Research Article Divergent Integrals in Elastostatics: General Considerations

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1 International Scholarly Research Network ISRN Applied Mathematics olume 2011, Article ID , 25 pages doi: /2011/ Research Article Divergent Integrals in Elastostatics: General Considerations.. Zozulya Materials Department, Centro de Investigacion Cientifica de Yucatan A.C., Calle 43, No. 130, Colonia Chuburná de Hidalgo, Mérida, YUC, Mexico Correspondence should be addressed to.. Zozulya, Received 4 April 2011; Accepted 31 May 2011 Academic Editors: S.-W. Chyuan and E. A. Navarro Copyright q Zozulya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This article considers weakly singular, singular, and hypersingular integrals, which arise when the boundary integral equation methods are used to solve problems in elastostatics. The main equations related to formulation of the boundary integral equation and the boundary element methods in 2D and 3D elastostatics are discussed in details. For their regularization, an approach based on the theory of distribution and the application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular, and hypersingular integrals, have been constructed. 1. Introduction A huge amount of publications is devoted to the boundary integral equation methods BIM and its application in science and engineering 1 5. One of the main problems arising in the numerical solution of the BIE by the boundary element method BEM is a calculation of the divergent integrals. In mathematics divergent integrals have established theoretical basis. For example, the weakly singular integrals are considered as improper integrals; the singular integrals are considered in the sense of Cauchy principal value P ; the hypersingular integrals had been considered by Hadamard as finite part integrals FP. Different methods have been developed for calculation of the divergent integrals. Usually different divergent integrals need different methods for their calculation. Analysis of the most known methods used for treatment of the different divergent integrals has been done in 3, 4, 6 8. The correct mathematical interpretation of the divergent integrals with different singularities has been done in terms of the theory of distributions generalized functions. The theory of distributions provides a unified approach for the study of the divergent integrals and integral operators with kernels containing different kind of singularities.

2 2 ISRN Applied Mathematics In our previous publications 9 19, approach based on the theory of distributions has been developed for regularization of the divergent integrals with different singularities. The above mentioned approach for regularization of the hypersingular integrals has been used for the first time in 11. Then it was further developed for static and dynamic problems of fracture mechanics in 15, 18, 19, respectively, and also in 9, 10, 13. The equations presented in 12, 14 can be applied for a wide class of divergent integral regularizations for example for 2D and 3D elasticity 5, 14. In the present paper, the approach for the divergent integral regularization based on the theory of distribution and Green s theorem is further developed. The equations that enable easy calculation of the weakly singular, singular, and hypersingular integrals for different shape functions are presented here. 2. Main Equations of Elastostatics Lets consider a homogeneous, lineally elastic body, which in three-dimensional Euclidean space R 3 occupies volume with smooth boundary. The region is an open bounded subset of the three-dimensional Euclidean space R 3 with a C 0,1 Lipschitzian regular boundary. The boundary contains two parts u and p such that u p and u p. On the part u are prescribed displacement u i x of the body points, and on the part p are prescribed traction p i x, respectively. The body may be affected by volume forces b i x. We assume that displacements of the body points and their gradients are small, so its stressstrain state is described by small strain deformation tensor ε ij x. The strain tensor and the displacement vector are connected by Cauchy relations ε ij 1 2 i u j j u i ), 2.1 where / x i is a derivative with respect to space coordinates x i. The components of the strain tensor must also satisfy the Saint-enant s relations 2 kl ε ij 2 il ε kj 2 kj ε il 2 ij ε kl. 2.2 From the balance of impulse and the moment of impulse lows follow that the stress tensor is symmetric one and satisfy the equations of equilibrium j σ ij b i 0 i, x. 2.3 Here and throughout the paper the summation convention applies to repeated indices. The tensor of deformation ε ij x and stress σ ij x are related by Hook s law σ ij c ijkl ε ij. 2.4 Here c ijkl are elastic modules. In the case of homogeneous anisotropic medium they are symmetric c ijkl c jikl c klij. 2.5

3 ISRN Applied Mathematics 3 and satisfy condition of ellipticity c ijkl ε ij ε kl α 1 ε ij ε ij, ε ij, α 1 > In the case of homogeneous isotropic medium, the elastic modules have the form c ijkl λδ ij δ kl μ δ ik δ jl δ il δ jk ), 2.7 where λ and μ are Lame constants, μ>0andλ> μ, δ ij is a Kronecker s symbol. Substituting stress tensor in 2.3 and using Hook s law 2.4 and Cauchy relations 2.1, weobtainthedifferential equations of equilibrium in the form of displacements which may be presented in the form A ij u j b i 0, x. 2.8 The differential operator A ij for homogeneous anisotropic medium has the form A ij c ikjl k l, 2.9 and for homogeneous isotropic medium has the form A ij μδ ij k k λ μ ) i j If the problem is defined in an infinite region, then solution of 2.8 must satisfy additional conditions at the infinity in the form u j x O r 1), σ ij x O r 2) for r, 2.11 where r x 2 1 x2 2 x2 3 is the distance in the three-dimensional Euclidian space. If the body occupied a finite region with the boundary, it is necessary to establish boundary conditions. We consider the mixed boundary conditions in the form u i x ϕ i x, x u, p i x σ ij x n j x P ij [ uj x ] ψ i x, x p The differential operator P ij : u j p i is called stress operator. It transforms the displacements into the tractions. For homogeneous anisotropic and isotropic medium, they have the forms P ij c ikjl n k l, P ij λn i k μ δ ij n n k i ), 2.13 respectively. Here n i are components of the outward normal vector, n n i i is a derivative in direction of the vector n x normal to the surface p.

