A simple finite element‐based framework for the analysis of elastic pseudo‐rigid bodies

This article advocates a general procedure for the numerical investigation of pseudo‐rigid bodies. The equations of motion for pseudo‐rigid bodies are shown to be mathematically equivalent to those corresponding to certain constant‐strain finite element approximations for general deformable continua. A straightforward algorithmic implementation is achieved in a classical finite element framework. Also, a penalty formulation is suggested for modelling contact between pseudo‐rigid bodies. Representative planar simulations using a non‐linear elastic model demonstrate the predictive capacity of the pseudo‐rigid theory, as well as the robustness of the proposed computational procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

A three‐dimensional C1 finite element for gradient elasticity

In gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C¹ continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C¹ finite elements and their perceived computational cost. In this context, the lack of three‐dimensional C¹ elements is of particular concern. In this paper we present a new C¹ hexahedral element which, to the best of our knowledge, is the first three‐dimensional C¹ element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C¹ elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good‐quality solutions. Copyright © 2008 John Wiley & Sons, Ltd.     

Mixed finite elements and Newton‐type linearizations for the solution of Richards' equation

We present the development of a two‐dimensional Mixed‐Hybrid Finite Element (MHFE) model for the solution of the non‐linear equation of variably saturated flow in groundwater on unstructured triangular meshes. By this approach the Darcy velocity is approximated using lowest‐order Raviart–Thomas (RT₀) elements and is ‘exactly’ mass conserving. Hybridization is used to overcome the ill‐conditioning of the mixed system. The scheme is globally first‐order in space. Nevertheless, numerical results employing non‐uniform meshes show second‐order accuracy of the pressure head and normal fluxes on specific grid points. The non‐linear systems of algebraic equations resulting from the MHFE discretization are solved using Picard or Newton iterations. Realistic sample tests show that the MHFE‐Newton approach achieves fast convergence in many situations, in particular, when a good initial guess is provided by either the Picard scheme or relaxation techniques. Copyright © 1999 John Wiley & Sons, Ltd.     

On the a priori and a posteriori error analysis of a two‐fold saddle‐point approach for nonlinear incompressible elasticity

In this paper, we reconsider the a priori and a posteriori error analysis of a new mixed finite element method for nonlinear incompressible elasticity with mixed boundary conditions. The approach, being based only on the fact that the resulting variational formulation becomes a two‐fold saddle‐point operator equation, simplifies the analysis and improves the results provided recently in a previous work. Thus, a well‐known generalization of the classical Babu?ka–Brezzi theory is applied to show the well‐posedness of the continuous and discrete formulations, and to derive the corresponding a priori error estimate. In particular, enriched PEERS subspaces are required for the solvability and stability of the associated Galerkin scheme. In addition, we use the Ritz projection operator to obtain a new reliable and quasi‐efficient a posteriori error estimate. Finally, several numerical results illustrating the good performance of the associated adaptive algorithm are presented. Copyright © 2006 John Wiley & Sons, Ltd.     

A Koiter‐Newton approach for nonlinear structural analysis

A new approach termed the Koiter‐Newton is presented for the numerical solution of a class of elastic nonlinear structural response problems. It is a combination of a reduction method inspired by Koiter's post‐buckling analysis and Newton arc‐length method so that it is accurate over the entire equilibrium path and also computationally efficient in the presence of buckling. Finite element implementation based on element independent co‐rotational formulation is used. Various numerical examples of buckling sensitive structures are presented to evaluate the performance of the method. The examples demonstrate that the method is robust and completely automatic and that it outperforms traditional path‐following techniques. This improved efficiency will open the door for the direct use of detailed nonlinear finite element models in the design optimization of next generation flight and launch vehicles. Copyright © 2013 John Wiley & Sons, Ltd.     

