Multiresolutional Techniques in Finite Element Method Solution of Eigenvalue Problem

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1 Multresolutonal Technques n Fnte Element Method Soluton of Egenvalue Problem Marcn Kamńs Char of Mechancs of Materals, Techncal Unversty of Łódź Al. Poltechn 6, Łodz, POLAND, tel/fax marcn@mm-lx.p.lodz.pl, marcna@p.lodz.pl Abstract. Computatonal analyss of undrectonal transent problems n multscale heterogeneous meda usng specally adopted homogenzaton technque and the Fnte Element Method s descrbed below. Multresolutonal homogenzaton beng the extenson of the classcal mcro-macro tradtonal approach s used to calculate effectve parameters of the composte. Effectveness of the method s compared aganst prevous technques thans to the FEM soluton of some engneerng problems wth real materal parameters and wth ther homogenzed values. Further computatonal studes are necessary n ths area, however applcaton of the multresolutonal technque s justfed by the natural multscale character of compostes. Introducton Wavelet analyss [] perfectly reflects the very demandng needs of composte materals computatonal modelng. It s due to the fact that wavelet functons le Haar, snusodal (harmonc), Gabor, Morlet or Daubeches, for nstance, relatng neghborng scales n the medum analysed can effcently model a varety of heterogenetes preservng compostes perodcty, for nstance. It s evdent now that wavelet technques may serve for analyss n the fnest scale by varous numercal technques [,4,5] as well as usng multresolutonal analyss (MRA) [3,5,6,8]. The frst method leads to the exponental ncrease of the total number of degrees of freedom n the model, because each new decomposton level almost doubles ths number, whle an applcaton of the homogenzaton method s connected wth determnaton of effectve materal parameters. Both methodologes are compared here n the context of egenvalue problem soluton for a smply supported lnear elastc Euler-Bernoull beam usng the Fnte Element Method (FEM) computatonal procedures. The correspondng comparson made for a transent heat transfer has been dscussed before n [5]. Homogenzaton of a composte s performed here through () smple spatal averagng of composte propertes, () two-scale classcal approach [7] as well as (3) thans to the multresolutonal technque based on the Haar wavelets. An applcaton of the symbolc pacage MAPLE guarantees an effcent ntegraton of algebrac formulas defnng effectve propertes for a composte wth materal propertes gven by some wavelet functons. M. Buba et al. (Eds.): ICCS 4, LNCS 337, pp. 7 78, 4. Sprnger-Verlag Berln Hedelberg 4

2 7 M. Kamńs Multresolutonal Homogenzaton Scheme MRA approach uses the algebrac transformaton between varous scales provded by the wavelet analyss to determne the fne-scale behavor and ntroduce t explctly nto the macroscopc equlbrum equatons. The followng relaton:... () defnes the herarchcal geometry of the scales and ths chan of subspaces s so defned that s fner than. Further, let us note that the man assumpton on j j+ general homogenzaton procedure for transent problems s a separate averagng of the coeffcents from the governng partal dfferental equaton responsble for a statc behavor and of the unsteady component. The problem can be homogenzed only f ts equlbrum can be expressed by the followng operator equaton: BT + u + λ = L( AT + v) () Ths equaton n the multscale notaton can be rewrtten at the gven scale j as ( j) ( j) ( j) ( j) ( j) ( j) B T + u + λ = L A T + v, (3) wth the recurrence relatons used j tmes to compute B ( j), A ( j), u ( j), v ( j). MRA homogenzaton theorem s obtaned as a lmt for j B T + u + λ = L ( A T + v ), (4) whch enables to elmnate nfnte number of the geometrcal scales wth the reduced coeffcents B,A. If the lmts defnng the matrces B and A exst, h h h h then there exst constant matrces B, A and forcng terms u, v, such that the reduced coeffcents and forcng terms are gven by B,A,u, v. The homogenzed coeffcents are equal to h h h A = A, B A A ~ = I (5), h h A ~ I A v h = + A ~ A ~ (6) u u I exp, where h h A ~ (7) = log I + ( I + B A ) A. As the example let us revew the statc equlbrum of elastc Euler-Bernoull beam d d E(x) u(x) = f (x) ; x [,], (8) dx dx where E(x), defnng materal propertes of the heterogeneous medum, vares arbtrarly on many scales. The unt nterval denotes here the Representatve Volume Element (RVE), called also the perodcty cell. Ths equaton can represent lnear elastc behavor of undrectonal structures as well as undrectonal heat conducton and other related physcal felds. A perodc structure wth a small parameter ε>,

