Chapter 3:Adaptive Concurrent Multilevel Model 93 o=6y)8 (Stress-strain), X 4=x(y)月 )(Strain-displacement). (3.11) x In(3.9)-(3.11),is a Y-periodic function representing the character- istic modes of deformation in the RVE and ,国=o,(c以np,ew)=+ (3.12) 2 ax dx, =(e,x are homogenized macroscopic stress and strain tensors,respectively,that are obtained by volumetric averaging.The asymptotic homogenization method provides good convergence characteristics with respect to certain norms,in addition to bounds on effective properties.Solutions of RVE boundary value problems with imposed unit macroscopic strains are used in the calculation of the anisotropic homogenized elasticity tensor C.The RVE boundaries are subjected to periodicity conditions,implying that all bound- ary nodes separated by the periods Yi,Y2,Y3 along the three orthogonal co- ordinate directions will follow the displacement constraints: 4(,x2,x3)=4,(x±k,2±k3Y2,x3±kY3),i=1,2,3. (3.13) Following macroscopic analysis with the homogenized moduli C(), numerical simulations of the RVE boundary value problems yield stresses and strains in the microstructural RVE. Limitations of the regularized problem in m Limitations in solving the regularized problem for variables in hetero- geneous microdomains arise from assumptions of relatively uniform mac- roscopic fields and periodicity of the RVE.In uncoupling the macroscopic problem in 2pom from the microscopic RVE problem in Y,it is assumed that the RVE has infinitesimal dimensions in comparison with the macro- scopic scale,i.e.,>0.While solutions of the problems in mapproach those for the actual domain in this limit,considerable differences may result when the scale factor is finite and the RVE solutions are not perio- dic.The occurrence of such errors is significant in regions of high local gradients,free edges,or discontinuities
0 1 0 1 ˆ ( ) ( ) (Stress–strain), ( ) ( ) (Strain–displacement). kl k ij ij t kl k i i l u x u u x σ σ χ ∂ = ∂ ∂ = ∂ y x y x (3.11) In (3.9)–(3.11), kl χi is a Y-periodic function representing the characteristic modes of deformation in the RVE and 0 0 1 () (,) , () ( ) 2 j i ij ij ij ij Y Y j i u u e e x x ε ε σ ⎧ ⎫ ⎪ ⎪ ∂ ∂ ∑ = = + = ⎨ ⎬ ⎪ ⎪ ∂ ∂ ⎩ ⎭ x x y x x, y (3.12) are homogenized macroscopic stress and strain tensors, respectively, that are obtained by volumetric averaging. The asymptotic homogenization method provides good convergence characteristics with respect to certain norms, in addition to bounds on effective properties. Solutions of RVE boundary value problems with imposed unit macroscopic strains are used in the calculation of the anisotropic homogenized elasticity tensor ( ) H C ijkl x . The RVE boundaries are subjected to periodicity conditions, implying that all boundary nodes separated by the periods Y1, Y2, Y3 along the three orthogonal coordinate directions will follow the displacement constraints: 1 2 3 1 11 2 2 2 3 3 3 ( , , ) ( , , ), 1,2,3. i i u x x x u x kY x kY x kY i =± ± ± = (3.13) ( ) H ijkl x , numerical simulations of the RVE boundary value problems yield stresses and strains in the microstructural RVE. Limitations of the regularized problem in Ω hom geneous microdomains arise from assumptions of relatively uniform macroscopic fields and periodicity of the RVE. In uncoupling the macroscopic problem in Ω hom from the microscopic RVE problem in Y, it is assumed that the RVE has infinitesimal dimensions in comparison with the macroscopic scale, i.e., ε → 0 . While solutions of the problems in Ω hom approach those for the actual domain Ω het in this limit, considerable differences may ε gradients, free edges, or discontinuities. result when the scale factor is finite and the RVE solutions are not periodic. The occurrence of such errors is significant in regions of high local Limitations in solving the regularized problem for variables in heteroChapter 3: Adaptive Concurrent Multilevel Model 93 Following macroscopic analysis with the homogenized modul i C
