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1 Copyrght c 2006 Tech Scence Press CMES, vol.6, no., pp.-3, 2006 An Alternatve Approach to Boundary Element Methods va the ourer Transform aban M. E. Duddeck Abstract: In general, the use of Boundary Element Methods (BEM) s restrcted to physcal cases for whch a fundamental soluton can be obtaned. or smple dfferental operators (e.g. sotropc elastcty) these specal solutons are known n ther explct form. Hence, the realzaton of the BEM s straght forward. or more complcated problems (e.g. ansotropc materals), we can only construct the fundamental soluton numercally. Ths s normally done before the actual problem s tackled; the values of the fundamental solutons are stored n a table and all values needed later are nterpolated from these entres. The drawbacks of ths approach le n the hgh amount of storage capacty, whch s requred, and n numercally errors due to nterpolaton especally near the sngularty of the fundamental soluton. Hence, an alternatve BEM, the ourer BEM, was proposed n Duddeck (2002) whch s based on boundary ntegral equatons (BIE) obtaned va ourer transform. It can be appled to all problems as long as the dfferental operator s lnear and has constant coeffcents. The frst step to derve the ourer transformed BIE conssts n a rgorous mathematcal formulaton va dstrbuton theory, whch was developed by Schwartz (950/5) at the end of the 940s and whch s stll the mathematcal bass for the treatment of partal dfferental equatons, e.g. Hörmander (990). In the context of BEM, ths theory offers a straghtforward approach towards the dscusson of sngulartes normally encountered n the BIE. Dstrbuton theory s able to handle all knd of sngulartes (umps, weak, strong and hyper sngular values) occurrng n the BEM formulatons and t s the adequate approach for the dscusson of the correspondng ntegratons. In fact t can be shown by ths approach, cf. Duddeck (2002), that all strong and hyper sngular components are vanshng. In addton, the dstrbutontheory enlarges the applcablty of ourer transform leadng to alternatve formulatons for lnear dfferental equa- Queen Mary College, London Unversty, Department of Engneerng, Mle End Road, London E 4NS, UK, f.duddeck@qmul.ac.uk tons wth constant coeffcents. All dfferentatons are converted to multplcatons; the dfferental operator becomes a smple algebrac expresson, whch can easly be nverted. Ths nverse dfferental operator s the ourer transform of the fundamental soluton. In the approach dscussed here, ths ourer fundamental soluton and not the fundamental soluton tself s taken for the computaton of all entres to the BEM-matrces. Based on Parsevals formula, whch states the equvalence of energy expressons n the ourer and the orgnal space, alternatve BIE can be derved n the ourer space leadng to the same entres for the matrces. Thus for the ourer BEM, every term should be establshed n the ourer space. Because a Galerkn approach leads to symmetrc matrces and does not requre a second ntegraton n the ourer BEM, ths approach was preferred to the conventonal collocaton BEM. The tral and the test functons can be easly transformed to the ourer space as long as they are defned on straght elements. Otherwse a numercal approach can be selected. In ths paper, the method s appled to thn plate problems accordng to Krchhoff s theory. The dfferental operator s of fourth order leadng to hghly sngular ntegral equatons. Although these sngulartes are qute complex, t can be shown easly that all strong and hyper sngular terms are vanshng n both, the orgnal and the ourer transformed space. In the small example, all ntegrals were solved analytcally, thus - n contrast to other publcatons, e.g. Maucher and Hartmann (999) no numercal errors,.e. artfcal oscllatons, are occurrng at the corners of a rectangular plate. keyword: Boundary Element Methods, ourer Transform, undamental Solutons, Krchhoff Plates. Introducton In the lterature, BEM models are mostly dscussed for sotropc plates. Collocaton approaches can be found for example n Antes and Panagotopoulos (992); Beskos

