A minimal remeshing finite element method for crack growth is presented. Discontinuous enrichment functions are added to the finite element approximation to account for the presence of the crack. This method allows the crack to be arbitrarily aligned within the mesh. For severely curved cracks, remeshing may be needed but only away from the crack tip where remeshing is much easier. Results are presented for a wide range of two-dimensional crack problems showing excellent accuracy. Copyright © 1999 John Wiley & Sons, Ltd.

Elastic crack growth in finite elements with minimal remeshing

Meshless methods: An overview and recent developments

A new continuous reproducing kernel interpolation function which explores the attractive features of the flexible time-frequency and space-wave number localization of a window function is developed. This method is motivated by the theory of wavelets and also has the desirable attributes of the recently proposed smooth particle hydrodynamics (SPH) methods, moving least squares methods (MLSM), diffuse element methods (DEM) and element-free Galerkin methods (EFGM). The proposed method maintains the advantages of the free Lagrange or SPH methods; however, because of the addition of a correction function, it gives much more accurate results. Therefore it is called the reproducing kernel particle method (RKPM). In computer implementation RKPM is shown to be more efficient than DEM and EFGM. Moreover, if the window function is C∞, the solution and its derivatives are also C∞ in the entire domain. Theoretical analysis and numerical experiments on the 1D diffusion equation reveal the stability conditions and the effect of the dilation parameter on the unusually high convergence rates of the proposed method. Two-dimensional examples of advection-diffusion equations and compressible Euler equations are also presented together with 2D multiple-scale decompositions.

Reproducing kernel particle methods

A new finite element method is presented that features the ability to include in the finite element space knowledge about the partial differential equation being solved This new method can therefore be more efficient than the usual finite element methods An additional feature of the partition-of-unity method is that finite element spaces of any desired regularity can be constructed very easily This paper includes a convergence proof of this method and illustrates its efficiency by an application to the Helmholtz equation for high wave numbers The basic estimates for a posteriori error estimation for this new method are also proved © 1997 by John Wiley & Sons, Ltd

The Partition of Unity Method

A local symmetric weak form (LSWF) for linear potential problems is developed, and a truly meshless method, based on the LSWF and the moving least squares approximation, is presented for solving potential problems with high accuracy. The essential boundary conditions in the present formulation are imposed by a penalty method. The present method does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the “energy”. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No post-smoothing technique is required for computing the derivatives of the unknown variable, since the original solution, using the moving least squares approximation, is already smooth enough. Several numerical examples are presented in the paper. In the example problems dealing with Laplace & Poisson's equations, high rates of convergence with mesh refinement for the Sobolev norms ||·||0 and ||·||1 have been found, and the values of the unknown variable and its derivatives are quite accurate. In essence, the present meshless method based on the LSWF is found to be a simple, efficient, and attractive method with a great potential in engineering applications.

A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

An element-free Galerkin method which is applicable to arbitrary shapes but requires only nodal data is applied to elasticity and heat conduction problems. In this method, moving least-squares interpolants are used to construct the trial and test functions for the variational principle (weak form); the dependent variable and its gradient are continuous in the entire domain. In contrast to an earlier formulation by Nayroles and coworkers, certain key differences are introduced in the implementation to increase its accuracy. The numerical examples in this paper show that with these modifications, the method does not exhibit any volumetric locking, the rate of convergence can exceed that of finite elements significantly and a high resolution of localized steep gradients can be achieved. The moving least-squares interpolants and the choices of the weight function are also discussed in this paper.

Element‐free Galerkin methods

During rock failure and instability, cracks usually appear as microcracks in local areas and then expand into significant macroscopic cracks. In this study, the whole process of rock deformation and instability under uniaxial loading is investigated with standard rock specimens, and acoustic emission (AE) and digital image correlation (DIC) technology are introduced to explore the process of rock failure and instability. AE technology is used to identify the location of crack propagation caused by microcracks and large cracks, and DIC is used to measure the crack propagation at different locations. Results show that the evolution of accumulated energy is closely related to the change in stress. When the specimen approaches failure, a “y” shaped localization zone is formed, and the evolution path is consistent with the through-through path of the crack, which better reflects the propagation law of the crack in the rock. The spatial distribution of the AE location event and energy density is consistent with the evolution path of the localization zone. The deformation value of the deformation field is closely related to the initiation and evolution of the deformation localization zone. On the basis of density-based spatial clustering of applications with a noise-clustering algorithm, AE positioning events are further processed and projected into the digital image of the deformation field, and the results of clustering projection are in good agreement with the deformation localization zone. Results show that AE and DIC coupling localization techniques can effectively identify the fracture process zone and fracture mechanism of rock, providing a new technical means for further studying the mechanical properties of rock materials.

https://www.mdpi.com/2071-1050/14/22/15037/pdf?version=1668418313

L. Gu

Papers

Element‐free Galerkin methods

Coupling Study of Deformation Field Evolution and Acoustic Emission Response Characteristics in Rock Failure and Instability Process