94 S.Ghosh 3.3.2 Multiple Levels Coupling Multiple Scales In the multilevel methodology developed in Ghosh et al.[35,62,64],the overall heterogeneous computational domain is adaptively decomposed into a set of nonintersecting open subdomains,which may each belong to one of (domain for microscopic analysis)om(regularized domain for macroscopic analysis),or to a combination thereof.The resulting com- putational domain may be expressed as the union of subdomains be- longing to different levels expressed as -a"uUa"uUoUUa.where-0 2”n2"=0,22n22=0,2n2=0k≠1， (3.14) and2°ng"=0,2°n22=0,2'n22=0, 2n2=0k,1. Here the superscripts 10,11,and 12 correspond to level-0,level-1,or level-2 subdomains in the computational hierarchy;and superscripts tr cor- respond to the transition region between level-0/I and level-2 subdomains. Computations in different levels require different algorithmic treatments. The number of levels may not exactly correspond to the number of scales, even though they are connected to individual scales.The constituent sub- domains,e.g.,need not be contiguous and may occupy disjoint loca- tions in However,certain restrictions apply with respect to sharing of contiguous subdomain boundaries.If and represent boundaries of the corresponding level subdomains,then, .=Vk,1.Also has the same characteristics as or since and have compatible displacements. .=0 k,l,i.e.and 0 are not contiguous or may not share common edges. .=Vk,1.Also has the same characteristics as since 2 and have compatible displacements. om=omk,1.Also the interfaces of om and 2"are not compatible in general,and hence special constraint conditions need to be developed for
3.3.2 Multiple Levels Coupling Multiple Scales In the multilevel methodology developed in Ghosh et al. [35, 62, 64], the overall heterogeneous computational domain is adaptively decomposed into a set of nonintersecting open subdomains, which may each belong to one of Ω het (domain for microscopic analysis) Ω hom (regularized domain for macroscopic analysis), or to a combination thereof. The resulting computational domain Ω het may be expressed as the union of subdomains belonging to different levels expressed as 0 t 1 2 r 10 11 12 tr 10 10 het 1111 11 11 12 12 tr tr 10 11 10 12 11 12 11 tr , where 0, 0, 0, 0 , 0, 0, 0 ,. N N N N k k k k kl kkkk k l k l kl kl kl kl k l k l k l Ω Ω Ω Ω Ω ΩΩ ΩΩ Ω Ω ΩΩ ΩΩ ΩΩ Ω Ω ==== =∪∪∪ ∩ = ∩ = ∩ = ∩ = ∀≠ ∩= ∩= ∩ =∀ ∪∪∪∪ (3.14) Here the superscripts l0, l1, and l2 correspond to level-0, level-1, or level-2 subdomains in the computational hierarchy; and superscripts tr correspond to the transition region between level-0/1 and level-2 subdomains. Computations in different levels require different algorithmic treatments. The number of levels may not exactly correspond to the number of scales, even though they are connected to individual scales. The constituent subdomains, e.g., 10 Ω k need not be contiguous and may occupy disjoint locations in Ω het . However, certain restrictions apply with respect to sharing of contiguous subdomain boundaries. If 11 11 12 , , ∂∂∂ Ωkkk Ω Ω and tr Ωk ∂ represent boundaries of the corresponding level subdomains, then, • 10 11 10 11 , . kl kl ΩΩ Ω k l − ∂ ∩∂ = ∂ ∀ Also 10 11 Ω kl − ∂ has the same characteristics as 10 Ωk ∂ or 11 Ωl ∂ , since 10 Ωk ∂ and 11 Ωl ∂ have compatible displacements. • 10 12 0 , k l ∂ ∩∂ = ∀ Ω Ω k l , i.e., 10 Ω k ∂ and 12 Ωl ∂ are not contiguous or may not share common edges. • 12 tr 2 tr , l k l kl ΩΩΩ k l − ∂ ∩∂ = ∂ ∀ . Also l 2 tr Ωkl − ∂ has the same characteristics as 2 , l Ωk ∂ since 12 Ωk ∂ and tr Ωl ∂ have compatible displacements. • 10 / 1 tr 0 / 1 tr , l l l k lkl Ω ΩΩ k l − ∂ ∩∂ = ∂ ∀ . Also the interfaces of 10 / 1l Ωk ∂ and tr Ωl ∂ are not compatible in general, and hence special constraint conditions need to be developed for l l 0 / 1 tr Ωkl − ∂ . 94 S. Ghosh and Ω Ω ∩ = 0
