The Boundary Element Method

Abstract

There are several methods of deriving the BEM formulations: Cruse and Rizzo[l3] used Betti's reciprocal theorem; Brebbia and Dominguez[l8] introduced the weighted residual concept into the derivation, and Jeng and Wexler[l9] used a variational formulation similar to that used in the finite element method. The boundary element formulations may be divided into two different but closely related categories. The first and perhaps the most popular is the so-called "direct" formulation in which the unknown functions appearing in the formulation are the actual physical variables of the problem. For example in elasticity these unknown functions are displacement and traction fields. The other approach is called the "indirect" formulation in which the unknown functions are represented by fictitious source densities. Once these source densities are found the values of physical parameters can be obtained by simple integrations. In this book only the direct boundary element formulation will be described and readers interested in the indirect formulation should refer to other text books [20][21][22][23][24][25] on the subject. THE BOUNDARY ELEMENT FORMULATION IN ELASTICITYThe direct boundary element formulation for elastostatic problems can be derived from Betti's reciprocal work theorem for two self-equilibrated states (u,t,b) and (u*,t*,b*): u and u* are displacements; t and t* are tractions (i.e. t.=(J .. n., where n. are the components of the outward normal); and b and b*are body to~~J. Betti's r~ciprocal theorem (see Appendix A) can be expressed as J b;ujdO + J t;ujdr = J bju;dO + J tju;dr , o r o r (4.1) where repeated suffix summation is assumed, the domain 0 with boundary r and the domain 0* with boundary r* encompass the states (u,t,b) and (u*,t*,b*) respectively.As shown in figure 4.1, the problem under consideration in domain 0 is contained within a general region 0*, having the same elastic properties.The state of displacement u;, traction t~ and body force b~ is chosen to correspond to a known solution of the governing Navier's equation (2.7). The solution chosen is that due to a unit point force applied to the body, namely J.lu~ kk + ---l!...-11* ki + 8(X-X')e. = 0 , where X,X'eO*, 8(X-X') is the Dirac delta function and the unit vector component e j corresponds to a unit positive point force in the i direction applied at X'. In twodimensional problems e j is a force per unit thickness and in three-dimensional problems is a pure force. The displacement and traction field corresponding to the solution of (4.2) can be written as are the aisplacements and tractions in the j direction at a point X due fu a unit point force acting in the i direction at X'. The body force component b~ (force per unit volume) corresponds to a point force and is given by b~ = S(X-X')ej .The Dirac delta function S(X-X') has the property J g(X)S(X-X')dQ(X) = g(X') 0.Therefore the rust integral in (4.1) can be written as J b~ujdo. = uj(X')ej . ui(X') = J Uij(X',x)tj(x)dr(x) -J Tij(X',x)uj(x)dr(x) + J Uij(X',X)bj(X)dO(X) , r r 0 ~n wher...

Analysis of Crack Initiation and Propagation in a Piezoelectric Ceramic Using the Boundary Element Method

“…Thus, it is necessary for the macro scale to be simulated via a nonlinear boundary element formulation in order to exploit the local nonlinear behaviour of the material. The boundary element formulation is as follows [19]:…”

Laminate Element Method and Its Application to the Study of Guided Wave Resonance Phenomena in Layered Elastic Structures With Defects

“…In contrast to the classical fundamental-matrix solution g(x − ξ) for an infinite homogeneous elastic space [3], the matrix l(x, ξ) is derived for the pristine infinite layered structure as a whole. Their columns l j are displacement vectors u j (x, ξ) associated with the point sources δ(x − ξ)i j directed along the coordinate unit vectors i 1 and i 3 taken to be parallel to the axes x and z, respectively; ξ = (ξ 1 , ξ 3 ) is the point of LE source location.…”

“…General-purpose and specific finite-element computational codes have proven their efficiency for calculating wave propagation in elastic structures with complex geometry [2]. At the same time, with lengthy laminate waveguides, analytically based methods, such as boundary integral equation (BIE) technique and its derivations, e.g., boundary element method (BEM) or method of fundamental solutions (MFS), may serve as an efficient alternative [3]. They allow one to reduce the problem's dimension and obtain the results in a physically clear form of GW asymptotic expressions.…”

Recent developments in the evaluation of the 3D fundamental solution and its derivatives for transversely isotropic elastic materials

“…The above closed-form expressions of the fundamental solution and its derivatives can be directly used to deduce expressions of the boundary integral kernels T ik , D ijk and S ijk in the Somigliana displacement and stress identities [24,25,26], after application of the constitutive law, as shown by Távara et. al.…”