3 Multresolutonal Technques n Fnte Element Method Soluton 73 tendng to, relatng the lengths of the perodcty cell and the entre composte, s consdered n a classcal approach. The dsplacements are expanded as () () () u(x) = u (x, y) + ε u (x, y) + ε u (x, y) +..., (9) where u (x, y) are also perodc; the coordnate x s ntroduced for macro scale, whle y - n mcro scale. Introducng these expansons nto classcal Hooe's law, the homogenzed elastc modulus s obtaned as [6] dy E =. () E(y) The method called multresolutonal starts from the followng decomposton: d v(x) u(x) = dx E(x) () d v(x) = f (x) dx to determne the homogenzed coeffcent E (eff) constant for x [,]. Therefore x u(x) u() E(t) u(t) = + dt. () v(x) v() v(t) f (t) The reducton algorthm between multple scales of the composte conssts n determnaton of such effectve tensors B, A, p and q, such that x u(x) u(t) I + B + q + λ = A + p v(x) v(t) dt, (3) where I s an dentty matrx. In our case we apply C C dt ( t ) dt B = ; A =, C = ; C = (4) E(t) E(t) Furthermore, for f(x)= there holds p = q =, whle, n a general case, B and A do not depend on p and q. 3 Multresolutonal Fnte Element Method Let us consder the governng equaton e u = f, x (5) wth u =, x Γ. (6) u Varatonal formulaton of ths problem for the multscale medum at the scale s gven as e u ϕ d + γu ϕ d = fϕ dγ, x. (7) Γ

4 74 M. Kamńs Soluton of the problem must be found recursvely by usng some transformaton between the neghborng scales. Hence, the followng nonsngular n x n wavelet transform matrx W s ntroduced []: T W = T, (8) I and T ψ = W ϕ (9). T s a two-scale transform between the scales - and, such that ϕ T = T ϕ ψ, () wth j j ψ = ϕ (), j=,,n N denotes here the total number of the FEM nodal ponts at the scale. Let us llustrate the wavelet-based FEM dea usng the example of D lnear two-noded fnte element wth the shape functons [9] N T ( ξ) N = = N, () ( + ξ) where N s vald for ξ=- and N for ξ= n local coordnates system of ths element. The scale effect s ntroduced on the element level by nsertng new extra degrees of freedom at each new scale. Then, the scale corresponds to frst extra multscale DOF per the orgnal fnte element, scale next two addtonal multscale DOFs and etc. It may be generally characterzed as ψ ( ξ) = ψ ( ( + ξ) j ) (3) where j ξ j + (4) j + ξ j + The value of defnes the actual scale. The reconstructon algorthm starts from the orgnal soluton for the orgnal mesh. Next, the new scales are ntroduced usng the formula + + j N old = N new + + j + + j u = u = N u + ψ. (5) The wavelet algorthm for stffness matrx reconstructon starts at scale wth the smallest ran stffness matrx e K =, (6) h where h s the node spacng parameter. Then, the dagonal components of the stffness matrx for any > are equal to

5 Multresolutonal Technques n Fnte Element Method Soluton 75 K J e K =. (7) h It should be underlned that the FEM so modfed reflects perfectly the needs of computatonal modelng of multscale heterogeneous meda. The reconstructon algorthm can be appled for such n, whch assure a suffcent mesh zoom on the smallest scale n the composte. 4 Fnte Element Method Equatons of the Problem The followng varatonal equaton s proposed to study dynamc equlbrum for the lnear elastc system: ρ u δu d + C ε δε d = ρf δu d + t δu d (8) jl j l σ and u represents dsplacements of the system wth elastc propertes and mass densty defned by the elastcty tensor C (x) and ρ=ρ(x); the vector t denotes the jl stress boundary condtons mposed on. Analogous equaton for the σ homogenzed medum has the followng form: ρ ( eff ) u δu d + C ε δε d = ρ f δu d + jl j l t δu d σ ( ) where all materal propertes of the real system are replaced wth the effectve parameters. As t s nown [9], classcal FEM dscretzaton returns the followng equatons for real heterogeneous and homogenzed systems are obtaned: M q + C q + K q = Q, M q + C q + K q = Q (3). αβ β αβ β αβ β α αβ β αβ β αβ β α The R.H.S. vector equals to for free vbratons and then an egenvalue problem s solved usng the followng matrx equatons: K M ω Φ =, K M (3) ω Φ =. αβ ( α) αβ βγ αβ ( α) αβ βγ (9) 5 Computatonal Illustraton Frst, smply supported perodc composte beam s analyzed, where Young modulus E(x) and mass densty n the perodcty cell are gven by the followng wavelets:.e9; x.5 x h (x) =, x m (x) = + exp, σ=-.4; (3) 3.E9.5 < x πσ σ σ E(x) =. h(x) +.E9 m(x). (33) ~ h(x) ; x.5 ~ =, ρ ( x) =.5 h(x) +.5 m(x). (34) ;.5 < x

