Equivalent continuum modeling for non‐linear wave propagation in jointed media

This paper describes a methodology to build equivalent continuum (ECC) models to represent jointed media for non‐linear wave propagation. In the present work, we study the response of randomly jointed rock media both to quasi‐static and dynamic loadings. We have found that, unlike in the quasi‐static case where no relaxation is observed, during dynamic loading, the deviatoric stress drops in the plastic wave after reaching its peak. Such stress relaxation phenomenon is calculated in jointed media, although the model used for intact rock blocks is rate‐independent. This is a strong indication that EC models for jointed rock should include some rate‐dependence when used in wave propagation simulations. Copyright © 2010 John Wiley & Sons, Ltd.     

Time discontinuous Galerkin methods with energy decaying correction for non‐linear elastodynamics

In this paper a new time discontinuous Galerkin (TDG) formulation for non‐linear elastodynamics is presented. The new formulation embeds an energy correction which ensures truly energy decaying, thus allowing to achieve unconditional stability that, as shown in the paper, is not guaranteed by the classical TDG formulation. The resulting method is simple and easily implementable into existing finite element codes. Moreover, it inherits the desirable higher‐order accuracy and high‐frequency dissipation properties of the classical formulation. Numerical results illustrate the very good performance of the proposed formulation. Copyright © 2010 John Wiley & Sons, Ltd.     

Discontinuous Galerkin methods for non‐linear elasticity

This paper presents the formulation and a partial analysis of a class of discontinuous Galerkin methods for quasistatic non‐linear elasticity problems. These methods are endowed with several salient features. The equations that define the numerical scheme are the Euler–Lagrange equations of a one‐field variational principle, a trait that provides an elegant and simple derivation of the method. In consonance with general discontinuous Galerkin formulations, it is possible within this framework to choose different numerical fluxes. Numerical evidence suggests the absence of locking at near‐incompressible conditions in the finite deformations regime when piecewise linear elements are adopted. Finally, a conceivable surprising characteristic is that, as demonstrated with numerical examples, these methods provide a given accuracy level for a comparable, and often lower, computational cost than conforming formulations. Stabilization is occasionally needed for discontinuous Galerkin methods in linear elliptic problems. In this paper we propose a sufficient condition for the stability of each linearized non‐linear elastic problem that naturally includes material and geometric parameters; the latter needed to account for buckling. We then prove that when a similar condition is satisfied by the discrete problem, the method provides stable linearized deformed configurations upon the addition of a standard stabilization term. We conclude by discussing the complexity of the implementation, and propose a computationally efficient approach that avoids looping over both elements and element faces. Several numerical examples are then presented in two and three dimensions that illustrate the performance of a selected discontinuous Galerkin method within the class. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical modelling of non‐linear electroelasticity

The numerical modelling of non‐linear electroelasticity is presented in this work. Based on well‐established basic equations of non‐linear electroelasticity a variational formulation is built and the finite element method is employed to solve the non‐linear electro‐mechanical coupling problem. Numerical examples are presented to show the accuracy of the implemented formulation. Copyright © 2006 John Wiley & Sons, Ltd.     

A dual algorithm for the topology optimization of non‐linear elastic structures

Dual algorithms are ideally suited for the purpose of topology optimization since they work in the space of Lagrange multipliers associated with the constraints. To date, dual algorithms have been applied only for linear structures. Here we extend this methodology to the case of non‐linear structures. The perimeter constraint is used to make the topology problem well‐posed. We show that the proposed algorithm yields a value of perimeter that is close to that specified by the user. We also address the issue of manufacturability of these designs, by proposing a variant of the standard dual algorithm, which generates designs that are two‐dimensional although the loading and the geometry are three‐dimensional. Copyright © 2008 John Wiley & Sons, Ltd.     

