A formulation for electrostatic‐mechanical contact and its numerical solution

The progress in advanced technology fields requires more and more sophisticated formulations to consider contact problems properly. This paper is devoted to the development of a new constitutive model for electrostatic‐mechanical contacts, based on a micro–macro approach to describe the contact behaviour. The electric‐mechanical contact constitutive law is obtained considering the real microscopic shape of the contacting surfaces, the microscopic behaviour of force transmission and current flow. Some thermo‐mechanical macroscopic models based on microscopic characterizations have already been developed to compute the normal and tangential contact stiffness and the thermal contact resistance. On the basis of such macroscopic models, a similar model, suitable for the electric‐mechanical field, is developed. With reference to the thermal constriction resistance the electric contact resistance is studied, assuming a flux tube around each contacting asperity, and choosing a suitable geometry for its narrowing at the contact zone. The contact element geometry is based on well known theoretical and experimental micro‐mechanical laws, suitably adapted for the FEM formulation. The macroscopic stiffness matrix is calculated on the basis of the microscopic laws and it is continuously updated as a function of the changes in the mechanical and electric significant parameters. A consistent linearization of the set of equations is developed to improve the computational speed, within the framework of implicit methods. Copyright © 2005 John Wiley & Sons, Ltd.     

An X‐FEM approach for large sliding contact along discontinuities

The extended finite element method (X‐FEM) has been developed to minimize requirements on the mesh design in a problem with a displacement discontinuity. This advantage, however, still remains limited to the small deformation hypothesis when considering sliding discontinuities. The approach presented in this paper proposes to couple X‐FEM with a Lagrangian large sliding frictionless contact algorithm. A new hybrid X‐FEM contact element was developed with a contact search algorithm allowing for an update of contacting surfaces pairing. The stability of the contact formulation is ensured by an algorithm for fulfilling Ladyzhenskaya‐Babuska‐Brezzi (LBB) condition. Several 2D simple examples are presented in this paper in order to prove its efficiency and stability. Copyright © 2008 John Wiley & Sons, Ltd.     

Finite rotation quadrilateral element for multi‐layer beams

The paper presents the finite rotations beam equations derived on use of the generalized Reissner hypothesis with a scalar parameter for the transverse extension. The beam strain and change of curvature measures are obtained from the right stretch strain, and the virtual work is given for Biot‐type stress and couple resultants. The strain energy for the first‐order isotropic elastic material is assumed in terms of the right stretch strain, and constitutive equations for the beam stress and couple resultants are derived. Two finite rotation elements are developed from the derived beam equations: a beam element with the transverse stretch and a quadrilateral element. First, the beam element with the uniformly under‐integrated tangent operator is developed. Next, the formula linking the middle‐line variables and the interface variables of the beam is introduced consistently with the generalized Reissner kinematics. Linearization of this formula is performed, and the derived tangent operator is used to convert the two‐node beam element to a four‐node quadrilateral. Both the finite elements have been tested on several numerical examples, some of highly non‐linear characteristics, and their accuracy is very good. It has been established that the quadrilateral element, which is intended for applications to multi‐layer beams, performs very well for high elemental aspect ratios, and can therefore be applied to modelling of very thin layers. Copyright © 1999 John Wiley & Sons, Ltd.     

Fully coupled non‐linear analysis of piezoelectric solids involving domain switching

Domain switching is the cause of significant non‐linearity in the response of piezoelectric materials to mechanical and electrical effects. In this paper, the response of piezoelectric solids is formulated by coupling thermal, electrical, and mechanical effects. The constitutive equations are non‐linear. Moreover, due to the domain switching phenomenon, the resulting governing equations become highly non‐linear. The corresponding non‐linear finite element equations are derived and solved by using an incremental technique. The developed formulation is first verified against a number of benchmark problems for which a closed‐form solution exists. Next, a cantilever beam made of PZT‐4 is studied to evaluate the effect of domain switching on the overall force–displacement response of the beam. A number of interesting observations are made with respect to the extent of non‐linearity and its progressive spread as the load on the beam increases. Copyright © 2002 John Wiley & Sons, Ltd.     

Frame‐indifferent beam finite elements based upon the geometrically exact beam theory

This paper describes a new beam finite element formulation based upon the geometrically exact beam theory. In contrast to many previously proposed beam finite element formulations the present discretization approach retains the frame‐indifference (or objectivity) of the underlying beam theory. The space interpolation of rotational degrees‐of‐freedom is circumvented by the introduction of a reparameterization of the weak form corresponding to the equations of motion of the geometrically exact beam theory. Copyright © 2002 John Wiley & Sons, Ltd.     

