Finite elements for materials with strain gradient effects

A finite element implementation is reported of the Fleck–Hutchinson phenomenological strain gradient theory. This theory fits within the Toupin–Mindlin framework and deals with first‐order strain gradients and the associated work‐conjugate higher‐order stresses. In conventional displacement‐based approaches, the interpolation of displacement requires C¹‐continuity in order to ensure convergence of the finite element procedure for higher‐order theories. Mixed‐type finite elements are developed herein for the Fleck–Hutchinson theory; these elements use standard C⁰‐continuous shape functions and can achieve the same convergence as C¹ elements. These C⁰ elements use displacements and displacement gradients as nodal degrees of freedom. Kinematic constraints between displacement gradients are enforced via the Lagrange multiplier method. The elements developed all pass a patch test. The resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated. Copyright © 1999 John Wiley & Sons, Ltd.     

Finite element formulations of strain gradient theory for microstructures and the C0–1 patch test

Based on finite element formulations for the strain gradient theory of microstructures, a convergence criterion for the C^0–1 patch test is introduced, and a new approach to devise strain gradient finite elements that can pass the C^0–1 patch test is proposed. The displacement functions of several plane triangular elements, which satisfy the C⁰ continuity and weak C¹ continuity conditions are evaluated by the C^0–1 patch test. The difference between the proposed C^0–1 patch test and the C⁰ constant stress and C¹ constant curvature patch tests is elucidated. An 18-DOF plane strain gradient triangular element (RCT9+RT9), which passes the C^0–1 patch test and has no spurious zero energy modes, is proposed. Numerical examples are employed to examine the performance of the proposed element by carrying out the C^0–1 patch test and eigenvalue test. The proposed element is found to be without spurious zero energy modes, and it possesses higher accuracy compared with other strain gradient elements. Copyright © 2004 John Wiley & Sons, Ltd.     

A C<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript> continuous generalized finite element formulation applied to laminated Kirchhoff plate model

A generalized finite element method based on a partition of unity (POU) with smooth approximation functions is investigated in this paper for modeling laminated plates under Kirchhoff hypothesis. The shape functions are built from the product of a Shepard POU and enrichment functions. The Shepard functions have a smoothness degree directly related to the weight functions adopted for their evaluation. The weight functions at a point are built as products of C ^∞ edge functions of the distance of such a point to each of the cloud boundaries. Different edge functions are investigated to generate C ^k functions. The POU together with polynomial global enrichment functions build the approximation subspace. The formulation implemented in this paper is aimed at the general case of laminated plates composed of anisotropic layers. A detailed convergence analysis is presented and the integrability of these functions is also discussed.     

The generalized finite element method: an example of its implementation and illustration of its performance

The generalized finite element method (GFEM) was introduced in Reference 1 as a combination of the standard FEM and the partition of unity method. The standard mapped polynomial finite element spaces are augmented by adding special functions which reflect the known information about the boundary value problem and the input data (the geometry of the domain, the loads, and the boundary conditions). The special functions are multiplied with the partition of unity corresponding to the standard linear vertex shape functions and are pasted to the existing finite element basis to construct a conforming approximation. The essential boundary conditions can be imposed exactly as in the standard FEM. Adaptive numerical quadrature is used to ensure that the errors in integration do not affect the accuracy of the approximation. This paper gives an example of how the GFEM can be developed for the Laplacian in domains with multiple elliptical voids and illustrates implementation issues and the superior accuracy of the GFEM versus the standard FEM. Copyright © 2000 John Wiley & Sons, Ltd.     

A refined nonconforming quadrilateral element for couple stress/strain gradient elasticity

C^0?1 patch test (Int. J. Numer. Meth. Engng 2004; 61 :433–454) proposed by Soh and Chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress/strain gradient elasticity. The C^0?1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations. To pass the C^0?1 patch test, the element displacement functions used to calculate strains must satisfy C⁰ continuity (or weak C⁰ continuity) and quadratic completeness. In this paper, a 24‐DOF (degrees of freedom) quadrilateral element (CQ12+RDKQ) for the couple stress/strain gradient elasticity is developed by combining the refined thin plate element RDKQ and the nonconforming element CQ12. The element RDKQ, which satisfies weak C¹ continuity, is used to calculate strain gradients, whereas strains are computed by the element CQ12, which is established based on an extended variational functional and satisfies weak C⁰ continuity and quadratic completeness. Numerical examples show that the element (CQ12+RDKQ) passes the C^0?1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects. Copyright © 2010 John Wiley & Sons, Ltd.     

