Numerical methods for modeling heterogeneous materials 2

5.2 Element formulation5.2.1 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 discont

at macro-scale only It is necessary to use other methods for modeling the ite materials at micro-scale The extended finite element method (XFEM) is onepossibility.

compos-The XFEM is a well known method that is useful in modeling structures withdiscontinuities and/or singularities Perhaps the most important advantage of thismethod is that it can model discontinuities without conforming the mesh with thediscontinuities [3, 49], which is a challenge in the conventional FEM The XFEM im-proves the versatility of the conventional FEM by introducing a local enrichment ofthe approximating space The method has wide applications such as crack propaga-tions [62], material surfaces [96] or structures with voids [61], etc

5.2 Element formulation

5.2.1 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,

where φ is called a level set function φ is assumed to have positive values inside

captured implicitly by the level set function φ

An important example of such a function φ would be the signed distance function

φ(x) = ± min

In case the shape of a void or an inclusion is circular, the formulation of φ would be

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 isthe minimum value of all level set functions of each shape Figure 5-2 shows a levelfunction for the case of multiple elliptical shapes

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(b)

(c)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

(b)

(c)

5.2.2 General formulation

In XFEM, the presence of discontinuities is taken into account by the enrichmentfunctions at the region nearby these entities Consider a two dimensional domain

that assembles the elements and J be the set of nodes that need to be enriched Thedisplacements field in the XFEM can be approximated by a general form

in which the first term is identical to the conventional or the standard FEM

X

j∈J

The approximation 5.4 can be easily implemented in the conventional FE

enriched nodes Similar to the conventional FEM, we need to find the solution of thedisplacement u, which satisfies the principal of minimum potential energy

2Z

stationary of the potential energy 5.9 gives the familiar equilibrium equation

5.2.3 Choice of enriched nodes

The level set function φ is used to track the evolution of discontinuities and then todetermine the nodes that need to be enriched All the elements of the mesh will bescanned over to determine the elements that are cut by the discontinuities For a

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 enrichedelement 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 thatdiscontinuity, i.e strong or weak discontinuity Strong discontinuity is a discontinuity

in the solution such as displacement, etc Strong discontinuities are usually used tomodel cracks or holes The enrichment function for strong discontinuities is Heavisidefunction [67]

Weak discontinuity is a discontinuity in derivatives of the solution such as stress andstrain Weak discontinuities are used to model the interfaces between materials orphases of materials The ”ramp” function, or the so-called abs-enrichment, is used asthe enrichment function for this type of discontinuity

In this study, only the weak discontinuities are considered Hence the abs-enrichmentwill be used as the enrichment function We shall even use the weak discontinuity tomodel strong discontinuities by a ”penalty” approach It is well-known that model-ing a closed strong discontinuity without proper treatments will render the stiffnessmatrix singular However, such a situation will not appear if the discontinuity is aweak discontinuity rather than a strong discontinuity

5.2.5 Implementation

One of the tricky parts of the XFEM implementation is the calculation of the integralinvolving the stiffness matrix calculation of enriched elements For each enrichedelement, it is necessary to identify the points that correspond to the zero level setalong the edges of the element (if exist) The element then needs to be divided intosub-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 abs-enrichment function is considered, the integration order does not need to be high A

5.3.1 A unit cell containing a circular inclusion

This example is for validation purpose and also for a comparison between the XFEMand the VCFEM Consider the first example described in the previous chapter (Sec-

tion 4.4.2) which corresponds to a quarter of a bi-unit cell containing a circular

were applied at the left and bottom edges of the model and uniformly distributedunit tensile loads were applied at the top and the right edges The level set functionrepresenting 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 riched The number of enriched nodes were 208 Therefore, the number of additionaldegree of freedoms (DOF) was 208 × 2 = 416

are those with thick line

The same FE model as in Section 4.4.2, which consisted of 5203 nodes and 5074elements 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 ofillustration, the enriched elements were split to highlight the jump of the stress andthe change of the displacement across the interface of the two materials It is clearlyshown that the XFEM solution was in good agreement with the conventional FEMsolution in both the displacement and the stress despite of the coarser mesh Lookingback at Figure 4-8, we can see that the stress from XFEM solution was better than

that of VCFEM solution Though the computation of the XFEM is more expensivethan that of the VCFEM, the XFEM is clearly a better choice when we need accuratestress 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

