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14 Li and Liu: Meshfree and particle methods and applications Appl Mech Rev vol 55, no 1, January 2002 are partitions of unity, and in most cases the linear completeness, or consistency is also enforced a priori; there is no compatibility issue left to be tested, unlike the incompatible finite element shape function. However, if there are not enough quadrature points in a compact support, or quadrature points are not evenly distributed, spurious modes may occur. Today, quadrature integration is one of the two major shortcomings the cost of meshfree methods is the other left when meshfree methods compared with finite element methods. Beissel and Belytschko 237 proposed a stabilized nodal integration procedure by adding a residual of the equilibrium equation to the potential energy functional to avoid use of quadrature integration. However, adding the additional term in potential energy means sacrificing variational consistency, hence accuracy of the formulation. Gauss quadrature integration error via different set-up of background cells as well as quadrature point distribution is studied in 238. Itis found that if the background cell does not match with the compact support of the meshfree interpolant, considerable integration error may rise. The simplest remedy is the local, self-similar support integration. Assume the meshfree shape function is compactly supported, and the support for each and every particle is similar in shape, eg a circular region in 2D, a sphere in 3D. Take the Element Free Galerkin EFG method for example Belytschko et al 59,63. For linear elastostatics, the stiffness matrix is K IJ B I t DB J d (79) where is the problem domain. If all the shape functions have the same shape of compact support a 3D sphere in this case, the above integration can be rewritten as K IJ I B I t DB J d (80) where I is the support of particle I. Because all shape functions are compactly supported, the integrals in the rest of domain, ie / I , vanish. And we only need to evaluate K IJ within I and Iu . Since every I ,(I1,...,n) has the same shape, once a quadrature rule is fixed for one compact support, it will be the same for the rest of compact supports as well. We can then integrate the weak form locally from one compact support to another compact support. Therefore, it is free of the background cell or any implicit mesh. Note that this is different from the global domain quadrature integration, since in our case compact supports are overlapped with each other. This local quadrature idea is extended by Atluri and his colleagues to form new meshfree formulations 80–83,239– 242. The first formulation proposed by Atluri et al is called the local boundary integral equation LBIE. Consider a boundary value problem of Poisson’s equation 239. One can form a boundary integral equation for a chosen subdomain s note that s has nothing to do with a particle’s compact support, uy ux ũ* s n x,yd u s n xũ*x,yd ũ*x,ypxd (81) s where ũ* is the Green’s function ũ*x,y 1 2 ln r 0 r . (82) For each particle in the domain , one can form a local boundary integral equation 81. Letting u h (x) i i (x)d i , one may obtain the following algebraic equations N i u i j1 where K* ij su K*d ij j f i * (83) ũ*x,y i j n d ũ* j sq n x.y id ũ* j Ls n x,y id (84) f i * ũ*x,y i sq q¯d ū ũ* su n x,yd ũ*x,y i pxd. (85) s Those local boundary integrals and local domain integrals can be integrated by fixed quadrature rules. Sladek et al presented a detailed account on how to deal with singularity in numerical integrations 243. The obvious advantage of this formulation is that it does not need to enforce the essential boundary condition. Nevertheless, this formulation relies on a Green’s function, and it is limited to a handful of linear problems. Subsequently, Atluri et al 80,81 formed a local Petrov- Galerkin formulation MLPG with meshfree interpolant in the same local region s For linear elastostatics problem 49, they form N local petrov-Galerkin weak forms. Each of them around a distinct particle I is, N j1 where K ij d j f j (86) K IJ B I v T DB J s d v I J su d v I NDB J d su (87) f I v I st t¯d v I su ūd v I bd (88) s Again, s is not the compact support I , however, certain conditions must be imposed to s , such that K ij 0 at least for some ji. In practical implementation, the trial func-
