Conventional, Hybrid And SimplifiedBoundary Element Methods

Conventional, hybrid and simplified boundary element methods N. A. Dumont, M. F. F. Oliveira & R. A. P. Chaves Civil Engineering Department, Pontifícia Universidade do Rio de Janeiro – PUC-Rio, 22453-900 Rio de Janeiro, Brazil Abstract This paper introduces a brief, although insightful, comparative analysis of the conventional, collocation boundary element method (CBEM), the hybrid stress boundary element method (HSBEM) and the hybrid displacement boundary element method (HDBEM), not only investigating the mechanical properties of the resulting matrix equations, but mainly redefining a series of concepts in both HDBEM and CBEM that hadn’t been properly considered by previous authors. This is not a review paper, but rather a theoretical investigation of three methods, with many physical considerations and some innovations that point toward mesh-reduced formulations adequate for the numerical modelling of complex problems, such as in the case of material non-homogeneity and timedependency. 1 Introduction The hybrid stress boundary element method (HSBEM) was introduced in 1987 on the basis of the Hellinger-Reissner potential, as a generalization of Pian’s hybrid finite element method [1]. This two-field formulation makes use of fundamental solutions to interpolate the stress field in the domain of an elastic body, which ends up discretized as a superelement with arbitrary shape and arbitrary number of degrees of freedom located along the boundary. A few years after the introduction of the HSBEM, De Figueiredo and Brebbia proposed a variational counterpart, properly called the hybrid displacement boundary element method – HDBEM [2]. The HDBEM is equally consistent and presents the same computational characteristics of the HSBEM, although based on a different (three-field) variational principle. The present paper discusses these Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 24 Boundary Elements XXVI methods briefly as well as the conventional, collocation boundary element method (CBEM), not only investigating the spectral properties of the resulting matrix equations, but mainly redefining a series of concepts in both HDBEM and CBEM that hadn’t been properly considered by previous authors [3, 4]. A theoretical consequence of these developments is also presented – the simplified hybrid stress boundary element method (HSSBEM) – which has been successfully applied to a wide class of problems in the last years [5, 6]. Moreover, another simplifying formulation derived from the HDBEM is presented for the first time: the simplified hybrid displacement boundary element method (HSDBEM – a counterpart of the HSSBEM). A conceptually intriguing, further simplified version (HMRBEM) that combines HSSBEM and HSDBEM and closes a circle of theoretical achievements – pointing toward a mesh-reduced formulation – is also conceived [7, 8]. Although the simplified methods outlined are particularly advantageous in case of time-dependent problems and nonhomogeneous materials, as well as for the post-processing of results at internal points, the concepts outlined have to be restricted to the basic equations of elastostatics, owing to space limitations. Moreover, the developments are restricted to simply connected, finite domains, although the extension to general problems is straightforward. Also, an adequate literature review has to be left to a future publication. 2 Fundamental solutions as domain interpolation functions Consider an elastic body submitted to body forces bi in the domain Ω and traction forces t i on part Γσ of the boundary. Moreover, the displacements ui are known on the complementary part Γu of Γ . One is looking for an adequate approximation of the stress field that satisfies equilibrium in the domain, σ ji , j +bi = 0 in Ω (1) also satisfying following boundary equilibrium and compatibility equations: σ jiη j = ti along Γσ , ui = ui on Γu (2) A convenient approximate field solution σ ijf of the partial differential eqn (1) may be formulated in terms of a superposition of two types of fields, ∗ σ ijf = σ ij∗ + σ ijb = σ ijm pm∗ + σ ijb (3) in which σ is an arbitrary particular solution of eqn (1), b ij σ bji , j +bi = 0 (4) and σ is expressed as a sum of fundamental solutions, ∗ ij ∗ ∗ σ ij∗ = σ ijm pm∗ such that σ ∗ji , j = σ ijm , j pm∗ = 0 in Ω Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 (5) Boundary Elements XXVI 25 as a property of a fundamental solution, for a domain Ω that does not include the points of application of pm∗ . If some domain Ω 0 should comprise the point of application of a concentrated force pm∗ , then ∫ Ω0 - 1 if i and m refer to the same degree of freedom ∗ , j dΩ = −δ im =  σ ijm (6) 0 otherwise From the stresses in eqn (5) one derives the boundary traction forces ∗ σ ijm η j pm∗ ≡ pim∗ pm∗ (7) in which η j are the director cosines of the outward normal to the boundary. The displacements corresponding to the field solution σ ijf of eqn (3) are uif = ui∗ + uib = (uim∗ + uisr C sm ) pm∗ + uib (8) * The fundamental solution, as characterized by the superscript “ ”, is usually given in the literature by the function uim∗ alone, implicitly related to unitary forces pm∗ . The complete representation of eqn (8) is both mathematically and physically more adequate, since it is stated for an arbitrary (not unitary) concentrated force pm∗ and a term is added to take into account the arbitrary rigid body displacements, as denoted by the superscript “r”. In the rigid body displacement functions uisr , “s” refers to the rigid body displacement being interpolated, correlated to the concentrated forces pm∗ through some arbitrary matrix Csm of constants. The aim of this short outline was to introduce the terminology needed in the paper. As presented above, one is dealing with Green’s functions, the singularity of which is required to formulate an integral statement, as the Somigliana’s identity, basis of the CBEM. For the development of variational methods, on the other hand, a fundamental solution may be based on nonsingular (polynomial) functions, as in Pian’s hybrid finite element method or in the Trefftz methods, in general. The use of singular functions simplifies the whole formulation and ensures that the resultant matrix equations are well conditioned. In the following sections, one refers to four main groups of equations, according to the columns of Table 1, given for compactness and a better understanding of the way the various methods outlined in this paper hang together, as they are referred to in the rows. 3 Conventional BEM (CBEM) To derive the traditional boundary element method, one assumes that both displacements ui and traction forces ti are approximated along the boundary in Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 26 Boundary Elements XXVI terms of interpolation functions multiplying some nodal parameters d n and traction force intensities tA , respectively: ui = uin d n and ti = tiA t A along Γ (9) Using Somigliana’s identity properly written for the fundamental solution – characterized by “*” in eqn (8) and regardless any amount of rigid body displacements uisr Csm – taken as weighting functions, one expresses the matrix equation for the analysis of a general elastostatics problem as eqn C(1a), in which H ≡ H mn and G ≡ GmA , as defined in eqns C(2a-b), are traditional double layer potential and single layer potential matrices. d ≡ d n and t ≡ t A are the nodal displacement and traction force parameters introduced in eqn (9). Finally, d b ≡ d nb and t b ≡ t Ab = σ bjiη j are boundary nodal displacement and traction force parameters obtained as an arbitrary (as simple and convenient as possible) particular solution of the equilibrium differential eqn (4), for applied body forces bi , provided that this solution can be obtained [4]. Let the columns of a rectangular matrix W ≡ Wns be an orthogonal basis of the nodal displacements d related to rigid body displacements. For a finite domain, it follows necessarily from the definition of H in eqn C(2a) the orthogonality of eqn C(3a), which is a feature related to the physical nature of the fundamental solution. As a consequence, one can also define a rectangular matrix V according to eqn C(3b). Equation C(1a) is not a consistent set of equations, owing to the weighted-residual character of the CBEM, as G T V ≠ 0 . In despite of that, adequate consideration of boundary conditions expressed in terms of known values of the elements of d and t in eqn C(1a) enables the complete solution of a given problem. For the sake of expressing a stiffness-type relation starting from eqn C(1a), define a vector p − p b of