4 4 ISRN Applied Mathematics 3. Integral Representations for Displacements and Traction In order to establish integral representations for the displacements and tractions, let us consider bilinear form which depends on two fields of the strain tensor that correspond to two fields of the displacements u and u a u, u c ijkl ε ij u ε kl u. 3.1 Obviously that a u, u a u, u, a u, u σ ij u ε ij u α 1 ε ij u. 3.2 Integrating the equality 3.1 over the volume and applying the Gauss-Ostrogradskii formula, we will obtain a u, u d σ ij u ε kl u d σ ij n j u i ds u i jσ ij d. 3.3 Taking into account that A ij u j j σ ij and p i σ ij n j P ij u j, we will find the first Betti s theorem in the form u i A iju j d a u, u d u i P ] ij[ uj ds. 3.4 We will replace u i and u i in 3.4 and subtract resulting equation from 3.4. Because the form 3.1 is symmetrical one, we will obtain the second Betti s theorem in the form ) [ ] u i A iju j u i A ij u j d u i P ij u j u i P ] ij[ ) uj ds. 3.5 Taking into the account the definition of differential operator A ij given in 2.8 and the definition of differential operator P ij given in 2.12, we obtain relation bi u i b i u ) i d p i u i p i u ) i ds, 3.6 which is called the Betti s reciprocal theorem. This theorem is usually used for obtaining integral representations for the displacements and traction vectors. To do that, we consider solution of the elliptic partial differential equation 2.8 in an infinite space for the body force b i x δ ijδ x y A ij U kj x y δ ki x y 0, x, y R Now considering that u i x U ij x y, ] p i x P ij [u j x W ij x, y, 3.8

5 ISRN Applied Mathematics 5 from 3.6 we obtain the integral representation for the displacements vector u i y pi x U ji x y u j x W ji x, y ) ds b i x U ji x y d, 3.9 which is called Somigliana s identity. The kernels U ji x y and W ji x, y are called fundamental solutions for elastostatics. Applying to 3.9 the differential operator P ij, we will find integral representation for the traction in the form p i y pi x K ji x, y u j x F ji x, y ) ds b i x K ji x, y d The kernels K ji x, y and F ji x, y may be obtained applying the differential operator P ij to the kernels U ji x y and W ji x, y, respectively. The integral representations 3.9 and 3.10 are usually used for direct formulation of the boundary integral equations in elastostatics. 4. Fundamental Solutions In order to find the fundamental solutions U ij x y for the differential operator A ij,we consider the differential equations of elastostatics in the form displacements 3.7. Solutions of these equations are called the fundamental solutions. In 3D elastostatics, they have the form U ij x y 1 3 4υ δij i r j r ) πμ 1 υ r In 2D elastostatics they have the form U ij x y 1 3 4υ δ ij ln 1 ) 8πμ 1 υ r ir j r. 4.2 Here r x 1 y 1 2 x 2 y 2 2 x 3 y 3 2 and r x 1 y 1 2 x 2 y 2 2 for 3D and 2D case, respectively, i r r/ x i r/ y i x i y i /r. The kernels W ij x, y from 3.9 may be obtained by applying to U ij x y differential operator P ik, x λn i x k μ δ ik n j x j n k x i, 4.3 as it is shown here W ij x, y λn i x k U kj x, y μn k x k U ij x, y i U kj x, y. 4.4

6 6 ISRN Applied Mathematics Then after some transformations and simplifications, the expression for the kernels W ij x, y will have the following form W ij x, y 1 4π 1 υ r α nk x k r 1 2υ δ ij β i r j r ) 1 2υ n i x j r n j x i r )). 4.5 The kernels K ij x, y and F ij x, y from 3.10 may be obtained by applying differential operator P ik, y λn i y k μ δ ik n j y j n k y i 4.6 to U ij x y and W ij x, y, respectively K ij x, y λn i y k U jk x, y μn k y k U ji x, y i U jk x, y, F ij x, y λn i y k W jk x, y μn k y k W ji x, y i W jk x, y. 4.7 Then after some transformations and simplifications the expression for the kernels K ij x, y will have the form K ij x, y 1 4π 1 υ r α nk y k r 1 2υ δ ij β i r j r ) 1 2υ n i y j r n j y i r )), 4.8 and for the kernels, F ij x, y will have the form F ij x, y μ βnk x 2απ 1 υ r β k r 1 2υ n i y j r υ δ ij n k y k r n j y i r ) γn k y k r i r j r ) βν n i x n k y k r j r n k x n k y i r j r ) 1 2υ βn j x n k y k r i r n k x n k y δ ij n i x n j y ) 1 4υ n j x n i y ). 4.9 In , α 2, 1, β 3, 2andγ 5, 4 in 3D and 2D cases, respectively. The kernels contain different kind singularities; therefore, corresponding integrals are divergent. Here we will investigate there singularities and develop methods of divergent integrals calculation. 5. Singularities, Boundary Properties, and Boundary Integral Equations Simple observation shows that kernels in the integral representations 3.9 and 3.10 tend to infinity when r 0. More detailed analysis of gives us the following results.

7 ISRN Applied Mathematics 7 In the 3D case with x y, U ij x y r 1, W ij x, y r 2, K ij x, y r 2, F ij x, y r In the 2D case with x y, U ij x y ln r 1), W ij x, y r 1, K ij x, y r 1, F ij x, y r In order to investigate these functions and integrals with singular kernels, definition and classification of the integrals with various singularities will be presented here. Definition 5.1. Let one considers two points with coordinated x, y R n where n 3orn 2 and region with smooth boundary of the class C 0,1. The boundary integrals of the types G x, y r α ϕ x ds, α > 0, 5.3 where G x, y is a finite function in R n and ϕ x is a finite function in,areweakly singular for α<n 1, singular for α n 1, and strongly singular of hypersingular for α>n 1. Definition 5.2. Let we consider two points with coordinated x, y R n where n 3orn 2 and region with smooth boundary of the class C 0,1. The boundary integrals of the types G x, y ln r ϕ y dy, 5.4 where G x, y is a finite function in R n and ϕ x is a finite function in, are weakly singular. The integrals with singularities can not be considered in usual Riemann or Lebegue sense. In order to such integrals have sense, it is necessary special consideration of them. We will apply this definition of the integrals from 3.9 and Definition 5.3. Integrals in 3.9 with kernels U ij x y are weakly singular and must be considered as improper W.S. p i x U ij x y ds lim p i x U ij x y ds. ε 0 \ ε 5.5 Here ε is a part of the boundary, projection of which on tangential plane is contained in the circle C ε x of the radio ε with center in the point x.