2 2 Copyrght c 2006 Tech Scence Press CMES, vol.6, no., pp.-3, 2006 (99). The Galerkn BIE were presented n rang and Bonnet (998). Plates on Wnkler foundatons are treated by a collocaton method n Jahn (998). All these approaches were developed for sotropc plates n the orgnal space. Ansotropc plates are regarded for example n hao (995) where a collocaton approach was chosen and the dfferental equaton of fourth order was converted nto two PDEs of second order. Hence, a fundamental soluton can easly be found. The dsadvantage s the resultng vectoral character of the equaton. The ourer BEM proposed n Duddeck (2002) generalzes the tradtonal Galerkn BEM such that physcal cases where the fundamental soluton s not known explctly can be treated. rst applcatons n lnear elastcty and heat transfer can be found n Duddeck and Gesenhofer (2002); Duddeck (200). Here, the method s transferred to sotropc and ansotropc plate problems. In the frst part, the model for the bendng of plates s analyzed whch s due to Krchhoff and whch s vald for thn plates where shear deformatons can be neglected. In the second part, the fundamental soluton for ansotropc Krchhoff plates s gven n the ourer space. Applcatons to ansotropc cases wll be publshed n a separate paper. 2 The prncple of ourer BEM 2. The Galerkn BIE for the Posson equaton Because the ourer BEM approach s rather new and establshes a fundamental dfferent vew on BEM, the man prncples are demonstrated frst for the smple example of the Posson equaton n an n-dmensonal bounded doman Ω R n wth a boundary Ω = Γ u Γ t : Δu(x)= f (x), Δ = n 2 / x 2,x Ω, u(x)=u Γ (x), x Γ u Γ; t(x)=t Γ (x), x Γ t Γ. () u s the unknown quantty, f denotes known volume sources, and u Γ (x),t Γ (x) are the known boundary values. The correspondng BIE s, e.g. Bonnet (999): κ(x)u(x)= f (y)u(x y)dy Ω N t t φ t(y)u(x y)dγ y Γ N u u Γ φ u(y)a tu(x y)dγ y. (2) κ s the free term, At = = ν the boundary dfferental operator wth as the Nabla symbol and ν as the normal vector of the -th boundary element. φ t,φ u are the -th test and tral functons for u and t = At u and t,u are the known and unknown coeffcents of the dscretzaton. U s the fundamental soluton. The Galerkn BIE s the obtaned va a wegthng Γ φ t (x)κ(x)u(x)dγ x = N t t Γ N u u Γ Γ φ t (x) φ t (x) φ t (y)u(x y)dγ y dγ x Γ φ t (x) Γ Ω f (y)u(x y)dydγ x φ u(y)a tu(x y)dγ y dγ x. (3) The nner ntegral s a convoluton (..), whch leads n the ourer space to a multplcaton (all ntegrals are extended to nfnty; values outsde the orgnal support are kept to be zero). The outer ntegral s a scalar product <.,.>, whch s equal to a smlar scalar product n the ourer space. We have the two man theorems known for the ourer transform, e.g. Hörmander (990): u(x) φ (x) û( x)φ ( x); (4) φ (x),u(x) = (2π) n φ (x),û( x). (5) The symbol lnks an expresson n the orgnal space to the correspondng term n the ourer transformed space. Thus, the transform of (3) leads to BIE, whch refer only to the transformed fundamental soluton Û: φ t (x),û( x) N t N u = φ t (x), f( x)û( x) t φ t (x), φ t ( x)û( x) u φ t (x), φ u( x) A tû( x). (6)

