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Application of Multi-Region Trefftz Method to Elasticity

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Application of Multi-Region Trefftz Method to Elasticity Application of Multi-Region Trefftz Method to Elasticity J. Sladek1,V.Sladek1,V.Kompis2, R. Van Keer3 Abstract: This paper presents an application of a direct where unknowns are fictitious parameters. That is the main Trefftz method with domain decomposition to the two- reason, why in this paper a direct formulation is used inspite dimensional elasticity problem. Trefftz functions are substi- of the more frequently occurred indirect formulation in the lit- tuted into Betti’s reciprocity theorem to derive the boundary erature [Leitao (1998); Zielinski and Herrera (1987); Herrera integral equations for each subdomain. The values of displace- (1995)]. If the number of Trefftz functions used as test func- ments and tractions on subdomain interfaces are tailored by tions is high the final system of algebraic equations can be ill- continuity and equilibrium conditions, respectively. Since Tr- posed. To overcome this difficulty the whole domain is subdi- efftz functions are regular, much less requirements are put on vided into smaller subregions. For the same purpose the spe- numerical integration than in the traditional boundary integral cial Trefftz finite element formulations are frequently used by method. Then, the method can be utilized to analyse also very many researchers [Jirousek and Leon (1977); Jirousek (1978); narrow domains. Linear elements are used for modelling of Jirousek (1987); Freitas and Ji (1996); Freitas (1998); Jirousek the boundary geometry and approximation of boundary quan- and Zielinski (1993); Kompis and Bury (1999)]. On interfaces tities. Numerical results for a rectangular plate with varying of subregions displacement continuity and equilibrium of trac- aspect ratio and cantilever beam are presented. tions have to be satisfied. Adding such restriction conditions at nodal points to the discretized integral equations, we create keyword: plynomial Trefftz functions, direct formulation, a complete set of algebraic equations for the unknown values linear approximation, boundary integral equation at nodal points. Boundary integral formulations combined with Trefftz func- 1 Introduction tions as test functions can be used for the computation of boundary unknowns. An integral representation of displace- The effort to predict more effectively the response of the com- ments at internal points is not available in such a formula- plex problems of continuum mechanics motivates the authors tion. If such quantities are required, the integral representation to seek for formulations in which the governing equations in- with the fundamental (singular) solution has to be used [Balas, side the approximated domain are identically satisfied. Such Sladek and Sladek (1989)]. analytically derived functions are called Trefftz functions [Tr- Numerical tests were carried out on a rectangular plate with efftz (1926)] which can be found in the form of polynomials, varying side aspect ratio and a cantilever beam. Legendre, harmonic, Bessel, Hankel, singular Kupradze func- tions, etc. [Zielinski (1995)]. The main characteristic of the Trefftz method is the use of trial 2 Boundary integral equations functions that satisfy the governing differential equations in a For linear isotropic problems of elasticity, the governing equa- domain. The method can be classified into the indirect and tions are force equilibrium equations expressed in displace- the direct formulations [Jin, Cheung and Zienkiewicz (1990); ments, the Lame-Navier equations in domain Kita, Kamiya and Iao (1999)]. In the indirect formulation the trial function is given as a superposition of Trefftz func- λ · · = ; ; µ u ; µu X (1) tions. Then, the unknown parameters in the trial function are j ji i jj i determined so that boundary conditions have to be satisfied. In the direct formulation the Trefftz functions are considered where Xi is a body force vector. Recall that in two-dimensional as a weighting (test) function in Betti’s reciprocity theorem to problems under plane stress conditions special care should be derive boundary integral equations. Boundary values are ap- paid to the association of the coefficients with the material pa- proximated by standard polynomials within elements on dis- rameters. In general, cretized boundary. In this approach, unknown values have ´ ν ν a physical interpretation in contrast to the indirect approach, 2µ ; plane stress ; ν = ; λ = ν ν 1 · 1 2 ν ; otherwise 1 Institute of Construction and Architecture, Slovak Academy of Sciences, 84220 Bratislava, Slovakia. 