Maria Specovius-Neugebauer

Serguei A. Nazarov

Susanne Langer

Comptes Rendus Mécanique (CRM)

e.itéruit desle explicitecature à lathC. R. Mecanique 332 (2004) 591–596Artificial boundary conditions on polyhedral truncation surfacesfor three-dimensional elasticity systemsSusanne Langera, Serguei A. Nazarovb, Maria Specovius-Neugebaueraa Fachbereich 17, Mathematik/Informatik, Universityof Kassel, Heinrich-Plett-Str. 40, 34132 Kassel, Germanyb Institute of Mechanical Engeneering Problems, V.O. Bol’schoy Pr. 61, 199178 St. Petersburg, RussiaReceived 5 January 2004 ; accepted after revision 23 March 2004Available online 1 June 2004Presented by Évariste Sanchez-PalenciaAbstractFor a three-dimensional exterior problem in the framework of anisotrpic elasticity, artificial boundary conditions arconstructed on a polyhedral truncation surface. These conditions do not need an explicit formula for the fundamental matrixAn approach to adapt the shape of truncation surfaces to the shape of the enclosed cavity is discussed.To cite this article:S. Langer et al., C. R. Mecanique 332 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméConditions aux limites artificielles sur des surfaces polyhédrales de troncature pour des systèmes d’élastictridimensionalle. Pour un problème extérieur en trois dimensions dans le cadre de l’élasticité anisitrope, on constconditions au bord artificielles sur une surface de troncature polyhédrale. Ces conditions ne nécessitent pas une formupour la matrice fondamentale. On étudie ensuite une méthode permettant d’adapter la forme de la surface de tronforme de cavité.Pour citer cet article : S. Langer et al., C. R. Mecanique 332 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Keywords:Computational solid mechanics; Elasticity; Polyhedral truncation surfaces; Artificial boundary conditionsMots-clés :Mécanique des solides numérique ; Élasticité ; Surfaces polyhédrales de troncature ; Conditions aux limites artificielles1. Statement of the problemLet Ω = R3\G be a homogeneous anisotropic elastic space with the cavityG bounded by a piecewise smooclosed surface. Introducing the matricesd(x) =(1 0 0 0 αx3 −αx20 1 0 −αx3 0 αx10 0 1 αx2 −αx1 0), D(x) =(x1 0 0 0 αx3 αx20 x2 0 αx3 0 αx10 0 x3 αx2 αx1 0)(1)E-mail addresses:langer@mathematik.uni-kassel.de (S. Langer), serna@snark.ipme.ru (S.A. Nazarov),specovi@mathematik.uni-kassel.de (M. Specovius-Neugebauer).1631-0721/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crme.2004.03.011592 S. Langer et al. / C. R. Mecanique 332 (2004) 591–596nslumns whileingiileonussapacityfatives oflygralreexplicitersewhereα = 2−1/2 and stands for transposition, we formulate an exterior elasticity problem inΩ as followsL(∇x)u(x) := D(−∇x)AD(∇x)u(x) = 0, x ∈ ΩN(x,∇x)u(x) := D(n(x))AD(∇x)u(x) = g(x), x ∈ ∂Ω = ∂G (2)Hereu = (u1, u2, u3) is the displacement vector,n the outward unit normal (all vectors are treated as columin R3), the 6×6-matrixA is symmetric and positive definite, and contains the elastic moduli. Note that the covectorsD(∇x )u andAD(∇x)u (of height 6) can be understood as the representatives of strains and stressethe factorsα in (1) force the norms of the columns inR6 to coincide with the natural norms of the correspondtensors. It is known that, for any surface loadingg = (g1, g2, g3) ∈ Hl−1/2(∂G)3 (the Sobolev–Slobodetskspace) withl ∈ N = {1,2, . . .