An accelerated FFT algorithm for thermoelastic and non‐linear composites

A fast numerical algorithm to compute the local and overall responses of non‐linear composite materials is developed. This alternative formulation allows us to improve the convergence of the existing method of Moulinec and Suquet (e.g. Comput. Meth. Appl. Mech. Eng. 1998; 157 (1–2):69–94). In the present method, a non‐linear elastic (or conducting) material is replaced by infinitely many locally linear thermoelastic materials with moduli that depend on the values of the local fields. This makes it possible to use the advantages of an algorithm developed by Eyre and Milton (Eur. Phys. J. Appl. Phys. 1999; 6 (1):41–47), which has faster convergence. The method is applied to compute the local fields as well as the effective response of non‐linear conducting and elastic periodic composites. Copyright © 2008 John Wiley & Sons, Ltd.     

A refined non‐linear non‐conforming triangular plate/shell element

A refined non‐conforming triangular plate/shell element for geometric non‐linear analysis of plates/shells using the total Lagrangian/updated Lagrangian approach is constructed in this paper based on the refined non‐conforming element method for geometric non‐linear analysis. The Allman's triangular plane element with vertex degrees of freedom and the refined triangular plate‐bending element RT9 are used to construct the present element. Numerical examples demonstrate that the accuracy of the new element is quite high in the geometric non‐linear analysis of plates/shells. Copyright © 2003 John Wiley & Sons, Ltd.     

A generalized model of non‐linear viscoelasticity: numerical issues and applications

In this paper, we consider a non‐linear viscoelastic model with internal variable, thoroughly analyzed by Le Tallecit et al. (Comput. Methods Appl. Mech. Engrg 1993; 109 :233–258). Our aim is to study here the implementation in three dimensions of a generalized version of this model. Computational results will be analyzed to validate our model on toy problems without geometric complexity, for which pseudo‐analytical solutions are known. At the end, we present a three‐dimensional numerical simulation on a mechanical device. Copyright © 2011 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.     

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.     

Computational techniques for optimization of problems involving non‐linear transient simulations

Many optimization problems in engineering require coupling a mathematical programming process to a numerical simulation. When the latter is non‐linear, the resulting computer time may become unaffordably large because three sequential procedures are nested: the outer loop is associated to the optimization process, the middle one corresponds to the time marching scheme and the innermost loop is required for solving iteratively the non‐linear system of equations at each time step. We propose four techniques for reducing CPU time. First, derive the initial values of state variables at each time (innermost loop) from those computed at the previous optimization iteration (outermost loop). Second, select time increment on the basis of those used for the previous optimization iteration. Third, define convergence criteria for the simulation problem on the basis of the optimization process, so that they are only as stringent as really needed. Finally, computations associated to the optimization are shown to be greatly reduced by adopting Newton–Raphson, or a variant, for solving the simulation problem. The effectiveness of these techniques is illustrated through application to three examples involving automatic calibration of non‐linear groundwater flow problems. The total number of iterations is reduced by a factor ranging between 1·7 and 4·6. Copyright © 1999 John Wiley & Sons, Ltd.     

Neuro‐genetic algorithm for non‐linear active control of structures

In a companion paper, a new non‐linear control model was presented for active control of three‐dimensional (3D) building structures including geometrical and material non‐linearities, coupling action between lateral and torsional motions, and actuator dynamics (Int. J. Numer. Meth. Engng; DOI: 10.1002/nme.2195 ). A dynamic fuzzy wavelet neuroemulator was presented for predicting the structural response in future time steps. In this paper, a new neuro‐genetic algorithm or controller is presented for finding the optimal control forces. The control algorithm does not need the pre‐training required in a neural network‐based controller, which improves the efficiency of the general control methodology significantly. Two 3D steel building structures, a 12‐story structure with vertical setbacks and an 8‐story structure with plan irregularity, are used to validate the neuro‐genetic control algorithm under three different seismic excitations. Numerical validations demonstrate that the new control methodology significantly reduces the displacements of buildings subjected to various seismic excitations including structures with plan and elevation irregularities. Copyright © 2008 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.     