8 8 ISRN Applied Mathematics Definition 5.4. Integrals in 3.9 and 3.10 with kernels W ij x, y and K ij x, y are singular and must be considered in sense of the Cauchy principal values as P.. u i x W ij x, y ds lim u i x W ij x, y ds, ε 0 \ r<ε P.. p i x K ij x, y ds lim p i x K ij x, y ds. ε 0 \ r<ε 5.6 Here r <ε is a part of the boundary, projection of which on tangential plane is the circle C ε x of the radio ε with center in the point x. Definition 5.5. Integrals in 3.10 with kernels F ij x, y are hypersingular and must be considered in sense of the Hadamard finite part as F.P. u i x F ji x y ds lim ε 0 \ r<ε ) f j x u i x W ji x y ds 2u j x. 5.7 r <ε Here functions f j x are chosen from the condition of the limit existence. Singular character of the channels in 3.9 and 3.10 determine boundary properties of the corresponding potentials. Analysis of these formulae show that the boundary potentials, with the kernels U ij x y, are weakly singular, and, therefore, they are continuous everywhere in the R n and, therefore, may be continuously extended on the boundary. The potentials with the kernels W ij x, y and K ij x, y contain singular kernels, and they jump when crossing the boundary. The potential with the kernels F ij x, y contain hypersingular kernels. They continuously cross the boundary. Boundary properties of these potentials have been studied extensively in 1, 2, 5, 8, 20 and so forth. Summary of these studies may be expressed by the equations ± p i x U ji x y ds) p i x U ji x y ds) 0, ) ± u i x W ji x, y ds u j y u i x W ji x, y ds), ) ± p i x K ji x, y ds ± p j y p i x K ji x, y ds), ± 0 u i x F ji x y ds) u i x F ji x y ds). 5.8 The symbols ± and denote that two equalities, one with the top sign and the other with the bottom sign, are considered. The up index 0 points out that the direct value of the corresponding potentials on the surface should be taken.

9 ISRN Applied Mathematics 9 Now using integral representations for displacements and traction and boundary properties of the potentials, we can get boundary integral equations for elastostatics. Tending y in 3.9 and 3.10 to the boundary and taking into consideration boundary properties of the potentials 5.8, we obtain representation of the displacements and traction vectors on the boundary surface. On the smooth parts of the boundary, they have the following form ± 1 2 u i y 1 2 p i y pj x U ij x y u j x W ij x, y ) ds pj x K ij x, y u j x F ij x, y ) ds p j x U ij x y d, p j x K ij x, y d. 5.9 The plus and minus signs in these equations are used for the interior and exterior problems, respectively. Together with boundary conditions, they are used for compositing the BIE for the problems of elastostatics. The BIE are usually solved numerical transforming them into discrete system of finite dimensional equations. Such method is called the boundary element method BEM. 6. Projection Method and the BEM Equations The main idea of the BEM consists in approximation of the BIE and further solution of that approximated finite dimensional BE system of equations. The mathematical essence of this approach is the so-called projection method. Let us outline some results from mathematical theory of the projection methods related to the approximation of the BIE. For more information, one can refer to 4, 21. We consider two Banach spaces X and Y and functional equation in those spaces A u f, u D A X, f R A Y. 6.1 Here A : X Y is the linear operator mapping from Banach space X in Banach space Y, D A is a domain, and R A is a range of the operator A. The equation 6.1 is named the exact equation, and its solution is the exact solution. We denote L X, Y Banach space of the linear operators mapping from X in Y. Let in X and Y act sequences of projection operators P h and P such that h P 2 h P h, P h X X h, X h X, ) P 2 h P h, P h Y Y h, Y h Y, 6.2 where X h and Y h are finite dimension subspaces of the Banach spaces X and Y; h R 1 is a parameter of discretization. Now we consider operator A h L X h, Y h mapping in finite dimensional subspaces X h and Y h and an approximate equation A h u h f h, A h P h A P h, u h P h u, f h P h f. 6.3

10 10 ISRN Applied Mathematics Solution u h of 6.3 is the approximate solutions of 6.1. The general scheme of the approached equations construction 6.3 is illustrated by the following diagram. X D A A R A Y P h P h 6.4 X h D A h A h R Ah Y h Now let us consider operator A h L X h, Y h mapping in finite-dimensional subspaces X h and Y h and an approximate equation. Existence of the exact solution, convergence of the approximate solution to the exact one, and stability of the approximations are the main problems which arrived in application of the projection methods. In order to solve these problems, we have to formulate them mathematically. We assume that projection operators P h and P converge to identity operators in X and h Y, respectively. It means that lim h u u X 0 h 0 u X, lim P h f f Y 0 f Y. h Definition 6.1. Let conditions 6.5 be satisfied and stating from some h h 0 > 0 for any f Y, 6.3 has unique solution u h. In this case if lim A h u h A u Y 0, h then the solution of the approximate problem 6.3 converges to the exact solution 6.1. It means that the projective method presented on diagram 6.4 is applicable to the initial problem 6.1. Definition 6.2. Let for some sequence of the operators {A h } mapping from X h into Y h there is a constant γ>0 such that stating from some h h 0 > 0 A h u h A u Y γ u h X u h X h 6.7 then for sequence of the operators {A h }, the condition of stability of the approximate solution is satisfied. Conditions are very important for formulation conditions of existence, convergence and stability of the approximate solution. These conditions contain the following theorem 21, 22.