}, there exists the unique solutionu ∈ Hl+1loc ( Ω)3 which decays as|x| tends to infinity.This solution has the asymptotic formu(x) = (d(−∇x)F (x))b + (D(−∇x)F (x))a + ũ(x) (3)wherea, b are columns inR6, F denotes the fundamental(3×3)-matrix for the operatorL(∇x) in R3 (the Kelvintensor in the isotropic case) and the remainderũ fulfils the estimates∣∣∇kx ũ(x)∣∣  ck(1+ |x|)−k−3, k ∈ N0 = N ∪ {0} (4)outside a neighborhood ofG. We focus on computation of the polarization matrixP , which is an intrinsic integraattribute of a defect in a solid (see, e.g., [1,2]). The columnsP 1, . . . ,P 6 of P appear as coefficients in threpresentation (3) for special right-hand sides in problem (2). Namely, letZj denote the unique decaying solutito problem (2) with the right-hand sidegj (x) = D(n(x))Aej , j = 1, . . . ,6, ej is thej -th unit vector inR6. (5)We emphasize that the equalities∫∂Gd(x)gj (x)dsx = 0 ∈ R6, j = 1, . . . ,6 (6)are valid, which lead tob = 0 in representation (3) ofZj (see, e.g., [2]). Thus, we obtain the relations:Zj(x) = (D(−∇x)F (x))Pj + Z̃j (x) (7)SinceD(∇x)D(x) equals the 6× 6 unit-matrix, the differenceD(x)ej − Zj(x) satisfies the homogeneoproblem (2) but has a linear growth as|x| → ∞ (see (1)).The polarization matrix is always symmetric and positive definite in the case mes3 G > 0 (see, e.g., [2]); it enjoythe 4-rank tensor properties after being rewritten in a proper form. Similarly to the classical harmonic cand the virtual mass tensor,P appears as a key object in miscellaneousasymptotic formulae, as increments opotential energy and eigenfrequences due to formation of a void, damage tensors and topological derivshape functionals (cf. [3–6]), to mention a few. At the same time, the polarization matrix has been computed onfor an isotropic space and canonical shapes ofG such as a ball and a penny-shape crack. By virtue of the interepresentationP = −Ames3 G −(A,D(n)Z)∂G(8)whereZ = (Z1, . . . ,Z6) (see [6], p. 178), it is a fair approach to calculateP by changingZj in (8) for anapproximate solutionZj,R in a truncated domainΩR with appropriate artificial boundary conditions (ABC), heR is a truncation parameter which will be specified later on. However, ABC are usually constructed via anformula for the fundamental matrixF while, as shown in [7], such formulae are still available only for a transvisotropic elastic space.S. Langer et al. / C. R. Mecanique 332 (2004) 591–596 593nalen’sionin (7), acyIn this Note we modify the approach [8], used to construct ABC on truncation spheres, fora wide class ofpolyhedral surfacesand in Section 2 we derive second order differential ABC withoutany specification of thefundamental matrix.Moreover, since the shape of the truncation surfaceΓR is no longer fixed, we discuss iSection 3 an adaptation ofΓR to the shape of the cavityG in order to simplify the preparation of data for numericschemes.2. Derivation and justification of ABCLet T be a polyhedron with the sidesΣ1, . . . ,ΣJ which aretangent to the unit sphereS2. We denoteΓ = ∂Tand forR  R0, we putTR = {x: R−1x ∈ T }, ΓR = {x: R−1x ∈ Γ }while the boundR0 is chosen such thatG ⊂ TR0. For each sideΣj , we introduce the Cartesian coordinates(yj , zj ) = Θjx (9)where the axiszj is perpendicular toΣj , yj = (yj1, yj2) are coordinates on the planeΠj parallel toΣj , andΘjis an orthogonal 3× 3-matrix. In coordinates (9) the operatorL(∇x) takes the formD(−∇(yj ,zj ))AjD(∇(yj ,zj ))(see remark below for an exact expression of the matrixAj ). The inclusionv ∈ H 1(ΓR) means thatv ∈ H 1(ΣjR)for j = 1, . . . , J , moreover, the traces ofv on ΣiR andΣkR coincide on the edgesΣiR ∩ ΣkR of the polyhedralsurfaceΓR.Lemma 2.1.For the derivativesFki = ∂F k/∂xi of fundamental matrix columns, there holds the identity−(NFki,V )ΓR= b(Fki,V ;ΓR) := 1RJ∑j=1(AjDjRΘjF ki,DjRΘjV)ΣjR∀V ∈ H 1(ΓR)3 (10)whereΘj is taken from(9), andDjR(yj ,∇yj) = D(R∇yj ,−2− (yj )∇yj ) (11)is a differential operator on the planeΠj .We return to the general problem (2), but with boundary datag fulfilling condition (6); recall that this impliesb = 0 in (3). We look for an optimal approximation of the solutionu by a solutionuR to problem (2) restricted tothe truncated domainΩR = Ω ∩ TR. Identity (10) becomes a key tool for creating ABC. Indeed, in the Greformula(LuR,v)ΩR+ (NuR,v)∂G= a(uR, v;ΩR) − (NuR,v)ΓR (12)we replace the term(NuR, v)ΓR by −b(uR, v;ΓR) and obtain the variational formulation of the approximatproblema(uR, v;ΩR) + b(uR, v;ΓR) = (g, v)∂G ∀v ∈H1(ΩR)3 (13)The function spaceH1(ΩR)3 consists of vector functionsv ∈ H 1(ΩR)3 