nodal forces that are equivalent in terms of virtual work to the traction force parameters t − t b on the boundary, eqn C(1b), in which LT ≡ LmA is given in eqn C(2c). Then, eqn C(4) follows from eqns C(1a-b), with the definition of the stiffness matrix K C . In this equation – and in subsequent equations in the text – subscripts s and d denote that in the evaluation of matrices L and G one is considering either single or double nodes, in which case these matrices are rectangular. As a consequence, the brackets in G (d−1) characterize a generalized inverse, the expression of which is not important in the present outline [3, 4]. This inverse is always present as part of the product LTd G (d−1) , which is a square matrix. In fact, it is worth recalling the meaning of the matrices involved, particularly in which respects the nodal points at which the physical parameters pm∗ ≡ p ∗ (for the fundamental solution, whether a weighting function, as in this topic, or an interpolating field, as in the variational formulations of the following sections), d, p − p b and t − t b are supposed to act. In fact, while p ∗ , d Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 Boundary Elements XXVI 27 and p − p b are understood as nodal attributes, the traction force intensity G parameters t − t b are surface segment attributes, for which a normal direction η should be unequivocally attached, whether or not corner nodes are considered. Then, matrices L and G are rectangular, in general, as dimension t − t b ≥ dimension p − p b = dimension d = dimension p ∗ , although always rank L = rank G . When not explicitly indicated, either single or double node considerations apply in the evaluation of L and G. 4 Hybrid stress BEM (HSBEM) The governing matrix equations of this two-field formulation, as based on the Hellinger-Reissner potential [1], are given in eqns HS(1a-b). In these equations, the concentrated forces p ∗ ≡ pm∗ , physically attached to the same degrees of freedom as d − d b and p − p b , are primary unknowns of the problem, by means of which the state of displacements and stresses are described in the domain, according to eqns (8) and (3), considering the particular solution of eqn (4). Also, matrix H, as already defined in eqn C(2a), carries out a kinematic transformation in eqn HS(1a) and, as a transpose matrix, a contragradient equilibrium transformation in eqn HS(1b). Moreover, F ≡ Fmn of eqn HS(2a) is a flexibility matrix that is symmetric by construction and may be evaluated directly for all coefficients, except when m and n refer to the same nodal point. ∗ ∗ In eqn HS(2a), U mn ≡ U nm ≡ U ∗ are the values of the displacements ui∗ obtained in the direction of a given degree of freedom n for a unity concentrated force pm∗ applied at the degree of freedom m, according to eqn (8). Observe that eqn HS(1b) is consistent, according to eqn C(3a) and the orthogonality property W T (p − p b ) = 0 , provided that also eqn HS(3b) is satisfied. Then it follows, considering eqn C(3b) and requiring complete consistency of eqn HS(1a), that F has to be singular, eqn HS(3a). This equation is a means for the evaluation of the coefficients about the main diagonal of F, when the degrees of freedom m and n refer to the same nodal point, as they cannot be given by integration as in eqn HS(2a). Eliminating p∗ in eqns HS(1ab), for which a generalized inversion of the singular flexibility matrix has to be carried out [1, 4], one arrives at the nodal equilibrium equation of eqn HS(4), where K S is a stiffness matrix that is by construction symmetric, positive semidefinite, as K S W = 0 . 5 Simplified hybrid stress BEM (HSSBEM) Once eqn HS(4) is solved for the unknown subsets of displacements and forces d and p , one evaluates p∗ from eqn HS(1a), thus being able to express stresses and displacements at an interior point by means of eqns (3) and (8), provided that Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 28 Boundary Elements XXVI the matrix of constants Csm is known. For that sake, one may reason that, if eqn (8) is valid for interior points, it should also be valid for the nodal points themselves, that is, in both matrix and indicial notation, ∗ d = (U ∗ + WC)p∗ + d b or d n = (U nm + Wns Csm ) pm∗ + d nb (10) as uisr in eqn (8) is defined such that, at the nodal points, it corresponds to W ≡ Wns , which is in turn an orthogonal matrix: W T W = I . In this equation also ∗ appears the symmetric square matrix U nm ≡ U ∗ , already introduced in eqn HS(2a). Supposing that the coefficients about the main diagonal of this matrix have been evaluated in some way, one may pre-multiply eqn (10) by W T and solve for the product C sm pm∗ ≡ Cp ∗ , a vector of rigid body displacements required in both eqns (8) and (10): C sm pm∗ ≡ Cp ∗ = W T (d − d b ) − W T U ∗p ∗ (11) This expression, substituted back into eqn (10), yields the useful eqn HSS(1a), which may replace eqn HS(1a) with advantage and become, together with eqn HS(1b), the core of the so-called simplified hybrid stress boundary element method (HSSBEM) [5, 6], as the computationally intensive flexibility matrix F gives place to the displacement matrix U ∗ , the coefficients of which require no integration to be evaluated. As demonstrated elsewhere [5] and not repeated herein, for the sake of brevity, the coefficients about the main diagonal of matrix U ∗ may be obtained by resorting to the spectral property of eqn HSS(3), with the matrix of constants C obtained directly using the integral statement that the term in brackets in eqn (8) is orthogonal to rigid body displacements [4, 5]. Compare eqn HSS(3) with eqn HS(3a). It is conceptually enriching to pre-multiply eqn HSS(1a) by H, arriving at eqn HSS(1b), and compare the result with eqn HS(1a). Equation HS(2b) becomes evident, with increasing resemblance as one improves the mesh discretization of the numerical model. As a matter of fact, the main diagonal coefficients of U ∗ may be obtained by imposing the equality in eqn HS(2b) for the off-diagonal coefficients of F , in terms of least squares, with subsequent evaluation of the main diagonal coefficients of F [8]. This procedure is alternative to the one represented by eqn HSS(3). Although eqns HS(2b) and HSS(3) lead to different values of the main diagonal coefficients of U ∗ , this affects the final results obtained from K SS only within the expected numerical discretization errors. Formally, one writes from eqns HSS(1a-b), HSS(3) and HS(1b) the stiffness relation of eqn HSS(4), where K SS is the stiffness matrix related to this simplified formulation. Compare with K S in eqn HS(4). Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 Boundary Elements XXVI 6 29 Hybrid displacement BEM (HDBEM) The governing matrix equations of this three-field variational formulation are given in eqns HD(1a-c) [4], in which all matrices have been already defined in Sections 3 and 4, whether considering single or double nodes for L and G. Solving for p∗ and t − t b in equations above yields eqn HD(4), where K D is a stiffness matrix that is by construction symmetric, positive semidefinite, as K D W = 0 . In this formulation, the coefficients about the main diagonal of F are evaluated by requiring that K D is orthogonal to rigid body displacements. For this task, define the rectangular matrix Y of eqn HD(3b). Then, eqn HD(3a) is the criterion needed for the determination of the elements about the main diagonal of matrix F. For the inverse of G in these equations, see the remarks in the end of Section 3. 7 Simplified hybrid displacement BEM (HSDBEM) Starting with similar considerations as the ones in the beginning of Section 5, one may reason that, if eqn (3) is valid for interior points, it should also be valid on the boundary, in terms of traction forces σ jiη j measured at the nodal points of element segments, given in matrix form as eqn HSD(1), in which T∗ is a rectangular matrix directly obtainable, except for the nodes coinciding with the point of application of the concentrated forces p∗ . This equation may replace eqn HD(1a) in the triple set of equations of the hybrid displacement formulation of Section 6, thus circumventing the computationally intensive evaluation of matrix F. Solving for p∗ and t − t b in eqns HD(1b-c) and HSD(1) yields eqn HSD(4), where K SD is the stiffness matrix related to this simplified formulation. Observe that in the product LTd T∗ it is compulsory to consider double nodes. Comparing K SD with K D in eqn HD(4), one obtains that, according to eqns HD(3a-b), a criterion for the evaluation of the coefficients of T∗ for nodes coinciding with the point of application of the concentrated forces p∗ is given in eqn