Research Article  Open Access
WonTak Hong, "A NonConvex Partition of Unity and Stress Analysis of a Cracked Elastic Medium", Advances in Mathematical Physics, vol. 2017, Article ID 9574341, 9 pages, 2017. https://doi.org/10.1155/2017/9574341
A NonConvex Partition of Unity and Stress Analysis of a Cracked Elastic Medium
Abstract
A stress analysis using a meshfree method on a cracked elastic medium needs a partition of unity for a nonconvex domain whether it is defined explicitly or implicitly. Constructing such partition of unity is a nontrivial task when we choose to create a partition of unity explicitly. We further extend the idea of the almost everywhere partition of unity and apply it to linear elasticity problem. We use a special mapping to build a partition of unity on a nonconvex domain. The partition of unity that we use has a unique feature: the mapped partition of unity has a curved shape in the physical coordinate system. This novel feature is especially useful when the enrichment function has polar form, , because we can partition the physical domain in radial and angular directions to perform a highly accurate numerical integration to deal with edgecracked singularity. The numerical test shows that we obtain a highly accurate result without refining the background mesh.
1. Introduction
Many meshfree methods [1–9] showed great success to solve challenging problems. Whether it has explicitly defined a partition of unity function or not, most of the meshfree methods can be categorized as a partition of unity method. Some of them [2–5, 8, 10, 11] use partition of unity explicitly by utilizing background mesh. The partition of unity function allows researchers to use a priori knowledge of the solution and enables enriching the solution space with special functions [8, 12–14]. The main interest of such enrichment was focused on crack modeling [1, 2, 7, 8, 15, 16]. The partition of unity enrichment with discontinuous functions allows to model discontinuity in the domain such as a crack in an elastic medium. Thus, using the Heaviside enrichment function, researchers could avoid nonconvex partitioning at all, but an accurate numerical integration becomes challenging because of the discontinuous enrichment [16–21]. The discontinuous enrichment, however, was an ingenious tool to model dynamic crack propagation, since it can be replaced easily without changing the given background mesh [2]. Thus, engineers and researchers could save time by enrichment and avoid remeshing the entire domain. On the other hand, there has been different effort to deal with a nonconvex domain. Instead of enriching with discontinuous function, a mapping technique has been used to partition the given convex domain and coined almost everywhere partition of unity [22] because of failing partition of unity at some points at the boundary. The later effort focused more on accuracy and error estimation of elliptic boundary value problem, but elasticity equation has not yet been explored.
The goal of this study is performing accurate stress analysis on the cracked elastic medium with the almost everywhere partition of unity. It has been well known that the singularity can cause poor performance of finite element approximation for the elliptic boundary value problems [23–26], including elasticity problems. The quality of enriched meshfree method depends on the accuracy of the numerical integration [15]. The challenge for elasticity equation is the nonsmooth integrand due to the enrichment. Thus, to tackle the difficulty related to the numerical integration after enrichment, we use a mapping technique rather than using a discontinuous enrichment because the later requires integrating discontinuous function, which complicates the numerical integration. Another reason that we favor partitioning nonconvex domain explicitly over discontinuous enrichment is the matrix condition number. Most of the study with discontinuous enrichment uses finite element mesh as the partition of unity function, and it could result in unexpected linear dependency with polynomial enrichment [3, 27, 28]. We use a flattop partition of unity function to avoid linear dependency and thus keep the growth of matrix condition number reasonable with the polynomial and nonpolynomial enrichment.
With the almost everywhere partition of unity as defined in [22], it still has difficulty in dealing with nonsmooth enrichment function that is given in polar form because the supports of the partition of unity functions are quadrangular. Thus, upon defining a curved annular shaped partition of unity on a nonconvex domain, we can integrate the enriched function not only in radial direction but also in angular direction. As a result, with the help of mapped numerical quadrature, we obtain highly accurate computed stresses.