11 ISRN Applied Mathematics 11 Theorem 6.3. Let the following conditions be satisfied: 1 the projection operators P h and P converge to identity operators in the Banach spaces X h and Y, respectively, as it stated in 6.5 ; 2 the sequence of approximate operators {A h } converges to A on each exact solution; 3 the condition of stability 6.7 is satisfied for the sequence of operators {A h }. Then the following consequences take place: 1 the exact solution exists and it is unique; 2 for all enough small h exists a unique solution u h X h of the approximate equation 6.3 ; 3 the sequence of the approximate equation {u h } converges to the exact one and takes place in the estimation u h P h u Xh γ 1 P h A u A hp h u Yh. 6.8 Thus, using a projective method instead of the exact solution of 6.1 in functional space X, we can find the solution of the approximate equation 6.3 in finite dimensional space X h. The functional spaces {X, X h } and {Y, Y h } are related by means of the projection operators P h L X, X h and P h L Y, Y h, respectively. It is also important to construct inverse operator P 1 L X h h, X which maps the finite dimensional space X h into the initial functional space X. Such operator refers to as the operator of interpolation. Because of X h X, the interpolation operator is not unique; moreover, for any two functional spaces X h and X,it is possible to construct infinite set of interpolation operators. Let us apply the projection method to the BIE of elastostatics and construct corresponding finite dimensional BE equations. It is known 21, 23 that integral operators in 5.9 map between two functional spaces X and H 1/2 that are trace of displacements and traction on the boundary of the region in the following way: U ij p j W ij u j K ij p j F ij u j Uij x, y p j x ) ds : Y X, Wij x, y u j x ) ds : X Y, Kij x, y p j x ) ds : Y X, Fij x, y u j x ) ds : X Y. We have to construct finite dimensional functional spaces that correspond to X and Y and the corresponding projection operators. To construct finite dimensional functional spaces, we shall apply approximation by finite functions and splitting into finite elements 6.9 N n, n k, if n / k n 1

12 12 ISRN Applied Mathematics Because is the boundary of the region, these elements are called boundary elements. On each boundary element, we shall choose Q nodes of interpolation. Local projection operators act from functional X n and Y n to the finite directional ones X q n and Y q n P u q : X n X q n x n, P p q : Y n Y q n x n Global projection operators are defined as the sum of the local projection operators N P u nq P u q, n 1 N P p nq P p q. n They map X and Y to finite dimensional interpolations spaces ) N P u nq : X X q n n 1 ) N P p nq : Y Y q n n 1 x, x The local projection operators P u n and P p n establish correspondence between vectors of displacements and traction and their value on the nodes of interpolation of the boundary elements n P u q u i x { u n ) } i xq,q 1,...,Q x n, P p q p i x { p n ) } i xq,q 1,...,Q x n Similarly for the global operators, we have P u nq u i x { u n ) } i xq,q 1,...,Q; n 1,...,N x, P p nq p i x { p n ) } 6.15 i xq,q 1,...,Q; n 1,...,N x. Let us construct local interpolation operators P u q 1 and P p q 1. For this purpose, we will introduce systems of shape functions ϕ nq x and ψ nq x in the finite dimensional functional spaces X q n and Y q n. Then the vectors of displacements and traction on the boundary element n will be represented approximately in the form u i x p i x Q q 1 Q p n i q 1 u n ) i xq ϕnq x, x n, xq ) ϕnq x, x n, 6.16

13 ISRN Applied Mathematics 13 and on the whole crack surface in the form u i x p i x N Q n 1 q 1 N Q p n i n 1 q 1 u n ) i xq ϕnq x, x xq ) ϕnq x, x N n, n 1 N n. n Finite-dimensional analogies for the integral operators 6.9 are operators which map the finite-dimensional functional spaces X q N n 1 n and Y q N n 1 n, from one to another U nq ij P p nq U ij P u nq : Y q n X q n, W nq ij K nq ij P p nq W ij P u nq : X q n Y q n, P p nq K ij P u nq : Y q n X q n, 6.18 F nq ij P p nq F ij P u nq : X q n Y q n. Note that in contrast to differential operators, the integral operators are global, and they are defined in the entire space, that is, at every boundary element. Substitution of the expressions 6.17 in 2.8 gives us the finite-dimensional representations for the vectors of displacements and traction on the boundary in the form 1 2 um i y r 1 2 pm i y r N Q U n ji n 1 q 1 N Q K n ji n 1 q 1 yr, x q ) p n j yr, x q ) p n j xq ) W n ji ) ) xr, x q u n j xq Ui f, y, n, ) ) ) yq F n ji yr, x q u n j xq Ki f, y, n, 6.19 where U n ) ji yr, x q U ji y r, x ψ nq x ds, n K n ) ji yr, x q K ji y r, x ψ nq x ds, n W n ) ji yr, x q W ji y r, x ϕ nq x ds, n F n ) ji yr, x q F ji y r, x ϕ nq x ds. n 6.20 The volume potentials U i f, y, n and K i f, y, n depend on discretization of the domain. More detailed information on transition from the BIE to the BEM equations can be found in 1, 2, 4, 5, Boundary Elements and Approximation The BEM can be treated as the approximate method for the BIE solution, which includes approximation of the functions that belong to some functional space by discrete finite model.

