A finite element formulation for elastoplasticity based on a three-field variational equation

A three-field variational equation, which expresses the momentum balance equation, the plastic consistency condition, and the dilatational constitutive equation in a weak form, is proposed as a basis for finite element computations in hardening elastoplasticity. The finite element formulation includes algorithms for the integration of the elastoplastic rate constitutive equations which are similar to members of the “return mapping” family of algorithms employed in displacement formulations, except that the proposed algorithms are not required to explicitly satisfy the plastic consistency condition at the end of each time step. This condition is imposed globally by the inclusion of a variational equation that suitably constrains the solution. The plastic incompressibility constraint is also treated in an appropriate variational sense. Solution of the nonlinear finite element equations is obtained by use of Newton's method and details of the linearization of the variational equation are given. The formulation is developed for an associative von Mises plasticity model with general nonlinear isotropic and kinematic strain hardening. A number of numerical test examples are provided.     

Finite element formulation for dynamics of planar flexible multi-beam system

In some previous geometric nonlinear finite element formulations, due to the use of axial displacement, the contribution of all the elements lying between the reference node of zero axial displacement and the element to the foreshortening effect should be taken into account. In this paper, a finite element formulation is proposed based on geometric nonlinear elastic theory and finite element technique. The coupling deformation terms of an arbitrary point only relate to the nodal coordinates of the element at which the point is located. Based on Hamilton principle, dynamic equations of elastic beams undergoing large overall motions are derived. To investigate the effect of coupling deformation terms on system dynamic characters and reduce the dynamic equations, a complete dynamic model and three reduced models of hub-beam are prospected. When the Cartesian deformation coordinates are adopted, the results indicate that the terms related to the coupling deformation in the inertia forces of dynamic equations have small effect on system dynamic behavior and may be neglected, whereas the terms related to coupling deformation in the elastic forces are important for system dynamic behavior and should be considered in dynamic equation. Numerical examples of the rotating beam and flexible beam system are carried out to demonstrate the accuracy and validity of this dynamic model. Furthermore, it is shown that a small number of finite elements are needed to obtain a stable solution using the present coupling finite element formulation.     

Rosenbrock-type methods applied to finite element computations within finite strain viscoelasticity

In this article time-adaptive high-order Rosenbrock-type methods are applied to the system of differential–algebraic equations which results from the space-discretization using finite elements based on a constitutive model of finite strain viscoelasticity. It is shown that in this smooth problem more efficient finite element computations result in comparison to classical finite element approaches since the time integration on the basis of Rosenbrock-type methods does not lead to a system of non-linear equations. In other words, all aspects of implicit finite elements as local iterations on Gauss-point level and global equilibrium iterations do not occur. The first introduction to this approach proposed by Hartmann and Wensch [22] is extended here to the case of finite strain applications, where the geometrical non-linear deformation has an essential contribution to the non-linearities. Additionally, a clear decomposition into local (element or Gauss-point) work and global computational work using the Schur-complement is introduced to exploit the classical finite element character. Moreover, the extension to the reaction force computation, which is different to the classical approach, and the influence to mixed element formulations, here, the three-field formulation for displacements, pressure and dilatation, are discussed. The performance of various Rosenbrock-type methods is investigated and shows that for low accuracy requirements as in order one methods, the proposal yields a drastic reduction of the computational time.     

Discontinuous subgrid formulations for transport problems

In this paper we develop two discontinuous Galerkin formulations within the framework of the two-scale subgrid method for solving advection–diffusion-reaction equations. We reformulate, using broken spaces, the nonlinear subgrid scale (NSGS) finite element model in which a nonlinear eddy viscosity term is introduced only to the subgrid scales of a finite element mesh. Here, two new subgrid formulations are built by introducing subgrid stabilized terms either at the element level or on the edges by means of the residual of the approximated resolved scale solution inside each element and the jump of the subgrid solution across interelement edges. The amount of subgrid viscosity is scaled by the resolved scale solution at the element level, yielding a self adaptive method so that no additional stabilization parameter is required. Numerical experiments are conducted in order to demonstrate the behavior of the proposed methodology in comparison with some discontinuous Galerkin methods.     

On various formulations of large amplitude free vibrations of beams

Various finite element formulations of large amplitude free vibrations of beams with immovably supported ends are discussed in this paper. Analytical formulation based on the Rayleigh-Ritz method is also presented. Numerical results of the analytical approach are seen to be in good agreement with some of these finite element formulations. Mixed finite element formulations based on two methods are derived to study the large amplitude free vibrations of beams. The mixed finite element methods also show good agreement with the analytical and the above finite element formulations. Various points of view raised from time to time on the applicability of these formulations can now be clarified through these formulations and the numerical results. The weakness of the so-called improved Ritz-type finite element model in predicting the nonlinear frequency ratio is highlighted through various results of the above formulations. As a typical example, a hinged-hinged beam on immovable ends is considered for all the above formulations and the nonlinear frequencies show a good agreement amongst themselves at all amplitude levels.     

