Effects of approximations in analyses of beams of open thin‐walled cross‐section—part I: Flexural–torsional stability

In formulating a finite element model for the flexural–torsional stability and 3‐D non‐linear analyses of thin‐walled beams, a rotation matrix is usually used to obtain the non‐linear strain–displacement relationships. Because of the coupling between displacements, twist rotations and their derivatives, the components of the rotation matrix are both lengthy and complicated. To facilitate the formulation, approximations have been used to simplify the rotation matrix. A simplified small rotation matrix is often used in the formulation of finite element models for the flexural–torsional stability analysis of thin‐walled beams of open cross‐section. However, the approximations in the small rotation matrix may lead to the loss of some significant terms in the stability stiffness matrix. Without these terms, a finite element line model may predict the incorrect flexural–torsional buckling load of a beam. This paper investigates the effects of approximations in the elastic flexural–torsional stability analysis of thin‐walled beams, while a companion paper investigates the effects of approximations in the 3‐D non‐linear analysis. It is found that a finite element line model based on a small rotation matrix may predict incorrect elastic flexural–torsional buckling loads of beams. To perform a correct flexural–torsional stability analysis of thin‐walled beams, modification of the model is needed, or a finite element model based on a second‐order rotation matrix can be used. Copyright © 2001 John Wiley & Sons, Ltd.     

Non‐linear analysis of thin‐walled members of open cross‐section

The paper presents a means of determining the non‐linear stiffness matrices from expressions for the first and second variation of the Total Potential of a thin‐walled open section finite element that lead to non‐linear stiffness equations. These non‐linear equations can be solved for moderate to large displacements. The variations of the Total Potential have been developed elsewhere by the authors, and their contribution to the various non‐linear matrices is stated herein. It is shown that the method of solution of the non‐linear stiffness matrices is problem dependent. The finite element procedure is used to study non‐linear torsion that illustrates torsional hardening, and the Newton–Raphson method is deployed for this study. However, it is shown that this solution strategy is unsuitable for the second example, namely that of the post‐buckling response of a cantilever, and a direct iteration method is described. The good agreement for both of these problems with the work of independent researchers validates the non‐linear finite element method of analysis. Copyright © 1999 John Wiley & Sons, Ltd.     

Design sensitivity analysis and optimization of non‐linear transient dynamics. Part I—sizing design

A continuum‐based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non‐linear structural systems with elastic–plastic material and large deformation. The methodology is aimed for applications in non‐linear dynamic problems, such as crashworthiness design. The first‐order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct differentiation method and updated Lagrangian formulation are used since they are more appropriate for the path‐dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central difference method and the finite element method are used to discretize the temporal and spatial domains, respectively. The Hughes–Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate‐based incrementally objective stress integration scheme are used to handle the finite strain and rotation. An elastic–plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the finite difference results. Design optimization is carried out using the design sensitivity information. Copyright © 2000 John Wiley & Sons, Ltd.     

c‐Type method of unified CAMG and FEA. Part 1: Beam and arch mega‐elements—3D linear and 2D non‐linear

Computer‐aided mesh generation (CAMG) dictated solely by the minimal key set of requirements of geometry, material, loading and support condition can produce ‘mega‐sized’, arbitrary‐shaped distorted elements. However, this may result in substantial cost saving and reduced bookkeeping for the subsequent finite element analysis (FEA) and reduced engineering manpower requirement for final quality assurance. A method, denoted as c‐type, has been proposed by constructively defining a finite element space whereby the above hurdles may be overcome with a minimal number of hyper‐sized elements. Bezier (and de Boor) control vectors are used as the generalized displacements and the Bernstein polynomials (and B‐splines) as the elemental basis functions. A concomitant idea of coerced parametry and inter‐element continuity on demand unifies modelling and finite element method. The c‐type method may introduce additional control, namely, an inter‐element continuity condition to the existing h‐type and p‐type methods. Adaptation of the c‐type method to existing commercial and general‐purpose computer programs based on a conventional displacement‐based finite element method is straightforward. The c‐type method with associated subdivision technique can be easily made into a hierarchic adaptive computer method with a suitable a posteriori error analysis. In this context, a summary of a geometrically exact non‐linear formulation for the two‐dimensional curved beams/arches is presented. Several beam problems ranging from truly three‐dimensional tortuous linear curved beams to geometrically extremely non‐linear two‐dimensional arches are solved to establish numerical efficiency of the method. Incremental Lagrangian curvilinear formulation may be extended to overcome rotational singularity in 3D geometric non‐linearity and to treat general material non‐linearity. Copyright © 2003 John Wiley & Sons, Ltd.     