14 14 ISRN Applied Mathematics This model comprises finite number of values of the considered functions which are used for approximation of these functions by the shape functions determined on small subdomains called boundary elements. In this senses the BEM is closely related to a finite element method where the functions also belong to corresponding functional spaces and are approximated by finite model. Below we shall speak about finite element approximations and finite elements FE, keeping in mind that boundary elements are their specific case. It is important to mention that the local approximation of the considered function on one FE can be done independently from other FEs. It means that it is possible to approximate function on an FE by means of its values on the nodes independently of the place occupied considered the FE in the FE model and how behave the function on other FEs. Hence, it is possible to create the catalogue of various FE or BE with arbitrary node values interpolation function. Then from this catalogue can be chosen FEs which are necessary for approximation of the function and domain of its definition. The same FE can be used for discrete models of various functions or physical fields by determination of the necessary position of nodes in the model and further definition of the node values of the function or physical field. Thus, finite models of an area and its boundary do not not depend on functions and physical fields for which this area can be a domain of definition. Let us consider how to construct an FE model of an area R n and a BE model of its boundary R n 1 n 2, 3. Wefixinthearea finite number of points x q g 1,...,Q ; these points refer to as global node points q {x q : g 1,...,Q}. We shall divide the area into finite number of subareas n n 1,...,N which are FEs. They have to satisfy the following conditions: n m, N m/ n, m, n 1, 2,...,N, n. n On each FE we introduce a local coordinate system ξ. The nodal points x q n in the local system of coordinates we designate by ξ q. They are coordinates of nodal points in the local coordinate system. Local and global coordinate are related in the following way: N x q Λ n ξn. q n Functions Λ n depend on position of the nodal points in the FE and BE. They join individual FE together in a FE model. Borders of the FEs and position of the nodal points should be such that, after joining together, separate elements form discrete model of the area. Having constructed FE model of the area, we shall consider approximation of the function f x that belong to some functional space. The FE model of the area is the domain of function which should be approximated. We denote function f x on the FE n by f n x. Then N f x f n x. n On each FE, the local functions f n x may be represented in the form Q f n x f n x q ϕ nq ξ, q 1 7.4

15 ISRN Applied Mathematics 15 where ϕ nq ξ are interpolation polynomials or nodal functions of the FE with number n. In the nodal point with coordinates x q, they are equal to 1, and in other nodal points are equal to zero. Taking into account 7.3 and 7.4, the global approximation of the function f x looks like N Q f x f n ) x q ϕnq ξ. n 1 q If the nodal point q belongs to several Fes, it is considered in these sums only once. The FE and BE elements can be of various form and sizes, their surfaces can be curvilinear. The curvilinear FEs are very important in BEM because the boundary surface is usually curvilinear. But it is more convenient to use standard FE, whose surfaces coincide with coordinate planes of the local coordinates system. Mathematically it means that it is necessary to establish relation between local coordinates ξ i, in which the element has a simple appearance, and global x i where the FE represents more complex figure. Local coordinates ξ i should be functions of global ξ i x 1,x 2,x 3 ones, and on the contrary, global coordinates should be functions of x i ξ 1,ξ 2,ξ 3 ones. In order to these maps be one-to-one, it is necessary and sufficient that the Jacobians of transformations be nonzero x i J det ξ j / 0, J 1 ξ i det x j / The differential elements along coordinate axes are related by ) xi dx i dξ j, ξ j ) ξi dξ i dx j, x j dx J ξ dξ, dξ J 1 x dx. 7.7 The volume element in the R 3 is transformed under the formula d dx 1 dx 2 dx 3 J ξ dξ 1 dξ 2 dξ 3, 7.8 and the area element in the R 2 is transformed under the formula x α da dx 1 dx 2 det ξ β dξ 1dξ 2, α,β 1, The differential of the surface located in the R 3 is defined by the expression ds n 2 1 n2 2 n2 3) 1/2dξ1 dξ 2, 7.10

16 16 ISRN Applied Mathematics where n 1 x 1 ξ 1 x 3 ξ 2 x 2 ξ 2 x 3 ξ 1, n 2 x 3 ξ 1 x 1 ξ 2 x 1 ξ 1 x 3 ξ 2, 7.11 n 3 x 1 ξ 1 x 2 ξ 2 x 2 ξ 1 x 1 ξ 2. The element of length of a contour in the R 2 is defined by the expression [ dx1 ) 2 ) 2 ] 1/2 dx2 dl dξ dξ 1 dξ 1 It is important to point attention that it is quite enough to consider standard FE which can be transformed to the necessary form by suitable transformation of coordinates. The FE approximation has to be linear independent and compact in the corresponding functional space. We consider here some examples of the FE approximation, which are frequently used in the BEM Piecewise Constant Approximation The piecewise constant approximation is the simplest one. Interpolation functions in this case do not depend on the FE form and the dimension of the domain. They have the form 1 x n, ϕ nq x x / n. This approximation is linear independent and compact in the functional spaces in which it is not necessary to approximate the derivative of the function Piecewise Linear Approximation Interpolation functions in this case change linearly inside the FE and depend on its form and dimension of the domain. Therefore, interpolation functions are called shape functions. In 1D case, the FEs are linear, and their shape functions are ϕ nq 1 2 ) 1 ξ q 1 ξ 1, ξ q i ± In 2D case for the rectangular FE, shape functions are ϕ nq 1 4 ) ) 1 ξ q 1 ξ 1 1 ξ q 2 ξ 2, ξ q i ±

17 ISRN Applied Mathematics 17 For the triangular FE, it is convenient to introduce system of coordinates which connected with the Cartesian by relation x i ξ q x q i, ξq 1, i 1, 2, q 1, 2, For the triangular FE, the interpolation functions are ϕ nq ξ q The local coordinates ξ q are also called area coordinates Piecewise Quadratic Approximation Interpolation functions in this case change quadratically inside the FE and depend on its form and dimension of the domain. In 1D case, the FEs are bilinear, and their shape functions are ϕ nq 1 2 ξq 1 ξ 1 ) 1 ξ q q 1 ξ ) 1 1, 2, at the end nodes and ϕ nq 1 ξ at the midpoint node. In 2D case for the rectangular FE, shape functions are ϕ nq 1 4 ) ) ) 1 ξ q 1 ξ 1 1 ξ q 2 ξ 2 ξ q 1 ξ 1 ξ q q 2 ξ ) 2 1 1,..., at the angular nodes and ϕ nq 1 2 ϕ nq 1 2 ) ) 1 ξ ξ q 2 ξ 2 ) ) 1 ξ ξ q 1 ξ 1 ξ q 1 0, ξ q at the side nodes. For the triangular FE, the interpolation functions are ϕ nq 2ξ q 1 ) ) ξ q q 1,..., at the angular nodes and ϕ nq 4ξ 1 ξ at the side nodes.