3 ourer BEM 3 It s emphaszed here, that the matrx entres obtaned va these ourer BIE are dentcal to those obtaned normally wth the standard BEM approach. The other BIE are treated n the same manner. Thus there s no dfference between the two algebrac systems, that obtaned va tradtonal BEM and va the ourer BEM. or the dsplacement BIE dscussed here, we obtan, cf. Bonnet (999), x 2 h f x Ku u = u H u t G u u ; (7) where the entres can be evaluated n the ourer space u = φ (2π) n t (x), f( x)û( x) ; Hu = φ (2π) n t (x), φ t( x)û( x) ; G u = φ (2π) n t (x), φ u( x) A ; tû( x) Ku = φ (2π) n t (x), φ ( x)û( x) ; or n the orgnal space u = φt (x), f (x) U(x) ; Hu = φt (x),φ t(x) U(x) ; G u = φ t (x),φ u(x) AtU(x) ; Ku = φt (x),φ (x) U(x) ; The latter can be used only f the fundamental soluton U s known whle the frst can be computed n all cases as long as the dfferental operators s lnear and has constant coeffcents. The evaluatons n the ourer space consst of only one ntegraton of oscllant ntegrands and the formulatons n the orgnal space requre a double ntegraton. In the followng, ths prncple s transferred to thn plate theory. 3 The sotropc thn plate The mdsurface of the plate wth a unform thckness h s stuated n the (x,x 2 )-plane, see g.. w(x);x = x,x 2 denotes the out-of-plane bendng dsplacement. We defne the unt outward normal ν =(ν,ν 2 ) T and the unt tangent τ =(τ,τ 2 ) T =(ν 2,ν ) T. The moment m kl and m2 q 2 m 22 x 3 m q gure : Plate forces and moments m 2 the shear components q k are constructed from the stress tensor σ kl by h/2 m kl = σ kl (x,x 3 )x 3 dx 3, q k = h/2 h/2 h/2 σ k3 (x,x 3 )dx 3. (8) The moment m kl can be related to the vertcal dsplacement w by: m kl = K klmn mn w and the shear forces q k by: m kl = K klmn x m x n ŵ (9) q k = l m kl = D kll w q k = x l m kl = D x k x l x l ŵ; wth the stffness of the plate as: Eh 3 K klmn = D[( ν)δ km δ ln νδ kl δ mn ], D = 2( ν 2 ), where summaton has to be appled for repeated ndces k,l,m,n =,2. E,ν,D are the Young s modulus for elastcty, the Posson s coeffcent, and the flexural rgdty of the plate, respectvely. δ kl s Kronecker s symbol. The ourer transform s done wth respect to all coordnates. (.) denotes a quantty n the ourer space, thus x k are the wave numbers. kl s the short notaton for the partal dfferental operator 2 /( x k x l ). The dfferental equaton for the bendng of the sotropc Krchhoff plate s DΔΔw = f D( x 2 x 2 2) 2 ŵ = f, (0)

4 4 Copyrght c 2006 Tech Scence Press CMES, vol.6, no., pp.-3, 2006 wth f as transversal load per unt area and Δ as the Laplace operator. Thus the dfferental operator s n the ourer space x 2 P ( x)=d( x 2 x2 2 )2. () The transformed fundamental soluton Ŵ s obtaned as the nverse of P ( x) : Ŵ( x)= D x 4 = D( x 2, (2) x2 2 )2 wth x = x 2 x The boundary dfferental operators The boundary quanttes whch are relevant n the context of BEM are the deflecton w, the normal slope ϕ ν,the normal bendng moment m ν, and the equvalent Krchhoff shear q ν = ν k q k dm T /ds (we assume that the normal vector ν s pecewse constant) w ŵ; (3) ϕ ν = ν k k w ϕ ν = ν k x k ŵ; m ν = ν k ν l m kl mν = ν k ν l m kl ; q ν = ν k q k τ k k m lm ν l τ m q ν = ν k q k τ k x k m lm ν l τ m, where the twstng moment for the Krchhoff shear s defned as m T = m kl ν k τ l, and the dfferentaton wth respect to the arc length s of the boundary s d/ds = τ k k. At the -th corner, there s a partcular corner force f c whch s related to a ump of the twstng moment m T around the corner f c(x = x c) = m T (x = x c) m T (x = x c). (4) Ths corner force s for arbtrary angles fc = A c w f c = A cŵ, (5) ν ν gure 2 : Defnton of the normal vectors ν,ν for the corner term. wth the dfferental operator (see g. 2 for the defnton of ν,ν ) Ac =(ν)d(ν ν 2 ν ν 2 )( 22 ) 2D(ν ν ν 2 ν 2 ν ν ν 2 ν 2 ) 2. A c = ( ν)d(ν ν 2 ν ν 2 )( x2 x 2 2) 2D(ν ν ν 2 ν 2 ν ν ν 2 ν 2 ) x x 2. Thus, the three boundary dfferental operators Aφ,m,q for a boundary element wth the normal ν and the corner dfferental operator Ac are defned as follows: or ϕ ν : A ϕ = ν k k for m ν : A m = K klmnν k ν l mn for q ν : and for f c : A c see (5) x A ϕ = ν k x k ; (6) A q = Dν k kll K klmn τ pν kτ l mnp A m = K klmnν k ν l x m x n ; A q = Dν k x k x l x l K klmn τ pν kτ l x m x n x p ; A c see (5). The boundary quantty and ts transform are obtaned va Ak w or A kŵ (k = ϕ,m,q,c).