A finite‐strain gradient‐inelastic beam theory and a corresponding force‐based frame element formulation

This paper extends the gradient‐inelastic (GI) beam theory, introduced by the authors to simulate material softening phenomena, to further account for geometric nonlinearities and formulates a corresponding force‐based (FB) frame element computational formulation. Geometric nonlinearities are considered via a rigorously derived finite‐strain beam formulation, which is shown to coincide with Reissner's geometrically nonlinear beam formulation. The resulting finite‐strain GI beam theory: (i) accounts for large strains and rotations, unlike the majority of geometrically nonlinear beam formulations used in structural modeling that consider small strains and moderate rotations; (ii) ensures spatial continuity and boundedness of the finite section strain field during material softening via the gradient nonlocality relations, eliminating strain singularities in beams with softening materials; and (iii) decouples the gradient nonlocality relations from the constitutive relations, allowing use of any material model. On the basis of the proposed finite‐strain GI beam theory, an exact FB frame element formulation is derived, which is particularly novel in that it: (a) expresses the compatibility relations in terms of total strains/displacements, as opposed to strain/displacement rates that introduce accumulated computational error during their numerical time integration, and (b) directly integrates the strain‐displacement equations via a composite two‐point integration method derived from a cubic Hermite interpolating polynomial to calculate the displacement field over the element length and, thus, address the coupling between equilibrium and strain‐displacement equations. This approach achieves high accuracy and mesh convergence rate and avoids polynomial interpolations of individual section fields, which often lead to instabilities with mesh refinements. The FB formulation is then integrated into a corotational framework and is used to study the response of structures, simultaneously accounting for geometric nonlinearities and material softening. The FB formulation is further extended to capture member buckling triggered by minor perturbations/imperfections of the initial member geometry.     

Exact static element stiffness matrices of non‐symmetric thin‐walled curved beams

An evaluation procedure of exact static stiffness matrices for curved beams with non‐symmetric thin‐walled cross section are rigorously presented for the static analysis. Higher‐order differential equations for a uniform curved beam element are first transformed into a set of the first‐order simultaneous ordinary differential equations by introducing 14 displacement parameters where displacement modes corresponding to zero eigenvalues are suitably taken into account. This numerical technique is then accomplished via a generalized linear eigenvalue problem with non‐symmetric matrices. Next, the displacement functions of displacement parameters are exactly calculated by determining general solutions of simultaneous non‐homogeneous differential equations. Finally an exact stiffness matrix is evaluated using force–deformation relationships. In order to demonstrate the validity and effectiveness of this method, displacements and normal stresses of cantilever thin‐walled curved beams subjected to tip loads are evaluated and compared with those by thin‐walled curved beam elements as well as shell elements. Copyright © 2004 John Wiley & Sons, Ltd.     

Vertical dynamic responses of a simply supported bridge subjected to a moving train with two‐wheelset vehicles using modal analysis method

The vertical dynamic responses of a simply supported bridge subjected to a moving train are investigated by means of the modal analysis method. Each vehicle of train is modelled as a four‐degree‐of‐freedom mass–spring–damper multi‐rigid body system with a car body and two wheelsets. The bridge, together with track, is modelled as a simply supported Bernoulli–Euler beam. The deflection of the beam is described by superimposing modes. The train and the beam are regarded as an entire dynamic system, in which the contact forces between wheelset and beam are considered as internal forces. The equations of vertical motion in matrix form with time‐dependent coefficients for this system are directly derived from the Hamilton's principle. The equations of motion are solved by Wilson‐θ method to obtain the dynamic responses for both the support beam and the moving train. Compared with the results previous reported, good agreement between the proposed method and the finite element method is obtained. Finally, the effects of beam mode number, vehicle number, beam top surface, and train velocity on the dynamic responses of the entire train and bridge coupling system are studied, and the dynamic responses of beam are given under the train moving with resonant velocity. Copyright © 2005 John Wiley & Sons, Ltd.     

