condition along S3 is imposed by means of a Lagrange multiplier function, λ, replacing the normal derivative. The function λ is expanded in terms of standard, polynomial Nλ λj M j , where Nλ represents the total number of the unknown discrete Lagrange basis functions as λ = ∂∂nu¯ = j=1 multipliers (or, equivalently, the total number of Lagrange-multiplier nodes) along S3 , and M j are (piecewise) polynomials of degree p defined on a uniform partition of S3 characterized by the meshwidth h. (We note that the number of Lagrange multipliers Nλ satisfies Nλ = O(1/h).) The functions M j are used to weight the Dirichlet condition along the corresponding boundary segment S3 . We thus obtain the following system of Nα +Nλ discretized equations: ∂W i ∂W i dS − dS = − u¯ W i g(r, θ) dS, i = 1, 2, . . . , Nα , (3) λW i − u ¯ ∂n ∂n S4 S4 S3 j u¯M dS = f (r, θ)M j dS, j = 1, 2, . . . , Nλ . (4) S3
S3
It is easily shown that the above linear system is symmetric and nonsingular, provided Nα > Nλ . Now, with H k (Ω), k ∈ R denoting the usual Sobolev spaces with associated norms ·k,Ω (cf. [1]), we are able to ∂u ∈ H k (S3 ) for some k ≥ 1, then there exist positive constants C and β ∈ (0, 1), independent of Nα show that if ∂n and Nλ , such that
∂u Nα m −k − λ , (5) ≤ C N β + h p u − u ¯1,Ω + α ∂n −1/2,S3 where m = min{k, p + 1}. Moreover, since the error between the exact coefficients αi and approximate coefficients α ¯ i satisfies |αi − α ¯ i | ≤ C u − u¯0,Ω , we have ¯ i | ≤ Cβ Nα , |αi − α
(6)
which shows that the method approximates these coefficients at an exponential rate as Nα → ∞. Finally, we refer the interested reader to [4]–[5] where the method is applied to problems from solid and fluid mechanics for which the governing equation is a fourth order elliptic boundary value problem.
References [1] R. A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975. [2] M. Arad, Z. Yosibash, G. Ben-Dor and A. Yakhot, Comparing the flux intensity factors by a boundary element method for elliptic equations with singularities, Commun. Numer. Meth. Eng., 14, 657–670 (1998). [3] I. Babuˇska, A. Miller, The post-processing approach in the finite element method – Part 2: The calculation of stress intensity factors, Int. J. Numer. Meth. Eng., 20, 1111–1129 (1984). [4] M. Elliotis, G. Georgiou and C. Xenophontos, Solution of the planar Newtonian stick-slip problem with the singular function boundary integral method, Int. J. Numer. Meth. Fluids, 48, 1001–1021 (2005). [5] M. Elliotis, G. Georgiou and C. Xenophontos, The singular function boundary integral method for a two-dimensional fracture problem, Engineering Analysis with Boundary Elements, 30, 100–106 (2006). [6] B.A. Szab´ o and Z. Yosibash, Numerical analysis of singularities in two dimensions. Part 2: Computation of generalized flux/stress intensity factors, Int. J. Numer. Methods Engin. 39, 409–434 (1996). [7] C. Xenophontos, M. Elliotis and G. Georgiou, The singular function boundary integral method for elliptic problems with singularities, SIAM Journal of Scientific Computing, 28(2), 517–532 (2006).
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