B‐bar FEMs for anisotropic elasticity

Anisotropic elastic materials, such as the homogenized model of a fiber‐reinforced matrix, can display near rigidity under certain applied stress–the resulting strains are small compared with the strains that would occur for other stresses of comparable magnitude. The anisotropic material could be rigid under hydrostatic pressure if the material were incompressible, as in isotropic elasticity, but also for other stresses. Some commonly used finite elements are effective in dealing with incompressibility, but are ill‐equipped to handle materials that lock under non‐hydrostatic stress states (e.g., uniformly reduced serendipity and Q1/Q0 B‐bar hexahedra). The failure of the original B‐bar method is attributed to the assumption that the mode of deformation to be relieved is one of near incompressibility. The remedy proposed here is based on the spectral decomposition of the compliance matrix of the material. The spectrum can be interpreted to separate nearly‐rigid and flexible modes of stress and strain, which leads naturally to a generalized selective reduced integration. Furthermore, the spectral decomposition also enables a three‐field elasticity formulation that results in a B‐bar method that is effective for general anisotropic materials with an arbitrary nearly‐rigid mode of deformation.Copyright © 2014 John Wiley & Sons, Ltd.     

An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain‐gradient theory of elasticity

An advanced boundary element method (BEM) for solving two‐ (2D) and three‐dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form‐II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Interface dynamic stiffness matrix approach for three‐dimensional transient multi‐region boundary element analysis

A novel substructuring method is developed for the coupling of boundary element and finite element subdomains in order to model three‐dimensional multi‐region elastodynamic problems in the time domain. The proposed procedure is based on the interface stiffness matrix approach for static multi‐region problems using variational principles together with the concept of Duhamel integrals. Unit impulses are applied at the boundary of each region in order to evaluate the impulse response matrices of the Duhamel (convolution) integrals. Although the method is not restricted to a special discretization technique, the regions are discretized using the boundary element method combined with the convolution quadrature method. This results in a time‐domain methodology with the advantages of performing computations in the Laplace domain, which produces very accurate and stable results as verified on test examples. In addition, the assembly of the boundary element regions and the coupling to finite elements are greatly simplified and more efficient. Finally, practical applications in the area of soil–structure interaction and tunneling problems are shown. Copyright © 2009 John Wiley & Sons, Ltd.     

Axial symmetric elasticity analysis in non‐homogeneous bodies under gravitational load by triple‐reciprocity boundary element method

In general, internal cells are required to solve elasticity problems by involving a gravitational load in non‐homogeneous bodies with variable mass density when using a conventional boundary element method (BEM). Then, the effect of mesh reduction is not achieved and one of the main merits of the BEM, which is the simplicity of data preparation, is lost. In this study, it is shown that the domain cells can be avoided by using the triple‐reciprocity BEM formulation, where the density of domain integral is expressed in terms of other fields that are represented by boundary densities and/or source densities at isolated interior points. Utilizing the rotational symmetry, the triple‐reciprocity BEM formulation is developed for axially symmetric elasticity problems in non‐homogeneous bodies under gravitational force. A new computer program was developed and applied to solve several test problems. Copyright © 2008 John Wiley & Sons, Ltd.     

A family of second‐order boundary dampers for finite element analysis of two and three‐dimensional problems in transient exterior acoustics

Numerical modelling of exterior acoustics problems involving infinite medium requires truncation of the medium at a finite distance from the obstacle or the structure and use of non‐reflecting boundary condition at this truncation surface to simulate the asymptotic behaviour of radiated waves at far field. In the context of the finite element method, Bayliss–Gunzburger–Turkel (BGT) boundary conditions are well suited since they are local in both space and time. These conditions involve ‘damper’ operators of various orders, which work on acoustic pressure p and they have been used in time harmonic problems widely and in transient problems in a limited way. Alternative forms of second‐order BGT operators, which work on ṗ (time derivative of p) had been suggested in an earlier paper for 3D problems but they were neither implemented nor validated. This paper presents detailed formulations of these second‐order dampers both for 2D and 3D problems, implements them in a finite element code and validates them using appropriate example problems. The developed code is capable of handling exterior acoustics problems involving both Dirichlet and Neumann boundary conditions. Copyright © 2000 John Wiley & Sons, Ltd.     

