98 S.Ghosh The regularized laminate problem is subsequently analyzed using the h-and hp-adapted level-0 finite element codes,subjected to constant axial strain =1.0 in the out-of-plane direction.While the analytical trans- verse stresso is approximately two orders lower compared to the lead- ing order stressit exhibits a singularity of the formC near the interface-free edge juncture 4(x/h=4).Here r is the distance from the singular point at the free edge and C.is a constant along each radial line at a fixed angle depending on material properties.The exponent A has been evaluated to be 0.9629358 in [6]from traction and displacement con- tinuity at the material interface and traction free conditions on edges.The initial mesh consists of 200 QUAD4 elements.Adaptations are performed in each element until the element error meets the criterion25 The h-and hp-adapted mesh are shown in Figs.3.2c and 3.3a,respectively. Following iterative cycles,the converged h-adapted mesh consists of 1,664 elements with 3,282 degrees of freedom,while the converged hp-adapted mesh consists of 344 elements with 1,834 degrees of freedom.The small- est element size in both cases is 0.0025he,where he is the initial element size. TOTAL ELEMENT。I剩 TOTL68F.3拉粉 性tt#tt出 (a) (b) (c) Fig.3.3.(a)Unidirectional composite laminate subjected to out-of-plane loading; (b)a representative volume element of the microstructure,with a single fiber in a square matrix;(c)FE model with h-adapted mesh
The regularized laminate problem is subsequently analyzed using the h- and hp-adapted level-0 finite element codes, subjected to constant axial strain 1.0 zz ε = in the out-of-plane direction. While the analytical transverse stress σ yy is approximately two orders lower compared to the leading order stress σ zz , it exhibits a singularity of the form 1 yy s C rλ σ + = near the interface-free edge juncture A (x/h = 4). Here r is the distance from the singular point at the free edge and Cs is a constant along each radial line at a fixed angle θ, depending on material properties. The exponent λ has been evaluated to be 0.9629358 in [6] from traction and displacement continuity at the material interface and traction free conditions on edges. The initial mesh consists of 200 QUAD4 elements. Adaptations are performed max 0.25 k φ φ ≤ . The h- and hp-adapted mesh are shown in Figs. 3.2c and 3.3a, respectively. Following iterative cycles, the converged h-adapted mesh consists of 1,664 elements with 3,282 degrees of freedom, while the converged hp-adapted mesh consists of 344 elements with 1,834 degrees of freedom. The smallest element size in both cases is 0.0025he, where he is the initial element size. (a) (b) (c) Fig. 3.3. (a) Unidirectional composite laminate subjected to out-of-plane loading; (b) a representative volume element of the microstructure, with a single fiber in a square matrix; (c) FE model with h-adapted mesh in each element until the element error meets the criterion 98 S. Ghosh
Chapter 3:Adaptive Concurrent Multilevel Model 99 TOTAL ELEMENT =344 TOTAL DOF=1834 0P=1 口P=2 日P=3 Heterogeneous Material Monolithic Material (a) (KieIn3uis jo u3uans) 0 h adaptation -2 hp adaptation ..hp adaptation with additional local p enrichment Analytical 3 0.5 1 1.5 2 2.5 3 r(radial distance from singular point) (b) Fig.3.4.(a)hp-adapted meshes in the regularized domain ,(b)convergence of the strength of singularity for the h-and hp-adapted meshes The strength of the singularity A controls the rate of convergence and its value may be determined in the course of the adaptive refinements.The value of Ais obtained by evaluatingat two different values ofr close
(a) (b) Fig. 3.4. (a) hp-adapted meshes in the regularized domain l 0 Ω , (b) convergence of the strength of singularity for the h- and hp-adapted meshes The strength of the singularityλ controls the rate of convergence and its value may be determined in the course of the adaptive refinements. The value of λ is obtained by evaluating σ yy at two different values of r close Chapter 3: Adaptive Concurrent Multilevel Model 99