18 18 ISRN Applied Mathematics In order to construct system of algebraic equation for the BEM 6.18, we have to represent global coordinates as function of local ones. The most convenient way to do that is to use the interpolation function ϕ nq defined in In this case, the global coordinates have the form x i x q i ϕ nq ξ. Here x q i is the global coordinate of nodal points for the FE number n. 8. Regularization of the Divergent Integrals 7.24 As it was mentioned above in order to solve the BIE 5.9 numerically, we have to transform them to the finite dimensional BE equations In order to do that transform, we have to calculate integrals One of the main problems that occur in this situation is the presence of the divergent integrals. They can not be calculated in traditional way, for example, numerically using quadrature formulas. For example, the integrals with the kernels U ij x y are weakly singular WS. They have to be considered as improper integrals. The integrals with the kernels W ij x, y and K ij x, y are singular. They have to be considered in the sense of Cauchy as principal values Ps. The integrals with the kernels F ij x, y are hypersingular. They have to be considered in the sense of Hadamard as finite parts FPs. Traditional approach to the divergent integral calculation may be found in 1, 2, 6, 8. Following 11 16, we will develop here and apply the method of the divergent integral calculation based on the theory of distribution. This approach consists in application of the second Green theorem and transformation of divergent integrals into regular ones. In 12 we have developed formulas for regularization of the divergent integrals r m in the form a F.P. a F.P. ϕ x r m k 1 dx i 0 ϕ x k 1 d 1 i 1 rm i 0 1 i 1 d i P i d k 1 i ϕ x x a a dx i 1 k r m k dx k 1 i x a a 1 k P k d k ϕ x, r m k dx k [ Δ k i 1 P i ϕ x n r P ] i m 2i r nδ k i 1 ϕ x ds m 2i 1 r m 2k Δk 1 ϕ x d, for 1D and 2D cases, respectively. We will demonstrate here how this approach works in the problems of elastostatics D Divergent Integrals In 2D elastostatics after introducing local system of coordinates and simplification, all divergent integrals can be presented in the form 8.1 J 0 b a ϕ x ln 1 x dx, J k b a ϕ x xk dx, k 1, Here ϕ x is the smooth function that depend on the shape of the BE and the interpolation polynomials.

19 ISRN Applied Mathematics 19 We consider first the weakly singular integral J 0. Because of logarithmical singularity, we can not use formula 8.1. Therefore, we start from the formula for integration by parts in the form b a dg x b dx ϕ x dx ϕ x g x b a a dϕ x dx g x dx. 8.3 Let in this formula be g x x ln 1/x, dg x /dx ln 1/x, then we obtain b J 0 W.S. ϕ x ln 1 a x dx ϕ x x ln 1 x b a b dϕ x a dx x ln 1 dx. 8.4 x Obviously the integral on the left is divergent and on the right is regular one. For the linear BE and piecewise constant approximation ϕ x 1, and we get J 0 W.S. b a ln x y dx b y ) ln b y a y ) ln a y a<y<b ). 8.5 For the singular integral J 1 regularization, we will also use the formula for integration by parts 8.3. Let in this formula g x ln 1/x, dg x /dx 1/x, then we obtain b J 1 P.. a ϕ x dx x dϕ x dx x ln 1 x ϕ x ln 1 ) b x a b d 2 ϕ x x ln 1 dx. 8.6 a dx 2 x Here also the integral on the left is divergent and on the right is regular one. For the linear BE and piecewise constant approximation ϕ x 1, and we get b dx J 1 P.. a x y ln b y a y, ) a<y<b. 8.7 Finally for the hypersingular integral J 2 regularization, we will use formula 8.1 and the above-obtained result for singular J 1 regularization. Finally we get b J 2 F.P. a ϕ x d 2 ϕ x dx x ln 1 x 2 dx 2 x dϕ x dx ln 1 x ϕ x x ) b a b a d 3 ϕ x x ln 1 dx. 8.8 dx 3 x Here also the integral on the left is divergent and on the right is regular one. For the linear BE and piecewise constant approximation ϕ x 1, and we get J 2 F.P. b a dx 1 1 ) 2 ) ) x y b y a y a<y<b ). 8.9

20 20 ISRN Applied Mathematics From this equation can be found Hadamard s example of a function that is positive every were in the integration region, but its integral is a negative one F.P. a a dy y 2, a > a D Divergent Integrals In 3D elastostatics after introducing local system of coordinates and simplification, all divergent integrals can be presented in the form J l,m k Sn x l 1 xm 2 r k ϕ x ds, l, m 0, 1, 2,k 3, 4, Here ϕ x is the smooth function that depends on shape of the BE and interpolation polynomials Integrals with Kernels of the Type r k,k 1, 2, 3 From 8.1 for k 1, we get regularization for weakly singular integral J 0,0 1 W.S. [ ϕ x d ϕ x r ] n r 2r r nϕ x ds rδϕ x d Here r n x α y α n α, and the summation convention applies to repeated indices α 1, 2. The integral on the left is divergent and on the right are regular ones. For the linear BE and piecewise constant approximation ϕ x 1 and circular area, we can calculate this integral analytically. Introducing polar, coordinates we will get J 0,0 1 1 r 2π n r dl rdϕ πr. 2 S n r 0 r 8.13 In order to regularize singular integral, we will use, relation 1/r 2 1/2 Δ ln r 2. Then in thesamewayweget ϕ x r d 1 ϕ x 2r ) n ln r ln r r 2 n ϕ x ds 1 ln r 2 Δϕ x d The volume integral on the right is weakly singular. Taking into account the relation ln r 2 r 2 /6 Δ ln r 4, we obtain regularization for this weakly singular integral ln r 2 Δϕ x d 1 ) 2Δϕ x r n ln r 3 r 2 ln r 4 n Δϕ x ds r 2 ln r 4 Δ 2 ϕ x d. 6

