Keywords: meshless, element-free, crack propagation, quadrature.

Meshless and in general Partition of Unity methods are receiving attention in computational mechanics research because of some advantages they have compared to the classical finite element method, namely higher rates of convergence, ability to model discontinuities by basis enrichment, insensitivity to distortion in large displacement problems, construction of solutions with any desired degree of continuity. In these methods the discretization is purely nodal, and the finite element concept of connectivity between elements is not introduced. Consequently only a cloud of nodes needs to be generated, and local refinement is made increasing the node density in the region of interest. Given the shape functions constructed on a given nodal arrangement, the weak form of the equilibrium equations (or in general the variational principle for the problem under consideration) require quadratures over the domain. This is a major task in the application of these methods. In fact the shape functions are very complex compared to the finite element ones and they are therefore much difficult to integrate. Different approaches have been used for performing this task:

• introduction of integration cells in the domain, i.e. of an underlying mesh used only for the quadrature;
• nodal integration approaches, requiring proper modifications of the variational principle [1,2];
• modification of the variational principle from a global to a series of local ones restricted to the support of the weight functions [3,4].

In the paper a new and different approach is introduced [5,6], based on the partition of unity property of the shape functions stating that the sum of the shape functions at any point is equal to . Introducing the sum of the shape functions as a unit weight in the quadrature of the variational form, the quadrature over the whole domain can be transformed into a sum of integrals over the weight function supports. The general problem of the quadrature over the whole domain is therefore converted into the sum of integrals over a standard domain which is generally a square or a circle, depending on the choice of the support of the weight functions.

This method, called here Partition of Unity Quadrature (PUQ) does not require therefore the subdivision of the whole domain into smaller integration cells and the modification of the variational principle is not needed. Moreover its range of applicability is not limited to meshless methods, but it can be seen as a general approach for computing integrals over arbitrary domains.

The crack propagation problem has been analyzed in literature [7] using standard cell quadrature, while nodal integration [1], modified variational principles [3,4] and partition of unity quadrature [6,5] have actually been applied on patch tests and on some classical benchmark problems. In the analysis of the crack problem this requires, at each crack tip advancement, the redefinition of the integration cells and Gauss points around the tip. The present paper is a starting point for investigating the application of the partition of unity quadrature in crack propagation problems. As the crack advances the PUQ does not require any modification of an underlying quadrature mesh, and it is therefore well suited for this class of problems. Although no crack propagation numerical examples are presented, the PUQ has been tested for the evaluation of the stress intensity factors compared to standard Gaussian quadrature on subcells, as their correct evaluation can be regarded as the starting point for the prediction of the crack trajectory. The presented examples illustrate the effectiveness of the PUQ, even if further research work is needed for improving its numerical efficiency.

1

S. Beissel, T. Belytschko, "Nodal integration of the element free Galerkin method", Comp. Meth. Appl. Mech. Engrg., 139, 49-74, 1996. doi:10.1016/S0045-7825(96)01079-1

2

J.S. Chen, C.T. Wu, S. Yoon and Y. You. "A stabilized conforming nodal integration for Galerkin meshfree methods", to appear, Int. J. Num. Meth. Engrg.

3

S.N. Atluri, T. Zhu, "A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics", Comp. Mech., 22, 117-127, 1998. doi:10.1007/s004660050346

4

S.N. Atluri, T. Zhu, "New concepts in meshless methods", Int. J. Num. Meth. Engrg., 47, 537-556, 2000. doi:10.1002/(SICI)1097-0207(20000110/30)47:1/3<537::AID-NME783>3.0.CO;2-E

5

A. Carpinteri, G. Ferro and G. Ventura, "Partition of Unity Weighted Quadrature in Meshless Methods", European Conference on Computational Mechanics ECCM2001, Cracow, PL, June 2001.

6

A. Carpinteri, G. Ferro and G. Ventura, "The Partition of Unity Quadrature in Meshless Methods", accepted for publication on Int. J. Num. Meth. Engrg. doi:10.1002/nme.455

7

T. Belytschko, L. Gu, Y.Y. Lu, "Fracture and crack growth by element-free Galerkin methods", Model. Simul. Mater. Sci. Engrg., 2, 519-534, 1994. doi:10.1088/0965-0393/2/3A/007

purchase the full-text of this paper (price £20)

go to the previous paper
go to the next paper
return to the table of contents
return to the book description
purchase this book (price £122 +P&P)