Stability and robustness issues in numerical modeling of material failure with the strong discontinuity approach

Robustness and stability of the Continuum Strong Discontinuity Approach (CSDA) to material failure are addressed. After identification of lack of symmetry of the finite element formulation and material softening in the constitutive model as possible causes of loss of robustness, two remedies are proposed: (1) the use of an specific symmetric version of the elementary enriched (E-FEM) finite element with embedded discontinuities and (2) a new implicit-explicit integration of the internal variable, in the constitutive model, which renders the tangent constitutive algorithmic operator positive definite and constant. The combination of both developments leads to finite element formulations with constant, in the time step, and non-singular tangent structural stiffness, allowing dramatic improvements in terms of robustness and computational costs. After assessing the convergence and stability properties of the new strategies, three-dimensional numerical simulations of failure problems illustrate the performance of the proposed procedures.     

Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study

In this study, the effect of the centrifugal forces on the eigenvalue solution obtained using two different nonlinear finite element formulations is examined. Both formulations can correctly describe arbitrary rigid body displacements and can be used in the large deformation analysis. The first formulation is based on the geometrically exact beam theory, which assumes that the cross section does not deform in its own plane and remains plane after deformation. The second formulation, the absolute nodal coordinate formulation (ANCF), relaxes this assumption and introduces modes that couple the deformation of the cross section and the axial and bending deformations. In the absolute nodal coordinate formulation, four different models are developed; a beam model based on a general continuum mechanics approach, a beam model based on an elastic line approach, a beam model based on an elastic line approach combined with the Hellinger–Reissner principle, and a plate model based on a general continuum mechanics approach. The use of the general continuum mechanics approach leads to a model that includes the ANCF coupled deformation modes. Because of these modes, the continuum mechanics model differs from the models based on the elastic line approach. In both the geometrically exact beam and the absolute nodal coordinate formulations, the centrifugal forces are formulated in terms of the element nodal coordinates. The effect of the centrifugal forces on the flap and lag modes of the rotating beam is examined, and the results obtained using the two formulations are compared for different values of the beam angular velocity. The numerical comparative study presented in this investigation shows that when the effect of some ANCF coupled deformation modes is neglected, the eigenvalue solutions obtained using the geometrically exact beam and the absolute nodal coordinate formulations are in a good agreement. The results also show that as the effect of the centrifugal forces, which tend to increase the beam stiffness, increases, the effect of the ANCF coupled deformation modes on the computed eigenvalues becomes less significant. It is shown in this paper that when the effect of the Poisson ration is neglected, the eigenvalue solution obtained using the absolute nodal coordinate formulation based on a general continuum mechanics approach is in a good agreement with the solution obtained using the geometrically exact beam model.     

Dynamic response simulation for reinforced concrete slabs

Present investigation comprises development of a new finite element numerical formulation for nonlinear transient dynamic analysis of reinforced concrete slab structures. Depending on many experimental data, new material constitutive relationships for concrete material have been formulated. A regression analysis of available experimental data in the SPSS-statistical program has been employed for formulating the proposed material finite element models, and the appropriateness of the models are confirmed through the histograms and measured indices of determination. Concrete slab structures were analyzed using eight-node serendipity degenerated plate elements. The constitutive models of the nonlinear materials are introduced to take into account the nonlinear stress–strain relationships of concrete. For studying the stress profile of the concrete slab through its thickness, a layered approach is adopted. Elastic perfectly plastic and strain hardening plasticity approaches have been employed to model the compressive behavior of concrete. Assumptions for strain rate effect were included in dynamic analysis by supposing the dynamic yield function as a function of the strain rate, in addition to be the total plastic strain. The yield condition is formulated in terms of the first two stress invariants. Geometrical nonlinearity was considered in analysis as a mathematical model based on the total lagrangian approach taking into account Von Karman assumptions. Implicit Newmark with corrector–predictor algorithm was used for time integration solution of the equation of the motion for slab structures. An incremental and iterative procedure is adopted to trace the entire response of the structure; a displacement convergence criterion is adopted in the present study. A computer program coded in FORTRAN has been developed and used for the dynamic analysis of reinforced concrete slabs. The numerical results show good agreement with other published studies’ results which include deflections.     

Comparison of finite element and pendulum models for simulation of sloshing

The pendulum model is a cost effective tool for the simulation of sloshing. However, the accuracy and applicability of the model has not been well established. In this article, we compare the simulation results obtained from the pendulum model and a more complicated finite element model for sloshing of liquids in tanker trucks. In the pendulum model, we assume that the liquid in the tanker is a point mass oscillating like a frictionless pendulum subjected to an external acceleration. In the finite element model, we solve the full Navier-Stokes equations written for two fluids to obtain the location and motion of the free surface. Stabilized finite element formulations are used in these complex 3D simulations. These finite element formulations are implemented in parallel using the message-passing interface libraries. The numerical example includes the simulation of sloshing in tanker trucks during turning.     