5 ourer BEM 5 or any well-posed problem half of the boundary data must be gven. or each par of dual varables (dual n the sense that the pars lead to work terms) ether one of these two varables or the relaton between both (a Robn type boundary condton) must be prescrbed. There are four dfferent types of boundary condtons : w(x)=w Γ (x) for x Γ w ; ϕ ν (x)=ϕ νγ (x) for x Γ ϕ ; m ν (x)=m νγ (x) for x Γ m ; q ν (x)=q νγ (x) for x Γ q. (7) In addton, we have to prescrbe for each corner pont x c ether the ump of the twstng moment fc(x c) or the dsplacement w(x c ) at ths pont. or establshng the BIE, the followng dervatves of the fundamental soluton W are requred, cf. Beskos (99) for the orgnal space: The fundamental slope Φ ν (wth lnr0 2 = ) Φ ν = AϕW = ν k x k 4πD ln x Φ ν = A ϕŵ = ν k x k D( x 2, (8) x2 2 )2 the fundamental normal moment: M ν = AmW = [2( ν)ln x 8π (3 ν)(ν x ) 2 ( 3ν)(τ x ) 2] M ν = A mŵ = K klmnν k ν l x m x n D( x 2, (9) x2 2 )2 and the fundamental Krchhoff shear Q ν = AqW = [ 2ν x 4π x (ν)(ν x κ x )((ν x ) 2 (τ x ) 2 ) ] (20) Q ν = A qŵ = ν k x k x l x l ( x 2 x2 2 )2 K klmnτ pν k τ l x m x n x p D( x 2 x2 2 )2. The fundamental corner force s c = A cw c = A cŵ, (2) The evaluaton of the dervatves of the fundamental soluton n the ourer space s very easy compared to that n the orgnal space. 3.2 The symmetrc Galerkn BIE The Somglana dentty as a weak form equvalent to (0) s, cf. rang and Bonnet (998); Jahn (998), κ(x)w(x)= f (y)w(x y)dω Ω q ν (y)w(x y)dγ y m ν (y)φ ν (x y)dγ y Γ Γ ϕ ν (y)m ν (x y)dγ y w(y)q ν (x y)dγ y Γ f c (y c)w(x y c) Γ w(y c) c (x y c). (22) Accordng to Beskos (99), the free term s κ = ψ/(2π),.e. the percentage of the total angle 2π. The Galerkn verson s obtaned by addtonal weghtng wth test functons φ q (q s the dual varable of w). It s n dstrbutonal notaton (<.,.>denotes the scalar product and s the symbol for convoluton) φ q,w χ = φ q, f W φ q,q ν W φ q,m ν Φ ν = φq,ϕ ν M ν φ q,w Q ν φ q, f c (y c )W(x y c ) φ q,w(y c) c (x y c). (23) The boundary factor κ s obtaned mplctely by w χ = χ(x)w(x) wth χ as the cutoff-dstrbuton of the doman, cf. Duddeck (2002), χ(x) :=... x Ω κ(x)... x Γ 0... x / Ω = Ω Γ. (24) We ntroduce now the dscretzatons for all boundary quanttes. Because of the hgh order of the dfferental operator (order four) we have to respect certan contnuty requrements for the tral functons. The approxmaton of the deflecton w must be C at the nodes (the tangental dervatve must be contnuous) and the tral functons for the slope should be C 0,.e. contnuous. Therefore, the boundary values are approxmated by Hermte polynomals, cf. rang and Bonnet (998), (.) = d(.)/ds = τ k k(.) s the tangental dervatve, w(x) w φ w(x)p φ ϕ(x). (25) The functons φ w,φ ϕ are constructed from the tral functons for the reference element (L e s the length of the -th

6 6 Copyrght c 2006 Tech Scence Press CMES, vol.6, no., pp.-3, 2006 element) φ 0 w = φ 0 ϕ = x 2 (3 2x ) (x ) 2 ( 2x ) 0 x 2 (x )L e x (x ) 2 L e 0 for the ( )-th element for the -th element otherwse for the ( )-th element for the -th element otherwse. These functons have the nterpolaton propertes The transformatons of the lnear tral functons are φ 0 m,q = x H(x )H( x )δ(x 2 ) φ 0 m,q = x e x e x x 2 φ 0 m,q =( x )H(x )H( x )δ(x 2 ) φ 0 m,q = ex x x 2. (30) φ 0 w(x = x l )=δ kl d ds φ0 w(x = x l )=0 (26) φ 0 ϕ(x = x l d )=0 ds φ0 ϕ(x = x l )=δ kl. The other quanttes of the boundary,.e. m ν and q ν,are approxmated by pecewse lnear polynomals φ 0 m = φ0 q = x for the ( )-th element x for the -th element 0 otherwse (27) These tral functons are n dstrbutonal notaton (H s the Heavsde-dstrbuton and δ denotes the Dracdstrbuton): φ 0 w = x 2 (3 2x )H(x )H( x )δ(x 2 ) φ 0 w = 2 6 x x 3 ex 6 x e x 2e x x 4 ; φ 0 w =(x ) 2 ( 2x )H(x )H( x )δ(x 2 ) φ 0 w = 2 x3 6 x 6 x e x 2 x e x x 4. (28) or the slope tral functons we get φ 0 ϕ = x 2 (x )L e H(x )H( x )δ(x 2 ) φ 0 ϕ = L e 6 x 2 ex 4 x e x 6e x 2 x x 4 ; φ 0 ϕ = x (x ) 2 L eh(x )H( x )δ(x 2 ) φ 0 ϕ = L e 6 x 2 4 x 2 x e x 6e x. (29) x 4 Wth the excepton of the deflecton tself, all boundary quanttes are dependent on the normal vector, they are dscontnuous at corner ponts. Hence we defne two dfferent nodal values at these corners. The treatment of the corner force f c and the corner deflecton w c n the ourer space s enabled by defnng partcular corner tral functons φ c = δ(x x c) f c = f δ(x x c ) f c = f e <x c,x> ; (3) w c = w δ(x x c) ŵ c = w e <x c,x>. The contrbutons of the corner terms to (23) can be wrtten as φ q, f c (y c)w(x y c) = f φq,φ c W (2π) 2 f φ q (x), φ c ( x)ŵ( x). (32) These dscretzatons result fnally n the dscretzed Galerkn BIE φ q,w χ = φ q, f χ W N m m φq,φ m Φ ν N ϕ N q q φ q,φ q W p φq,φ N w ϕ M ν w φq,φ w Q ν f φ q,φ c W w φ q,φ c c. (33) wth φq,w χ = φq,χp q ; pq s the polynomal defned for the test functon φq. The kernels of ths BIE can be

7 ourer BEM 7 hghly sngular, cf. rang and Bonnet (998) for the regularzaton. A more formal presentaton of ths Galerkn BIE s φ q,w χ = φ q, f χ W N m m φq,φ m AϕW N ϕ p φ q,φ ϕ A mw N w w φq,φ w AqW f φ q,φ c W N q q φ q,φ q W w φ q,φ c A c W, (34) whch fnds ts ourer equvalent n (after cancellng the factor (2π) 2 ) φ q(x),ŵ χ = φ q(x), f χ Ŵ N q q φ q(x), φ qŵ N m φ m q(x), φ ma ϕŵ N ϕ φ p q(x), φ ϕamŵ N w φ w q(x), φ wa qŵ f φ q(x), φ cŵ φ w q(x), φ cacŵ. (35) or a symmetrc Galerkn method, addtonal BIEs are requred whch wll be gven n the followng for the ourer space. If needed they can be transferred easly to the orgnal space. The BIE for the normal slope s obtaned by applyng the adont of Aϕ = A ϕ on (35) and by choosng the dual test functon. Here we need a moment test functon φm. The BIE s then φ m(x), A ϕŵ χ = φ m(x), f χ AϕŴ N q N m N ϕ N w φ q m(x), φ qaϕŵ φ m m(x), φ maϕ A ϕŵ φ p m (x), φ ϕaϕ A mŵ φ w m(x), φ waϕ A qŵ φ f m(x), φ caϕŵ φ w m (x), φ caϕ A cŵ. (36) The other BIE are obtaned analoguously. By usng the operator notaton ntroduced n (6), the followng scheme can be establshed for the total system φ (x), B ŵ χ = φ (x), f B χ Ŵ φ u (x), φ ÂŴ, (37) wth ( ) φ = φ q, φ m, φ ϕ, φ w, φ c, φ c φ = ( φ q, φ m, φ ϕ, φ w, φ c, φ ) c u = ( q,m,p,w,f,w ), the matrx  s equal to: I A ϕ Am A q I A c A ϕ Aϕ A ϕ A m A ϕ Aq A ϕ A ϕ Ac A ϕ A m A ϕ A m Am A m A q A m A m Ac Am whle we get for B: ( I A ϕ Am A q I A q I A c Aϕ A q A ϕ Aϕ A c A m A q Am A m A c Aq A q A q Aq A ; c A q I A c Ac A c Ac Ac A q ). A c

8 8 Copyrght c 2006 Tech Scence Press CMES, vol.6, no., pp.-3, 2006 gure 3 : An example of a non-vanshng ntegrand leadng to a strong sngularty (left) and the correspondng term f all terms, especally the free term on the left-hand sde of the BIE, are taken nto account. I s the dentty operator such that IŴ = Ŵ. Thetransformed terms n the operator matrx  of (37) are obtaned by smple multplcaton. The scalar product on the left-hand sde of (37) can be used for regularzaton purposes n the orgnal as well as n the transformed space. The dfferentaton Bw χ = B{w(x)χ(x)} should be done carefully takng nto account that the product of the deflecton wth the cutoff-dstrbuton of the doman has to be evaluated. 