2. Preliminaries
Defining a partition of unity on a given domain is the most important part of a meshfree method. Among many partition of unity functions, we adopt the following standard definitions and theorems introduced in [28].
Definition 1. For a point finite open covering of a domain , a family of Lipschitz functions, , is called a partition of unity subordinate to the covering if for each and there exists a number such that for each with the following condition:
Definition 2. For integers , we define a piecewise polynomial function bywhere whose coefficients are inductively constructed by the following recursion formula:
Let us define and The superscript and stand for right and left simple polynomials, respectively. Then, with these simple polynomials, we construct a piecewise polynomial function with flattop property in one dimension as follows:The above partition of unity with flattop is exactly the same as the convolution of the characteristic function and the scaled window function defined by for , 0 for . is a constant that makes [28]. The most obvious way to construct a higher dimension flattop partition of unity function, although its support is rather restricted to be rectangular, is using a tensor product of onedimensional partition of unity functions [28]:Other studies [11, 29] generalized the tensor product idea to a quadrangular shape. In this study, we use a partition of unity for annular shape background mesh to deal with nonconvex domain.
3. The Linear Elasticity Equations
Let be an elastic medium in with boundary . The displacement vectors are denoted by and stress tensor are denoted by . Let us denote the strain tensor . Then the straindisplacement and stressstrain relations are given byrespectively, where is the following differential matrix:and , is a symmetric positive definite matrix of material constants. For an isotropic elastic medium, the matrix is given aswhen plane stress is the quantity of interest orwhen plane strain is the quantity of interest. Here, and are defined as follows: is the modulus of elasticity and is Poisson’s ratio.
The equilibrium equations of elasticity are given bywhere is the vector that describes body force per unit area.
Using the stressstrain relation, (6) and (7), we can express (12) in terms of the displacement vector . Let us consider the following system of partial differential equations in terms of the displacement vector:subject to the following boundary conditions,where , is an outward unit normal vector to the traction boundary , and
Let . Then the variational equation of the linear elasticity equation, (13) and (14), becomes as follows.
Find the vector such that , on , andwhere is the strain energy of the displacement vector .
Let us denote the basis functions defined on by The components of the displacement vector in terms of basis functions are given as the following forms:where are called the amplitudes of the basis functions . LetThen the displacement vector can be written as Substituting (20) into (17), we have the following.
Lemma 3. The bilinear form on , which is the common support of and , becomes one of the following four equations:
4. A MeshFree Approximation for the Linear Elasticity Equation on an EdgeCracked Domain
We define the meshfree approximation space as follows:where is the partition of unity function and is the local approximation function that has polynomial reproducing order . Standard version of finite element method has polynomial reproducing order . We refer to [28] for different local approximation functions that can be used. In this study, we use Lagrange interpolating polynomials of degree and some singular functions for the local approximation function.
Using the meshfree approximation space , we state the meshfree approximation of the linear elasticity equation as follows.
Find the vector such that , on , andwhere the bilinear forms and are defined as in (17) and (18).
We define the enriched meshfree approximation space as follows:where is a special function that is enriched on the patches with , which contain the crack tip.
Then the enriched meshfree approximation of the linear elasticity equation becomes as follows.
Find the vector such that , on , andwhere the bilinear forms and are defined as in (24).
Using Lemma 3, we see that the bilinear form on the common support can be obtained using the following integral:If we use , (27) becomes either one of the four following equations:where and are global basis number and and are index functions depending on the global basis number. If is polynomial, then the integrands in (28) are piecewise polynomial so we can accurately evaluate the integral using numerical quadratures.
If is an enrichment of type , then the integrals in (29)–(31) have singular integrands of type or . However, if we use the change of variable given in (33), these terms are changed to and , respectively. Thus, the stiffness calculation given in (27) is regular on the coordinate and we could perform highly accurate numerical integration.
