Multiscale Enrichment based on Partition of Unity - PDF document

1 Multiscale Enrichment based on Partition
Multiscale Enrichment based on Partition of Unity Jacob Fish and Zheng Yuan Departments of Civil, Mechanical and Aerospace Engineerinh Rensselaer Polytechnic Institute Troy, NY 12180, USA fishj@rpi.edu Abstract 1. Introduction This manuscript is concerned with developing a system The method advocated in this paper belongs to the category of methods employing hierarchical decomposition of the approximation space in the form of , where and cuuu=+ f c f One of the salient features of the mathematical homogenization theory is that the fine scale solution is completely described by the coarse scale and that the influence functions can be precomputed at a material point. The microstructure in the unit cell could be periodic or random, but the solution has to be locally periodic, i.e., at the neighboring points in the coarse mesh, homologous by periodicity (same ˆ y ), the value of a response function is the same, but at points corresponding to different points in the coarse scale (different ), the

2 value of the function can be different.
value of the function can be different. x For these assumptions to be valid a unit cell has to “experience” a constant variation of 0ij over its domain . This assumption breaks down in the high gradient regions, such as in the vicinity of cracks or cutouts or where the characteristic size of the unit cell is comparable to the coarse scale features. To allow for variation of the coarse scale solution over the unit cell domain, MEPU removes the dependency of the fine scale functions on the coarse scale solution by replacing 0ij and ()ikl y in Eq. (1) with an independent set of degrees-of-freedom ija and the influence functions defined over the local supports ()()iklN x x , respectively. It is convenient to replace a pair of subscripts in and kl ikl kla denoting the enrichment mode by a single upper case Roman subscripts, A, i.e. and ikliA klAaa . The resulting solution approximation states 0()()()iiiiAuuNdNa A xxx (3) where summation convention is employed for the repeated indices; Greek

3 subscripts denote finite element nodes.
subscripts denote finite element nodes. The discrete system of equations is obtained using a standard Galerkin method. MEPU exploits the special structure of the influence functions by utilizing Homogenization-Like Integration (HLI) scheme. The accuracy of the integration scheme is studied in Section 3. Let be a typical integral to be evaluated over the element domain , then the Homogenization-Like Integration scheme states: eIfd= e 111IIngaussngaussIIIIIIIfIJfdWJfWJfd===== (4) where is the biunit parent domain, the Jacobean, and ngauss the number of quadrature points. J The HLI scheme schematically depicted in Figure 1, centers a unit cell at each of the coarse scale Gauss points. The value of the integrand at a coarse scale Gauss point is replaced by its integral over the unit cell domain normalized by the volume of the unit cell. For coarse scale elements, encompassing numerous unit cells there is a significant cost savings compared to the direct numerical integration over all unit cells contained in

4 the coarse scale element. Moreover, dire
the coarse scale element. Moreover, direct numerical integration difficulties caused by intersecting coarse scale element boundaries with unit cell boundaries are automatically resolved by HLI. Compared to the classical homogenization theory, there is an overhead associated with: (i) additional degrees-of-freedom, , which cannot be condensed out on the element level, and (ii) the need for integration over all the coarse scale Gauss points as opposed to a single integration over the unit cell domain required by the classical homogenization theory for linear problems. Ideally, MEPU should be used in the critical regions only, whereas the classical homogenization should be employed elsewhere. The issue of transitioning between the two regions is discussed in Section 4. Aa 11- 1 I e Figure 1: Homogenization-like integration scheme Remark 1: MEPU can be generalized to quasi-continuum as follows. The starting point is the asymptotic expansion 01()(,)iiiuuuxx y and 1(,)()()iiAu 0A x xyy where y is a discrete

5 variable denoting position of atoms. Fo
variable denoting position of atoms. For more details and extension to multiple temporal scales we refer to [49]. The discrete influence function can be obtained by minimizing potential energy in a unit cell,, with respect to the discrete influence function 0(()()())iiAAu+xy 0x ()iA y subjected to periodic boundary conditions ˆ()()iAiA=+ y y y on while keeping the coarse scale fields fixed over the unit cell domain. Once the influence functions are computed, the enriched quasi-continuum interpolation is given by Eq.(3), and the formulation proceeds along the lines described in [47, 48]. 3. Analysis of the integration scheme Let be the domain of coarse scale elements in 1D such that where represents the number of unit cells in and is the size of the unit cell. Consider a typical integrand of the form where [0,1e= ] 1ucn= ucn e ()()2012ccxcxgx++ ( ) gx is a -periodic function such that () ucgxdxn== (5) xy Figure 5: Configuration for the two-dimensional continuum fracture problem We consid