3.3 Regularzaton As n the tradtonal BEM, the ntegrals of the ourer BIE can be sngular. Due to the fact that the ourer transform shfts local sngulartes to global sngulartes and vce versa, the procedures developed n the standard approach cannot be appled drectly (a local sngularty s a sngularty due to a sngle pont and a global sngularty s a sngularty due to an ntegraton of a non-vanshng ntegrand at nfnty). As n the tradtonal approach, weak, strong and hyper sngular values are encountered n the ourer BEM. By the means of a rgorous dstrbutonal dscusson, t was shown n Duddeck (2002) that the free terms of the left-hand sde of the BIE cancel out wth all strong and hyper sngular terms on the rght-hand sde. The nonvanshng ntegrands have only to be evaluated together. gure 3 shows an example of a non-vanshng ntegrand (left) and the correspondng vanshng term f all contrbutons are summed up (rght). 4 Example for sotropc thn plates 4. Clamped square plate or a clamped square plate wth Ω =[0,] [0,] and wth two elements at each sde, the system (37) can be reduced because of w = 0,ϕ ν = 0 on the total boundary and due to w c = 0 for all corner ponts. The rght-hand sde of the ourer BIE s φ T q(x) I Aϕ I φ m(x) A φ c(x) ϕ Aϕ A ϕ A N q ϕ φ qŵ N m φ I mŵ. I φ cŵ A ϕ or the volume forces f we get φ T q(x) φ m(x) φ c(x) I A ϕ I f χ Ŵ. (38) The free term on the left-hand sde s zero because of w = ϕ ν = 0 along the boundary. Lnear tral and test functons are chosen for the normal moment and the Krchhoff shear forces at the boundary. or the corner forces we have the four tral functons φ c = δ(x )δ(x 2 ) φ 2 c = δ(x )δ(x 2 ) φ 3 c = δ(x )δ(x 2 ) φ 4 c = δ(x )δ(x 2 ) φ 2 c = ; φ 2 c = e x ; φ 3 c = e (x x 2 ) ; φ 4 c = e x 2. Therefore, we get the followng system of equatons:

9 ourer BEM 9 gure 4 : The clamped thn plate under unform loadng (4 4elements) or =...N q : 0 = φ q, f χ Ŵ N q N m φ m q, φ ma ϕŵ for =...N m : 0 = φ m, f χ AϕŴ N m q φ q, φ qŵ N q φ m m, φ maϕ A ϕŵ for =... : 0 = φ c, f χ Ŵ N q N m φ m c, φ ma ϕŵ f c φ q, φ cŵ ; φ q m, φ qaϕŵ q φ c, φ qŵ f c φ m, φ caϕŵ f c φ c, φ cŵ. ; A unform loadng f = H(x )H(2 x )H(x 2 )H(2 x 2 ) f = ( e2x )( e 2x 2) (39) x x 2 was appled. All ntegratons were computed analytcally n the ourer space. or the moment as well as for the shear force, the values at the corner are theoretcal zero whch s approxmatvely fulflled even by a coarse mesh. The dsplacements w, the slope ϕ = w, the moment m, and the shear force q ν n the nteror are shown n g.4. As a second example, a clamped plate (a a = 2 2m)s regarded subected to a sngle unt pont load P = nthe center. The results are gven n g.5. The maxmum vertcal deflecton n the center of the plate drectly under the load s w max = m, whch can be compared to the analytcal value gven by Tmoshenko and Wonowsky- Kreger (959) of w max,analyt. = Pa 2 /D = (the stffness s assumed to be D = ). or the coarse grd of 4 4 elements, ths result s reasonable. It s emphaszed here, that these results, whch repre-

10 0 Copyrght c 2006 Tech Scence Press CMES, vol.6, no., pp.-3, 2006 gure 5 : The clamped thn plate subected to a sngle Drac force at the center (4 4 elements) sent the real physcal values are obtaned drectly n the ourer space wthout any nverse ourer transform. 4.2 Verfcaton of the example The results of the example of a clamped square plate are verfed by comparng the matrx entres obtaned va the newly establshed ourer BEM approach wth those orgnatng from a standard BEM evaluaton to be found n rang and Bonnet (998). An example of a lnear test and tral functon s n the orgnal and the ourer space: φ q =( x )H(x )H( x )δ(x 2 ) φ q = ex x x 2 (40) The fundamental solutons used n both approaches are (r 0 s an arbtrary term), cf. Duddeck (2002); rang and Bonnet (998): W(x)= x 2 8πD (ln x ln r 0 ) Ŵ( x)= D( x 2. (4) x2 2 )2 An exemplary analytcal ntegraton of one entry of the matrx leads n the sngle layer case for the orgnal space to(wehavechosenr 0 = e) H = φ q,φ q W = ( x ) ( y )(x y ) 2 8Dπ 0 0 ( ) (x y ) ln 2 dy dx e = [ 3 Dπ 52 3x2 92 x3 ln x 24 0 ln x 2 92 x ln x 2 48 x3 ln x x4 ln x x 288 x4 ln x x x2 ln x 2 32 ] 5 dx = 52Dπ.