An implicit–explicit solution method for electro‐osmotic flow through three‐dimensional micro‐channels

This paper presents a comprehensive finite‐element modelling approach to electro‐osmotic flows on unstructured meshes. The non‐linear equation governing the electric potential is solved using an iterative algorithm. The employed algorithm is based on a preconditioned GMRES scheme. The linear Laplace equation governing the external electric potential is solved using a standard pre‐conditioned conjugate gradient solver. The coupled fluid dynamics equations are solved using a fractional step‐based, fully explicit, artificial compressibility scheme. This combination of an implicit approach to the electric potential equations and an explicit discretization to the Navier–Stokes equations is one of the best ways of solving the coupled equations in a memory‐efficient manner. The local time‐stepping approach used in the solution of the fluid flow equations accelerates the solution to a steady state faster than by using a global time‐stepping approach. The fully explicit form and the fractional stages of the fluid dynamics equations make the system memory efficient and free of pressure instability. In addition to these advantages, the proposed method is suitable for use on both structured and unstructured meshes with a highly non‐uniform distribution of element sizes. The accuracy of the proposed procedure is demonstrated by solving a basic micro‐channel flow problem and comparing the results against an analytical solution. The comparisons show excellent agreement between the numerical and analytical data. In addition to the benchmark solution, we have also presented results for flow through a fully three‐dimensional rectangular channel to further demonstrate the application of the presented method. Copyright © 2007 John Wiley & Sons, Ltd.     

Cyclic steady states of nonlinear electro‐mechanical devices excited at resonance

We present an efficient numerical method to solve for cyclic steady states of nonlinear electro‐mechanical devices excited at resonance. Many electro‐mechanical systems are designed to operate at resonance, where the ramp‐up simulation to steady state is computationally very expensive – especially when low damping is present. The proposed method relies on a Newton–Krylov shooting scheme for the direct calculation of the cyclic steady state, as opposed to a naïve transient time‐stepping from zero initial conditions. We use a recently developed high‐order Eulerian–Lagrangian finite element method in combination with an energy‐preserving dynamic contact algorithm in order to solve the coupled electro‐mechanical boundary value problem. The nonlinear coupled equations are evolved by means of an operator split of the mechanical and electrical problem with an explicit as well as implicit approach. The presented benchmark examples include the first three fundamental modes of a vibrating nanotube, as well as a micro‐electro‐mechanical disk resonator in dynamic steady contact. For the examples discussed, we observe power law computational speed‐ups of the form S  = 0.6·ξ  ^? 0.8, where ξ is the linear damping ratio of the corresponding resonance frequency. Copyright © 2016 John Wiley & Sons, Ltd.     

Formulation of equations of motion of finite element form for vehicle–track–bridge interaction system with two types of vehicle model

Vehicle, track and bridge are considered as an entire system in this paper. Two types of vertical vehicle model are described. One is a one foot mass–spring–damper system having two‐degree‐of‐freedom, and the other is four‐wheelset mass–spring–damper system with two‐layer suspension systems possessing 10‐degree‐of‐freedom. For the latter vehicle model, the eccentric load of car body is taken into account. The rails and the bridge deck are modelled as an elastic Bernoulli–Euler upper beam with finite length and a simply supported Bernoulli–Euler lower beam, respectively, while the elasticity and damping properties of the rail bed are represented by continuous springs and dampers. The dynamic contact forces between the moving vehicle and rails are considered as internal forces, so it is not necessary to calculate the internal forces for setting up the equations of motion of the vehicle–track–bridge interaction system. The two types of equations of motion of finite element form for the entire system are derived by means of the principle of a stationary value of total potential energy of dynamic system. The proposed method can set up directly the equations of motion for sophisticated system, and these equations can be solved by step‐by‐step integration method, to obtain simultaneously the dynamic responses of vehicle, of track and of bridge. Illustration examples are given. Copyright 2004 © John Wiley & Sons, Ltd.     

Decomposition contact response (DCR) for explicit finite element dynamics

We propose a new explicit contact algorithm for finite element discretized solids and shells with smooth and non‐smooth geometries. The equations of motion are integrated in time with a predictor‐corrector‐type algorithm. After each predictor step, the impenetrability constraints and the exchange of momenta between the impacting bodies are considered and enforced independently. The geometrically inadmissible penetrations are removed using closest point projections or similar updates. Penetration is measured using the signed volume of intersection described by the contacting surface elements, which is well‐defined for both smooth and non‐smooth geometries. For computing the instantaneous velocity changes that occur during the impact event, we introduce the decomposition contact response method. This enables the closed‐form solution of the jump equations at impact, and applies to non‐frictional as well as frictional contact, as exemplified by the Coulomb frictional model. The overall algorithm has excellent momentum and energy conservation characteristics, as several numerical examples demonstrate. Copyright © 2005 John Wiley & Sons, Ltd.     

Monolithic modelling of electro‐mechanical coupling in micro‐structures

The purpose of the present work is to model and to simulate the coupling between the electric and mechanical fields. A new finite element approach is proposed to model strong electro‐mechanical coupling in micro‐structures with capacitive effect. The proposed approach is based on a monolithic formulation: the electric and the mechanical fields are solved simultaneously in the same formulation. This method provides a tangent stiffness matrix for the total coupled problem which allows to determine accurately the pull‐in voltage and the natural frequency of electro‐mechanical systems such as MEMs. To illustrate the methodology results are shown for the analysis of a micro‐bridge. Copyright © 2005 John Wiley & Sons, Ltd.     