(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 FGMwhose properties gradually changed along the y direction The plate was made of

two constituent materials whose properties were Em = 1, νm = 0.3 for the matrix

bottom edge of the specimen and sparser gradually along the y direction to the topedge The sizes of the inclusions were the same The specimen was subjected to aunit uniform tensile load applied at the top edge and the bottom edge was clamped.The XFEM mesh was uniform as shown in Figure 5-7 It consisted of 16129elements and 16384 nodes The same FEM model as in Section 4.4.7 was used as areference for validation

elements are those with thick line

Figure 5-8 and Figure 5-9 show the comparisons of the displacement and the Mises stress between two solutions For the sake of illustration, the enriched elementswere split to highlight the jump of the stress and the change of the displacementacross the interface of the two materials Again it is clearly shown that the XFEMsolution was in good agreement with the conventional FEM solution in both thedisplacement and the stress despite of the coarser mesh Looking back at Figure4-23, we can see that the stress from the XFEM solution was better than that of the

von-VCFEM solution Unlike in the von-VCFEM solution, the stresses near the inclusionswere captured correctly We can see that the reciprocal effects between adjacentinclusions are displayed clearly in Figure 5-9.

5.3.3 A unit cell containing holes

This example presents that the XFEM can also be used to model porous structures.Consider a quarter of a bi-unit cell containing 4 elliptical holes of semimajor and

method Each hole was considered as an inclusion with very small Young modulus

were modeled as an “extremely soft” inclusion Symmetric boundary conditions wereapplied at the left and bottom edges of the model and unit tensile loads were applied

at the right and top edges Both VCFEM and XFEM were used to solve this problem

Figure 5-9: Von-mises stress distribution of the FGM specimen: FEM (left) and XFEM(right)

The VCFEM model consisted of 21 nodes and four 8−node elements The XFEMmesh was a uniform mesh containing 64 × 64 elements and 3969 nodes as shown inFigures 5-11 In this mesh, 694 elements were cut by the holes and were enriched.The number of enriched nodes was 464 The number of additional DOFs was thus

464 × 2 = 928 The level set function for this XFEM model was already shown inFigure 5-2

It is clearly shown in Figure 5-12 that the solution of the VCFEM blows up since

Figure 5-11: XFEM mesh of the porous unit cell: ◦ indicates enriched nodes, and enrichedelements are those with thick line

the stiffness matrix becomes singular due to the spurious mode

On the other hand, the XFEM model gave good results The solution obtainedfrom XFEM were compared with a reference FEM model consisting of 4366 nodes and

4145 elements It is clearly shown in Figure 5-14 and Figure 5-13 that our XFEM andthe FEM solution are in good agreement for both the displacement and the stress.For the sake of illustration, enriched elements are split to represent the jump ofthe stress and the change of displacement across the holes and the porous parts ofthe XFEM model are also removed

5.3.4 A specimen made of porous material

Consider a quarter of a unit square specimen containing 95 circular holes The radius

of each circular hole was chosen so that the volume of the total porous inclusions

is equal to 30% of the total volume of the specimen The material properties were

−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

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

(b)

solu-tion(right)

Figure 5-14: Von-mises stress distribution of the porous unit cell: FEM (left) and XFEM(right)

andwas considered as an inclusion with very small Young modulus The properties of the

to a uniform tensile load on the top

A uniform mesh containing 256×256 elements and 65536 nodes was used as shown

in Figures 5-15 18320 elements were cut by the inclusions and thus were enriched.The number of enriched nodes was 12240 Therefore, the number of additional DOFswas 12240 × 2 = 24480

The solution obtained from XFEM was compared with a reference FEM modelconsisted of 61367 nodes and 58294 elements It is clearly shown in Figure 5-16 andFigure 5-17 that the XFEM displacement and stress are in good agreement with theFEM solution Although the holes are modeled by the penalty method, the XFEMhas properly captured the displacement of both the outer boundary and the inner

(a) (b)

indicates enriched nodes, and enriched elements are those with thick line

hole boundaries, as shown in Figure 5-16 Although the sizes of the XFEM and FEMmodel were almost equivalent to each other, it was noticeably that the time to solvethe XFEM model was faster than the FEM, partly because of the fact that meshingwas not required Noting that, the VCFEM was not capable of solving this problemsince the holes cause singularity due to the spurious mode

For the sake of illustration, the enriched elements were split to represent the jump

of the stress and the change of displacement across the two materials Besides, theporous parts of the XFEM model were removed