21 ISRN Applied Mathematics 21 For the linear BE and piecewise constant approximation ϕ x 1 and circular area, we can calculate singular integral in 8.14 analytically. Introducing polar coordinates, we will get J 0,0 2 S n r n ln r 2π r ln r dl rdϕ 2π ln r r 2 0 r 2 Finally from 8.1 for k 3, we get regularization for hupersingular integral [ ϕ x d Δϕ x r n r 3 2r ϕ x r n r 1 ] 3 r nϕ x r n Δϕ x ds rδ 2 ϕ x d For the linear BE and piecewise constant approximation ϕ x 1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get J 0,0 3 S n r 2π n dl r3 0 r r 3 rdϕ 2π r Now using 8.1, any divergent integral with the kernels of the type 1/r k for any positive integer k can be calculated Integrals with Kernels of the Type x 2 α/r k, k 3, 4, 5 The weakly singular integral with kernel x 2 α/r 3 is calculated taking into account the equation x 2 α/r 3 1/3 2/r Δ x 2 α/r. Itiseasytoshowthat J 2,0 3 W.S. ϕ x x2 1 r d W.S. ϕ x d 1 r 3 W.S. ϕ x Δ x2 1 r d The first integral here is already calculated in The second one may be presented in the form [ ϕ x Δ x2 1 2n1 x 1 d ϕ x x ) ] 1 2 r n x2 1 r r r 3 r x 2 1 nϕ x ds r Δϕ x d Combining the last two equations, finally we will get J 2,0 3 1 [ 2n1 x 1 ϕ x x 1 2 r n r ) ) ] n x r r 3 2r r r n ϕ x ds 1 ) x r r Δϕ x d. 8.21

22 22 ISRN Applied Mathematics For the linear BE and piecewise constant approximation ϕ x 1 and circular area we can calculate this integral analytically. Introducing polar coordinates, we will get J 2,0 3 1 x 2 1 r n 2x 1n 1 2r ) n dl 3 S n r 3 r r 1 3 2π 0 r cos ϕ ) 2r r 3 2π rdϕ 0 2r cos ϕ ) 2 r 2π ) r rdϕ 2 0 r rdϕ πr The singular integral with kernel x 2 α/r 4 is calculated taking into account that Δx 2 α/r 4 1/4 2/r 2 Δx 2 α/r 2. In this case, J 2,0 4 P.. ϕ x x2 1 r d 1 ) 2 ϕ x 2 4 r 2 Δx2 1 d 1 ϕ x r 2 2 r d 1 ϕ x Δ x r d The first integral here is already calculated in The second one may be presented in the form ϕ x Δ x 1 2 d r2 [ 2n1 x 1 ϕ x 2x 1 r 2 r 4 2 r n ) x ] r x 1 nϕ x ds 2 r Δϕ x d Combining the last two equations, finally we will get J 2,0 4 1 [ 2 x1 r n ϕ x n 1x 1 r ) n ln r ln r 2 x ) ] r 4 r 2 r 2 2 2r 2 n ϕ x ds 1 ln r 2 x ) 1 2 Δϕ x d. 4 r For the linear BE and piecewise constant approximation ϕ x 1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get J 2,0 4 1 [ 2 x1 r n x 1n 1 r ] n ln r 2π ) 2r r cos ϕ 2π dl rdϕ 2 S n r 4 r 2 r 2 0 r 4 0 r cos ϕ ) 2 rdϕ r 2 2π r ln r 0 r 2 rdϕ π ln r.

23 ISRN Applied Mathematics 23 The hypersingular integral with kernel x 2 α/r 5 is calculated taking into account that x 2 α/r 5 1/3 2/r 3 Δ x 2 α/r 3. In this case, it is easy to show that J 2,0 5 F.P. ϕ x x2 1 r d 1 ) 2 ϕ x 5 3 r 3 Δx2 1 d 2 ϕ x r 3 3 r d 1 ϕ x Δ x r d The first integral here is already calculated in The second one may be presented in the form [ ϕ x Δ x2 1 2n1 x 1 d ϕ x 3x 1 r3 r 3 r 5 2 r n ) ] x2 1 r nϕ x ds 3 x 2 1 r 3 Δϕ x d Combining the last two equations, finally we will get J 2,0 5 2 n1 x 1 ϕ x r n 3 r 3 r 3x ) 1 2 r n 3 2r 5 2 ) 1 3 W.S. r x2 1 Δϕ x d. r 3 ) ) 1 r x2 1 r 3 n ϕ x ds 8.29 The volume integral here is weakly singular. Its regularization may be done using 8.12 and For the linear BE and piecewise constant approximation ϕ x 1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get J 2,0 2rn 5 S n 3r 2x 1 2 r n x ) 1n 1 dl 2 3 3r 5 r 3 3 2π 0 r r rdϕ 2 2π ) 2r r cos ϕ rdϕ r 5 2π 0 r cos ϕ ) 2 rdϕ r π r Integrals with Kernels of the Type x 1 x 2 /r k,k 3, 4, 5 The weakly singular integral with kernel x 1 x 2 /r 3 is calculated using 8.1. Taking into account that x 1 x 2 /r 3 1/3Δ x 1 x 2 /r, itiseasytoshowthat J 1,1 3 W.S. 1 3 Here r x 1 n 2 x 2 n 1. ϕ x x 1x 2 d 1 [ x1 x 2 r n ϕ x r 3 3 r 3 x 1 x 2 Δϕ x d. r r ) r x ] 1x 2 n ϕ x ds r 8.31