Coupled three‐layered analysis of smart piezoelectric beams with different electric boundary conditions

This paper presents a theoretical and finite element (FE) formulation of a three‐layered smart beam with two piezoelectric layers acting as sensors or actuators. For the definition of the mechanical model a partial layerwise theory is considered for the approximation of the displacement field of the core and piezoelectric face layers. An electrical model for different electric boundary conditions (EBC), namely, electroded layers with either closed‐ or open‐circuit electrodes with electric potential prescribed or layers without electrodes, is considered. Using a variational formulation, the direct piezoelectric effect for the different EBC is physically incorporated into the mechanical model through appropriate approximations of the electric field in the axial and transverse directions. An FE model of a three‐layered smart beam with different EBC is proposed considering a fully coupled electro‐mechanical theory through the use of effective stiffness parameters and a modified static condensation. FE solutions of the quasi‐static electrical and mechanical actuations and natural frequencies are presented. Comparisons with numerical FE and analytical solutions available in the literature demonstrate the representativeness of the developed theory and the effectiveness of the proposed FE model for different EBC. Copyright © 2005 John Wiley & Sons, Ltd.     

Crack face contact in X‐FEM using a segment‐to‐segment approach

In this study, we propose a segment‐to‐segment contact formulation (mortar‐based) that uses Lagrange's multipliers to establish the contact between crack faces when modeled with the extended finite element method (X‐FEM) in 2D problems. It is shown that, in general, inaccuracies arise when the contact is formulated following a point‐to‐point approach. This is due to the non‐linear character of the X‐FEM interpolation along the crack faces that leads to crack face interpenetration. However, the segment‐to‐segment approach optimizes the fulfilment of the contact constraints along the whole crack segment, and in practice the contact is modeled precisely. Convergence studies for mesh sequences have been performed, showing the advantages of the proposed methodology. Copyright © 2009 John Wiley & Sons, Ltd.     

Penalty function method for combined finite–discrete element systems comprising large number of separate bodies

Large‐scale discrete element simulations, the combined finite–discrete element method, DDA as well as a whole range of related methods, involve contact of a large number of separate bodies. In the context of the combined finite–discrete element method, each of these bodies is represented by a single discrete element which is then discretized into finite elements. The combined finite–discrete element method thus also involves algorithms dealing with fracture and fragmentation of individual discrete elements which result in ever changing topology and size of the problem. All these require complex algorithmic procedures and significant computational resources, especially in terms of CPU time. In this context, it is also necessary to have an efficient and robust algorithm for handling mechanical contact. In this work, a contact algorithm based on the penalty function method and incorporating contact kinematics preserving energy balance, is proposed and implemented into the combined finite element code. Copyright © 2000 John Wiley & Sons, Ltd.     

Object‐oriented finite element analysis of thermo‐hydro‐mechanical (THM) problems in porous media

The design, implementation and application of a concept for object‐oriented in finite element analysis of multi‐field problems is presented in this paper. The basic idea of this concept is that the underlying governing equations of porous media mechanics can be classified into different types of partial differential equations (PDEs). In principle, similar types of PDEs for diverse physical problems differ only in material coefficients. Local element matrices and vectors arising from the finite element discretization of the PDEs are categorized into several types, regardless of which physical problem they belong to (i.e. fluid flow, mass and heat transport or deformation processes). Element (ELE) objects are introduced to carry out the local assembly of the algebraic equations. The object‐orientation includes a strict encapsulation of geometrical (GEO), topological (MSH), process‐related (FEM) data and methods of element objects. Geometric entities of an element such as nodes, edges, faces and neighbours are abstracted into corresponding geometric element objects (ELE–GEO). The relationships among these geometric entities form the topology of element meshes (ELE–MSH). Finite element objects (ELE–FEM) are presented for the local element calculations, in which each classification type of the matrices and vectors is computed by a unique function. These element functions are able to deal with different element types (lines, triangles, quadrilaterals, tetrahedra, prisms, hexahedra) by automatically choosing the related element interpolation functions. For each process of a multi‐field problem, only a single instance of the finite element object is required. The element objects provide a flexible coding environment for multi‐field problems with different element types. Here, the C++ implementations of the objects are given and described in detail. The efficiency of the new element objects is demonstrated by several test cases dealing with thermo‐hydro‐mechanical (THM) coupled problems for geotechnical applications. Copyright © 2006 John Wiley & Sons, Ltd.