Numerical methods for modeling heterogeneous materials 2

Chapter Extended Finite Element Method 5.1 Introduction In the previous chapter, we have studied the VCFEM in applying for composite materials. It shows that the VCFEM cannot produce the stress solution correctly, especially when either the inclusion size or the difference between two materials’ properties is large. Thus the VCFEM is advantageous in analyzing composite materials at macro-scale only. It is necessary to use other methods for modeling the composite materials at micro-scale. The extended finite element method (XFEM) is one possibility. The XFEM is a well known method that is useful in modeling structures with discontinuities and/or singularities. Perhaps the most important advantage of this method is that it can model discontinuities without conforming the mesh with the discontinuities [3, 49], which is a challenge in the conventional FEM. The XFEM improves the versatility of the conventional FEM by introducing a local enrichment of the approximating space. The method has wide applications such as crack propagations [62], material surfaces [96] or structures with voids [61], etc. 99 5.2 5.2.1 Element formulation Level set method The level set method, which is a numerical method for tracking interfaces and shapes [97], is usually used in the XFEM to capture the discontinuities. In two dimensions, a close curve Γint can be represented as the zero-level set of a function φ as follows: Γ = {x ∈ R2 : φ(x) = 0} (5.1) where φ is called a level set function. φ is assumed to have positive values inside the domain bounded by Γ and negative values outside. Hence the curve Γint can be captured implicitly by the level set function φ. An important example of such a function φ would be the signed distance function φ(x) = ± x − xΓ xΓ ∈Γ (5.2) In case the shape of a void or an inclusion is circular, the formulation of φ would be φ = x − xc − rc (5.3) where xc and rc is the center and the radius of the circular inclusion or void. Figure 5-1 shows an example of a level set function for the case of a circular shape. If there are several discontinuities, the value of level set function at one point is the minimum value of all level set functions of each shape. Figure 5-2 shows a level function for the case of multiple elliptical shapes. 100 0.5 0.9 0.4 0.8 0.3 0.7 0.2 0.6 0.1 0.5 0.4 −0.1 0.3 −0.2 0.2 −0.3 0.1 −0.4 (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −0.5 (b) (c) Figure 5-1: Level set (a) zero level set (b) level set contour (c) level set function 101 0.5 0.9 0.4 0.8 0.3 0.7 0.2 0.6 0.1 0.5 0.4 −0.1 0.3 −0.2 0.2 −0.3 0.1 −0.4 (a) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 −0.5 (b) (c) Figure 5-2: Level set (a) zero level set (b) level set contour (c) level set function 102 5.2.2 General formulation In XFEM, the presence of discontinuities is taken into account by the enrichment functions at the region nearby these entities. Consider a two dimensional domain Ω ⊂ R2 , which is partitioned in to N el finite elements. Let I be the set of all nodes that assembles the elements and J be the set of nodes that need to be enriched. The displacements field in the XFEM can be approximated by a general form u(x) = N i (x)ui + i∈I M j (x)ψj (x)aj (5.4) j∈J stdrd. FE approx. enrichment in which the first term is identical to the conventional or the standard FEM approximation and the additional term stands for the enrichment. N i and ui are the standard FE shape functions and unknown nodal displacements respectively; ψj are the enrichment functions; aj are the unknown variables corresponding to the enriched nodes; M j are also the FE shape functions, which are not necessarily the same as N i but need to form a partition of unity, e.g. Mj = (5.5) j∈J The approximation 5.4 can be easily implemented in the conventional FE formulation if the unknown variables aj are treated as an additional degrees of freedom at enriched nodes. Similar to the conventional FEM, we need to find the solution of the displacement u, which satisfies the principal of