5. Numerical Examples
We consider two examples. We use partition of unity function defined in (4) throughout this section. Also, we fix the polynomial reproducing order, in (23), when comparing the enriched meshfree solution with the finite element solution without enrichment.
Example 1. Let us consider the equation of elasticity on a domain with a crack along the negative axis. Assume that Young’s modulus and Poisson’s ratio . We impose the following stresses along all boundaries of the given domain:The displacement vector is fixed at the crack tip and component of the displacement vector, , is fixed at to prevent rigid body motion.
Let us illustrate how to construct a partition of unity on a nonconvex domain. We consider the background mesh pictured in Figure 1(a). We use eight patches as shown in Figure 1(a). For each patch , we construct a partition of unity function that corresponds to the patch . The support of the partition of unity functions is shown in Figure 1(b). Note that the support overlaps. The thin strips that are enclosed by dotted lines are the overlapping region of the partition of unity functions. For example, the support of the third partition of unity functions, , is the shaded region in Figure 2. Note that the partition of unity functions does not overlap along the negative axis where the crack is located.
(a)
(b)
Let be the coordinate transformation:Then, as shown in Figure 2, through the mapping , the partition of unity on the cracked domain in the coordinate system is generated by using the partition of unity of the reference domain in the rectangular coordinate system. The four outer patches , are mapped to quadrangles , with one curved side.
Let the partition of unity function on the reference coordinate system be , where . This reference partition of unity functions can be easily constructed by using the tensor product of onedimensional flattop partition of unity function (4); Then the partition of unity functions on the physical coordinate system is given bywhere .
Next, we enrich the following four singular functions in the patch that contains the crack tip.A prior knowledge to the solution behavior or a numerical experiment could provide an optimal or nearly optimal choice for the enrichment functions.
In order to show the effectiveness of the enrichment, we compare the enriched meshfree solution with plain meshfree solution with polynomial reproducing order four. The plain meshfree solution with polynomial reproducing order four is comparable to the standard version finite element methods with . Figure 3 shows the meshfree solution with and without enrichment along with true stress near the crack tip, . We clearly see the effectiveness of the enrichment in handling crack singularities. There is virtually no difference between the true stress and the computed stress obtained by enriched meshfree method.
(a)
(b)
(c)
Example 2. Let us consider the equations of elasticity on a domain shown in Figure 4, which is isotropic with material constants: and . The boundary conditions are given as follows: , along , , along , , along .
We use Mesh I and Mesh II, shown in Figures 5(a) and 5(b), for the version of finite element method. MAM (Method of Auxiliary Mapping) [30] is powerful tool that can be used with version finite element method when the problem has known singularity. MAM uses the same Mesh I but special mappings are used on the shaded regions; see Figure 5(c). The given domain, Figure 4, is essentially the same as Example 1, except the crack lying in the positive axis instead of negative axis. We use the background mesh given in Figure 5(d) to construct partition of unity functions. The eight enrichment patches are shaded. For this example, we use only one singular enrichment function, . The degrees of freedom are compared in Table 1 for four different methods. We see that the degree of freedom for the enriched meshfree method is the smallest, except , and has the simplest mesh setup.

(a)
(b)
(c)
(d)
We calculate the strain energy of computed solution by , where is the bilinear form defined in (17). The computed energy for four different methods is summarized in Table 2. The true energy for this problem is and can be obtained by the extrapolation technique [24]. We calculate the relative error in energy norm as follows:Table 3 lists the computed relative errors in percent. The enriched meshfree method is not only robust but also highly efficient at handling singularity arising in elasticity problem on a cracked domain. With the right enrichment function, we see that the approximate solution of the enriched meshfree method converges much faster than MAM with fewer degrees of freedom.