Dynamic analysis of fixed cracks in composites by the extended finite element method

This paper is dedicated to simulation of dynamic analysis of fixed cracks in orthotropic media using an extended finite element method. This work is in fact an extension to dynamic problems of the recently developed orthotropic extended finite element method for fracture analysis of composites. In this method, the Heaviside and near-tip enrichment functions are used in the framework of the partition of unity for modeling crack discontinuity and crack-tip singularities within the classical finite element method. In this procedure, elements that include a crack are not required to conform to crack edges. Therefore, mesh generation can be performed without any need to comply to crack edges and the method is capable of modeling the crack propagation without any remeshing. To determine the fracture properties, mixed-mode dynamic stress intensity factors (DSIFs) are evaluated by means of domain separation integral (J^′-integral) method. Results of the proposed method are compared with other available analytical and computational results.     

Developing new enrichment functions for crack simulation in orthotropic media by the extended finite element method

New enrichment functions are proposed for crack modelling in orthotropic media using the extended finite element method (XFEM). In this method, Heaviside and near‐tip functions are utilized in the framework of the partition of unity method for modelling discontinuities in the classical finite element method. In this procedure, by using meshless based ideas, elements containing a crack are not required to conform to crack edges. Therefore, mesh generation is directly performed ignoring the existence of any crack while the method remains capable of extending the crack without any remeshing requirement. Furthermore, the type of elements around the crack‐tip remains the same as other parts of the finite element model and the number of nodes and consequently degrees of freedom are reduced considerably in comparison to the classical finite element method. Mixed‐mode stress intensity factors (SIFs) are evaluated to determine the fracture properties of domain and to compare the proposed approach with other available methods. In this paper, the interaction integral (M‐integral) is adopted, which is considered as one of the most accurate numerical methods for calculating stress intensity factors. Copyright © 2006 John Wiley & Sons, Ltd.     

UNIFIED CONSTRUCTION OF FINITE ELEMENT AND FINITE VOLUME DISCRETIZATIONS FOR COMPRESSIBLE FLOWS

A framework for the construction of node-centred schemes to solve the compressible Euler and Navier–Stokes equations is presented. The metric quantities are derived by exploiting some properties of C⁰ finite element shape functions. The resulting algorithm allows to implement both artificial diffusion and one-dimensional upwind-type discretizations. The proposed methodology adopts a uniform data structure for diverse grid topologies (structured, unstructured and hybrid) and different element shapes, thus easing code development and maintenance. The final schemes are well suited to run on vector/parallel computer architectures. In the case of linear elements, the equivalence of the proposed method with a particular finite volume formulation is demonstrated.     

A 3‐node C0 triangular element for the modified couple stress theory based on the smoothed finite element method

In this paper, a 3‐node C⁰ triangular element for the modified couple stress theory is proposed. Unlike the classical continuum theory, the second‐order derivative of displacement is included in the weak form of the equilibrium equations. Thus, the first‐order derivative of displacement, such as the rotation, should be approximated by a continuous function. In the proposed element, the derivative of the displacement is defined at a node using the node‐based smoothed finite element method. The derivative fields, continuous between elements and linear in an element, are approximated with the shape functions in element. Both the displacement field and the derivative field of displacement are expressed in terms of the displacement degree of freedom only. The element stiffness matrix is calculated using the newly defined derivative field. The performance of the proposed element is evaluated through various numerical examples.     

The meshless finite element method

A meshless method is presented which has the advantages of the good meshless methods concerning the ease of introduction of node connectivity in a bounded time of order n, and the condition that the shape functions depend only on the node positions. Furthermore, the method proposed also shares several of the advantages of the finite element method such as: (a) the simplicity of the shape functions in a large part of the domain; (b) C⁰ continuity between elements, which allows the treatment of material discontinuities, and (c) ease of introduction of the boundary conditions. Copyright © 2003 John Wiley & Sons, Ltd.     