An adaptive stabilization strategy for enhanced strain methods in non‐linear elasticity

This paper proposes and analyzes an adaptive stabilization strategy for enhanced strain (ES) methods applied to quasistatic non‐linear elasticity problems. The approach is formulated for any type of enhancements or material models, and it is distinguished by the fact that the stabilization term is solution dependent. The stabilization strategy is first constructed for general linearized elasticity problems, and then extended to the non‐linear elastic regime via an incremental variational principle. A heuristic choice of the stabilization parameters is proposed, which in the numerical examples proved to provide stable approximations for a large range of deformations, different problems and material models. We also provide explicit lower bounds for the stabilization parameters that guarantee that the method will be stable. These are not advocated, since they are generally larger than the ones based on heuristics, and hence prone to deteriorate the locking‐free behavior of ES methods. Numerical examples with two different non‐linear elastic models in thin geometries and incompressible situations show that the method remains stable and locking free over a large range of deformations. Finally, the method is strongly based on earlier developments for discontinuous Galerkin methods, and hence throughout the paper we offer a perspective about the similarities between the two. Copyright © 2009 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.     

Numerical method for non‐linear wave and diffusion equations by the variational iteration method

This paper applies He's variational iteration to the wave equations in an infinite one‐dimensional medium and some non‐linear diffusion equations. A suitable choice of an initial solution can lead to the needed exact solution by a few iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

An explicit discontinuous Galerkin method for non‐linear solid dynamics: Formulation,parallel implementation and scalability properties

An explicit‐dynamics spatially discontinuous Galerkin (DG) formulation for non‐linear solid dynamics is proposed and implemented for parallel computation. DG methods have particular appeal in problems involving complex material response, e.g. non‐local behavior and failure, as, even in the presence of discontinuities, they provide a rigorous means of ensuring both consistency and stability. In the proposed method, these are guaranteed: the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation. The semi‐discrete system of ordinary differential equations is integrated in time using a conventional second‐order central‐difference explicit scheme. A stability criterion for the time integration algorithm, accounting for the influence of the DG discretization stability, is derived for the equivalent linearized system. This approach naturally lends itself to efficient parallel implementation. The resulting DG computational framework is implemented in three dimensions via specialized interface elements. The versatility, robustness and scalability of the overall computational approach are all demonstrated in problems involving stress‐wave propagation and large plastic deformations. Copyright © 2007 John Wiley & Sons, Ltd.     

A rotation‐free corotational plane beam element for non‐linear analyses

This paper presents a plane beam element without rotational degrees of freedom that can be used for the analysis of non‐linear problems. The element is based on two main ideas. First, a corotational approach is adopted, which means that the kinematics of the element is decomposed into a rigid body motion part and a deformational part. Next, in the deformational part, the local nodal rotations are extrapolated as a function of the local displacements of the two nodes of the element and the first nodes to the left and right of the element. Six numerical applications are presented in order to assess the performance of the formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non‐linear solids and saturated porous media

A time‐discontinuous Galerkin finite element method (DGFEM) for dynamics and wave propagation in non‐linear solids and saturated porous media is presented. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite's polynomial) and linear interpolations for both displacements and velocities, in the time domain is particularly proposed. Consequently, continuity of the displacement vector at each discrete time instant is exactly ensured, whereas discontinuity of the velocity vector at the discrete time levels still remains. The computational cost is then obviously saved, particularly in the materially non‐linear problems, as compared with that required for the existing DGFEM. Both the implicit and explicit algorithms are developed to solve the derived formulations for linear and materially non‐linear problems. Numerical results illustrate good performance of the present method in eliminating spurious numerical oscillations and in providing much more accurate solutions over the traditional Galerkin finite element method using the Newmark algorithm in the time domain. Copyright © 2003 John Wiley & Sons, Ltd.