Nguyen Dinh Du, Msc 

[email protected]

1. Tinh Quoc Bui, Dam Quang Vo, Chuanzeng Zhang, Du Dinh Nguyen. A consecutive-interpolation quadrilateral element (CQ4): Formulation and applications.

An efficient, smooth and accurate quadrilateral element with four-node based on the consecutiveinterpolation procedure (CIP) is formulated. The CIP is developed recently by Zheng et al. (Acta Mech Sin 26 (2010) 265–278) for triangular element with three-node. In this setting the approximation functionshandle both nodal values and averaged nodal gradients as interpolation conditions. Two stages of the interpolation are required; the primary stage is carried out using the same procedure of the standard finite element method (FEM), and the interpolation is further reproduced in the secondary step according to both nodal values and average nodal gradients derived from the previous interpolation. The new consecutive-interpolation quadrilateral element with four-node (CQ4) deserves many desirable characteristics of an efficient numerical method, which involves continuous nodal gradients, continuous nodal stresses without smoothing operation, higher-order polynomial basis, without increasing the degree of freedom of the system, straightforward to implement in an existing FEM computer code, etc. Four benchmark and two practical examples are considered for the stress analysis of elastic structures in two-dimension to show the accuracy and the efficiency of the new element. Detailed comparison and some other aspects including the convergence rate, volumetric locking, computational efficiency, insensitivity to the mesh, etc. are investigated. Numerical results substantially indicate that the consecutive-interpolation finite element method (CFEM) with notable features pertains to high accuracy, convergence rate, and efficiency as compared with the standard FEM.

2. An extended consecutive-interpolation quadrilateral element (XCQ4) applied to linear elastic fracture mechanics.

A novel extended 4-node quadrilateral finite element (XCQ4) based on a consecutive-interpolation procedure (CIP) with continuous nodal stress for accurately modeling singular stress fields near crack tips of two-dimensional (2D) cracks in solids is presented. In contrast to the traditional approaches, the approximation functions constructed based on the CIP involve both nodal values and averaged nodal gradients as interpolation conditions. Our objective is to exhibit a pioneering extension of the recently developed CQ4 element enhanced by enrichment to precisely model 2D elastic crack problems, taking advantages of the strengths and making use all the desirable features of both techniques, the CIP and the local enriched partition of unity method. The stress intensity factors (SIFs) are estimated using the interaction integral. The accuracy and performance of the proposed XCQ4 and its numerical properties are illustrated by numerical examples, considering both single and mixed-mode problems with complicated configurations. Compared with reference solutions available in the literature and the conventional XQ4 results, it is found that the accuracy of the XCQ4 is high. Studies on the convergence rate of the SIFs in relative errors also reveal a better performance of the XCQ4 over the classical XQ4. The fracture parameters are found to be stable for different areas of integration paths around the crack tip. Further applications of the developed XCQ4 to other complex problems are potential.

3. Analysis of 2-dimensional transient problems for linear elastic and piezoelectric structures using the consecutive-interpolation quadrilateral element (CQ4).

The recently developed consecutive-interpolation procedure (CIP) based 4-node quadrilateral element (CQ4) is used to study free and forced 2-dimensional vibrations of elastic and piezoelectric structures of complex geometries. For some of these problems results have also been computed with the standard quadrilateral element under the constraint that the two approaches have the same number of degrees of freedom. In each case it is found that the numerical solution using the CQ4 element has better accuracy than that found with the standard quadrilateral element, and these solutions agree well with results available in the literature. The main difference between the CIP basis functions and those for the traditional 4-node quadrilateral element is that the former incorporate both nodal values and averaged gradients of the trial solution at the nodes and the latter only the nodal values. However, for both sets of basis functions only nodal values appear as unknowns in the matrix formulation of the problem.

4. Dynamic stationary crack analysis of isotropic solids and anisotropic composites by enhanced local enriched consecutive-interpolation elements.