2 Faculty of Mech. Engn., University of Zilina, 01026 Zilina, Slovakia. where ν is Poisson’s ratio and µ is the shear modulus associ- 3 ´ · µ Dept. Mathematical Analysis, University of Gent, B-9000 Gent, Belgium. ated with the Young modulus as E = 2µ 1 µ . 2 Copyright ­c 2000 Tech Science Press CMES, vol.1, no.4, pp.1-8, 2000 An equivalent to the Lame-Navier equation is the force equi- It makes the numerical integrations much easier in the Tre- librium in terms of stresses fftz boundary integral equation formulation than in the conven- tional formulation based on singular boundary integral equa- σ = : ij; j Xi 0 (2) tions. On the other hand, the numerical stability is more ques- tionable in the former case because of a higher condition num- Γ = Γ [ Γ µ Γ Γ Γ ´ Let u and t denote parts of the boundary u t ber of the discretized BIE. with prescribed displacements and tractions, respectively. The boundary integral equation (8) can be used only for the Traction vector ti is defined as a scalar product of stress tensor computation of unknown boundary quantities. If we are in- σ ij and outward normal vector n j. The weak formulation of terested in displacement and stress values in the interior of eq. (2) can be expressed in weighted residual form the body, the Somigliana identity and integral representation Z Z for stresses, respectively, has to be utilized for the evalu- ´σ µ Ω · ´ µ Γ W1i ij; j Xi d W2i ui ui d ation. Fundamental solutions corresponding to the Lame- Ω Γu Z Navier governing equation (1), with the Dirac delta function ´ µ Γ = ; ´ µ W1i ti ti d 0 (3) distribution of body forces, are denoted here as Uij x y . Γ t Then, the governing equations can be written as where prescribed quantities are denoted by overbar and ui, ti λ · ´ µ· ´ µ=δ δ ´ µ ; ; are trial functions. Weight functions are chosen in the follow- µ U jm; ji x y µUim jj x y im x y (9) ing form with the fundamental solutions in two dimensions being given £ £ ; = ; = as W1i ui W2i ti (4) ¨ © £ £ 1 ´ µ= ν µ δ · ; where u and t are the displacements and tractions corre- ´ ; i i Uij r 4 3 lnr ij r;ir j π ´ νµ sponding to the weighting field. 8 µ 1 1 ∂r ´ ; µ= νµδ · ] ´ Applying the Gauss-Green formula to the domain integral in [ ; Tij x y 1 2 ij 2r;ir j π ´ νµ ∂ eq. (3), we have 4 µ 1 r n © ´ νµ´ µ : ; 1 2 r; n r n (10) Z Z Z i j j i £ £ £ ´σ µ Ω = Γ Γ ui ij; j Xi d ui ti d ti ui d Ω Γ Γ The Somigliana identity has the form [Balas, Sladek and Z Z £ £ Sladek (1989) σ Ω Ω : · u d u X d (5) ij; j i i i Γ Ω Z Z µ= ´ µ ´ µ Γ ´ µ ´ ; µ Γ ui ´y t j x Uij x y d x u j x Tij x y d x Substituting eq. (5) into (3), with taking into account the pre- Γ Γ scribed boundary conditions, one obtains Z ´ µ ´ µ Ω : · Xj x Uij x y d x (11) Z Z Z Z Ω £ £ £ £ Γ · σ Ω = Ω : u t dΓ t u d u d X u d (6) i i i i ij; j i i i Γ Γ Ω Ω If an interior point y is lying very close to the boundary, the second integral on the r.h.s. of eq. (11) is nearly singular. The If the weighting field is selected as a Trefftz function the ho- accurate evaluation of such integral requires special attention. mogeneous governing equation is satisfied It is well known that the integral representation (11) can be σ£ regularized ( see e.g. [Balas, Sladek and Sladek (1989)] ) as = 0(7) ij; j Z Z ¼ ¼ Ω ´ µ= ´ζ µ· ´ µ ´ µ Γ [ ´ µ ´ζ µ] ¢ in Ω ,whereΩ . ui y ui t j x Uij x y d x u j x u j Γ Γ Finally, one obtains the boundary integral equation Z ; µ Γ · ´ µ ´ µ Ω ; Tij´x y d x Xj x Uij x y d x (12) Z Z Z Ω £ £ £ Γ Γ = Ω : ui ti d ti ui d Xiui d (8) Γ Γ Ω Γ where ζ ¾ can be selected as the nearest boundary point to Ω The boundary integral equation (8) relates the boundary dis- y ¾ . placements and traction vectors. Then, the unprescribed quan- Similarly, one can derive a regularized integral representation tities can be computed from that equation. In contrast to the for stresses [Balas, Sladek and Sladek (1989)]. Note that in the conventional boundary integral equations the free term is miss- integral representation (12) the strong singularity of the kernel Γ ing here. The governing equation for the fundamental solution Tij (as y approaches ζ ¾ ) is smoothed by the vanishing factor µ ´ζ µ contains the Dirac delta function which causes a singular be- u j ´x u j in the numerical integration. A more advanced haviour of the fundamental solution. Here, Trefftz functions cancellation of divergent terms is discussed elsewhere [Sladek are nonsingular and all the integrals exist in a regular sense. and Sladek (1998)]. Application of multi-region Trefftz method to elasticity 3 ; ;::: The Trefftz formulation of the boundary integral equations for b = 1 2 ,whereNq and Ne is the number of boundary supplemented with integral representations based on singular and domain elements, respectively.
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