Efficient non‐linear model reduction via a least‐squares Petrov–Galerkin projection and compressive tensor approximations

A Petrov–Galerkin projection method is proposed for reducing the dimension of a discrete non‐linear static or dynamic computational model in view of enabling its processing in real time. The right reduced‐order basis is chosen to be invariant and is constructed using the Proper Orthogonal Decomposition method. The left reduced‐order basis is selected to minimize the two‐norm of the residual arising at each Newton iteration. Thus, this basis is iteration‐dependent, enables capturing of non‐linearities, and leads to the globally convergent Gauss–Newton method. To avoid the significant computational cost of assembling the reduced‐order operators, the residual and action of the Jacobian on the right reduced‐order basis are each approximated by the product of an invariant, large‐scale matrix, and an iteration‐dependent, smaller one. The invariant matrix is computed using a data compression procedure that meets proposed consistency requirements. The iteration‐dependent matrix is computed to enable the least‐squares reconstruction of some entries of the approximated quantities. The results obtained for the solution of a turbulent flow problem and several non‐linear structural dynamics problems highlight the merit of the proposed consistency requirements. They also demonstrate the potential of this method to significantly reduce the computational cost associated with high‐dimensional non‐linear models while retaining their accuracy. Copyright © 2010 John Wiley & Sons, Ltd.     

Efficient non‐linear reanalysis of skeletal structures using combined approximations

Non‐linear reanalysis of large‐scale structures usually involves much computational effort, because the set of non‐linear equations must be solved repeatedly during the solution process. Various approximations that are often used for linear reanalysis are not sufficiently accurate for non‐linear problems. In this study, solution procedures based on the combined approximations approach are developed and compared in terms of efficiency and accuracy. Various path‐independent non‐linear analysis and reanalysis problems are considered, including material non‐linearity, geometric non‐linearity and buckling analysis. Numerical examples demonstrate the effectiveness of the procedures presented. It is shown that in various cases accurate results can be achieved efficiently. Copyright © 2007 John Wiley & Sons, Ltd.     

Three‐dimensional mixed‐mode (I and II) crack‐front fields in ductile thin plates — effects of T‐stress

It has been well‐established that the non‐singular T‐stress provides a first‐order estimate of geometry and loading mode (e.g. tension versus bending) effects on elastic–plastic crack‐front field under mode I loading conditions. The objective of this paper is to exam the T‐stress effect on three‐dimensional (3D) crack‐front fields under mixed‐mode (modes I and II) loading. To this end, detailed 3D small strain, elastic–plastic simulations are carried out using a 3D boundary layer (small‐scale yielding) formulation. Characteristics of near crack‐front fields are investigated for a wide range of T‐stresses (T/σ₀ = ?0.8, ?0.4, 0.0, 0.4, 0.8). The plastic zones and thickness and angular and radial variations of the stresses are studied, corresponding to two values of the remote elastic mixity parameters M_e = 0.3 and 0.7, under both low and high levels of applied loads. It is found that different T‐stresses have a significant effect on the plastic zones size and shapes, regardless of the mode mixity and load level. The thickness, angular and radial distributions of stresses are also affected markedly by T‐stress. It is important to include these effects when investigating the mixed‐mode ductile fracture failure process in thin‐walled structural components.     

On the computation of design derivatives for Huber–Mises plasticity with non‐linear hardening

This paper concerns design sensitivity analysis (DSA) for an elasto–plastic material, with material parameters depending on, or serving as, design variables. The considered constitutive model is Huber–Mises deviatoric plasticity with non‐linear isotropic/kinematic hardening, one which is applicable to metals. The standard radial return algorithm for linear hardening is generalized to account for non‐linear hardening functions. Two generalizations are presented; in both the non‐linearity is treated iteratively, but the iteration loop contains either a scalar equation or a group of tensorial equations. It is proven that the second formulation, which is the one used in some parallel codes, can be equivalently brought to a scalar form, more suitable for design differentiation. The design derivatives of both the algorithms are given explicitly, enabling thus calculation of the ‘explicit’ design derivative of stresses entering the global sensitivity equation. The paper addresses several issues related to the implementation and testing of the DSA module; among them the concept of verification tests, both outside and inside a FE code, as well as the data handling implied by the algorithm. The numerical tests, which are used for verification of the DSA module, are described. They shed light on (a) the accuracy of the design derivatives, by comparison with finite difference computations and (b) the effect of the finite element formulation on the design derivatives for an isochoric plastic flow. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical solution of non‐linear Klein–Gordon equations by variational iteration method

In this paper, numerical solution of non‐linear Klein–Gordon equations with power law non‐linearities are obtained by the new application of He's variational iteration method. Numerical illustrations that include non‐linear Klein–Gordon equations and non‐linear partial differential equations are investigated to show the pertinent features of the technique. Copyright © 2006 John Wiley & Sons, Ltd.     

Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

The consistency and stability of a Thomas–Gladwell family of multistage time‐stepping schemes for the solution of first‐order non‐linear differential equations are examined. It is shown that the consistency and stability conditions are less stringent than those derived for second‐order governing equations. Second‐order accuracy is achieved by approximating the solution and its derivative at the same location within the time step. Useful flexibility is available in the evaluation of the non‐linear coefficients and is exploited to develop a new non‐iterative modification of the Thomas–Gladwell method that is second‐order accurate and unconditionally stable. A case study from applied hydrogeology using the non‐linear Richards equation confirms the analytic convergence assessment and demonstrates the efficiency of the non‐iterative formulation. Copyright © 2004 John Wiley & Sons, Ltd.     

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.     

Improving stability by change of representation in time‐stepping analysis of non‐linear beams dynamics

In this paper we propose a change in the representation of the discrete motion equations in structural non‐linear dynamics to obtain an improvement in the stability of time numerical integrations. In particular, natural local state variables are indicated for a finite element approach to beam problems. The results, relative to Newmark approximations for the variations in the displacement and velocity vectors, show a significant increase in the range of stability of the time integration process and a reduction in the number of Newton iterations required in the time integration steps. The proposed method, further, preserves energy as well as the linear and angular momentum of the dynamical system. Copyright © 2006 John Wiley & Sons, Ltd.     

A discontinuous Galerkin formulation of non‐linear Kirchhoff–Love shells

Discontinuous Galerkin (DG) methods provide a means of weakly enforcing the continuity of the unknown‐field derivatives and have particular appeal in problems involving high‐order derivatives. This feature has previously been successfully exploited (Comput. Methods Appl. Mech. Eng. 2008; 197 :2901–2929) to develop a formulation of linear Kirchhoff–Love shells considering only the membrane and bending responses. In this proposed one‐field method—the displacements are the only unknowns, while the displacement field is continuous, the continuity in the displacement derivative between two elements is weakly enforced by recourse to a DG formulation. It is the purpose of the present paper to extend this formulation to finite deformations and non‐linear elastic behaviors. While the initial linear formulation was relying on the direct linear computation of the effective membrane stress and effective bending couple‐stress from the displacement field at the mid‐surface of the shell, the non‐linear formulation considered implies the evaluation of the general stress tensor across the shell thickness, leading to a reformulation of the internal forces of the shell. Nevertheless, since the interface terms resulting from the discontinuous Galerkin method involve only the resultant couple‐stress at the edges of the shells, the extension to non‐linear deformations is straightforward. Copyright © 2008 John Wiley & Sons, Ltd.     

A non‐linear interface element for 3D mesoscale analysis of brick‐masonry structures

This paper presents a novel interface element for the geometric and material non‐linear analysis of unreinforced brick‐masonry structures. In the proposed modelling approach, the blocks are modelled using 3D continuum solid elements, whereas the mortar and brick–mortar interfaces are modelled by means of the 2D non‐linear interface element. This enables the representation of any 3D arrangement for brick‐masonry, accounting for the in‐plane stacking mode and the through‐thickness geometry, and importantly it allows the investigation of both the in‐plane and the out‐of‐plane responses of unreinforced masonry panels. A co‐rotational approach is employed for the interface element, which shifts the treatment of geometric non‐linearity to the level of discrete entities, and enables the consideration of material non‐linearity within a simplified local framework employing first‐order kinematics. In this respect, the internal interface forces are modelled by means of elasto‐plastic material laws based on work‐softening plasticity and employing multi‐surface plasticity concepts. Following the presentation of the interface element formulation details, several experimental–numerical comparisons are provided for the in‐plane and out‐of‐plane static behaviours of brick‐masonry panels. The favourable results achieved demonstrate the accuracy and the significant potential of using the developed interface element for the non‐linear analysis of brick‐masonry structures under extreme loading conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

Implementation of a generalized exponential basis functions method for linear and non‐linear problems

In this paper, we address shortcomings of the method of exponential basis functions by extending it to general linear and non‐linear problems. In linear problems, the solution is approximated using a linear combination of exponential functions. The coefficients are calculated such that the homogenous form of equation is satisfied on some grid. To solve non‐linear problems, they are converted to into a succession of linear ones using a Newton–Kantorovich approach. The generalized exponential basis functions (GEBF) method developed can be implemented with greater ease compared with exponential basis functions, as all calculations can be performed using real numbers and no characteristic equation is needed. The details of an optimized implementation are described. We compare GEBF on some benchmark problems with methods in the literature, such as variants of the boundary element method, where GEBF shows a good performance. Also, in a 3D problem, we report the run time of the proposed method compared with that of Kratos, a parallel, highly optimized finite element code. The results show that in this example, to obtain the same level of error, much less computational effort is needed in the proposed method. Practical limitations might be encountered, however, for large problems because of dense matrix operations involved. Copyright © 2015 John Wiley & Sons, Ltd.     