Chapter 3:Adaptive Concurrent Multilevel Model 95 A few cycles in the iterative solution process are required to settle into an "optimal"distribution of the computational levels,even for linear elastic problems.The three levels of computational hierarchy,in the order of sequence of evolution are discussed next. Computational subdomain level-0 o Macroscopic analysis with homogenized properties is performed in the level-0 subdomain.Unless the microstructural morphology suggests strong nonperiodicity,the computational model can generally start with the as- sumption that=(mie all elements belong to the level-0 subdomain.This subdomain assumes relatively uniform defor- mation with"statistically"periodic microstructures,where the regularized problem formulation is nearly applicable.Upon establishing a representa- tive volume element Y(x)for the material at a point x,the asymptotic expansion-based homogenization method is implemented to yield an assumed orthotropic homogenized elasticity tensorCfrom(3.10). Components of C()for plane problems are calculated from the solu- tion of three separate boundary value problems of the RVE with periodic boundary conditions and imposed unit macroscopic strains given as e 22 e22 (3.15) 10 0 The homogenized elastic stiffness componentsCC Cand C are calculated from the volume averaged stresses according to (3.10).In the event that the elastic coefficient Cis needed,a fourth boundary value problem should be solved with ()=(00.0,1.Since the microstructure and the corresponding RVE can change from element to element(E)in the computational domain,each element Eto should be assigned its location specific RVE Y(x).Drastically different moduli in adjacent elements could lead to non- physical stress concentrations.Smoothing schemes may be needed for regularization in these regions for macroscopic analysis.However,switch- ing levels can enable a smooth transition from one RVE to another through the introduction of intermediate level-2 regions
an “optimal” distribution of the computational levels, even for linear elastic problems. The three levels of computational hierarchy, in the order of sequence of evolution are discussed next. Computational subdomain level-0 l0 Ω 11 11 11 22 22 22 12 12 12 10 0 0, 1, 0. 00 1 ee e ee e ee e ⎧ ⎫ ⎧⎫ ⎧ ⎫ ⎧⎫ ⎧ ⎫ ⎧⎫ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎨ ⎬ ⎨⎬ ⎨ ⎬ ⎨⎬ ⎨ ⎬ ⎨⎬ == = ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎪ ⎩ ⎭ ⎩⎭ ⎩ ⎭ ⎩⎭ ⎩ ⎭ ⎩⎭ (3.15) Macroscopic analysis with homogenized properties is performed in the level-0 subdomain. Unless the microstructural morphology suggests strong nonperiodicity, the computational model can generally start with the assumption that ( ) 0 0 0 het 1 hom l l N ΩΩ Ω Ω == ⊂ ∪k k = , i.e., all elements belong to the level-0 subdomain. This subdomain assumes relatively uniform deformation with “statistically” periodic microstructures, where the regularized problem formulation is nearly applicable. Upon establishing a representative volume element Y( ) x for the material at a point x, the asymptotic expansion-based homogenization method is implemented to yield an assumed orthotropic homogenized elasticity tensor ( ) H C ijkl x from (3.10). Components of ( ) H C ijkl x for plane problems are calculated from the solution of three separate boundary value problems of the RVE with periodic boundary conditions and imposed unit macroscopic strains given as The homogenized elastic stiffness components 1111 2222 , , H H C C 1212 , H C 1133 2233 1122 , , and HH H CC C are calculated from the volume averaged stresses Σ ij according to (3.10). In the event that the elastic coefficient 3333 H C is needed, a fourth boundary value problem should be solved with ( ) ( ) T T 11, 22, 12, 33 eeee = 0,0,0,1 . Since the microstructure and the corresponding RVE can change from element to element ( ) 0 0 l El ∈Ω in the computational domain, each element El 0 should be assigned its location specific RVE Y( ) x . Drastically different moduli in adjacent elements could lead to nonphysical stress concentrations. Smoothing schemes may be needed for regularization in these regions for macroscopic analysis. However, switching levels can enable a smooth transition from one RVE to another through the introduction of intermediate level-2 regions. Chapter 3: Adaptive Concurrent Multilevel Model 95 A few cycles in the iterative solution process are required to settle into
