The generalized product partition of unity for the meshless methods

doi:10.1016/j.jcp.2009.10.047

Journal of Computational Physics

Volume 229, Issue 5, 1 March 2010, Pages 1600-1620

https://doi.org/10.1016/j.jcp.2009.10.047 Get rights and content

Abstract

The partition of unity is an essential ingredient for meshless methods named by GFEM, PUFEM (partition of unity FEM), XFEM (extended FEM), RKPM (reproducing kernel particle method), RPPM (reproducing polynomial particle method), the method of hp clouds in the literature. There are two popular choices for partition of unity: a piecewise linear FEM mesh and the Shepard-type partition of unity. However, the partition of unity (PU) by a FEM mesh leads to the singular (or nearly singular) matrices and non-smooth approximation functions. The Shepard-type partition of unity requires lengthy computing time and its implementation is difficult. In order to alleviate these difficulties, Oh et al. introduced the smooth piecewise polynomial PU functions with flat-top, that lead to small matrix condition numbers, and almost everywhere partition of unity, that can handle essential boundary conditions. Nevertheless, we could not have the smooth closed form PU functions with flat-top for general polygonal patches (2D) and general polyhedral patches (3D). In this paper, we introduce one of the most simple and efficient partition of unity, called the (generalized) product partition of unity. The product PU functions constructed by this method are the closed form smooth piecewise polynomials with flat-top and could handle background meshes (general polygonal patches as well as general polyhedral patches) arising in practical applications of 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:

•
First, we construct a partition of unity on $R$ that consists of two simple smooth piecewise polynomials that look like the step functions.
•
Second, through the coordinate projection from $R^{d}, d = 2, 3,$ onto $R$ , we construct the partition of unity on $R^{d}$ 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).
•
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 $Ω \subset R^{2}$ as well as a domain $Ω \subset R^{3}$ . 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 $\bar{Ω}$ is the closure of $Ω \subset R^{d}$ , we define the vector space $C (\bar{Ω})$ to consist of all those functions $ϕ \in C^{m} (Ω)$ for which $D^{α} ϕ$ is bounded and uniformly continuous on $Ω$ for $| α | = α_{1} + \dots + α_{d} ⩽ m .$ In the following, a function $ϕ \in C^{m} (Ω)$ is said to be a $C^{m}$ -function. If $Ψ$ is a function defined on $Ω$ , we define the support of $Ψ$ as $supp Ψ = \bar{{x \in Ω | Ψ (x) \neq 0}} .$ For an integer $k ⩾ 0$ , we also use the usual Sobolev space denoted by $H^{k} (Ω)$ .

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 $δ$ , usually, $0.01 ⩽ δ ⩽ 0.1$ , 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 $C^{n - 1}$ -PU functions defined by $Ψ_{x}^{R} (x, y) = ψ_{0}^{R} (x) and Ψ_{x}^{L} (x, y) = ψ_{0}^{L} (x), for all (x, y) \in R^{2}$ that satisfy $Ψ_{x}^{R} (x, y) + Ψ_{x}^{L} (x, y) = 1, for all (x, y) \in R^{2} .$ In other words, two functions are the compositions of the coordinate projection, $(x, y) \to x$ , and $ψ_{0}^{R}, ψ_{0}^{L}$ , respectively. Moreover, observing $Ψ_{x}^{L} = 1 - Ψ_{x}^{R}$ may have an advantage on implementing these two basic functions. The graph of $Ψ_{x}^{R}$ is sketched in Fig. 1. The schematic diagram for $Ψ_{x}^{R}$ and $Ψ_{x}^{L}$ are shown in Fig. 2. That is, $\{\begin{matrix} Ψ_{x}^{L} \end{matrix}$

Patchwise reproducing polynomial particle methods (RPPM)

Consider a model problem $- Δ u + u = f in Ω,$ $u = g_{d} along Γ_{D},$ $\nabla u \cdot n = g_{t} along Γ_{N},$ where $Ω$ is a polygonal domain in $R^{d}$ , n is the outward normal vector along $Γ_{N}$ and $Γ_{D} \cup Γ_{N} = \partial Ω$ . We assume that the partition of $Ω$ into patches and the partition of unity for $Ω$ are those constructed in the previous section. Then the variational formulation of the model problem is: find $u \in H^{1} (Ω)$ such that $u = g_{d}$ on $Γ_{D}$ and $\int_{Ω} \nabla v \cdot \nabla ud Ω - \int_{Γ_{D}} v \nabla u \cdot n d Γ + \int_{Ω} uvd Ω = \int_{Ω} vfd Ω + \int_{Γ_{N}} {vg}_{t} d Γ,$ for all $v \in H^{1} (Ω)$ .

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 $δ$ of the non-flat-top parts, $ω^{nflt}$ , can not be too small for the convergence of meshless methods. This is because the error estimates proved in [21] are as follows: $‖ u - u^{approx} ‖_{L^{2} (Ω)} ⩽ \sqrt{κ} {(\sum_{I = 1}^{M} {(ε_{I}^{(0)})}^{2})}^{1 / 2} and ‖ \nabla (u - u^{approx})$

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¹: Supported in part by NSF Grant DMS-07-13097.

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The generalized product partition of unity for the meshless methods

Abstract

Introduction

Section snippets

Definitions

One-dimensional partition of unity functions

The generalized product partition of unity for two-dimensional domains

Patchwise reproducing polynomial particle methods (RPPM)

Concluding remarks

Comput. Meth. Appl. Mech. Eng.

Comput. Meth. Appl. Mech. Eng.

Comput. Meth. Appl. Mech. Eng.

J. Comput. Phys.

Comput. Meth. Appl. Mech. Eng.

Comput. Meth. Appl. Mech. Eng.

Comput. Meth. Appl. Mech. Eng.

Comput. Meth. Appl. Mech. Eng.

Comput. Meth. Appl. Mech. Eng.

The Meshless Method

Survey of Meshless and Generalized Finite Element Methods: A Unified Approach

The Finite Element Method for Elliptic Problems

An hp adaptive method using clouds

Comput. Meth. Appl. Mech. Eng.

A particle-partition of unity methods part VII: adaptivity

Meshfree Methods for Partial Differential Equations