11 ourer BEM The correspondng ntegral n the ourer space s computed as H = φ (2π) 2 q(x), φ q( x)ŵ( x) = φ (2π) 2 q(x)φ q( x)ŵ( x)dx R 2 = 2 2x sn x x 2 2cos x D4π 2 R 2 x 4 ( x4 2x2 x2 2 x4 2 ) dx dx 2 = 2πD R 5sgnx 2ex 2 4x 6 2 5e x 2 4x 6 2 [ 5ex 2 4x 6 2 3x 4 2 5sgnx 2 2x 7 2 5sgnx 2e x 2 4x 6 2 sgn x 2e x 2 4x 5 sgnx 2ex 2 2 4x = 52Dπ. 5ex 2 4x 7 2 5e x 2 4x 7 2 5sgnx 2e x 2 4x 7 2 ex 2 4x 5 5sgnx 2ex 2 2 4x 7 2 3sgnx 2 4x 5 2 ex 2 4x 5 2 ] dx 2 The kernel n the ourer space s hyper sngular and s regularzed as dscussed n secton 3.3,.e. the strong and hyper sngular parts cancel wth the free terms on the left-hand sde of the BIE. The result shown here s only the part orgnatng from the weak sngular and the regular parts. The two results are dentcal, thus the procedure after establshng the matrces can be taken from the standard BEM approach and t s not dscussed here. 5 Generalzaton and Outlook 5. The General Ansotropc Plate The stran-stress relatons for general ansotropc plates are wth the flexbltes a kl : ε = a σ a 2 σ 22 a 6 σ 2 ; ε 22 = a 2 σ a 22 σ 22 a 26 σ 2 ; ε 2 = a 6 /2σ a 26 /2σ 22 a 66 /2σ 2. (42) They are lnked to the correspondng rgdtes D kl by D = c (a 22 a 66 a 2 26); D 22 = c (a a 66 a 2 6); D 2 = c (a 6 a 26 a 2 a 66 ); D 66 = c (a a 22 a 2 2); D 6 = c (a 2 a 26 a 22 a 6 ); D 26 = c (a 2 a 6 a a 26 ); (43) wth = 2 c h 3 det a a 2 a 6 a 2 a 22 a 26 a 6 a 26 a 66. The bendng and twstng moments are m = (D D D 6 2 )w; m 2 = (D 6 D D 66 2 )w; m 22 = (D 2 D D 26 2 )w. (44) and the shear forces are q = [D 3D 6 2 (D 2 2D 66 ) 22 D ]; q 2 = [D 6 (D 2 2D 66 ) 2 3D D ]. The dfferental operator for general ansotropc thn plates s, cf. Albuquerque, Sollero, Venturn, and Alabad (2006); Lekhntsk (968), P ( )=[D 4D 6 2 2(D 2 2D 66 ) 22 4D D ] P ( x)=[d x 4 4D 6 x 3 x 2 2(D 2 2D 66 ) x 2 x 2 2 4D 26 x x 3 2 D 22 x 4 2]. Whch leads to the ourer fundamental soluton Ŵ =[D x 4 4D 6 x 3 x 2 2(D 2 2D 66 ) x 2 x 2 2 4D 26 x x 3 2 D 22 x 4 2]. (45) The relevant boundary operators are n the ourer space A ϕ = ν k x k; A m =(ν ν D 2ν ν 2D 6 ν 2ν 2D 2 ) x 2 2(ν ν D 6 2ν ν 2 D 66 ν 2 ν 2 D 26) x x 2 (ν ν D 2 2ν ν 2 D 26 ν 2 ν 2 D 22) x 2 2 ; A q = [ν D ( ν 2ν 2) ν ν 2ν 2D 2 2ν 2ν 2ν 2D 6 ] x 3 [ν ν ν 2D ν 2(2 ν 2ν 2)D 2 4ν D 6 2ν ν 2ν 2D 26 4ν 2ν 2ν 2D 66 ] x 2 x 2 [ν 2ν 2ν D 22 ν (2 ν ν )D 2 4ν 2D 26 2ν 2ν ν D 6 4ν ν ν D 66 ] x 2 2 x [ν 2D 22 ( ν ν ) ν 2ν ν D 2 2ν ν ν D 26 ] x 3 2;

12 2 Copyrght c 2006 Tech Scence Press CMES, vol.6, no., pp.-3, 2006 A c =(ν ν 2 ν ν 2 )[(D D 2 ) x 2 (D 2 D 22 ) x 2 2](ν ν ν 2 ν 2 ν ν ν 2 ν 2 )[D 6 x 2 D 26 x 2 2] [2(ν ν 2 ν ν 2 )(D 6 D 26 ) 4ν 2 ν 2 ν 2 ν 2 D 66] x x 2. Once more, the Galerkn BIE can be obtaned by nsertng these operators and the ansotropc fundamental soluton nto the Galerkn BIE of the sotropc case. 6 Summary A ourer transformed approach to solve BEM problems was presented n ths paper. It s based on the knowledge of only the ourer transformed fundamental soluton and not the fundamental solutons tself. Ths s advantageous n cases, where the dfferental operator s rather complex and no analytcal expresson for the fundamental soluton has been establshed. All equatons are defned and solved n the ourer transformed space. Ths leads to the same matrx entres as n the usual BEM approach. As