100 S.Ghosh to the singular point,and its convergence is shown in Fig.3.4b.For the same smallest element size,the h-adapted mesh reaches up to a value of A=0.66,whereas the hp-adapted mesh goes up to =0.78.Upon further enriching elements near the singular point by p-adaptation,reaches 0.89. Local and pollution errors in the regularized problem A posteriori error estimates based on elemental stresses or strain energy, e.g.,jumps in variables,their gradients,or element residuals,are local in nature.Babuska [6]and Oden [54]have introduced element pollution error as one that is produced due to residual forces in other contiguous and noncontiguous elements in the mesh.Pollution error can be significant with uniform meshes in problems containing singularities,and local error estimation methods are incapable of detecting them.Consequently,in domains consisting of cracks,free edges,laminate interfaces,etc.accurate element error estimates in the energy norm may benefit from the addition of pollution errors to the local errors,i.e., ler er+lerlpouin (3.20) with equidistribution of error estimates in the mesh,the pollution error is negligible.However in problems where the singularity exponent is less than half the order of interpolation p,i.e.,2<p,the pollution error is significant.The basic algorithm develops an equivalent residual as the sum of element-wise local and pollution residuals as B(cr.)(r)+B(er.where 2Be） +V-(EV+. (3.21) 2aw-之.br,V-vld0-nwer and g=Gx 'ng-gngE 0ctomhom (3.22) g6=t-ok'nk∈a2m∩T, Here is the polynomial subspace of V,is an enriched space approximation of.The major steps in the evaluation of the pollution error are given in [54,62]
to the singular point, and its convergence is shown in Fig. 3.4b. For the same smallest element size, the h-adapted mesh reaches up to a value of λ = 0.66, whereas the hp-adapted mesh goes up to λ = 0.78. Upon further enriching elements near the singular point by p-adaptation, λ reaches 0.89. Local and pollution errors in the regularized problem A posteriori error estimates based on elemental stresses or strain energy, e.g., jumps in variables, their gradients, or element residuals, are local in nature. Babuska [6] and Oden [54] have introduced element pollution error as one that is produced due to residual forces in other contiguous and noncontiguous elements in the mesh. Pollution error can be significant with uniform meshes in problems containing singularities, and local error estimation methods are incapable of detecting them. Consequently, in domains consisting of cracks, free edges, laminate interfaces, etc. accurate element error estimates in the energy norm may benefit from the addition of pollution errors to the local errors, i.e., local pollution ee e er er er = + (3.20) with equidistribution of error estimates in the mesh, the pollution error is negligible. However in problems where the singularity exponent λ is less than half the order of interpolation p, i.e., 2λ < p , the pollution error is significant. The basic algorithm develops an equivalent residual as the sum of element-wise local and pollution residuals as local poll local 1 1 H 1 poll H 1 1, ( , ) [ ( , ) ( , )], where ( , ) [ ( )] d d ˆ ( , ) [ ( : )] d d ˆ E E E k k E E j j N N h h kh h kh h kh h k k N h k h k h h k N N h kh h j h j h h k j j k B BB B V B V Ω Ω Ω Ω Ω Ω Ω Ω = = ∂ = ∂ = = ≠ = + = +∇⋅ ∇ + ⋅ ∂ ∀ ∈ = +∇ ∇ + ∂ ∀ ∈ ∑ ∑ ∑∫ ∫ ∑ ∑ ∫ ∫ er v er v er v er v f Eu v g v v er v f E u v g v v (3.21) ' ' ' ' hom hom k hom and = , = t . k k k k k k k k k k t Ω Ω Ω Γ ⎡ ⎤ ⋅ − ⋅ ∈∂ ∩∂ ⎣ ⎦ − ⋅ ∈∂ ∩ g σ n σ n g σ n (3.22) Here H V is the polynomial subspace of V , h V is an enriched space approximation of H V . The major steps in the evaluation of the pollution error are given in [54, 62]. 100 S. Ghosh
Chapter 3:Adaptive Concurrent Multilevel Model 101 Composite laminate subjected to out-of-plane loading The problem of composite laminate with free edge,similar to the one in Sect. 3.2.1 is studied to understand the effect of local and pollution errors.The top half of the laminate is a composite with 28.2%volume fraction of boron fiber in epoxy matrix with effective orthotropic homogenized properties: Eu(psi) E22(psi) E3s(psi) G2(psi) V V到 V2 0.99×106 0.99×10 17.2×10 0.27×10 0.43 0.29 0.29 The bottom half is monolithic epoxy material with properties Epoy=0.5x10psi andvpoy=0.34.Out-of-plane loading is simulated using generalized plane strain condition with prescribed s=0.1%.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. For a uniform mesh,the local error is concentrated near the intersection of the interface and the free edge region,whereas the pollution error is more diffused and occurs in bands,starting at points slightly away from the intersection point around the free edge.When h-adaptation is applied, the maximum local error reduces from 6.285x10 to 1.176x10,while the pollution error reduces from 5.115x10 to 3.105x10.This is shown in Fig.3.5.For this problem,the inclusion of pollution error in the total element error estimate is found to add little to the criteria for h-and hp- adaptation and only local error is considered henceforth. Micromechanical analysis with the Voronoi cell FEM Accurate micromechanical modeling of deformation and damage in com- plex heterogeneous microstructures requires very high resolution models. Micromechanical analysis in the multilevel computational framework is conducted by the Voronoi cell finite element model (VCFEM)developed by Ghosh et al.in [26,34,46,48-50]for accurate and efficient image- based modeling of nonuniform heterogeneous microstructures.Morpho- logical arbitrariness in dispersion,shape,and size of heterogeneities,as acquired from actual micrographs are readily modeled by this method.The VCFEM computational mesh results from tessellating the microstructure