96 S.Ghosh Level-0 mesh enrichment by h-and hp-adaptation Computational models in the level-0 subdomains are enhanced adaptively by selective h-or hp-mesh refinement strategy based on suitably chosen "error"criteria.Local enrichment through successive mesh refinement or interpolation function augmentation serves a dual purpose in the multilevel computational strategy.The first goal is to identify regions of high discretization "error"and improve convergence through mesh enhance- ment in a finite element subspace VV with the requirement Findusatisfying: aWv:E"Vu do=avdo+tvdr VeV (3.16) such thatspreset tolerance. The second is to identify regions of high modeling error due to limitations of the regularized problem in representing the heterogeneous domain and to zoom in on these regions to create higher resolution.These regions are generally characterized by large solution gradients and localization of regularized macroscopic variables.Element refinement is helpful in reducing the length-scale disparity between macroscopic elements in mand the local microstructure In [35,64],the h-adaptation procedure has been used to subdivide macroscopic elements into smaller elements in regions of high stress or strain gradients,while keeping the order of interpolation fixed.The rate of convergence of this method for nonsmooth domains is quite limited, especially with solution singularities,e.g.,in Szabo and Babuska [74].As a remedy,the hp-version of finite element refinement has been established [1].This method is capable of producing exponentially fast convergence in the finite element approximations to the energy norm for solutions of linear elliptic boundary value problems on nonsmooth domains,such as those with singularities.The rate of convergence of the hp-finite element model is estimated by the inequality u-u≤Chpm-, (3.17) where h is the mesh size,p is the order of interpolation polynomial,m cor- responds to the regularity of the solution,C is a constant and u=min(p,m-1).The parameter m dictates the distribution and sequence of h-and p-refinements in the hp-adaptation scheme.Smaller m leads to
Level-0 mesh enrichment by h- and hp-adaptation Computational models in the level-0 subdomains are enhanced adaptively by selective h- or hp-mesh refinement strategy based on suitably chosen “error” criteria. Local enrichment through successive mesh refinement or interpolation function augmentation serves a dual purpose in the multilevel computational strategy. The first goal is to identify regions of high V V adap ⊂ with the requirement Find H H adap adap , | , Γ =u uu u satisfying: hom hom H adap adap H H adap :: d d d such that preset tolerance. Ω Ω Γ t ∇ ∇ = ⋅ + ⋅ ∀∈ Ω ΩΓ − ≤ ∫ ∫∫ H vE u fv tv v V u u (3.16) The second is to identify regions of high modeling error due to limitations of the regularized problem in representing the heterogeneous domain and to zoom in on these regions to create higher resolution. These regions are generally characterized by large solution gradients and elements in Ω hom and the local microstructure Ω het . In [35, 64], the h-adaptation procedure has been used to subdivide macroscopic elements into smaller elements in regions of high stress or strain gradients, while keeping the order of interpolation fixed. The rate of convergence of this method for nonsmooth domains is quite limited, especially with solution singularities, e.g., in Szabo and Babuska [74]. As a remedy, the hp-version of finite element refinement has been established [1]. This method is capable of producing exponentially fast convergence in the finite element approximations to the energy norm for solutions of linear elliptic boundary value problems on nonsmooth domains, such as those with singularities. The rate of convergence of the hp-finite element model is estimated by the inequality ( 1) , hp m fe Ch p u µ − − u u − ≤ (3.17) where h is the mesh size, p is the order of interpolation polynomial, m corresponds to the regularity of the solution, C is a constant and µ = − min( , 1) p m . The parameter m dictates the distribution and sequence of h- and p-refinements in the hp-adaptation scheme. Smaller m leads to discretization “error” and improve convergence through mesh enhancement in a finite element subspace is helpful in reducing the length-scale disparity between macroscopic 96 S. Ghosh localization of regularized macroscopic variables. Element refinement