6. Concluding Remarks
We have used a mapping technique, given in (33), to deal with the crack singularity. In this study, we map the entire computational domain to the coordinate system to remove the singularity. A domain that has more general shape can be mapped using conformal mapping technique [17]; however, essential boundary condition imposition becomes nontrivial with the current almost everywhere partition of unity. Thus, the idea of the mapping method in this study can be more powerful if we can avoid global coordinate mapping so that the mapping can be used only locally where it is needed. To that end, a coupling method [31] or generalized product partition of unity [29] can be utilized together to localize the mapping technique. Also the same mapping technique can be used to improve singular integral given in [14].
Competing Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
References
 T. Belytschko, Y. Y. Lu, and L. Gu, “Elementfree galerkin methods,” International Journal for Numerical Methods in Engineering, vol. 37, no. 2, pp. 229–256, 1994. View at: Publisher Site  Google Scholar  MathSciNet
 N. Moës, J. Dolbow, and T. Belytschko, “A finite element method for crack growth without remeshing,” International Journal for Numerical Methods in Engineering, vol. 46, no. 1, pp. 131–150, 1999. View at: Publisher Site  Google Scholar
 I. Babuška, U. Banerjee, and J. E. Osborn, “Survey of meshless and generalized finite element methods: a unified approach,” Acta Numerica, vol. 12, pp. 1–125, 2003. View at: Publisher Site  Google Scholar  MathSciNet
 I. Babuska, V. Nistor, and N. Tarfulea, “Generalized finite element method for secondorder elliptic operators with Dirichlet boundary conditions,” Journal of Computational and Applied Mathematics, vol. 218, no. 1, pp. 175–183, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 T. Strouboulis, K. Copps, and I. Babuška, “The generalized finite element method,” Computer Methods in Applied Mechanics and Engineering, vol. 190, no. 3233, pp. 4081–4193, 2001. View at: Publisher Site  Google Scholar  MathSciNet
 S. De and K. J. Bathe, “The method of finite spheres,” Computational Mechanics, vol. 25, no. 4, pp. 329–345, 2000. View at: Publisher Site  Google Scholar  MathSciNet
 W. K. Liu, S. Jun, and Y. F. Zhang, “Reproducing kernel particle methods,” International Journal for Numerical Methods in Fluids, vol. 20, no. 89, pp. 1081–1106, 1995. View at: Publisher Site  Google Scholar  MathSciNet
 J. Dolbow, N. Moës, and T. Belytschko, “Discontinuous enrichment in finite elements with a partition of unity method,” Finite Elements in Analysis and Design, vol. 36, no. 3, pp. 235–260, 2000. View at: Publisher Site  Google Scholar
 H.S. Oh, J. G. Kim, and J. Jeong, “The closed form reproducing polynomial particle shape functions for meshfree particle methods,” Computer Methods in Applied Mechanics and Engineering, vol. 196, no. 3536, pp. 3435–3461, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. A. Duarte and J. T. Oden, “An $h$$p$ adaptive method using clouds,” Computer Methods in Applied Mechanics and Engineering, vol. 139, no. 1–4, pp. 237–262, 1996. View at: Publisher Site  Google Scholar  MathSciNet
 W.T. Hong and P.S. Lee, “Mesh based construction of flattop partition of unity functions,” Applied Mathematics and Computation, vol. 219, no. 16, pp. 8687–8704, 2013. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 I. Babuška, U. Banerjee, and J. E. Osborn, “Generalized finite element methods—main ideas, results and perspective,” International Journal of Computational Methods, vol. 1, no. 1, pp. 67–103, 2004. View at: Publisher Site  Google Scholar
 I. Babuška, U. Banerjee, and J. E. Osborn, “On principles for the selection of shape functions for the generalized finite element method,” Computer Methods in Applied Mechanics and Engineering, vol. 191, no. 4950, pp. 5595–5629, 2002. View at: Publisher Site  Google Scholar  MathSciNet
 W.T. Hong, “Enriched meshfree method for an accurate numerical solution of the Motz problem,” Advances in Mathematical Physics, vol. 2016, Article ID 6324754, 12 pages, 2016. View at: Publisher Site  Google Scholar  MathSciNet