The recently developed local enriched consecutive-interpolation 4-node quadrilateral element (XCQ4) is extended to study transient dynamic stress intensity factors (DSIFs) for isotropic solids and anisotropic composite materials containing stationary cracks. The XCQ4 involves both nodal values and averaged nodal gradients as interpolation conditions to smooth the distribution of unknown variables. The possible physical properties of enriched nodes in consecutive-interpolation procedure (CIP) would be thought as non-locality feature, which could improve the accuracy of results and eliminate the non-smooth stresses among inter-elements. In XCQ4, the crack is determined by level set function and enriched by Heaviside and crack-tip enrichment functions with special anisotropic enriched crack-tip functions. Timedependent discrete equations for dynamic cracks are solved by Newmark time integration scheme at
each time step without considering the effects of velocity-based global damping matrix. The proposed method is verified through a series of numerical examples of transient fracture in both isotropic and anisotropic materials. Numerical DSIFs are compared with reference solutions available in literature. The behavior of dynamic response is explored in specimens with complex configuration under step and sine loads.

5. Enhanced nodal gradient finite elements with new numerical integration schemes for 2D and 3D geometrically nonlinear analysis.

The consecutive-interpolation procedure (CIP) has been recently proposed as an enhanced technique for traditional finite element method (FEM) with various desirable properties such as continuous nodal gradients and higher accuracy without increasing the total number of degrees of freedom (DOFs). It is common knowledge that linear finite elements, e.g., four-node quadrilateral (Q4) or eight- node hexahedral (HH8) elements, are not highly suitable for geometrically nonlinear analysis. The elements with quadratic interpolation functions have to be used instead. In this paper, the CIP-enhanced four-node quadrilateral element (CQ4), and the CIP-enhanced eight-node hexahedral element (CHH8), are for the first time extended to investigate geometrically nonlinear problems of two- (2D) and three-dimensional (3D) structures. To further enhance the efficiency of the present approaches, novel numerical integration schemes based on the concept of mid-point rules, namely element mid-points (EM) and element mid-edges (EE) are integrated into the present CQ4 element. For CHH8, the 3D-version of EM (namely 3D-EM) and the element mid-faces (EF) scheme are investigated. The accuracy and computational efficiency of the two novel quadrature schemes in both regular and irregular (distorted) meshes are analyzed. Numerical results indicate that the new integration approaches perform more efficiently than the well-known Gaussian quadrature while gaining equivalent accuracy. The performance of the CIP-enhanced elements, which is examined through numerical experiments, is found to be equivalent to that of quadratic Lagrangian finite element counterparts, while having the same discretization with that by the linear finite elements. In addition, we also apply the present CQ4 and CHH8 elements associated with different numerical integration techniques to nearly incompressible materials.

6. Modeling the transient dynamic fracture and quasi-static crack growth in
cracked functionally graded composites by the extended four-node gradient
finite elements.

This paper presents numerical simulations of transient dynamic fracture behaviors and quasi-static crack growth in cracked functionally graded composites using the enhanced extended consecutive-interpolation quadrilateral element (XCQ4). The mechanical properties of functional composites are assumed to vary continuously in spatial coordinates. In terms of XCQ4 modeling, only one function of ramp type (either linear, quadratic or cubic order), alternatively to the usual four branch functions, is used as crack-tip enrichment, somewhat reducing the complexity in implementation. Another advantage of employing the ramp function, which is not derived from asymptotic solutions, lies in the less additional degrees of freedom (DOFs) required for the branch functions, saving the computational effort. The merits of the developed approach are demonstrated through our numerical experiments, which are devoted to both dynamic loading with stationary cracks and quasi-static crack growth simulation in FGMs composites. The accuracy of the developed model is verified by comparing the computed results with respect to reference solutions derived from analytical solution, other numerical methods, and experimental data.

7. Analysis of linear elastic fracture mechanics for cracked functionally
graded composite plate by enhanced local enriched consecutive-interpolation
elements.

This paper reports the application of consecutive-interpolation procedure into four-node quadrilateral elements for analysis of two-dimensional cracked solids made of functionally graded composite plate. Compared to standard finite element method, the recent developed consecutive-interpolation has been shown to possess many desirable features, such as higher accuracy and smooth nodal gradients it still satisfies the Kronecker-delta property and keeps the total number of degrees of freedom unchanged. The discontinuity in displacement fields along the crack faces and stress singularity around the crack tips are mathematically modeled using enrichment functions. The Heaviside function is employed to describe displacement jump, while four branch functions being developed from asymptotic stress fields are taken as basis functions to capture singularities. The interesting characteristic of functionall graded composite plate is the spatial variation of material properties which are intentionally designed to be served for particular purposes. Such variation has to be taken into account during the computation of Stress Intensity Factors (SIFs). Performance of the proposed approach is demonstrated and verified through various numerical examples, in which SIFs are compared with reference solutions derived from other methods available in literatures.