applcaton, a Galerkn BEM for sotropc thn plates was dscussed. urther work mght transfer ths method to orthotropc or general ansotropc plates wth or wthout Wnkler foundatons. Thck plates can as well be tackled as geometrcal and physcal non-lnear problems, cf. Duddeck (2002). Statc and dynamc cases may be ncluded. The range of studes found n the lterature, e.g. the recent works of Baz and Alabad (2006); Moraru (2006); Purbolaksono and Alabad (2005); Wen, Alabad, and Young (2002), can be enlarged. The rgorous dstrbutonal approach used for the ourer BEM shows clearly that all strong and hyper sngular entres cancel. Thus only weak sngular matrx entres have to be computed. To the authors opnon, ths s general the case n all such-lke problems. References Albuquerque, E. L.; Sollero, P.; Venturn, W. S.; Alabad, M. H. (2006): Boundary element analyss of ansotropc Krchhoff plates. Internatonal Journal of Solds and Structures, vol. 43, pp Antes, H.; Panagotopoulos, P. D. (992): The Boundary Integral Approach to Statc and Dynamc Contact Problems. Brkhäuser. Baz, P. M.; Alabad, M. H. (2006): Lnear Bucklng Analyss of Shear Deformable Shallow Shells by the Boundary Doman Element Method. CMES: Computer Modellng n Engneerng & Scences, vol. 3, no., pp Beskos, D. E.(Ed): Boundary Element Analyss of Plates and Shells. Sprnger. Bonnet, M. (999): Boundary Integral Equaton Methods for Solds and luds. Wley, New York. Duddeck,. M. E. (200): ourer-bem or what to do f no fundamental soluton s avalable? Meccanca, vol. 36, pp Duddeck,.M.E.(2002): ourer-bem Generalzaton of Boundary Element Methods by ourer Transform. Sprnger. Duddeck,. M. E.; Gesenhofer, M. (2002): A Generalzaton of BEM by ourer Transform. Computatonal Mechancs, vol. 28, pp rang, A.; Bonnet, M. (998): A Galerkn symmetrc and drect BIE method for Krchhoff elastc plates: ormulaton and mplementaton. Internatonal Journal of Numercal Methods n Engneerng, vol. 4, pp Hörmander, L. (990): The Analyss of Lnear Partal Dfferental Operators, volume I. Sprnger, 2nd edton. Jahn, P. (998): De Randelement-Methode für elastsch gebettete Platten. PhD thess, Unversty of Kassel, Germany, 998. Lekhntsk, S. G. (968): & Breach. Ansotropc Plates. Gordon Maucher, R.; Hartmann,. (999): Corner sngulartes of Krchhoff plates and the boundary element method. Computer Methods n Appled Mechancs and Engneerng, vol. 73, pp Moraru, G. (2006): BEM based on dscontnuous solutons n the theory of Krchhoff plates on an elastc foundaton. Engneerng Analyss wth Boundary Elements, vol. 30, pp Purbolaksono, J.; Alabad, M. (2005): Dual Boundary Element Method for Instablty Analyss of Cracked

13 ourer BEM 3 Plates. CMES: Computer Modellng n Engneerng & Scences, vol. 8, no., pp Schwartz, L. (950/5): Théore des Dstrbutons, volume I-II. Hermann & Ce, Pars. Tmoshenko, S.; Wonowsky-Kreger, S. (959): Theory of plates and shells. McGraw-Hll, New York, 2nd edton. Wen, P. H.; Alabad, M. H.; Young, A. (2002): Boundary Element Analyss of Curved Cracked Panels wth Mechancally astened Repar Patches. CMES: Computer Modellng n Engneerng & Scences, vol.3, no., pp. 0. hao, M. (995): Randelementmethode und Makroelemente für sotrope und ansotrope Krchhoffsche Platten. PhD thess, ETH urch, 995.

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