Ministerio de Ciencia y Tecnología PB96-1380Ministerio de Ciencia y Tecnología  PB96-1322-C03-0

Ariza Moreno, María del Pilar

Domínguez Abascal, José

idUS. Depósito de Investigación Universidad de Sevilla

See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/2469462A Direct Approach for Hypersingular Integral Evaluation in 3-D B.E.M.Article · March 1999Source: CiteSeerCITATIONS0READS203 authors, including:Some of the authors of this publication are also working on these related projects:Numerical methods in poroelasticity View projectPilar ArizaUniversidad de Sevilla38 PUBLICATIONS   477 CITATIONS   SEE PROFILEJose DominguezUniversidad de Sevilla107 PUBLICATIONS   2,913 CITATIONS   SEE PROFILEAll content following this page was uploaded by Pilar Ariza on 22 March 2014.The user has requested enhancement of the downloaded file.A Direct Approach for Hypersingular IntegralEvaluation in 3-D B.E.M.M.P. Ariza and J. Dominguez, Member ASCEEscuela Superior de Ingenieros, Universidad de Sevilla,e-mail: mpariza@cica.es and pepon@cica.es1 IntroductionThere are two main diﬃculties behind the use of Hypersingular Boundary Elements(HBE): One is the continuity conditions that must be satisﬁed by the density func-tions; and the other, the integration of the strongly singular and hypersingular kernelsobtained by diﬀerentiation of the classical ones.To satisfy the requirement that surface derivatives of displacement should be con-tinuous at the boundary collocation point, several strategies have been proposed (seefor instance, Martha et al.,1992). In the current paper a simple and eﬃcient approachis presented. It is an extension to three dimensions of the idea introduced by Gallegoand Dominguez (1996). The elements are continuous with the classical representationof the variables. However, collocation is done at points inside the elements where theC1,α continuity requirement is satisﬁed. Using this strategy the diﬃculties producedby the continuity requirements at collocation points are avoided and the number ofnodal unknowns remains the same as in the classical formulation.The second important diﬃculty of the HBE formulation is the integration of hy-persingular and strongly singular kernels. This problems has been dealt with fromdiﬀerent points of view and several regularization techniques have been proposed (seethe review paper by Tanaka et al.,1994).In this paper, Laplace’s potential problem is treated ﬁrst. The classical inte-gral representation is diﬀerentiated with respect to a general direction and taken tothe boundary to obtain a hypersingular BIE in terms of potential, ﬂux and surfacederivatives at the boundary collocation point. The hypersingular and strongly singu-lar integrals are regularized by subtracting two terms of Taylor’s series expansion ofthe potential at collocation point. The remaining hypersingular and strongly singularintegrals are analytically transformed into regular or weakly singular integrals whichcan be numerically computed without special diﬃculties. In a similar way, the hy-persingular BIE is obtained for elasticity taking to the boundary the traction integralrepresentation. The regularization is done by using two terms of the Taylor’s seriesexpansion of the displacement and one term of the traction expansion, at collocationpoint. All the remaining integrals containing hypersingular and strongly singularkernels are transformed also in this case into regular or weakly singular integrals byanalytical procedures.The method presented is general for boundary surfaces and boundary elementsof any shape. The regularization is done prior to any boundary discretization. Allthe subsequent integrations are performed analytically in a direct way without anychange of coordinates.2 Flux BIE in potential problemsConsider the integral representation of the potential u of a Laplace’s problem at anypoint y in the interior of a 3-D domain Ω bounded by a surface Γ with unit outwardnormal n (x) at boundary point x.Taking derivatives at point y along a directiondeﬁned by the unit vector N(y) and calling q(y) = (∂u/∂yi)Ni(y) one obtains aﬂux integral representation which contains hypersingular and strongly singular inte-grals. This representation can be transformed into a new boundary integral equationwhere all the hypersingular and strongly singular integrals have been transformedinto regular and/or weakly singular integrals.12q(y) +∫Γ{ViNi [u (x)− u(y)−u,k(y)(xk − yk)]−WiNi q (x)} dΓ +u,k(y)∫Γ− 34πr3r,iNi∂r∂n(xk − yk)dΓ + u,k(y)∫ΓNinkr,i4πr2dΓ +u(y)∮∂Γr×N4πr3dl+u,k(y)∮∂Γ14πr(ek ×N) dl =0 (1)The two line integrals extend over the contour of the boundary Γ when this bound-ary is open and are zero when it is closed.Explicit expressions for Vi and Wi can befound in Dominguez et al. (1999). Integral equation (1) can be discretized to obtaina BE formulation whose integrals over the elements can be handled without specialnumerical diﬃculties.3 Traction BIE in elasticityA similar approach can be followed for the hypersingular traction boundary integralequation in elasticity to obtain a general traction BIE in terms only of regular andweakly singular integrals:12pl(y) +∫Γ{s∗lmkNm [uk (x)− uk(y)−uk,h(y)(xh − yh)] −d∗lmkNm [pk (x)− pk(y)]} dΓ + (2)μ4π (1− υ) [uk(y)Ilk + uk,h(y)Jlhk + pk(y)Klk] = 0where uk and pk stand for the k component of displacement and traction vectors,respectively, d∗lmk and s∗lmk are linear combinations of derivatives of the fundamentalsolution displacements and tractions, respectively. Explicit expressions for d∗lmk ands∗lmk, and for the integrals Ilk, Jlhk and Klk can be found in Dominguez et al.