minimum potential energy Π= where T Ω uT T dΓ C dΩ − (5.6) Γ are the strains; C is the stiffness matrix of the constituent materials; T are the prescribed boundary tractions. The strains can be derived from Equation 5.4 as 103 follow = Bd (5.7) where d is the extended nodal displacement which is added by the unknown variables aj ; B is formed as follow   (M i ψi ),x  N i,x  Bi =  N i,y (M i ψi ),y   N i,y N i,x (M i ψi ),y (M i ψi ),x      (5.8) The formulation of B at unenriched nodes is the same in the standard FEM. Substituting Equation 5.4 and Equation 5.7 into 5.6 gives Π = dT Kd − dT F (5.9) where B Ti CB j dΩ K ij = Ω Fi P T T dΓ = Γ where P i ≡ N i for a non-enriched node, and P i ≡ M i ψi for an enriched node. The stationary of the potential energy 5.9 gives the familiar equilibrium equation Kd = F 5.2.3 (5.10) Choice of enriched nodes The level set function φ is used to track the evolution of discontinuities and then to determine the nodes that need to be enriched. All the elements of the mesh will be scanned over to determine the elements that are cut by the discontinuities. For a given element I, if there exist any two nodes i and j, (i, j ∈ I) such that φi φj < 0, 104 that element will be enriched. All of the nodes of that enriched element will be added to the set of nodes J that need to be enriched. Figure 5-3 illustrates an enriched element and its respecting enriched nodes. 5.2.4 Enrichment function The enrichment functions are usually functions of the level set functions. The choice of an enrichment function related to a discontinuity is generally based on the type of that discontinuity, i.e. strong or weak discontinuity. Strong discontinuity is a discontinuity in the solution such as displacement, etc. Strong discontinuities are usually used to model cracks or holes. The enrichment function for strong discontinuities is Heaviside function [67] ψ = H(φ) (5.11) Weak discontinuity is a discontinuity in derivatives of the solution such as stress and strain. Weak discontinuities are used to model the interfaces between materials or phases of materials. The ”ramp” function, or the so-called abs-enrichment, is used as the enrichment function for this type of discontinuity ψ = |φ(x)| (5.12) In this study, only the weak discontinuities are considered. Hence the abs-enrichment will be used as the enrichment function. We shall even use the weak discontinuity to model strong discontinuities by a ”penalty” approach. It is well-known that modeling a closed strong discontinuity without proper treatments will render the stiffness matrix singular. However, such a situation will not appear if the discontinuity is a weak discontinuity rather than a strong discontinuity. 105 5.2.5 Implementation One of the tricky parts of the XFEM implementation is the calculation of the integral involving the stiffness matrix calculation of enriched elements. For each enriched element, it is necessary to identify the points that correspond to the zero level set along the edges of the element (if exist). The element then needs to be divided into sub-triangle based on these zero-level set points using the Delaunay triangulation, and the Gaussian integration will be performed for each sub-triangle. Since only absenrichment function is considered, the integration order does not need to be high. A 3rd order integration is sufficient. Figure 5-3 illustrates this integration process.  ✂✁☎✄  ✝✁✞✄  ✠✟✡✄  ✂✆☎✄  ✠✟✡✄  ✝✁✞✄ Figure 5-3: Domain subdivision and integration points 5.3 5.3.1 Numerical results A unit cell containing a circular inclusion This example is for validation purpose and also for a comparison between the XFEM and the VCFEM. Consider the first example described in the previous chapter (Sec106 tion 4.4.2) which corresponds to a quarter of a bi-unit cell containing a circular inclusion of radius R = 0.2. The material properties were E1 = 1, ν1 = 0.3 for the matrix and E2 = 0.1, ν2 = 0.3 for the inclusion. Symmetric boundary conditions were applied at the left and bottom edges of the model and uniformly distributed unit