24 24 ISRN Applied Mathematics For the linear BE and piecewise constant approximation ϕ x 1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get J 1,1 3 1 [ x1 x 2 r n 3 S n r 3 r ] 2π dl r 0 r 3 cos ϕ sin ϕ 2π rdϕ r 3 0 2r cos ϕ sin ϕ rdϕ 0. r 8.32 The singular integral with kernel x 1 x 2 /r 4 is calculated using equation 8.1. Taking into account that x 1 x 2 /r 4 1/4 Δx 1 x 2 /r 2, it is easy to show that J 1,1 4 P ϕ x x 1x 2 d 1 [ 2x1 x 2 r n ϕ x r 4 4 r 4 x 1 x 2 Δϕ x d. r 2 r ) r 2 x ] 1x 2 r 2 n ϕ x ds 8.33 For the linear BE and piecewise constant approximation ϕ x 1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get J 1,1 4 1 [ 2x1 x 2 r n 4 S n r 4 r ] dl 1 2π 2r 3 cos ϕ sin ϕ rdϕ 1 2π 2r cos ϕ sin ϕ rdϕ 0. r r r The hypersingular integral with kernel x 1 x 2 /r 5 is calculated using 8.1. Taking into account that x 1 x 2 /r 5 1/3 Δx 1 x 2 /r 3,itiseasytoshowthat J 1,1 5 F.P. 1 3 ϕ x x [ 1x 2 x1 x 2 r n d ϕ x r ) r 5 r 5 3r 3 x 1 x 2 Δϕ x d. r 3 x ] 1x 2 r 3 n ϕ x ds 8.35 The volume integral here is weakly singular. Its regularization may be done using For the linear BE and piecewise constant approximation ϕ x 1 and circular area, we can calculate this integral analytically. Introducing polar coordinates, we will get [ J 1,1 x1 x 2 r n 5 r ] 2π dl S n r 5 3r 3 0 r 3 cos ϕ sin ϕ 2π 2r cos ϕ sin ϕ rdϕ rdϕ r 5 0 3r 3 9. Conclusions The method of regularization of the weakly singular, singular and hypersingular integrals, based on the second Green theorem, is developed here. These divergent integrals are presented when the boundary integral equation methods are used to solve problems in elastostatics. The equations that permit easy calculation of the weakly singular, singular, and hypersingular integrals for different shape functions are presented here. The divergent

25 ISRN Applied Mathematics 25 integrals over the circular domain have been calculated analytically. The main equations related to the boundary integral equation and boundary elements methods formulation in 2D and 3D elastostatics have been discussed in details. References 1 J. Balas, J. Sladek, and. Sladek, Stress Analysis by Boundary Element Methods, Elsevier, Amsterdam, The Netherlands, P. K. Banerjee, Boundary Element Method in Engineering Science, McGraw Hill, New York, NY, USA, S. W. Chyuan, Y. S. Liao, and J. T. Chen, An efficient method for solving electrostatic problems, Computing in Science & Engineering, vol. 5, no. 3, pp , A. N. Guz and.. Zozulya, Brittle Fracture of Constructive Materials under Dynamic Loading, Naukova Dumka, Kiev, Ukraine, S. Mukherjee and Y. X. Mukherjee, Boundary Methods: Elements, Contours, and Nodes, vol. 187 of Mechanical Engineering, CRC/Taylor & Francis, Boca Raton, Fla, USA, J. T. Chen and H. K. Hong, Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series, Applied Mechanics Reviews, vol. 52, no. 1, pp , H. K. Hong and J. T. Chen, Derivations of integral equations in elasticity, Journal of Engineering Mechanics Division, vol. 114, no. 6, pp , M. Tanaka,. Sladek, and J. Sladek, Regularization techniques applied to boundary element methods, Applied Mechanics Reviews, vol. 47, no. 10, pp , A. N. Guz and.. Zozulya, Fracture dynamics with allowance for crack edge contact interaction, International Journal of Nonlinear Sciences and Numerical Simulation, vol. 2, no. 3, pp , A. N. Guz and.. Zozulya, Elastodynamic unilateral contact problems with friction for bodies with cracks, International Applied Mechanics, vol. 38, no. 8, pp. 3 45, Zozulya, Integrals of Hadamard type in dynamic problems of crack theory, Doklady Akademii Nauk Ukrainskoj SSR Serija A, no. 2, pp , 1991 Russian Zozulya, Regularization of the divergent integrals. I. General consideration, Electronic Journal of Boundary Elements, vol. 4, no. 2, pp , Zozulya, Regularization of the divergent integrals. II. Application in fracture mechanics, Electronic Journal of Boundary Elements, vol. 4, no. 2, pp , Zozulya, The regularization of the divergent integrals in 2-D elastostatics, Electronic Journal of Boundary Elements, vol. 7, no. 2, pp , Zozulya, Regularization of hypersingular integrals in 3-D fracture mechanics: triangular BE, and piecewise-constant and piecewise-linear approximations, Engineering Analysis with Boundary Elements, vol. 34, no. 2, pp , Zozulya, Divergent integrals in elastostatics: regularization in 3-D case, Computer Modeling in Engineering & Sciences, vol. 70, no. 3, pp , Zozulya and P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics, Journal of the Chinese Institute of Engineers, vol. 22, no. 6, pp , Zozulya and A. N. Lukin, On the solution of three-dimensional problems of fracture mechanics by the method of boundary integral equations, International Applied Mechanics, vol. 34, no. 6, pp , Zozulya and. A. Men shikov, Solution of three-dimensional problems of the dynamic theory of elasticity for bodies with cracks using hypersingular integrals, International Applied Mechanics,vol. 36, no. 1, pp , L. Gaul, M. Kögl, and M. Wagner, Boundary Element Methods for Engineers and Scientists, Springer, Berlin, Germany, S. L. Sobolev, Applications of Functional Analysis in Mathematical Physics, vol. 7 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, A. Trenogin, Functional Analysis, Fizmatlit, Moscow, Russia, J. Nečas and I. Hlaváček, Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction, vol. 3 of Studies in Applied Mechanics, Elsevier Scientific, Amsterdam, The Netherlands, 1980.

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