Appl Mech Rev vol 55, no 1, January 2002 Li and Liu: Meshfree and particle methods and applications 15 Fig. 6 Geometry definition of a representative nodal domain tion’s support is I whereas the I-th weighting function’s support is denoted as Is . Note that all the integrations here are local; no background cell is needed. The term Petrov here indicates that one uses different trial and test weighting function even though they may be the same function but they have different support size, I Is ). This will result an unsymmetric stiffness matrix in general. Let Is I , and the trial function be the same as the weighting function. Then the above Petrov-Galerkin formulation becomes the conventional Bubnov-Galerkin formulation. In that case, it returns the local quadrature integration scheme we presented at the beginning. It is worth mentioning that if one choose Is as an n-dimensional sphere, the numerical integration may be carried out by Cubature, which is recently documented in details by De and Bathe 244. In order to completely eliminate quadrature points, Chen et al 72 proposed a so-called stabilized conforming nodal integration for meshfree Galerkin method. They first identify that for linear exactness in the Galerkin approximation, the shape functions have to be linearly consistent, and the domain integration has to be able to integrate the derivatives of shape functions to nullity for interior nodes and to meet traction equilibrium. The argument made by Chen et al is that for meshfree solution of a nodally integrated weak form to be stable and convergent, two conditions need to be satisfied: 1 derivatives of meshfree shape functions evaluated at the nodal point must be avoided and 2 nodal integration must satisfy integration constraints. It is shown in their study 245 that a direct integration introduces numerical instability due to rank deficiency in the stiffness matrix. To stabilize the nodal integration, they proposed a so-called smoothing stabilization technique. The basic idea is that one first integrates strain in a chosen neighborhood of the particle I, say I , to replace the strain at point I with the average strain, as illustrated in Fig. 6, provided the general triangulation is possible. Note that here I is not the compact support of the particle I (supp( I )), it is the Voronoi cell that contains the particle I. Then divergence theorem is used to replace the area, or volume integration around particle I by a contour integration of the Voronoi cell boundary. The contour integration is carried out by sampling the values at the vertices of the Voronoi cell. In the implementation 72,245, ¯ij h x I h ij xxx I d (89) where (x) is the characteristic function of small area I 1 xx I A I x I (90) 0 x I where A I meas( I ). Therefore, ¯ij h x I 1 2A I I u i h u h j x j x i d 1 2A I u h i n j u h j n i dS. (91) I Finally, employing an assumed strain method and integrating the weak form by a nodal integration, the meshfree discrete equation is obtained. It is shown that if linear basis functions are used in the construction of shape function, the strain smoothing of Eq. 91 in conjunction with the nodal integration of weak form will result the linear exactness in the Galerkin approximation. The main virtue of this approach is that it completely eliminates Gauss quadrature points, which is especially attractive in inelastic large deformation calculation with a Lagrangian formulation. 3.4 Applications One of the early incentives to develop meshfree Galerkin methods was its ability to simulate crack growth—a critical issue in computational fracture mechanics. Belytschko and his co-workers have systematically applied the EFG method to simulate crack growth/propagation problems 60,63,190– 192,235,246,247. Special techniques, such as the visibility criterion, are developed in modeling a discontinuous field 60,246. Subsequently, a partition of unity method is also exploited in crack growth simulation 220. It is fair to say that at least in 2D crack growth simulation meshfree Galerkin procedure offers considerable advantages over the traditional finite element methods, because remeshing is avoided. Meshfree simulation has been conducted by Li et al to simulate failure mode transition 248,249. The simulation has successfully replicated failure mode transition observed in Zhou-Rosakis-Ravichadran experiment 250, which is related to the early Kalthoff problem 251,252. Figure 7 shows a crack growth from a shear band. Another area where meshfree Galerkin methods have clear edge over finite element computations is its ability to handle large deformation problems. See Fig. 8. Chen and his co-workers proposed a concept of Lagrangian kernel and have been using RKPM to simulate several large deformation problems, such as metal forming, extrusion 253,254, large deformation of rubber materials 255,256, soil mechanics problem 257, shape design sensitivity and optimization, etc 71,258. Li226,259,260 and Jun 261 developed an explicit RKPM code to compute large deformation
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