6 76 M. Kamńs The composte specmen s dscretzed usng each tme 8 -noded lnear fnte elements wth untary nerta moments. The comparson starts from a collecton of the egenvalues reflectng dfferent homogenzaton technques gven n Tab.. Further, the egenvalues for heterogeneous beams are gven for st order wavelet projecton n Tab., for nd order projecton n Tab. 3, 3 rd order - n Tab. 4. The egenvalues computed for varous homogenzaton models approxmate the values computed for the real composte models wth dfferent accuracy - the weaest effcency s detected n case of spatally averaged composte and the dfference n relaton to the real structure results ncrease together wth the egenvalue number and the projectons order. The results obtaned thans to MRA projecton are closer to those relevant to MRA homogenzaton for a sngle RVE n composte; classcal homogenzaton s more effectve for ncreasng number of the cells n ths model. Table. Egenvalues for the smply supported homogenzed composte beams Egenvalue Spatal averagng Classcal approach Multresolutonal model,84 E 3,665 E 6,8 E,895 E3 5,864 E 9,965 E 3 9,59 E3,969 E3 5,45 E3 4 3,3 E4 9,383 E3,594 E4 5 7,4E4,9 E4 3,893 E4 6,535 E5 4,75 E4 8,7 E4 Table. Egenvalues for the smply supported composte beam, st order wavelet projecton 64 RVEs 3 RVEs 6 RVEs 8 RVEs 4 RVEs RVEs RVE 3,534 E 3,535 E 3,537 E 3,55 E 3,599 E 3,89 E 4,59 E 5,656 E 5,66 E 5,679 E 5,76 E 6,37 E 7,887 E,593 E3,864 E3,87 E3,89 E3,99 E3 3,5 E3 4,973 E3 7,37 E3 9,56 E3 9,87 E3 9,6 E3 9,88 E3,35 E4 4,867 E4 3,5 E4, E4,4 E4,75 E4,536 E4 3,758 E4 6,743 E4 6,4 E4 4,59 E4 4,67 E4 4,786 E4 5,655 E4 8,448 E4,347 E5,678 E5 Table 3. Egenvalues for the smply supported composte beam, nd order wavelet projecton 3 RVEs 6 RVEs 8 RVEs 4 RVEs RVEs RVE 3,636 E 3,639 E 3,65 E 3,73 E 3,936 E 4,64 E 5,83 E 5,84 E 5,95 E 6,39 E 8,6 E,63 E3,95 E3,975 E3 3,75 E3 3,65 E3 5,9 E3 7,4 E3 9,348 E3 9,48 E3, E4,334 E4 4,875 E4 3,53 E4,88 E4,34 E4,65 E4 3,846 E4 6,83 E4 6,9 E4 4,76 E4 4,9 E4 5,85 E4 8,64 E4,36 E5,69 E5 Table 4. Egenvalues for the smply supported composte beam, 3 rd order wavelet projecton 6 RVEs 8 RVEs 4 RVEs RVEs RVE 3,66 E 3,674 E 3,76 E 3,964 E 4,664 E 5,879 E 5,96 E 6,354 E 8, E,6 E3,993 E3 3,96 E3 3,637 E3 5,74 E3 7,479 E3 9,54 E3,7 E4,354 E4 4,876 E4 3,59 E4,355 E4,66 E4 3,93 E4 6,839 E4 6,34 E4 4,954 E4 5,857 E4 8,796 E4,373 E5,69 E5