HSD(3). Pre-multiplying eqn HSD(1) by G d and comparing with eqn HD(1a), eqn HD(2) becomes evident, with increasing resemblance as one improves the mesh discretization of the numerical model. As a matter of fact, the unevaluated coefficients of T∗ in eqn HSD(1) may be obtained by imposing the equality in eqn HD(2) for the off-diagonal coefficients of F , in terms of least squares, with subsequent evaluation of the main diagonal coefficients of F [8]. This procedure is alternative to the one represented by eqn HSD(3) and, although it leads to different values of T∗ , results in terms of K SD are affected only within the expected numerical discretization errors. Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 30 Boundary Elements XXVI 8 Mesh-reduced hybrid BEM (HMRBEM) The simplified stress and the simplified displacement formulations of Sections 5 and 7 may be combined in order to arrive at a formulation that seems to be extremely convenient for complex applications, as it will be briefly outlined. First of all, observe that one may select eqns HD(1c) and HS(1b) as the governing equations of a given problem, solve for p∗ and arrive at a stiffness relation with a matrix H T (G ( −1) ) T L that is the transpose of K C given in eqn C(4). The same procedure will be tried without explicit reference to G and H. In a first step, one pre-multiplies eqn HSS(1a) by L: L(I − WW T )U ∗p ∗ = L(I − WW T )(d − d b ) (12) Comparing this expression with eqn HD(1c), eqn HMR(2a) is inferred. Actually, this relation is written as L a U ∗ ≈ G Ta , in terms of admissible matrices L a and G a that adequately take into account only forces that are in balance and are, consequently, orthogonal to the rigid body displacements imbedded in eqn (12), whether considering single or double nodes [8]. This conceptually more consistent approach is too extensive to outline in this paper [3, 4]. Enforcing the equality in eqn HMR(2a) is an adequate means of evaluating the coefficients about the main diagonal of U ∗ , for which the evaluation of a few elements of G s is required. In a second step, pre-multiplying eqn HSD(1) by LTd yields eqn HMR(1b). Comparing eqn HMR(1b) with eqn HS(1b), one infers eqn HMR(2b), which is in principle an alternative means of evaluating, in terms of leastsquares, the coefficients of T∗ that cannot be given directly. Now, solving for p∗ in eqns HMR(1a-b) results in eqn HMR(4), where K MR is a stiffness matrix expressed in the frame of a mesh-reduced formulation, so understood that only L and the coefficients about the main diagonal of H and G require some integration to be obtained, as the off-diagonal coefficients of the matrices U ∗ and T∗ are evaluated directly at the boundary nodal points. (U ∗ ) ( −1) in eqn HMR(4) is a generalized inverse of U ∗ in eqn HMR(1a) [5, 8]. For an elastostatics problem with homogeneous materials, there is no advantage in using this technique, as compared with the previous procedures, particularly considering the CBEM and the HSSBEM. However, the effectiveness of the formulation represented by eqn HMR(4) becomes evident if one considers the application to a general transient problem with nonhomogeneous material, to be solved in a frequency-dependent approach, for instance [9]. In this case, matrices U ∗ and T∗ may be split into U ∗ = U ∗0 + U ∗n , T∗ = T0∗ + Tn∗ Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 (13) Table 1: Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 BEMs C Main matrix governing equations and transformations interrelating the various boundary element formulations outlined in the paper. 1 – Matrix governing equations a) H(d − d b ) = G(t − t b ) b) p − p b = LT (t − t b ) 2 – Matrix definitions / relations a) H mn ≡ ∫ p im∗ u in dΓ + δ mn Γ b) GmA ≡ ∫ uim∗ tiA dΓ Γ c) LmA ≡ ∫ uimtiA dΓ Γ HS a) Fp ∗ = H(d − d b ) b) H T p ∗ = p − p b a) PW⊥ U ∗p ∗ = PW⊥ (d − d b ) HSS b) HU ∗p ∗ = H(d − d b ) ∗ a) Fmn ≡ ∫ pim∗ uin∗ dΓ + U mn b) HU ∗ ≈ F HSD HMR T p = t −t ∗ ∗ a) HW = 0 b) H T V = 0 c) PW = WW T K C (d − d b ) = p − p b G d T∗ ≈ F d) PW⊥ = I − PW a) FV = 0 b) V Tp ∗ = 0 K S (d − d b ) = p − p b K S ≡ H T (F + VV T ) −1 H K SS (d − d b ) = p − p b (U ∗ + WC)V = 0 K SS ≡ H T (HU∗ + VV T )−1 H a) FY = 0 b) Y = (G ( −1) ) T LW K D (d − d b ) = p − p b T Y=0 ∗ b K C ≡ LTd G (d−1) H ≈ LTs G −s 1H K D ≡ LTG ( −1) F(G ( −1) ) T L K