An approximation technique for design sensitivity analysis of the critical load in non‐linear structures

A new technique of approximating design sensitivities of the critical load is presented in this paper. The technique results in stable and reliable estimations of design sensitivities at prebuckling points. Since taking derivatives of an approximated eigenvalue problem gives unstable sensitivities as the point approaches the critical load, the sensitivities are approximated directly from the exact sensitivity expressions. The sensitivities are approximated by applying two common approaches that are used in the critical load estimation and are called ‘one‐ and two‐point approximation’. The reliability and applicability of the proposed technique are demonstrated through several numerical examples of truss and beam structures. Two‐point approximation of design sensitivities gives better results than one‐point approximation. Copyright © 1999 John Wiley & Sons, Ltd.     

QALE‐FEM for numerical modelling of non‐linear interaction between 3D moored floating bodies and steep waves

This paper presents further development of the quasi arbitrary Lagrangian–Eulerian finite element method (QALE‐FEM) based on a fully non‐linear potential theory to numerically simulate non‐linear responses of 3D moored floating bodies to steep waves. In the QALE‐FEM (recently developed by the authors and applied to 2D floating bodies), the complex unstructured mesh is generated only once at the beginning of calculation and is moved to conform to the motion of boundaries at other time steps by using a robust spring analogy method specially suggested for these kind of problems, avoiding the necessity of high‐cost remeshing. In order to tackle challenges associated with 3D floating bodies, several new numerical techniques are developed in this paper. These include the technique for moving the mesh near body surfaces, the scheme for calculating velocity on 3D body surfaces and the modified semi‐implicit time integration method for floating bodies procedure (ISITIMFB‐M) that is more efficient for dealing with the full coupling between waves and bodies. Using the newly developed techniques and methods, various cases for 3D floating bodies with motions of up to six degrees of freedom (DoFs) are simulated. These include a SPAR platform, a barge‐type floating body and one or two Wigley Hulls in head seas or in oblique waves. For some selected cases, the numerical results are compared with experimental data available in the public domain and satisfactory agreements are achieved. Many results presented in this paper have not been found elsewhere to the best knowledge of the authors. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficient non‐linear solid–fluid interaction analysis by an iterative BEM/FEM coupling

An iterative coupling of finite element and boundary element methods for the time domain modelling of coupled fluid–solid systems is presented. While finite elements are used to model the solid, the adjacent fluid is represented by boundary elements. In order to perform the coupling of the two numerical methods, a successive renewal of the variables on the interface between the two subdomains is performed through an iterative procedure until the final convergence is achieved. In the case of local non‐linearities within the finite element subdomain, it is straightforward to perform the iterative coupling together with the iterations needed to solve the non‐linear system. In particular a more efficient and a more stable performance of the new coupling procedure is achieved by a special formulation that allows to use different time steps in each subdomain. Copyright © 2005 John Wiley & Sons, Ltd.     

On evaluation of shape sensitivities of non‐linear critical loads

The present paper focuses on the evaluation of the shape sensitivities of the limit and bifurcation loads of geometrically non‐linear structures. The analytical approach is applied for isoparametric elements, leading to exact results for a given mesh. Since this approach is difficult to apply to other element types, the semi‐analytical method has been widely used for shape sensitivity computation. This method combines ease of implementation with computational efficiency, but presents severe accuracy problems. Thus, a general procedure to improve the semi‐analytical sensitivities of the non‐linear critical loads is presented. The numerical examples show that this procedure leads to sensitivities with sufficient accuracy for shape optimization applications. Copyright © 2002 John Wiley & Sons, Ltd.     

Local boundary value problems for the error in FE approximation of non‐linear diffusion systems

In this work we investigate the a posteriori error estimation for a class of non‐linear, multicomponent diffusion operators, which includes the Stefan–Maxwell equations. The local error indicators for the global error are based on local boundary value problems, which are chosen to approximate either the global residual of the finite element approximation or the global linearized error equation. Using representative numerical examples, it is shown that the error indicators based on the latter approach are more accurate for estimating the global error for this problem class as the problem becomes more non‐linear, and can even produce better adaptive mesh refinement (AMR). In addition, we propose a new local error indicator for the error in output functionals that is accurate, inexpensive to compute, and is suitable for AMR, as demonstrated by numerical examples. Copyright © 2007 John Wiley & Sons, Ltd.