7 Multresolutonal Technques n Fnte Element Method Soluton 77 Free vbratons for and 3-bays perodc beams are solved usng classcal and homogenzaton-based FEM mplementaton. The untary nerta momentum s taen n all computatonal cases, ten perodcty cells compose each bay, whle materal propertes nserted n the numercal model are calculated by spatal averagng, classcal and multresolutonal homogenzaton schemes and compared aganst the real structure response. Frst egenvalues changes for all these beams are contaned n Fgs., the resultng values are mared on the vertcal axes, whle the number of egenvalue beng computed on the horzontal ones.,5e+,e+,5e+,e+ 5,3E+,5E+8 ω α e,av e,hom e,real e,wav α Fg.. Egenvalues progress for varous two-bays composte structures 9,E+ 8,E+ 7,E+ 6,E+ 5,E+ 4,E+ 3,E+,E+,E+,E+ ω α e,av e,hom e,real e,wav α Fg.. Egenvalues progress for varous three-bays composte structures Egenvalues obtaned for varous homogenzaton models approxmate the values computed for the real composte wth dfferent accuracy - the worst effcency n egenvalues modelng s detected n case of spatally averaged composte and the dfference n relaton to the real structure results ncrease together wth the egenvalue number. Wavelet-based and classcal homogenzaton methods gve more accurate results the frst method s better for smaller number of the bays, and classcal homogenzaton approach s recommended n case of ncreasng number of the bays and the RVEs. The justfcaton of ths observaton comes from the fact, that the wavelet functon s less mportant for the ncreasng number of the perodcty cells n

8 78 M. Kamńs the structure. Another nterestng result s that the effcency of approxmaton of the maxmum deflectons for a mult-bay perodc composte beam by the deflectons encountered for homogenzed systems ncrease together wth an ncrease of the total number of the bays. 6 Conclusons The most mportant result of the homogenzaton-based Fnte Element modelng of the perodc undrectonal compostes s that the real composte behavor s rather well approxmated by the homogenzed model response. MRA homogenzaton technque gvng more accurate approxmaton of the real structure behavor s decsvely more complcated n numercal mplementaton snce necessty of usage of the combned symbolc-fem approach. The technque ntroduces new opportuntes to calculate effectve parameters for the compostes wth materal propertes approxmated by varous wavelet functons. A satsfactory agreement between the real and homogenzed structures models enables the applcaton to other transent problems wth determnstc as well as stochastc materal parameters. Multresolutonal homogenzaton procedure has been establshed here usng the Haar bass to determne complete mathematcal equatons for homogenzed coeffcents and to mae mplementaton of the FEM-based homogenzaton analyss. As t was documented above, the Haar bass approxmaton gves suffcent approxmaton of varous mathematcal functons descrbng most of possble spatal dstrbutons of compostes physcal propertes. References. Al-Aghbar, M., Scarpa, F., Staszews, W.J., On the orthogonal wavelet transform for model reducton/synthess of structures. J. Sound & Vbr. 54(4), pp ,.. Chrston, M.A. and Roach, D.W., The numercal performance of wavelets for PDEs: the mult-scale fnte element, Comput. Mech., 5, pp. 3-44,. 3. Dorobantu M., Engqust B., Wavelet-based numercal homogenzaton, SIAM J. Numer. Anal., 35(), pp , Glbert, A.C., A comparson of multresoluton and classcal one-dmensonal homogenzaton schemes, Appl. & Comput. Harmonc Anal., 5, pp. -35, Kamńs, M., Multresolutonal homogenzaton technque n transent heat transfer for undrectonal compostes, Proc. 3 rd Int. Conf. Engneerng Computatonal Technology, Toppng, B.H.V. and Bttnar Z., Eds, Cvl-Comp Press,. 6. Kamńs, M., Wavelet-based fnte element elastodynamc analyss of composte beams, WCCM V, Mang, H.A., Rammerstorfer, F.G. and Eberhardstener, J., Eds, Venna. 7. Sanchez-Palenca, E., Non-homogeneous Meda and Vbraton Theory. Lecture Notes n Physcs, vol. 7, Sprnger-Verlag, Berln, Stenberg, B.Z. and McCoy, J.J., A multresoluton homogenzaton of modal analyss wth applcaton to layered meda, Math. Comput. & Smul., 5, pp , Zenewcz, O.C. and Taylor, R.L., The Fnte Element Method. Henemann-Butterworth,.