 Q. Z. Xiao and B. L. Karihaloo, “Improving the accuracy of XFEM crack tip fields using higher order quadrature and statically admissible stress recovery,” International Journal for Numerical Methods in Engineering, vol. 66, no. 9, pp. 1378–1410, 2006. View at: Publisher Site  Google Scholar  Zentralblatt MATH
 E. Chahine, P. Laborde, and Y. Renard, “Crack tip enrichment in the XFEM using a cutoff function,” International Journal for Numerical Methods in Engineering, vol. 75, no. 6, pp. 629–646, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 S. Natarajan, S. Bordas, and D. Mahapatra, “Numerical integration over arbitrary polygonal domains based on SchwarzChristoffel conformal mapping,” International Journal for Numerical Methods in Engineering, vol. 80, no. 1, pp. 103–134, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 S. Natarajan, D. R. Mahapatra, and S. P. Bordas, “Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework,” International Journal for Numerical Methods in Engineering, vol. 83, no. 3, pp. 269–294, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 G. Ventura, R. Gracie, and T. Belytschko, “Fast integration and weight function blending in the extended finite element method,” International Journal for Numerical Methods in Engineering, vol. 77, no. 1, pp. 1–29, 2009. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 S. E. Mousavi and N. Sukumar, “Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons,” Computational Mechanics, vol. 47, no. 5, pp. 535–554, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 Y. Sudhakar and W. A. Wall, “Quadrature schemes for arbitrary convex/concave volumes and integration of weak form in enriched partition of unity methods,” Computer Methods in Applied Mechanics and Engineering, vol. 258, pp. 39–54, 2013. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H.S. Oh and J. W. Jeong, “Almost everywhere partition of unity to deal with essential boundary conditions in meshless methods,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 41–44, pp. 3299–3312, 2009. View at: Publisher Site  Google Scholar  MathSciNet
 P. Ciarlet, Basis Error Estimates for Elliptic Problems, vol. 2, North Holland, 1991.
 B. Szabo and I. Babuska, Finite Element Analysis, A WileyInterscience Publication, John Wiley & Sons, New York, NY, USA, 1991. View at: MathSciNet
 T. R. Lucas and H. S. Oh, “The method of auxiliary mapping for the finite element solutions of elliptic problems containing singularities,” Journal of Computational Physics, vol. 108, no. 2, pp. 327–342, 1993. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H.S. Oh, C. Davis, J. G. Kim, and Y. Kwon, “Reproducing polynomial particle methods for boundary integral equations,” Computational Mechanics, vol. 48, no. 1, pp. 27–45, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 R. Tian, G. Yagawa, and H. Terasaka, “Linear dependence problems of partition of unitybased generalized FEMs,” Computer Methods in Applied Mechanics and Engineering, vol. 195, no. 37–40, pp. 4768–4782, 2006. View at: Publisher Site  Google Scholar  MathSciNet
 H.S. Oh, J. G. Kim, and W.T. Hong, “The piecewise polynomial partition of unity functions for the generalized finite element methods,” Computer Methods in Applied Mechanics and Engineering, vol. 197, no. 45–48, pp. 3702–3711, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 H.S. Oh, J. W. Jeong, and W. T. Hong, “The generalized product partition of unity for the meshless methods,” Journal of Computational Physics, vol. 229, no. 5, pp. 1600–1620, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 H. S. Oh and I. Babuŝka, “The method of auxiliary mapping for the finite element solutions of elasticity problems containing singularities,” Journal of Computational Physics, vol. 121, no. 2, pp. 193–212, 1995. View at: Publisher Site  Google Scholar  MathSciNet
 W.T. Hong and P.S. Lee, “Coupling flattop partition of unity method and finite element method,” Finite Elements in Analysis and Design, vol. 67, pp. 43–55, 2013. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
Copyright
Copyright © 2017 WonTak Hong. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.