Mechanically based models: Adaptive refinement for B‐spline finite element

This article presents two new methods for adaptive refinement of a B‐spline finite element solution within an integrated mechanically based computer aided engineering system. The proposed techniques for adaptively refining a B‐spline finite element solution are a local variant of np‐refinement and a local variant of h‐refinement. The key component in the np‐refinement is the linear co‐ordinate transformation introduced into the refined element. The transformation is constructed in such a way that the transformed nodal configuration of the refined element is identical to the nodal configuration of the neighbour elements. Therefore, the assembly proceeds as with classic finite elements, while the solution approximation conforms exactly along the inter‐element boundaries. For the h‐refinement, this transformation is introduced into a construction that merges the super element from the finite element world with the hierarchical B‐spline representation from the computational geometry. In the scope of developing sculptured surfaces, the proposed approach supports C⁰ as well as the Hermite B‐spline C¹ continuous shapes. For sculptured solids, C⁰ continuity only is considered in this article. The feasibility of the proposed methods in the scope of the geometric design is demonstrated by several examples of creating sculptured surfaces and volumetric solids. Numerical performance of the methods is demonstrated for a test case of the two‐dimensional Poisson equation. Copyright © 2003 John Wiley & Sons, Ltd.     

Analysis of three‐dimensional crack initiation and propagation using the extended finite element method

An Erratum has been published for this article in International Journal for Numerical Methods in Engineering 2005, 63(8): 1228. We present a new formulation and a numerical procedure for the quasi‐static analysis of three‐dimensional crack propagation in brittle and quasi‐brittle solids. The extended finite element method (XFEM) is combined with linear tetrahedral elements. A viscosity‐regularized continuum damage constitutive model is used and coupled with the XFEM formulation resulting in a regularized ‘crack‐band’ version of XFEM. The evolving discontinuity surface is discretized through a C⁰ surface formed by the union of the triangles and quadrilaterals that separate each cracked element in two. The element's properties allow a closed form integration and a particularly efficient implementation allowing large‐scale 3D problems to be studied. Several examples of crack propagation are shown, illustrating the good results that can be achieved. Copyright © 2005 John Wiley & Sons, Ltd.     

An accurate higher-order theory and C finite element for free vibration analysis of laminated composite and sandwich plates

At present, it is difficult to accurately predict natural frequencies of sandwich plates with soft core by using the C⁰ plate bending elements. Thus, the C¹ plate bending elements have to be employed to predict accurately dynamic response of such structures. This paper proposes an accurate higher-order C⁰ theory which is very different from other published higher-order theory satisfying the interlaminar stress continuity, as the first derivative of transverse displacement has been taken out from the in-plane displacement fields of the present theory. Therefore, the C⁰ interpolation functions is only required during its finite element implementation. Based on the Hamilton’s principle and Navier’s technique, analytical solutions to the natural frequency analysis of simply-supported laminated plates have been presented. To further extend the ranges of application of the proposed theory, an eight-node C⁰ continuous isoparametric element is used to model the proposed theory. Numerical results show the present C⁰ finite element can accurately predict the natural frequencies of sandwich plate with soft core, whereas other global higher-order theories are unsuitable for free vibration analysis of such soft-core structures.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

Robust and direct evaluation of J2 in linear elastic fracture mechanics with the X‐FEM

The aim of the present paper is to study the accuracy and the robustness of the evaluation of J_k‐integrals in linear elastic fracture mechanics using the extended finite element method (X‐FEM) approach. X‐FEM is a numerical method based on the partition of unity framework that allows the representation of discontinuity surfaces such as cracks, material inclusions or holes without meshing them explicitly. The main focus in this contribution is to compare various approaches for the numerical evaluation of the J₂‐integral. These approaches have been proposed in the context of both classical and enriched finite elements. However, their convergence and the robustness have not yet been studied, which are the goals of this contribution. It is shown that the approaches that were used previously within the enriched finite element context do not converge numerically and that this convergence can be recovered with an improved strategy that is proposed in this paper. Copyright © 2008 John Wiley & Sons, Ltd.     