Finite elements for elasticity with microstructure and gradient elasticity

We present a general finite element discretization of Mindlin's elasticity with microstructure. A total of 12 isoparametric elements are developed and presented, six for plane strain conditions and six for the general case of three‐dimensional deformation. All elements interpolate both the displacement and microdeformation fields. The minimum order of integration is determined for each element, and they are all shown to pass the single‐element test and the patch test. Numerical results for the benchmark problem of one‐dimensional deformation show good convergence to the closed‐form solution. The behaviour of all elements is also examined at the limiting case of vanishing relative deformation, where elasticity with microstructure degenerates to gradient elasticity. An appropriate parameter selection that enforces this degeneration in an approximate manner is presented, and numerical results are shown to provide good approximation to the respective displacements and strains of a gradient elastic solid. Copyright © 2007 John Wiley & Sons, Ltd.     

A numerical procedure to solve non‐linear kinematic problems in spatial mechanisms

This paper presents a numerical method to solve the forward position problem in spatial mechanisms. The method may be incorporated in a software for the kinematic analysis of mechanisms, where the procedure is systematic and can be easily implemented, achieving a high degree of automation in simulation. The procedure presents high computational efficiency, enabling its incorporation in the control loop to solve the forward position problem in the case of a velocity control scheme. Also, in this paper preliminary results on the convergence of the proposed procedure are shown, and efficiency results of the method applied to representative spatial mechanisms are presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Adaptive superposition of finite element meshes in non‐linear transient solid mechanics problems

An s‐adaptive finite element procedure is developed for the transient analysis of 2‐D solid mechanics problems with material non‐linearity due to progressive damage. The resulting adaptive method simultaneously estimates and controls both the spatial error and temporal error within user‐specified tolerances. The spatial error is quantified by the Zienkiewicz–Zhu error estimator and computed via superconvergent patch recovery, while the estimation of temporal error is based on the assumption of a linearly varying third‐order time derivatives of the displacement field in conjunction with direct numerical time integration. The distinguishing characteristic of the s‐adaptive procedure is the use of finite element mesh superposition (s‐refinement) to provide spatial adaptivity. Mesh superposition proves to be particularly advantageous in computationally demanding non‐linear transient problems since it is faster, simpler and more efficient than traditional h‐refinement schemes. Numerical examples are provided to demonstrate the performance characteristics of the s‐adaptive method for quasi‐static and transient problems with material non‐linearity. Copyright © 2007 John Wiley & Sons, Ltd.     

hp‐Generalized FEM and crack surface representation for non‐planar 3‐D cracks

A high‐order generalized finite element method (GFEM) for non‐planar three‐dimensional crack surfaces is presented. Discontinuous p‐hierarchical enrichment functions are applied to strongly graded tetrahedral meshes automatically created around crack fronts. The GFEM is able to model a crack arbitrarily located within a finite element (FE) mesh and thus the proposed method allows fully automated fracture analysis using an existing FE discretization without cracks. We also propose a crack surface representation that is independent of the underlying GFEM discretization and controlled only by the physics of the problem. The representation preserves continuity of the crack surface while being able to represent non‐planar, non‐smooth, crack surfaces inside of elements of any size. The proposed representation also provides support for the implementation of accurate, robust, and computationally efficient numerical integration of the weak form over elements cut by the crack surface. Numerical simulations using the proposed GFEM show high convergence rates of extracted stress intensity factors along non‐planar curved crack fronts and the robustness of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Semi‐analytical analysis for multi‐directional functionally graded plates: 3‐D elasticity solutions

Semi‐analytical 3‐D elasticity solutions are presented for orthotropic multi‐directional functionally graded plates using the differential quadrature method (DQM) based on the state‐space formalism. Material properties are assumed to vary not only through the thickness but also in the in‐plane directions following an exponential law. The graded in‐plane domain is solved numerically via the DQM, while exact solutions are sought for the thickness domain using the state‐space method. Convergence studies are performed, and the present hybrid semi‐analytical method is validated by comparing numerical results with the exact solutions for a conventional unidirectional functionally graded plate. Finally, effects of material gradient indices on the displacement and stress fields of the plates are investigated and discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Scaled boundary finite‐element analysis of a non‐homogeneous elastic half‐space