tensile loads were applied at the top and the right edges. The level set function representing the inclusion was shown in Figure 5-1. A uniform mesh containing 64 × 64 elements and 3969 nodes were used as shown in Figure 5-4. In this mesh, 312 elements were cut by the inclusions and thus were enriched. The number of enriched nodes were 208. Therefore, the number of additional degree of freedoms (DOF) was 208 × = 416. (a) (b) Figure 5-4: Meshes (a) FEM (b) XFEM: ◦ indicates enriched nodes, and enriched elements are those with thick line The same FE model as in Section 4.4.2, which consisted of 5203 nodes and 5074 elements was used as the reference. Figure 5-5 and Figure 5-6 show the comparisons of the displacement and the von-Mises stress of the two solutions. For the sake of illustration, the enriched elements were split to highlight the jump of the stress and the change of the displacement across the interface of the two materials. It is clearly shown that the XFEM solution was in good agreement with the conventional FEM solution in both the displacement and the stress despite of the coarser mesh. Looking back at Figure 4-8, we can see that the stress from XFEM solution was better than 107 that of VCFEM solution. Though the computation of the XFEM is more expensive than that of the VCFEM, the XFEM is clearly a better choice when we need accurate stress distribution. 1.6 1.6 1.4 1.4 1.2 1.2 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0.2 0.4 0.6 0.8 1.2 (a) FEM 1.4 1.6 0.2 0.4 0.6 0.8 1.2 1.4 1.6 (b) XFEM Figure 5-5: Total displacement of the unit cell containing one inclusion Figure 5-6: Von-mises stress distribution of the unit cell containing one inclusion: FEM (left) and XFEM (right) 5.3.2 A FGM specimen Consider a model corresponding to a FG plate under a tensile load that was solved by the VCFEM in Section 4.4.7 The square plate of unit size was made of a FGM whose properties gradually changed along the y direction. The plate was made of 108 Appendix C Interpolation Polynomial Matrices First order P, for 3-sided to 5-sided Voronoi cells:   x 0   0 y   P =  y x   0    0 −y −x Second order P, for 6-sided to 7-sided Voronoi cells:   x 0 y 2xy x 0   0 y   P =  y x 0 y 2 x y x2   0    0 −y −x 0 −y −2 x y −x2 130 Third order P, for 8-sided to 10-sided Voronoi cells:  x 0 y2 x y x2 0 y3 x y2  0 y  P =  y x 0 y 2 x y x2 0 ···  0  0 −y −x 0 −y −2 x y −x2 0 −y  3x y x 0   ··· y3 x y x y x3    −3 x y −3 x2 y −x3 Fourth order P, for 11-sided to 14-sided Voronoi cells:  P = x 0 y2 x y x2 0 y3 x y2  0 y   0 0 y x 0 y 2 x y x2 0 ···   0 −y −x 0 −y −2 x y −x2 0 −y 3x y ··· y3 x y x y x y x3 −3 x y −3 x2 y −x3 0 4xy 2x y y4 x y3 6x y −y −4 x y −3 x2 y x    x y x y x4    −4 x3 y −x4 131 Fifth order P, for 15-sided to 18-sided Voronoi cells:  P = x 0 y2 x y x2 0 y3 x y2  0 y   0 0 y x 0 y 2 x y x2 0 ···   0 −y −x 0 −y −2 x y −x2 0 −y 3 x2 y ··· y3 x3 x y x y x3 −3 x y −3 x2 y −x3 y 5 x y 10 x2 y ··· x4 y4 x y3 y5 0 x3 y x y4 −y −5 x y −2 x2 y x2 y x3 y y4 x y3 x4 x y x y x4 · · · −y −4 x y −3 x2 y −4 x3 y  x y x 0   x2 y 10 x3 y x4 y x5    −2 x3 y −5 x4 y −x5 −x4 132 Bibliography [1] ABAQUS 6.5. 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Pankaj, “Towards modelling of a trabecular bone,” Computers & Structures. 145 [...]... y2 2 x y x2 0 0 y3 3 x y2  1 0 0 y  P =  0 0 1 0 0 y x 0 0 y 2 2 x y x2 0 0 ···   0 1 0 0 −y −x 0 0 −y 2 2 x y −x2 0 0 −y 3  2 3 3x y x 0 0   ··· y3 3 x y 2 3 x 2 y x3    −3 x y 2 −3 x2 y −x3 0 Fourth order P, for 11-sided to 14-sided Voronoi cells:  P = x 0 0 y2 2 x y x2 0 0 y3 3 x y2  1 0 0 y   0 0 1 0 0 y x 0 0 y 2 2 x y x2 0 0 ···   0 1 0 0 −y −x 0 0 −y 2 2 x y −x2 0 0 −y 3 2. .. y 2 3 x 2 y x3 −3 x y 2 −3 x2 y −x3 0 4 0 0 4xy 3 0 2 2 2x y y4 2 x y3 6x y 3 −y 4 −4 x y 3 −3 x2 y 2 x 4  0 0   6 x 2 y 2 4 x 3 y x4    −4 x3 y −x4 0 131 Fifth order P, for 15-sided to 18-sided Voronoi cells:  P = x 0 0 y2 2 x y x2 0 0 y3 3 x y2  1 0 0 y   0 0 1 0 0 y x 0 0 y 2 2 x y x2 0 0 ···   0 1 0 0 −y −x 0 0 −y 2 2 x y −x2 0 0 −y 3 3 x2 y ··· y3 x3 0 3 x y 2 3 x 2 y x3 −3 x y 2. .. x3 0 3 x y 2 3 x 2 y x3 −3 x y 2 −3 x2 y −x3 0 y 5 5 x y 4 10 x2 y 3 ··· x4 0 0 0 0 y4 4 x y3 0 y5 0 0 0 2 x3 y 2 x y4 −y 5 −5 x y 4 2 x2 y 3 0 6 x2 y 2 2 x3 y y4 2 x y3 x4 0 0 6 x 2 y 2 4 x 3 y x4 · · · −y 4 −4 x y 3 −3 x2 y 2 −4 x3 y  4 5 x y x 0 0   2 x2 y 3 10 x3 y 2 5 x4 y x5    2 x3 y 2 −5 x4 y −x5 0 −x4 1 32 0 Bibliography [1] ABAQUS 6.5 ABAQUS Inc [2] S Ghosh and R L Malett, “Voronoi... properties were E1 = 1, ν1 = 0.3 for the matrix The holes was modeled by the penalty method Each 1 12 3 2. 