Composite laminate subjected to out-of-plane loading The problem of composite laminate with free edge, similar to the one in Sect. 3.2.1 is studied to understand the effect of local and pollution errors. The top half of the laminate is a composite with 28.2% volume fraction of boron fiber in epoxy matrix with effective orthotropic homogenized properties: E11 (psi) E22 (psi) E33 (psi) G12 (psi) ν 12 ν 31 ν 23 6 0.99 10 × 6 0.99 10 × 6 17.2 10 × 6 0.27 10 × 0.43 0.29 0.29 The bottom half is monolithic epoxy material with properties 6 Eepoxy = 0.5 10 psi and 0.34 × = ν epoxy . Out-of-plane loading is simulated using generalized plane strain condition with prescribed ε zz = 0.1% . 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. For a uniform mesh, the local error is concentrated near the intersection of the interface and the free edge region, whereas the pollution error is more diffused and occurs in bands, starting at points slightly away from the intersection point around the free edge. When h-adaptation is applied, the maximum local error reduces from 3 6.285 10− × to 4 1.176 10− × , while the pollution error reduces from 4 5.115 10− × to 5 3.105 10− × . This is shown in Fig. 3.5. For this problem, the inclusion of pollution error in the total element error estimate is found to add little to the criteria for h- and hpadaptation and only local error is considered henceforth. Micromechanical analysis with the Voronoi cell FEM Accurate micromechanical modeling of deformation and damage in complex heterogeneous microstructures requires very high resolution models. Micromechanical analysis in the multilevel computational framework is conducted by the Voronoi cell finite element model (VCFEM) developed by Ghosh et al. in [26, 34, 46, 48–50] for accurate and efficient imagebased modeling of nonuniform heterogeneous microstructures. Morphological arbitrariness in dispersion, shape, and size of heterogeneities, as acquired from actual micrographs are readily modeled by this method. The VCFEM computational mesh results from tessellating the microstructure Chapter 3: Adaptive Concurrent Multilevel Model 101
102 S.Ghosh Max 1.176E04 9.414E05 7.064E05 4.714E05 2.364E-05 Min. L1.416t07 (a) Max一 1.176E04 9414E05 7.084E05 4.714E05 2.364E05 Min. -1.416t0/ (b) Fig.3.5.Distribution of(a)local and(b)pollution error for the h-adapted mesh with dispersed heterogeneities into a network of N multisided Voronoi polygon or cell elements,i.e,as shown in Fig.3.8.Each Voronoi cell with embedded heterogeneities(particle,fiber, void,crack,etc.)represents the region of contiguity for the heterogeneity and is treated as an element in VCFEM.In this sense,a Voronoi cell ele- ment manifests the basic structure of the material microstructure and its evolution and is considerably larger than conventional FEM elements. Incorporation of known functional forms from analytical micromechanics substantially enhances its convergence.The VCFEM formulation is based on the assumed stress hybrid finite element method,and makes independ- ent assumptions of equilibrated stress fields in the interior of each
(a) (b) Fig. 3.5. Distribution of (a) local and (b) pollution error for the h-adapted mesh VC E Ω micro with dispersed heterogeneities into a network of NVCE multisided Voronoi polygon or cell elements, i.e., VCE VCE VCE micro 1 N e e Ω Ω= = ∪ , as shown in Fig. 3.8. Each Voronoi cell with embedded heterogeneities (particle, fiber, void, crack, etc.) represents the region of contiguity for the heterogeneity and is treated as an element in VCFEM. In this sense, a Voronoi cell element manifests the basic structure of the material microstructure and its evolution and is considerably larger than conventional FEM elements. Incorporation of known functional forms from analytical micromechanics substantially enhances its convergence. The VCFEM formulation is based on the assumed stress hybrid finite element method, and makes independent assumptions of equilibrated stress fields ( ) m c/ σ ij in the interior of each 102 S. Ghosh