5.3.5 A unit cell of a bone model

Since the level set method is used in the XFEM to capture discontinuities, the XFEMcan be used to model any arbitrary shape without explicitly describing the disconti-nuities’ interfaces As seen in the previous examples, the penalty method is capable ofmodeling voids/holes, it is possible to use the same approach to model other complexgeometrical structures The boundaries of the structure are modeled as discontinu-ities while the unwanted parts are modeled as voids as per the penalty method One

and XFEM (right) Displacement is scaled by a factor of 1/5

of such examples is the structure of trabecular bones The structure of bones can bedetermined from microscopic images Some samples of the micrographs of the bonesare illustrated in Figure 5-18.

In this example, a macro scale (1mm) of the micrograph of a trabecular bone(source: science.nasa.gov ) was chosen to be modeled by XFEM The micrograph wasfirst de-noised and then converted into a black and white image The de-noised andthe black and white conversion helped to create a smooth level set data that describesthe geometry Although the level set could be constructed directly from the sourcemicrograph, it would be less accurate to capture the transition due to the noise andblurriness (Figure 5-20) The level set function was then constructed from the blackand white image, as shown in Figure 5-19

The porous part of the bone was also modeled by XFEM using the penalty

inside the porous inclusions were modeled by the penalty method and the material

micrograph (c) black and white image (d) level set function

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 DOFswas 3861 × 2 = 7722

with thick line

The problem was also solved by using another finer uniform mesh containing 256elements and 65536 nodes In this case 11734 elements were cut by the inclusionsand were enriched The number of enriched nodes were 8026 Hence, the number ofadditional DOFs was 8026 × 2 = 16052 It is obvious that the number of enrichedelement was increased by a factor of two, and thus the additional complexity of theXFEM model was dependent on the boundary of the discontinuous shapes represented

by the level set

For the sake of illustration, enriched elements are split to represent the jump of thestress and the change of displacement across the two materials, and the porous parts

of the XFEM model are removed The two solutions for two meshes are shown in

Figure 5-22 The results for the two models were in good agreement, thus the obtainedsolution were converged It is noticeable that although the finer mesh model showssmoother boundaries, the coarser mesh model also gives acceptable result.

This particular example proves that the XFEM can be used to model porousmaterials effectively with low cost It is because the XFEM can handle the holeseasily without conforming the mesh to the discontinuities

and 256 × 256 XFEM mesh (right) Displacement is scaled by a factor of 1/100

In this chapter, various examples of the applications of the XFEM have been shown.These examples demonstrate that the XFEM is advantageous in modeling discontinu-ities such as inclusions and/or holes Since the XFEM is not required to conform themesh to the geometry of discontinuities, it can easily model any inclusion or hole witharbitrary shape Especially from Sections 5.3.3, 5.3.4 and 5.3.5 we can see that theXFEM can model porous structures efficiently by implementing the penalty methodfor porous parts This proves that the XFEM is more flexible than the VCFEM since

the VCFEM has not been able to handle porous materials yet.

Section 5.3.5 shows an interesting application of the XFEM in porous structures.The XFEM could model such a complicated structure of a unit cell of a trabecularbone as shown in Figure 5-19 with a mesh of only 128 × 128 elements Since the mesh

of the conventional FE model was required to conform with the boundaries of thebone, this mesh would be very fine Furthermore, the meshing process would be alsocomplicated since the boundaries were arbitrary Hence, special meshing tools havebeen being developed for simulating trabecular bones [98] Some other authors whoapplied the FEM for analyzing bones used equivalent finite-elements to represents thevoxels from micrographs [99–101] In this study, the XFEM was proved to be efficient

in modeling trabecular bone structures at micro-scale

Nevertheless, the XFEM also had disadvantages In general it was more expensivethan the VCFEM For composite materials, the VCFEM needed only one Voronoicell to take care of an inclusion while the XFEM required smaller elements near aninclusion to capture correctly the boundary of the inclusion Though it was notnecessary for the XFEM mesh to conform with an inclusion, the elements needed to

be small around the inclusion because the elements needed to be cut by the level set.However, the XFEM gave better stress result than the VCFEM due to the factthat the polynomials approximation of the VCFEM could not capture the complexstress field in general cases So the XFEM is better in modeling structure at micro-scale and the VCFEM is more suitable in modeling materials at macro-scale