Chapter 3:Adaptive Concurrent Multilevel Model 97 algebraic rates,while large m for smooth solutions yield exponential rate of convergence with successive p-refinements.The adaptation scheme follows the criteria:Perform p-refinement if p+2sm,and perform h-refinement if p+2 m. It is necessary to solve a sequence of element level regularized boundary value problems in 2 to estimate the local regularity parameter m.If(k)characterizes the error estimator in the FE space Y(k)for the kth element,using polynomials of order p+q(g is the enhancement), i.e., B)=-Vv:(E"Vut do+fvdo (3.18) gvdog Vvey(k). Where g=n+'nmnoctom and g=temr,is the approximate traction on a (3.19) Here(k)is interpreted as the finite element approximation to the true error e(k)=uuin element k,such that the total error is bounded by the sum of the element-wise error estimatorse.The parameter m is estimated by solving the local element boundary value problem in (3.18)for three successive values of g and solving for C mand from the approximate convergence criterion. A numerical example of the regularized problem Convergence of the hp-adaptive refinement is explored for a composite laminate (Fig.3.3a)in this example.The top half of the laminate (above A-A)consists of 30.7%volume fraction of silicon carbide fibers in an epoxy matrix with homogenized orthotropic elasticity matrix (in GPa) as CM=9.1.C=9.1,CM=.3.C=3.7,C=4.1,and C=104.2.The bottom half consists of a monolithic matrix material with properties Epoy=3.45 GPa,=0.35.Due to symmetry in the x= and yz planes,only one quarter of the laminate is modeled.Symmetric boundary conditions are employed on the surfaces x=0 and y=0,and the top and right surfaces are assumed to be traction free
algebraic rates, while large m for smooth solutions yield exponential rate of convergence with successive p-refinements. The adaptation scheme follows the criteria: Perform p-refinement if p m + 2 ; ≤ and perform h-refinement if p m + > 2 . It is necessary to solve a sequence of element level regularized boundary value problems in Ω hom to estimate the local regularity parameter m. If ( ) p q k φ + characterizes the error estimator in the FE space ( ) p q Y k + for the kth element, using polynomials of order p+q (q is the enhancement), i.e., hom hom hom H fem ( , ) :( d d d ( ), k k k k pq k p q B Y k Ω Ω Ω Ω Ω Ω + + ∂ =− ∇ ∇ + ⋅ + ⋅ ∂ ∀∈ ∫ ∫ ∫ H v v E u fv gv v φ (3.18) hom ' ' hom hom hom 1 where [ ] 2 and is the approximate traction on . k k k k kk k k k k t Ω Ω Ω Ω ∂ = ⋅ + ⋅ ∈∂ ∂ = ∈∂ Γ g σ n σ n g t ∩ ∩ (3.19) Here ( ) p q k + φ is interpreted as the finite element approximation to the true error fem ( ) H H e uu k = − in element k, such that the total error is bounded by the sum of the element-wise error estimators 2 2 k k e ≤ ∑ φ . The parameter m is estimated by solving the local element boundary value problem in (3.18) for three successive values of q and solving for , and k k C m φ from the approximate convergence criterion. A numerical example of the regularized problem Convergence of the hp-adaptive refinement is explored for a composite laminate (Fig. 3.3a) in this example. The top half of the laminate (above A-A) consists of 30.7% volume fraction of silicon carbide fibers in an epoxy matrix with homogenized orthotropic elasticity matrix (in GPa) as 1111 = 9.1, = 9.1, 2.3, 3.7, 2222 1212 1133 H H HH CC CC = = 2233 4.1 H C = , and 1122 104.2 H C = .The bottom half consists of a monolithic matrix material with properties Eepoxy = 3.45 GPa, epoxy ν = 0.35. Due to symmetry in the xz and yz planes, only one quarter of the laminate is modeled. Symmetric boundary conditions are employed on the surfaces x = 0 and y = 0, and the top and right surfaces are assumed to be traction free. Chapter 3: Adaptive Concurrent Multilevel Model 97