5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 Figure 5- 12: Displacement of the porous unit cell: VCFEM solution 2 2 1.8 1.8 1.6 1.6 1.4 1.4 1 .2 1 .2 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0 .2 0 .2 0 0 0 .2 0.4 0.6 0.8 1 (a) 1 .2 1.4 1.6 1.8 2 0 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 1.8 2 (b) Figure 5-13: Displacement... FIELD(NSECPT,NFIELD)=COORDS(3) C RETURN END 129 Appendix C Interpolation Polynomial Matrices First order P, for 3-sided to 5-sided Voronoi cells:   x 0 0   1 0 0 y   P =  0 0 1 0 0 y x      0 1 0 0 −y −x 0 Second order P, for 6-sided to 7-sided Voronoi cells:  2 2  x 0 0 y 2xy x 0 0   1 0 0 y   P =  0 0 1 0 0 y x 0 0 y 2 2 x y x2      0 1 0 0 −y −x 0 0 −y 2 2 x y −x2 0 130 Third order P, for 8-sided to... written in Fortran language The material definition in the main input file: *MATERIAL, NAME=FGM *ELASTIC, DEPENDENCIES=1 7.00000000E+010, 0.300,, -0. 025 00000 4 .27 000000E+011, 0.170,, 0. 025 00000 *EXPANSION, DEPENDENCIES=1 2. 34000000E-005,, -0. 025 00000 4.30000000E-006,, 0. 025 00000 *CONDUCTIVITY, DEPENDENCIES=1 2. 33000000E+0 02, , -0. 025 00000 6.50000000E+001,, 0. 025 00000 *DENSITY, DEPENDENCIES=1 2. 70700000E+003,,... the micrograph 118 bottom and subjected to a uniform distributing tensile load on the top A uniform mesh containing 128 × 128 elements and 16384 nodes were used as shown in Figures 5 -21 5586 elements were cut by the inclusions and thus were enriched The number of enriched nodes were 3861 Therefore, the number of additional DOFs was 3861 × 2 = 7 722 Figure 5 -21 : XFEM meshes: ◦ indicates enriched nodes, and... simulations of the heterogeneous materials In this study, the VCFEM was shown to be advantageous in predicting the displacements of heterogeneous models while the XFEM was advantageous in predicting the detailed stresses Hence a combination of 125 the two methods would take the advantages of the both methods The VCFEM may be used for homogenizing the effective material properties of heterogeneous materials at... “inclusion” were E2 = 0.001, which was thousand times smaller than the material of the matrix; and ν1 = 0.3 The specimen was clamped at the bottom and subjected to a uniform tensile load on the top A uniform mesh containing 25 6 25 6 elements and 65536 nodes was used as shown in Figures 5-15 18 320 elements were cut by the inclusions and thus were enriched The number of enriched nodes was 122 40 Therefore, the... result This particular example proves that the XFEM can be used to model porous materials effectively with low cost It is because the XFEM can handle the holes easily without conforming the mesh to the discontinuities (a) (b) Figure 5 -22 : Displacement and von-mises stress solutions for 128 × 128 XFEM mesh (left) and 25 6 × 25 6 XFEM mesh (right) Displacement is scaled by a factor of 1/100 5.4 Concluding . cell: VCFEM solution 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 1.8 2 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 1.8 2 (a) 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 1.8 2 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 1.8 2 (b) Figure 5-13: Displacement. accurate stress distribution. 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 (a) FEM 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 0 0 .2 0.4 0.6 0.8 1 1 .2 1.4 1.6 (b) XFEM Figure 5-5: Total. were E 1 = 1, ν 1 = 0.3 for the matrix. The holes was modeled by the penalty method. Each 1 12 −1 −0.5 0 0.5 1 1.5 2 2.5 3 3.5 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 Figure 5- 12: Displacement of the