6 er a unit cell with a square inclusion a
er a unit cell with a square inclusion as shown in Figure 6. The phase properties of the fine scale constituents are as follows. Inclusion : Young’s Modulus = 60GPa, Poisson’s ratio = 0.2. Matrix: Young’s Modulus = 2GPa, Poisson’s ratio = 0.2. The coarse scale domain contains 16 by 16 unit cells. The reference mesh is shown in Figure 6. a. Undeformed configuration b. Deformed configuration Figure 6: Reference mesh for the two-dimensional contimum fracture problem For comparison, three methods are investigated: (i) homogenization (HOMO), which uses standard finite element with homogenized properties, (ii) MEPU, and (iii) combination of HOMO and MEPU, where MEPU is used around the crack tip and HOMO elsewhere. Both structured and unstructured meshes are considered. All the meshes are shown in Figure 7. The enrichment functions, , are obtained using finite element solution of the corresponding unit cell problem (2). A c. Very Fine Mesh a. Coarse Mesh b. Fine Mesh Figure 8: Mesh re

7 finement for the two-dimensional contimu
finement for the two-dimensional contimum fracture problem Table 2: The relative error in mode-I stress intensity factor for two-dimensional contimum fracture problem Methods\Mesh Coarse Fine Very fine MEPU 76.11% 32.74% 0.36% MEPU_HOMO 74.48% 30.86% 1.10% HOMO 98.28% 34.72% 3.75% Figure 9: Convergence of mode-I stress intensity factor In the next set of numerical examples, we consider a three-dimensional fracture problem with two different unit cells. The former is a fibrous composite with the following phase properties. Fiber: Young’s modulus = 379GPa, Poisson’s ratio = 0.21. Matrix: Young’s modulus = 69GPa, Poisson’s ratio = 0.33. The problem configuration and the corresponding unit cell are shown in Figure 10. The latter is a satin weave [51]. The phase properties of the weave constituents are as follows. Blackglas Matrix: Young’s modulus = 69GPa, Poisson’s ratio = 0.33. Bundle phase: Young’s modulus = 679GPa, Poisson’s ratio = 0.21. The unit c

8 ell shown in Figure 11 is discretized wi
ell shown in Figure 11 is discretized with 1566 elements. In general, it is not feasible to use a discretization scheme with grid spacing comparable to the scale of spatial heterogeneities. To obtain the reference solution the satin weave problem The results of the potential energy and the energy release rate obtained using molecular mechanics (the reference solution), MEPU and Quasi-Continuum method based on the Born rule are summarized in Table 4. Table 4: Potential energy E and Potential energy release rate G of two dimensional atomic fracture problem Potential energy Potential energy release rate Methods E Error G Error QC 1.75 90.42% 0.041460 89.88% MEPU 0.925 0.65% 0.021139 3.18% REF 0.919 - 0.021835 - It can be seen that MEPU provides a very accurate solution for both the potential energy and the energy release rate, whereas quasi-continuum assuming Born rule behaves poorly for problems involving heterogeneous potentials. 6. Conclusions and Future Research Directions Mult

9 iscale Enrichment based on Partition of
iscale Enrichment based on Partition of Unity presented in this manuscript provides a considerable improvement over classical mathematical homogenization theory and quasi-continuum for continuum and discrete systems, respectively. One of its main attractions stems from the ease of implementation into most of the commercial software packages. Any commercial code that allows for adding user-defined elements with arbitrary number of degrees-of-freedom per node can be used for implementation. MEPU can be easily implemented into the quasi-continuum framework by replacing the Born rule with appropriate enrichment functions. Homogenization error estimators [52,53] can be utilized to identify the location where enrichment is needed. So far, the method has been studied assuming small deformations, harmonic potentials and steady state conditions. It remains to be seen whether the coarse scale continuity conditions set fourth in this paper are sufficient to avoid spurious wave reflections at the atomistic/continuum interfac

10 e or whether some sort of overlapping as
e or whether some sort of overlapping as described in [54] is needed. Wave dispersion is another issue, which needs a careful examination. Existing methodologies (see for example [55,56,57]) are based on continuum (or quasi-continuum) enrichment. It remain to be seen if higher order enrichment functions (18 functions in 3D corresponding to the gradients of the coarse scale strain field), which play a central role in the formulation of existing dispersive models, have to be added to enrich the element kinematics. With increase in the order of enrichment functions, the polynomial order of the coarse scale fields may have to be appropriately increased to let the unit cell “experience” higher order coarse fields over its domain. Thus, it may be advantageous to use different polynomial order of the shape (support) functions in the first and second term of Eq. (3). ijmnP These are just few of the issues that will be addressed in our future research work. Acknowledgment This work was supported by the N

11 ational Science Foundation under grants
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