Optimal Trajectory Planning for Flexible Link Manipulators with Large Deflection Using a New Displacements Approach

The main objective of the present paper is to determine the optimal trajectory of very flexible link manipulators in point-to-point motion using a new displacement approach. A new nonlinear finite element model for the dynamic analysis is employed to describe nonlinear modeling for three-dimensional flexible link manipulators, in which both the geometric elastic nonlinearity and the foreshortening effects are considered. In comparison to other large deformation formulations, the motion equations contain constant stiffness matrix because the terms arising from geometric elastic nonlinearity are moved from elastic forces to inertial, reactive and external forces, which are originally nonlinear. This makes the formulation particularly efficient in computational terms and numerically more stable than alternative geometrically nonlinear formulations based on lower-order terms. In this investigation, the computational method to solve the trajectory planning problem is based on the indirect solution of open-loop optimal control problem. The Pontryagin’s minimum principle is used to obtain the optimality conditions, which is lead to a standard form of a two-point boundary value problem. The proposed approach has been implemented and tested on a single-link very flexible arm and optimal paths with minimum effort and minimum vibration are obtained. The results illustrate the power and efficiency of the method to overcome the high nonlinearity nature of the problem.     

The dual reciprocity boundary element formulation for nonlinear diffusion problems

This paper presents an extension of the dual reciprocity boundary element method (DRBEM) to deal with nonlinear diffusion problems in which thermal conductivity, specific heat, and density coefficients are all functions of temperature. The DRBEM, recently applied to the solution of problems governed by parabolic and hyperbolic equations, consists in the transformation of the differential equation into an integral equation involving boundary integrals only, the solution of which is achieved by employing a standard boundary element discretization coupled with a two-level finite difference time integration scheme. Contrary to previous formulations for the diffusion equation, the dual reciprocity BEM utilizes the well-known fundamental solution to Laplace's equation, which is space-dependent only. This avoids complex time integrations that normally appear in formulations employing time-dependent fundamental solutions, and permits accurate numerical solutions to be obtained in an efficient way. For nonlinear problems, the integral of conductivity is introduced as a new variable to obtain a linear diffusion equation in the Kirchhoff transform space. This equation involves a modified time variable which is itself a function of position. The problem is solved in an iterative way by using an efficient Newton-Raphson technique which is shown to be rapidly convergent.     

Comparison of different formulations of 2D beam elements based on Bond Graph technique

Bond Graphs are well suited for modelling multibody systems. In this paper modelling of planar flexible beams undergoing large overall motions are studied based on finite element (FE) technique. Two well-known approaches are used – the co-rotational (CR) and absolute nodal coordinate (ANC) formulation. Two ANC formulations are analyzed – one in which elastic forces is described using classical beam theory in a local coordinate frame, and another based on a global continuum mechanics approach. Starting from these classical formulations velocity formulations are developed and used to develop Bond Graph FE components. The effect of gravity has been considered as well. These components can be put in libraries and used for systematic Bond Graph flexible body model development. It is shown that Bond Graph technique is capable of dealing with different flexible body formulations and can be used as a general approach in parallel to other modelling approaches. Models are developed and simulations are performed using the object oriented environment of BondSim. Owing to the object oriented approach, transformation from one to the other model is relatively simply. The results are illustrated by suitable examples and they confirm accuracy of the developed models. It was shown that the CR approach offers much better performance than the both ANC formulations.     

Quadratic and Higher-Order Constraints in Energy-Conserving Formulations of Flexible Multibody Systems

The treatment of constraints is considered here within the framework ofenergy-momentum conserving formulations for flexible multibody systems.Constraint equations of various types are an inherent component of multibodysystems, their treatment being one of the key performance features ofmathematical formulations and numerical solution schemes.Here we employ rotation-free inertial Cartesian coordinates of points tocharacterise such systems, producing a formulation which easily couples rigidbody dynamics with nonlinear finite element techniques for the flexiblebodies. This gives rise to additional internal constraints in rigid bodies topreserve distances. Constraints are enforced via a penalty method, which givesrise to a simple yet powerful formulation. Energy-momentum time integrationschemes enable robust long term simulations for highly nonlinear dynamicproblems.The main contribution of this paper focuses on the integration of constraintequations within energy-momentum conserving numerical schemes. It is shownthat the solution for constraints which may be expressed directly in terms ofquadratic invariants is fairly straightforward. Higher-order constraints mayalso be solved, however in this case for exact conservation an iterativeprocedure is needed in the integration scheme. This approach, together withsome simplified alternatives, is discussed.Representative numerical simulations are presented, comparing the performanceof various integration procedures in long-term simulations of practicalmultibody systems.     