such thatv|ΓR ∈ H 1(ΓR) and2−1a(uR,uR;ΩR) expresses the elastic energy stored by the bodyΩR , i.e.,a(u, v;Ω) = (AD(∇x)u,D(∇x)v)Ω (14)Since the above-mentioned change in (12), owing to (10), does not touch the detached asymptotic termdiscrepancy left in (13) byu|ΩR is only generated by the remainderũ, the decay (4) of which makes the discrepansmall. Thus, a technique developed in [8] leads to the following assertion.594 S. Langer et al. / C. R. Mecanique 332 (2004) 591–596gaffineal’tiction byn on thellipsoid,has tooseTheorem 2.2.For any g ∈ L2(∂G)3, there exists a unique solutionuR ∈ H1(ΩR)3 of problem(13) where thequadratic formsa andb are taken from(14)and(10). If g has zero mean value along∂G, the solutionuR and thesolution(3) of problem(2) are related by∥∥(1+ |x|)−ε∇x(u − uR);L2(ΩR)∥∥ + ∥∥(1+ |x|)−1−ε(u − uR);L2(ΩR)∥∥  CεR−ε−5/2∥∥g;L2(∂G)∥∥ (15)where the constantCε is independent ofg andR  R0, provided|ε| < 1/2.Since the special right-hand side (5) of problem (2) forZj verifies equality (6), we arrive at the followinassertion.Corollary 2.3. Let PR be the matrix calculated according to formula(6) with Zj changed for the solutionZj,Rof problem(13) whereg = gj is taken from(5). Then the inequality∥∥P − PR;R6×6∥∥  cεR−ε−5/2holds true withε ∈ (−1/2,1/2) and the constantcε depending onA andG.3. Affine transform for the elasticity systemEmploying an approach used in [9] in the framework of two-dimensional elasticity, we consider thetransformx → x = mx (16)wherem = (mij ) is a(3× 3)-matrix with detm = 1.By a direct calculation, we obtain that problem (2) in the new variablesx keeps the formD(−∇x)AD(∇x)u(x) = 0, x ∈ R3\GD(n(x))AD(∇x)u(x) = g(x), x ∈ ∂G (17)whereG = {x ∈ R3: x ∈ G}, u(x) = (m)−1u(x), g(x) = |(m)−1n(x)|−1mg(x), A = MAM and the matrixM of size(6× 6) can be written as followsm211 m212 m213√2m12m13√2m11m13√2m11m12m221 m222 m223√2m22m23√2m21m23√2m21m22m231 m232 m233√2m32m33√2m31m33√2m31m32√2m21m31√2m22m32√2m23m33 m23m32 + m22m33 m23m31 + m21m33 m22m31 + m21m32√2m11m31√2m12m32√2m13m33 m13m32 + m12m33 m13m31 + m11m33 m12m31 + m11m32√2m11m21√2m12m22√2m13m23 m13m22 + m12m23 m13m21 + m11m23 m12m21 + m11m22Remark 1. If mj = Θj is an orthogonal matrix as in (9), thenMj is orthogonal as well while the matrixAj in (10)is equal toMAM.We do not rewrite tensor and vector fields inx-coordinates! Instead of this, we introduce ‘nonphysicdisplacementsu and stressesAD(∇x)u(x) so that we immerse our original problem (2) into a ‘virtual elasworld’. We emphasize that if problem (17) is solved, real elastic fields can be reconstructed from the solusimple algebraic calculations. At the same time, using this transformation, one can avoid the assumptiopolyhedronT in Section 2. Indeed, one can choose a convex polyhedron, e.g., with sides tangent to an eand transform by (16) the ellipsoid into unit ball. Then one can either deal with problem (17), or onetransform back the ABC constructed for problem (17) inaccordance to (10). In this way, it is possible to choany parallelepiped{x: |xk| < Lk} as the polyhedronT . Then the matrices look as follows:m = diag{L−11 ,L−12 ,L−13 }, M = diag{L−21 ,L−22 ,L−23 ,L−12 L−13 ,L−13 L−11 ,L−11 L−12 } (18)S. Langer et al. / C. R. Mecanique 332 (2004) 591–596 595the-es ofcste andtheof theberhescriptioning thesianasticlyseekh.The corresponding ABC provide accuracy (15) while position and sizes of the parallelepiped can be adopted toshape of the cavityG.Proposition 3.1.The polarization matricesP and P, calculated for problems(2) and(17) respectively, are related byP= MPM (19)Let m1, m2 andm = m1m2 be the matrices of the affine transforms(16) while M1, M2 andM are found inaccordance with the formulabefore the remark. A direct calculation of the matrix products leads to the equalityM = M1M2This homomorphism property shows that the setA of symmetric and positive definite(6 × 6)-matricesA,which can play a role of elastic moduli matrix in the elasticity problem (2), can be divided