The extended finite element method with new crack‐tip enrichment functions for an interface crack between two dissimilar piezoelectric materials

This paper studies the static fracture problems of an interface crack in linear piezoelectric bimaterial by means of the extended finite element method (X‐FEM) with new crack‐tip enrichment functions. In the X‐FEM, crack modeling is facilitated by adding a discontinuous function and crack‐tip asymptotic functions to the classical finite element approximation within the framework of the partition of unity. In this work, the coupled effects of an elastic field and an electric field in piezoelectricity are considered. Corresponding to the two classes of singularities of the aforementioned interface crack problem, namely, ? class and κ class, two classes of crack‐tip enrichment functions are newly derived, and the former that exhibits oscillating feature at the crack tip is numerically investigated. Computation of the fracture parameter, i.e., the J‐integral, using the domain form of the contour integral, is presented. Excellent accuracy of the proposed formulation is demonstrated on benchmark interface crack problems through comparisons with analytical solutions and numerical results obtained by the classical FEM. Moreover, it is shown that the geometrical enrichment combining the mesh with local refinement is substantially better in terms of accuracy and efficiency. Copyright © 2015 John Wiley & Sons, Ltd.     

A method for creating a class of triangular C1 finite elements

Finite elements providing a C¹ continuous interpolation are useful in the numerical solution of problems where the underlying partial differential equation is of fourth order, such as beam and plate bending and deformation of strain‐gradient‐dependent materials. Although a few C¹ elements have been presented in the literature, their development has largely been heuristic, rather than the result of a rational design to a predetermined set of desirable element properties. Therefore, a general procedure for developing C¹ elements with particular desired properties is still lacking. This paper presents a methodology by which C¹ elements, such as the TUBA 3 element proposed by Argyris et al., can be constructed. In this method (which, to the best of our knowledge, is the first one of its kind), a class of finite elements is first constructed by requiring a polynomial interpolation and prescribing the geometry, the location of the nodes and the possible types of nodal DOFs. A set of necessary conditions is then imposed to obtain appropriate interpolations. Generic procedures are presented, which determine whether a given potential member of the element class meets the necessary conditions. The behaviour of the resulting elements is checked numerically using a benchmark problem in strain‐gradient elasticity. Copyright © 2011 John Wiley & Sons, Ltd.     

Isogeometric FEM‐BEM coupled structural‐acoustic analysis of shells using subdivision surfaces

We introduce a coupled finite and boundary element formulation for acoustic scattering analysis over thin‐shell structures. A triangular Loop subdivision surface discretisation is used for both geometry and analysis fields. The Kirchhoff‐Love shell equation is discretised with the finite element method and the Helmholtz equation for the acoustic field with the boundary element method. The use of the boundary element formulation allows the elegant handling of infinite domains and precludes the need for volumetric meshing. In the present work, the subdivision control meshes for the shell displacements and the acoustic pressures have the same resolution. The corresponding smooth subdivision basis functions have the C¹ continuity property required for the Kirchhoff‐Love formulation and are highly efficient for the acoustic field computations. We verify the proposed isogeometric formulation through a closed‐form solution of acoustic scattering over a thin‐shell sphere. Furthermore, we demonstrate the ability of the proposed approach to handle complex geometries with arbitrary topology that provides an integrated isogeometric design and analysis workflow for coupled structural‐acoustic analysis of shells.     

A partition of unity‐based ‘FE–Meshfree’ QUAD4 element for geometric non‐linear analysis

The recently published ‘FE–Meshfree’ QUAD4 element is extended to geometrical non‐linear analysis. The shape functions for this element are obtained by combining meshfree and finite element shape functions. The concept of partition of unity (PU) is employed for the purpose. The new shape functions inherit their higher order completeness properties from the meshfree shape functions and the mesh‐distortion tolerant compatibility properties from the finite element (FE) shape functions. Updated Lagrangian formulation is adopted for the non‐linear solution. Several numerical example problems are solved and the performance of the element is compared with that of the well‐known Q4, QM6 and Q8 elements. The results show that, for regular meshes, the performance of the element is comparable to that of QM6 and Q8 elements, and superior to that of Q4 element. For distorted meshes, the present element has better mesh‐distortion tolerance than Q4, QM6 and Q8 elements. Copyright © 2009 John Wiley & Sons, Ltd.