A review of predictive nonlinear theories for multiscale modeling of heterogeneous materials

http://cmss.kaist.ac.kr/cmss/papers/2012%20Phantom-node%20method%20for%20shell%20models%20with%20arbitrary%20cracks.pdf

Phantom-node method for shell models with arbitrary cracks

In the past two decades, meshfree methods have emerged into a new class of computational methods with considerable success. In addition, a significant amount of progress has been made in ad...

Meshfree Methods: Progress Made after 20 Years

Localisation studies have been carried out for an unconstrained, elasto-plastic, strain-softening Cosserat continuum. Because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement. A finite, constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions. Because of the existence of rotational degrees-of-freedom in a Cosserat continuum additional wave types arise and wave propagation becomes dispersive. This has been investigated analytically and numerically for an elastic Cosserat continuum and an excellent agreement has been found between both solutions.

Localisation in a Cosserat continuum under static and dynamic loading conditions

A comparative study on unfilled and filled crack propagation for rock-like brittle material

SUMMARY The consistency condition for the nodal derivatives in traditional meshfree Galerkin methods is only the differentiation of the approximation consistency (DAC). One missing part is the consistency between a nodal shape function and its derivatives in terms of the divergence theorem in numerical forms. In this paper, a consistency framework for the meshfree nodal derivatives including the DAC and the discrete divergence consistency (DDC) is proposed. The summation of the linear DDC over the whole computational domain leads to the so-called integration constraint in the literature. A three-point integration scheme using background triangle elements is developed, in which the corrected derivatives are computed by the satisfaction of the quadratic DDC. We prove that such smoothed derivatives also meet the quadratic DAC, and therefore, the proposed scheme possesses the quadratic consistency that leads to its name QC3. Numerical results show that QC3 is the only method that can pass both the linear and the quadratic patch tests and achieves the best performances for all the four examples in terms of stability, convergence, accuracy, and efficiency among all the tested methods. Particularly, it shows a huge improvement for the existing linearly consistent one-point integration method in some examples. Copyright © 2012 John Wiley & Sons, Ltd.

http://gs1.dlut.edu.cn/newVersion/Files/dsxx/2379.pdf

Second‐order accurate derivatives and integration schemes for meshfree methods

SUMMARY An approximate level set method for three-dimensional crack propagation is presented. In this method, the discontinuity surface in each cracked element is defined by element-local level sets (ELLSs). The local level sets are generated by a fitting procedure that meets the fracture directionality and its continuity with the adjacent element crack surfaces in a least-square sense. A simple iterative procedure is introduced to improve the consistency of the generated element crack surface with those of the adjacent cracked elements. The discrete discontinuity is treated by the phantom node method which is a simplified version of the extended finite element method (XFEM). The ELLS method and the phantom node technology are combined for the solution of dynamic fracture problems. Numerical examples for three-dimensional dynamic crack propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright q 2009 John Wiley & Sons, Ltd.

Element-local level set method for three-dimensional dynamic crack growth

We study several enrichment strategies for dynamic crack propagation in the context of the extended finite element method and the effect of different directional criteria on the crack path. A new enrichment method with a time dependent enrichment function is proposed. In contrast to previous approaches, it entails only one crack tip enrichment function. Results for stress intensity factors and crack paths for different enrichments and direction criteria are given.

Time dependent crack tip enrichment for dynamic crack propagation

An element-free Galerkin (EFG) method with linear, quadratic and cubic approximations, which can exactly, in a numerical sense, pass the corresponding patch tests is proposed and is named as consistent EFG (CEFG) method. The development of this method is based on the Hu–Washizu three-field variational principle. Numerical integration schemes with corrected nodal derivatives at quadrature points are proposed according to the satisfaction of the orthogonality condition between stress and strain difference. Thus, the method is variationally consistent. The consistency of the corrected nodal derivatives and the satisfaction of patch test conditions are theoretically proved and also numerically validated. Numerical results show that the proposed CEFG method greatly improves the numerical performance of the EFG method in terms of accuracy, convergence, efficiency and stability, especially the proposed cubic CEFG method, which shows exceptional accuracy and convergence. Copyright © 2014 John Wiley & Sons, Ltd.

Consistent element‐free Galerkin method

Efficient implementation of the element-free Galerkin (EFG) method for a phase-field model of linear elastic fracture mechanics is presented, in which the convenience of the meshfree method to construct high order approximation functions and to implement h-adaptivity is fully exploited. A second-order moving-least squares approximation for both displacement and phase field is employed. Domain integration of the weak forms is evaluated by the quadratically consistent 3-point integration scheme. The refinement criterion using maximum residual strain energy history is proposed and the insertion of nodes is based on the background mesh. Numerical results show that the developed method is more efficient than the standard finite element method (3-node triangle element) due to the proposed h-adaptivity. In comparison with the standard EFG method, the proposed consistent EFG method significantly improves the computational efficiency and accuracy. The advantage of the quadratic approximation is also demonstrated. In addition, the feasibility of extending the proposed method to 3D is validated by the modeling of a twisting crack.

Qinglin Duan

Papers

Second‐order accurate derivatives and integration schemes for meshfree methods

Element-local level set method for three-dimensional dynamic crack growth

Time dependent crack tip enrichment for dynamic crack propagation

Consistent element‐free Galerkin method

Adaptive consistent element-free Galerkin method for phase-field model of brittle fracture