SD (d − d b ) = p − p b K SD ≡ LTd T∗ (G ( −1) ) T L a) PW⊥ U ∗p ∗ = PW⊥ (d − d b ) a) L a U ∗ ≈ G Ta K MR (d − d b ) = p − p b b) LTd T∗p ∗ = p − p b b) LTd T∗ ≈ H T K MR ≡ LTd T∗ (U ∗ ) ( −1) Boundary Elements XXVI b) p − p b = LT (t − t b ) c) G Tp ∗ = L(d − d b ) 4 – Stiffness matrices Γ a) Fp ∗ = G(t − t b ) HD 3 – Orthogonal properties 31 32 Boundary Elements XXVI in which U ∗0 and T0∗ refer to the homogeneous, static part of the fundamental solution that is singular by construction, as already outlined, whereas U ∗n and Tn∗ refer to the remaining part of the solution, which, although apparently complicated, involve no singularities and can always be evaluated directly, thus saving computational time. 9 Conclusions For the sake of brevity, comparative numerical results could not be considered in this article. All three basic formulations as well as the derived simplified ones perform equivalently, in terms of accuracy, as numerically assessed by Oliveira [8]. Moreover, the matrices involved in the formulations present comparable spectral properties (such as orthogonality to rigid body displacements), provided that one introduces some more concepts that mainly affect the consistency of matrices G and L [3, 4, 8]. Use of the inconsistent matrix G, as done traditionally in the literature and in this paper, for simplicity, may lead to unreliable results, as a consequence of eventual ill conditioning related to uncontrolled amounts of rigid body displacements. All considerations were made for a finite, simply connected domain. Infinite, as well as multiply connected domains can be dealt with in a straightforward way [4, 5]. The main contribution of the present paper is Table 1, which is a summary of the basic equations of three conceptually different methods, showing that they are in fact intimately related, with the consequence that adequate combinations of their equations and matrices can be carried out, thus leading to more powerful, simplified formulations. According to this theoretical outline, one suggests that an adequate numerical tool for dealing with transient, non-homogeneous problems can be easily developed starting from any existing code implemented in the frame of the conventional, collocation boundary element method for the modelling of static problems with homogeneous materials. Acknowledgment This project was supported by the Brazilian agencies CNPq and FAPERJ. References [1] N.A. Dumont. The Hybrid Boundary Element Method: an Alliance Between Mechanical Consistency and Simplicity. Applied Mechanics Reviews, 42: no. 11, Part 2, S54-S63, 1989. [2] T.G.B. De Figueiredo. A New Boundary Element Formulation in Engineering. In: C.A. Brebbia, S. A. Orszag, eds., Lecture Notes in Engineering, Springer-Verlag, 1991. [3] N.A. Dumont. An Assessment of the Spectral Properties of the Matrix G Used in the Boundary Element Methods. Computational Mechanics, 22: Nr. 1, 32-41, 1998. Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6 Boundary Elements XXVI 33 [4] N.A. Dumont. Variationally-Based, Hybrid Boundary Element Methods, Computer Assisted Mechanics and Engineering Sciences, Vol 10 pp 407430, 2003. [5] R.A.P. Chaves. The Simplified Hybrid Boundary Element Method Applied to Time-Dependent Problems. Ph.D. Thesis (in Portuguese), PUC-Rio, Brazil, 2003. [6] N.A. Dumont, R.A.P. Chaves. Simplified Hybrid Boundary Element Method Applied to General Time-Dependent Problems. Computational Mechanics – New Frontiers for the New Millenium, Eds. S. Valliappan e N. Khalili, Elsevier Science Ltd, pp 1009-1018, 2001. [7] N.A. Dumont, M.F.F. Oliveira, R.A.P. Chaves. Mesh-Reduced Boundary Element Formulation for the General Analysis of Transient Problems and Non-Homogeneous Materials. Accepted IABEM 2004 – International Association of Boundary Element Methods Conference 2004, Minnesota, USA, 2004. [8] M.F.F. Oliveira. Conventional, Hybrid and Simplified Boundary Element Methods. M.Sc. Dissertation (in Portuguese), PUC-Rio, Brazil, 2004. [9] N.A. Dumont, R.A.P. Chaves. Transient Heat Conduction in Orthotropic Functionally Graded Materials by the Hybrid Boundary Element Methods. Accepted IABEM 2004 – International Association of Boundary Element Methods Conference 2004, Minnesota, USA, 2004. Boundary Elements XXVI, C. A. Brebbia (Editor) © 2004 WIT Press, www.witpress.com, ISBN 1-85312-708-6