The generalized product partition of unity for the meshless methods
Introduction
Meshless methods [1], [2], [4], [12], [13], [14], [15], [16], [17], [18], [27], [28] have several advantages over the conventional finite element method [3], [29]. However, they have some difficulties such as large matrix condition numbers (or singular stiffness matrix), complicated (or non-smooth) partition of unity (PU) functions, ineffectiveness in handling essential boundary conditions, lengthy computing time due to complicated PU functions, and so forth.
In order to alleviate these difficulties, Oh et al. [21], [22], [23], [24], [25], [26] introduced patchwise RPPM (reproducing polynomial particle method) with use of the convolution PU functions with flat-top. It was shown in [23] that the PU function with flat-top lead to the small matrix condition number. However, it is not easy to extend the two-dimensional construction of the convolution PU functions to the three-dimensional cases. Oh et al. [23] constructed smooth piecewise polynomial PU functions with flat-top in one-dimensional case. Obviously, the tensor product of these one-dimensional PU functions yields higher dimensional PU functions with flat-top. However, tensor products of intervals cannot make neither triangular, general quadrangular patches (2D), nor tetrahedral, pentahedral, general hexahedral patches, arising in background meshes for meshless methods.
In this paper, we introduce a (two and three dimensional) unified method constructing a partition of unity associated with background meshes. We call this simple method the generalized product method. The procedure of this method is as follows:
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First, we construct a partition of unity on that consists of two simple smooth piecewise polynomials that look like the step functions.
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Second, through the coordinate projection from onto , we construct the partition of unity on that consists of two smooth PU functions with flat-top. Via the proper affine transformations, two PU functions are copied onto each side (in 2-Dim case), or each face (in 3-Dim case), of a patch, say a polygonal (or polyhedral) patch Q. Then, exactly one of two PU functions planted on each face is identically 1 on most part of the patch Q (called the flat-top part of Q).
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Finally, The product of those PU functions corresponding to the sides (faces) of Q that are “1” on the flat-top part of Q become a smooth PU function with flat-top corresponding to Q.
After introducing some preliminary results in Section 2, the generalized product method to construct smooth partition of unity is introduced in Section 3. In the same section, we also prove that the generalized product method yields a partition of unity on a domain as well as a domain . In order to show the effectiveness of this method, two-dimensional and three-dimensional numerical tests are carried out in Section 4.
We claim that the generalized product method constructing a partition of unity makes meshless methods much more useful.
Section snippets
Definitions
In this section, we introduce definitions and terminologies that are used throughout this paper.
Let is the closure of , we define the vector space to consist of all those functions for which is bounded and uniformly continuous on for In the following, a function is said to be a -function. If is a function defined on , we define the support of asFor an integer , we also use the usual Sobolev space denoted by .
One-dimensional partition of unity functions
In this section, we briefly review one-dimensional partition of unity (PU) with flat-top. For details of this construction, we refer to Oh et al. [23], in which we showed that PU functions with flat-top lead to a small matrix condition number.
Throughout this paper, we reserve the small real number , usually, , for the width of non-flat-top part of the PU functions.
The generalized product partition of unity for two-dimensional domains
Using (9), (10), we obtain two basic two-dimensional -PU functions defined bythat satisfyIn other words, two functions are the compositions of the coordinate projection, , and , respectively. Moreover, observing may have an advantage on implementing these two basic functions. The graph of is sketched in Fig. 1. The schematic diagram for and are shown in Fig. 2. That is,
Patchwise reproducing polynomial particle methods (RPPM)
Consider a model problemwhere is a polygonal domain in , n is the outward normal vector along and . We assume that the partition of into patches and the partition of unity for are those constructed in the previous section. Then the variational formulation of the model problem is: find such that on andfor all .
Now, the patchwise RPPM [21], [22] for a numerical solution for
Concluding remarks
The proposed method constructing a highly smooth piecewise polynomial (closed form) partition of unity with flat-top is simple and efficient for the p-version type partition of unity FEM (PUFEM). However, it may have limitations to use for the adaptive meshless methods because the width of the non-flat-top parts, , can not be too small for the convergence of meshless methods. This is because the error estimates proved in [21] are as follows:
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Cited by (16)
Simulating stress wave with flat-top partition of unity based high-order discontinuous deformation analysis
2018, Engineering Analysis with Boundary ElementsCitation Excerpt :However, linear dependence problem has been observed when high-order polynomials are assigned as local approximation in the traditional finite element PU mesh [28–30]. Therefore, in this paper, the blocks in DDA method are discretized by flat-top PU mesh, which is linearly independent for high-order approximation [31–33]. Thus, in the flat-top PU based high-order DDA method, the stress wave within an intact block is numerically approximated by the flat-top PU mesh and high-order polynomials, and the interaction between blocks is simulated by the contact algorithm in DDA method.
Dynamic analysis with flat-top partition of unity-based discontinuous deformation analysis
2018, Computers and GeotechnicsCitation Excerpt :Therefore, the flat-top PU method has been coupled with other methods for numerical analyses in the continuous problem domain. For example, Oh et al. [5,6] solved Poisson’s equation on the continuous polygonal and polyhedral domains using the so-called generalized product partition of unity, i.e., partition of unity with flat-top. Oh et al. [7] applied flat-top PU-based mesh-free particle methods for approximating the displacements of continuous rectangular, triangular and circular plates.
Augmented Numerical Manifold Method with implementation of flat-top partition of unity
2015, Engineering Analysis with Boundary ElementsCitation Excerpt :Similar to [33], a shrink technique is necessary to create the flat-top region and decaying region. As an alternative, meshfree construction of a flat-top PU was proposed in [42–44], which allows for arbitrary geometric refinement of the patches and avoids the issue of mesh generation, but shares the same limitations with other meshfree-based PUs. This paper aims to break the barrier through proposing a new formulation of flat-top PU based on regular mesh.
Effects of the smoothness of partitions of unity on the quality of representation of singular enrichments for GFEM/XFEM stress approximations around brittle cracks
2015, Computer Methods in Applied Mechanics and EngineeringProof of linear independence of flat-top PU-based high-order approximation
2014, Engineering Analysis with Boundary ElementsCitation Excerpt :The method is effective in reducing the matrix condition number with higher order polynomial local approximations [14], and it was proved to be automatically L2-stable [15]. Meshfree construction of a flat-top PU was proposed in [16–18], which allows for arbitrary geometric refinement of the patches and avoids the issue of mesh generation, but encounters the difficulties in dealing with essential boundary conditions and high computational cost for reasonable accuracy in numerical integration. An alternative way is to construct flat-top PU based on finite element mesh.
Mesh based construction of flat-top partition of unity functions
2013, Applied Mathematics and ComputationCitation Excerpt :Therefore, with the proposed partition of unity function, we can fully control the mesh size h as well as the patch-wise enrichment order p on a conventional finite element mesh. The open covering is sometimes called clouds [11,18], spheres [7] or patches [9,20,21]. In this paper we will adopt the notion of patches.
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Supported in part by NSF Grant DMS-07-13097.