As a result of stresses experienced during and after the deposition phase, a soil strata of uniform material generally exhibits an increase in elastic stiffness with depth. The immediate settlement of foundations on deep soil deposits and the resultant stress state within the soil mass may be most accurately calculated if this increase in stiffness with depth is taken into account. This paper presents an axisymmetric formulation of the scaled boundary finite‐element method and incorporates non‐homogeneous elasticity into the method. The variation of Young's modulus (E) with depth (z) is assumed to take the form E=m_Ez^α, where m_E is a constant and αis the non‐homogeneity parameter. Results are presented and compared to analytical solutions for the settlement profiles of rigid and flexible circular footings on an elastic half‐space, under pure vertical load with αvarying between zero and one, and an example demonstrating the versatility and practicality of the method is also presented. Known analytical solutions are accurately represented and new insight regarding displacement fields in a non‐homogeneous elastic half‐space is gained. Copyright © 2003 John Wiley & Sons, Ltd.     

A posteriori error estimates and adaptivity for finite element solutions in finite elasticity

Methods for a posteriori error estimation for finite element solutions are well established and widely used in engineering practice for linear boundary value problems. In contrast here we are concerned with finite elasticity and error estimation and adaptivity in this context. In the paper a brief outline of continuum theory of finite elasticity is first given. Using the residuals in the equilibrium conditions the discretization error of the finite element solution is estimated both locally and globally. The proposed error estimator is physically interpreted in the energy sense. We then present and discuss the convergence behaviour of the discretization error in uniformly and adaptively refined finite element sequences.     

A meshfree‐enriched finite element method for compressible and near‐incompressible elasticity

In this paper, a two‐dimensional displacement‐based meshfree‐enriched FEM (ME‐FEM) is presented for the linear analysis of compressible and near‐incompressible planar elasticity. The ME‐FEM element is established by injecting a first‐order convex meshfree approximation into a low‐order finite element with an additional node. The convex meshfree approximation is constructed using the generalized meshfree approximation method and it possesses the Kronecker‐delta property on the element boundaries. The gradient matrix of ME‐FEM element satisfies the integration constraint for nodal integration and the resultant ME‐FEM formulation is shown to pass the constant stress test for the compressible media. The ME‐FEM interpolation is an element‐wise meshfree interpolation and is proven to be discrete divergence‐free in the incompressible limit. To prevent possible pressure oscillation in the near‐incompressible problems, an area‐weighted strain smoothing scheme incorporated with the divergence‐free ME‐FEM interpolation is introduced to provide the smoothing on strains and pressure. With this smoothed strain field, the discrete equations are derived based on a modified Hu–Washizu variational principle. Several numerical examples are presented to demonstrate the effectiveness of the proposed method for the compressible and near‐incompressible problems. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient computation of order and mode of corner singularities in 3D‐elasticity

A general numerical procedure is presented for the efficient computation of corner singularities, which appear in the case of non‐smooth domains in three‐dimensional linear elasticity. For obtaining the order and mode of singularity, a neighbourhood of the singular point is considered with only local boundary conditions. The weak formulation of the problem is approximated by a Galerkin–Petrov finite element method. A quadratic eigenvalue problem ( P +λ Q +λ² R ) u = 0 is obtained, with explicitly analytically defined matrices P , Q , R . Moreover, the three matrices are found to have optimal structure, so that P , R are symmetric and Q is skew symmetric, which can serve as an advantage in the following solution process. On this foundation a powerful iterative solution technique based on the Arnoldi method is submitted. For not too large systems this technique needs only one direct factorization of the banded matrix P for finding all eigenvalues in the interval ?e(λ)∈(?0.5,1.0) (no eigenpairs can be ‘lost’) as well as the corresponding eigenvectors, which is a great improvement in comparison with the normally used determinant method. For large systems a variant of the algorithm with an incomplete factorization of P is implemented to avoid the appearance of too much fill‐in. To illustrate the effectiveness of the present method several new numerical results are presented. In general, they show the dependence of the singular exponent on different geometrical parameters and the material properties. Copyright © 2001 John Wiley & Sons, Ltd.