into classalgebraically equivalent matrices. For the algebraic equivalent matricesA andA, any attribute and characteristiof problem (2) are transformed with the help of elementary algebraic operations into the attribucharacteristics of problem (17) and vice versa. In particular, the fundamental matrixF(x) of the operatorL(∇x) =D(−∇x)AD(∇x) ∈ R3 takes the formF(x) = (m)−1F (m−1x)m−1 (20)As it was mentioned, the fundamental matrixF(x) is known for a transverse isotropic elastic space whilecorresponding matrixA contains 5 arbitrary constants. Thus, formula (20) gives an exact expressionfundamental matrixF(x) for an elastic material with 5+ (9− 1− 3) = 10 constants. Here 9 stands for the numof entries of the matrixm, 1 for the normalization factor causing detm = 1, and 3 corresponds to rotations of tspace which, of course, cannot influence elastic properties. Unfortunately, the authors do not know a deof the classF of elastic materials which are algebraically equivalent to transverse isotropic materials formclassT and, due to (20), have an explicit formula for the fundamental matrix. In any case,F i much wider thanT(take, e.g., matrices (18)).In R3 arbitrary anisotropic material has 21− 3 = 18 free constants (3 is used, e.g., to fix the Cartecoordinates). Based on the fact that the matrixA = MAM gains 5= 9− 3− 1 constants fromm, we formulateConjecture 3.2.Any anisotropic material is algebraically equivalent to an elastic material with a plane of elsymmetry.It is known (see, e.g., [9]) that in the two-dimensional case any material (5= 6 − 1 constants) is algebraicalequivalent to an orthotropic material, where two Young moduli coincide (3= 5− (4− 1− 1) constants).AcknowledgementThe work of the second author was supported by grants of the DFG and of the Institut franco-rusA.M. Ljapunov d’informatique et de mathématiques appliquées.References[1] A.B. Movchan, S.A. Nazarov, I.S. Zorin, Application of the elastic polarization tensor in the problems of the crack mechanics, MTverd. Tela 6 (1988) 128–134 (in Russian).596 S. Langer et al. / C. R. Mecanique 332 (2004) 591–59600)ndyerd. Tela 3estnik[2] S.A. Nazarov, The damage tensor and measures. 1. Asymptotic analysis of anisotropic media with defects, Mekh. Tverd. Tela 3 (20113–124;English translation: Mech. Solids 35 (3) (2000) 96–105.[3] V.G. Maz’ya, S.A. Nazarov, The asymptotic behavior of energy integrals under small perturbations ofthe boundary near corner points aconical points, Trudy Moskov. Mat. Obshch. 50 (1987) 79–129;English translation: Trans. Moscow Math. Soc. 50 (1988) 77–127.[4] I.V. Kamotskii, S.A. Nazarov, Spectral problems in singularly perturbed domains and selfadjoint extensions of differential operators, TrudSt.-Petersburg Mat. Obshch. 6 (1998) 151–212;English translation: Trans. Am. Math. Soc. Ser. 2 199 (2000) 127–181.[5] S.A. Nazarov, The damage tensor and measures. 3. Characteristics of damage associated with an invariant integral, Mekh. Tv(2001) 78–87;English translation: Mech. Solids 36 (3) (2001) 65–73.[6] S.A. Nazarov, J. Sokolowski, Asymptotic analysisof shape functionals, J. Math. Pures Appl. 82 (2) (2003) 125–196.[7] E. Kröner, Das Fundamentalintegralder anisotropen elastischen Differentialgleichungen, Z. Phys. 136 (1953) 402–410.[8] S.A. Nazarov, M. Specovius-Neugebauer, Artificial boundary conditions for Petrovsky-systems, Math. Methods Appl. Sci., in press.[9] A.A. Kulikov, S.A. Nazarov, M.A. Narbut, Linear transformations for the plane problem of anisotropic theory of elasticity, VSt.-Petersburg Univ. 2 (2000) 91–95 (in Russian).

Artificial boundary conditions on polyhedral truncation surfaces for three-dimensional elasticity systems

https://comptes-rendus.academie-sciences.fr/mecanique/item/10.1016/j.crme.2004.03.011.pdf

Artificial boundary conditions on polyhedral truncation surfaces for three-dimensional elasticity systems

Abstract

Similar works

Full text

Available Versions

Comptes Rendus Mécanique (CRM)