Moments-method analysis of a finite array of arbitrary shaped microstrip patch radiating elements

A full-wave model for the analysis of a finite array of arbitrary shaped, probe fed, microstrip elements is presented. The method of moments is used first to obtain a space-domain solution to the mixed potential integral equation (MPIE) for the isolated element in terms of linear ramp basis functions over triangular subdomains. A second order matrix equation for the array problem is then set up using characteristic current modes as basis functions. These converge rapidly and have orthogonality properties such that only the intermode coupling between separated elements need be computed—a relatively simple operation. Predicted and measured performance characteristics for a 37-element 8-GHz array of dual-mode patches are given which demonstrate the accuracy obtainable with this approach.     

Finite element modeling within an integrated geometric modeling environment: Part II—Attribute specification,domain differences,and indirect element types

This is the second of a two part paper that addresses the integration of finite element modeling and geometric modeling. Instead of considering the integration of currently available systems, this paper addresses both modeling techniques in general terms and identifies the functions that are needed to integrate them, taking full advantage of the capabilities of both. A set of geometric communication operators are identified and defined for use in carrying out this integration process. Part I [1] considered the integration of geometric modeling and finite element mesh generation. This part considers the remaining areas of the specification of finite element analysis attribute information, accounting for domain differences between the geometric and finite element models and the generation of finite element models using element types that are of a lesser dimension than the geometric entity they represent.     

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised-α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised-α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.     

Large deflection analysis of thin elastic structures by the assumed stress hybrid finite element method

Two assumed stress hybrid finite element models for analyzing the large deflection, linear elastic, static behavior of structures have been developed: a consistent model which satisfies the entire stress equilibrium equation and an inconsistent model which satisfies only the linear portion of this equation. These models are derived for two separate coordinate frames: a stationary system and a convected, updated system. Throughout the development “correction” terms are maintained in all the functionals to minimize the drifting of the approximate solution from the true solution. These correction terms correspond to checks on the stress equilibrium and compatibility in the reference state. Utilizing a tangent stiffness approach various incremental and incremental-iterative solution procedures are used. The actual applications utilize flat and shallow elements to analyze the large deflection (moderate rotation), small strain behavior of thin, linearly elastic beams and shells. For the beam problems a shallow curved, two node, six degree of freedom element is used. For the shell analysis flat and shallow shell elements are used. They are three node, fifteen degree of freedom, triangular elements. Example studies include comparisons of the consistent and inconsistent models, the flat and shallow elements, the two coordinate systems, the effectiveness of the correction terms and solution procedures, and the adequacy of the models and methods. The results demonstrate that the consistent and inconsistent models yield essentially the same results. Overall, the models yield satisfactory results with simple elements.     

Finite element modeling within an integrated geometric modeling environment: part I—Mesh generation

This is the first of a two part paper that addresses the integration of finite element modeling and geometric modeling. Instead of considering the integration of currently available systems, this paper addresses both modeling techniques in general terms and identifies the functions that are needed to integrate them, taking full advantage of the capabilities of both. A set of geometric communication operators are identified and defined for use in carrying out this integration process. This first part considers the integration of geometric modeling and mesh generation, whereas the second part considers the specification of analysis attributes, accounting for domain differences between the geometric and finite element models and generating finite element models using element types that are of a lesser dimension than the geometry they represent.     

Frame element with lateral deformable supports: Formulations and numerical validation

This paper presents the theory and the numerical validation of three different formulations of nonlinear frame elements with nonlinear lateral deformable supports. The governing differential equations of the problem are derived first and the three different finite element formulations are then presented. The first model follows a displacement-based formulation, which is based on the virtual displacement principle. The second one follows the force-based formulation, which is based on the virtual force principle. The third model follows the Hellinger–Reissner mixed formulation, which is based on the two-field mixed variational principle. The selection of the displacement and force interpolation functions for the different formulations is discussed. Tonti’s diagrams are used to conveniently represent the equations governing both the strong and the weak forms of the problem. The general matrix equations of the three formulations are presented, with some details on the issues regarding the elements’ implementations in a general-purpose finite element program. The convergence, accuracy, and computational times of the three elements are studied through a numerical example. The distinctive element characteristics in terms of force and deformation discontinuities between adjacent elements are discussed. The capability of the proposed frame models to trace the softening response due to softening of the foundation is also investigated. Overall, the force-based and the mixed models are much more accurate than the displacement-based model and require very few elements to reach the converged solution. The force-based element is slightly more accurate than the mixed model, but it is more prone to numerical instabilities as it involves inverting the element flexibility matrix.