(1999).This BIE is valid for open or closed, plane or curved boundaries and can be discretizedin a straightforward manner using a boundary element formulation.4 Boundary element formulationThe two facts which make the discretization of the tractions BIE diﬀerent to thediscretization of the classical BIE are: (1), the boundary displacement uk must satisfythe Holder continuity condition uk ∈ C1,α, at y; and (2), derivatives, at y, of thedisplacement components exist in the boundary integral equation. As a consequenceof the ﬁrst of this two characteristics, one should not use standard linear or quadraticelements with collocation at nodal points located at the contour of the element. Dueto the second, one can not use constant elements.Following the idea of Gallego and Domi´nguez(1996) for two dimensions, the bound-ary integral equation is discretized into quadratic surface elements with six or ninenodes located at their usual position to represent the geometry and the boundaryvariables. However, the collocation is not done at the contour nodes; i.e. at ξ1,ξ2 = ±1, but at certain points close to the nodes, inside the element.It should be noticed that elements of this type are continuous Co since the bound-ary variables are written as usual in terms of their values at nodes located at thecontour of the element. Only the collocation points are shifted to the interior ofthe element. Tractions and displacements at each collocation point are expressed interms of nodal values and shape function values at collocation point. The displace-ment derivatives are expressed in terms of nodal displacements and shape functionderivatives at collocation point.As a consequence of the collocation strategy, one may have two or more equationsfor each nodal component, obtained by collocation at as many points as elementscontain the node. These equations are added up to yield only one per nodal com-ponent. This Multiple Collocation Approach (MCA) is only used for the nodes onone crack surface except for those inside an element and nodes at the crack frontwhere no collocation at all is required (Δuk = 0). The integration process can beimplemented in a way such that the integrals corresponding to multiple collocationpoints are computed simultaneously.It is also important to mention that since the regularization process leads toexpressions of the boundary integrals with more terms than the original hypersingularand strongly singular integrals, the regular expressions of the integral will only beused over a part of the surface Γ close to the collocation point whereas the originalexpressions of the integrals are used in the rest of the boundary where they are non-singular.5 Numerical exampleTo show the robustness and simplicity of the present approach, a 3-D fracture me-chanics problem is analyzed.. A penny shape crack subject to internal pressure andincluded in an inﬁnite elastic domain is studied (shear modulus, μ = 106Pa, PoissonRatio, υ = 0.25). Only one surface of the crack is discretized into quadratic nine nodequadrilateral elements (Figure 1). The elements at the crack front are quarter-point.Figure 1: Boundary Element discretization for penny shape crack.The computed Crack Opening Displacements (COD) at nodes along a radius areshown in Figure 2. The Stress Intensity Factor evaluated from the COD at the ﬁrstrow of nodes from the crack front (quarter point nodes) KI=1.989σ√aπis within a1% range from the known analytical solution for this parameter.               	      	Figure 2: Crack Opening Displacement along a radius line.6 ConclusionsA formulation which permits the numerical treatment of potential derivative andtraction BIE without special diﬃculties has been presented in this paper.One of themain issues is to contribute to transform hypersingular formulations of the BEM insomething which is as clear, general and easy to handle as the classical formulation.Ideas such as Hadamart Finite Part or Cauchy Principal Values are not needed. Allthe integrals can be numerically evaluated in an easy way. The boundary of theproblem can be curved or plane, closed or open.A simple BE discretization strategy is adopted to fulﬁl the regularity conditionwithout increasing the number of equations or introducing more complicated contin-uous elements. The boundary approximation is done in the usual way with nodeslocated in most cases on the contour of the elements. Thus, the displacement andtraction (potential and potential derivative) continuity over the boundary is retained,contrary to non-conforming elements. The collocation points are located inside theelement with multiple collocation for nodes between two or more elements. The gen-erality of the current integration procedure allows for the use of special crack elementssuch as quarter point.7 AcknowledgmentsThis work was supported by the Comisio´n Interministerial de Ciencia y Tecnolog´iaof Spain. (PB96-1380 and PB96-1322-C03-01). The ﬁnancial support is gratefullyacknowledged.8 References1. Dominguez,J., Ariza, M.P. and Gallego,R. (1999),“Flux and Traction Bound-ary Elements without Hypersingular or Strongly-singular Integrals”, Int. J. Numer.Meth. Eng., (in press).2. Gallego, R. and Domi´nguez, J. (1996) “Hypersingular BEM for TransientElastodynamics”, Int. J. Numer. Meth. Eng., 39, 1681-1705.3. Martha,L. F., Gray, L. J.and Ingraﬀea,A. R. (1992) “Three-Dimensional Frac-ture Simulation with a Single-Domain Direct Boundary Element Formulation”, Int.J. Numer. Meth. Eng., 35, 1907-1921.4. Tanaka,M., Sladek V. and Sladek, J. (1994) “Regularization Techniques Ap-plied to Boundary Element Methods”, Appl. Mech. Reviews, 47,457-499.View publication stats

A Direct Approach for Hypersingular Integral Evaluation in 3-D B.E.M.

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A Direct Approach for Hypersingular Integral Evaluation in 3-D B.E.M.

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