Two-field Variational Formulation Research Articles

Weak solvability via bipotentials for contact problems with power-law friction

We study a bilateral contact model with power-law friction and a regularized version of it as well, a regularization of the friction law being considered. The behavior of the material is described by means of a subdifferential inclusion. We deliver a two-field variational formulation via bipotentials for the original problem and also for the regularized version. Subsequently, we study the solvability of the variational formulations we propose, based on abstract auxiliary results. Some properties of the solutions, paying attention to the data dependence, are also investigated. Finally, we deliver a convergence result.

Two-field variational formulations for a class of nonlinear mechanical models

A nonlinear boundary value problem arising from continuum mechanics is considered. The nonlinearity of the model arises from the constitutive law which is described by means of the subdifferential of a convex constitutive map. A bipotential , related to the constitutive map and its Fenchel conjugate, is considered. Exploring the possibility to rewrite the constitutive law as a law governed by the bipotential , a two-field variational formulation involving a variable convex set is proposed. Subsequently, we obtain existence and uniqueness results. Some properties of the solution are also discussed.

Monolithic formulation of electromechanical systems within the context of hybrid finite elements

In electromechanical devices, a strong coupling exists between the electromagnetic and displacement field. Due to this strong interaction, a need arises to develop a robust, fully coupled scheme for modeling electromechanical phenomena. With this goal in view, we present a monolithic numerical scheme for modeling fully coupled electromechanical systems. It is shown in the literature that for structural problems, hybrid elements that are based on a two-field variational formulation are less susceptible to locking and provide a robust numerical strategy especially for shell-type structures. Hence, we extend our monolithic formulation to the hybrid finite element framework. Our monolithic formulation is based on a total Lagrangian framework, where the eddy current and structural equations are solved on the reference configuration. Consistent linearization is performed to ensure a quadratic rate of convergence. The efficacy of the presented algorithm, and especially that of the hybrid formulation is demonstrated with the help of numerical examples.

Conservation Properties of the Trapezoidal Rule in Linear Time Domain Analysis of Acoustics and Structures

The trapezoidal rule, which is a special case of the Newmark family of algorithms, is one of the most widely used methods for transient hyperbolic problems. In this work, we show that this rule conserves linear and angular momenta and energy in the case of undamped linear elastodynamics problems, and an “energy-like measure” in the case of undamped acoustic problems. These conservation properties, thus, provide a rational basis for using this algorithm. In linear elastodynamics problems, variants of the trapezoidal rule that incorporate “high-frequency” dissipation are often used, since the higher frequencies, which are not approximated properly by the standard displacement-based approach, often result in unphysical behavior. Instead of modifying the trapezoidal algorithm, we propose using a hybrid finite element framework for constructing the stiffness matrix. Hybrid finite elements, which are based on a two-field variational formulation involving displacement and stresses, are known to approximate the eigenvalues much more accurately than the standard displacement-based approach, thereby either bypassing or reducing the need for high-frequency dissipation. We show this by means of several examples, where we compare the numerical solutions obtained using the displacement-based and hybrid approaches against analytical solutions.

A hybrid finite element formulation for flexible multibody dynamics

This work deals with the transient analysis of flexible multibody systems within a hybrid finite element framework. Hybrid finite elements are based on a two-field variational formulation in which the displacements and stresses are interpolated separately yielding very good coarse mesh accuracy. Most of the literature on flexible multibody systems uses beam-theory-based formulations. In contrast, the use of hybrid finite elements uses continuum-based elements, thus avoiding the problems associated with rotational degrees of freedom. In particular, any given three-dimensional constitutive relations can be directly used within the framework of this formulation. Since the coarse mesh accuracy as compared to a conventional displacement-based formulation is very high, the scheme is cost effective as well. A general formulation is developed for the constrained motion of a given point on a line manifold, using a total Lagrangian method. The multipoint constraint equations are implemented using Lagrange multipliers. Various kinds of joints such as cylindrical, prismatic, and screw joints are implemented within this general framework. Hinge joints such as spherical, universal, and revolute joints are obtained simply by using shared nodes between the bodies. In addition to joints, the formulation and implementation details for a DC motor actuator and for prescribed relative rotation are also presented. Several example problems illustrate the efficacy of the developed formulation.

Improved hybrid elements for structural analysis

Hybrid elements, which are based on a two-field variational formulation with the displacements and stresses interpolated separately, are known to deliver very high accuracy, and to alleviate to a large extent problems of locking that plague standard displacement-based formulations. The choice of the stress interpolation functions is of course critical in ensuring the high accuracy and robustness of the method. Generally, an attempt is made to keep the stress interpolation to the minimum number of terms that will ensure that the stiffness matrix has no spurious zero-energy modes, since it is known that the stiffness increases with the increase in the number of terms. Although using such a strategy of keeping the number of interpolation terms to a minimum works very well in static problems, it results either in instabilities or fails to converge in transient problems. This is because choosing the stress interpolation functions merely on the basis of removing spurious energy modes can violate some basic principles that interpolation functions should obey. In this work, we address the issue of choosing the interpolation functions based on such basic principles of interpolation theory and mechanics. Although this procedure results in the use of more number of terms than the minimum (and hence in slightly increased stiffness) in many elements, we show that the performance continues to be far superior to displacement-based formulations, and, more importantly, that it also results in considerably increased robustness.

A nonlocal model with strain-based damage

A thermodynamically consistent formulation of nonlocal damage in the framework of the internal variable theories of inelastic behaviours of associative type is presented. The damage behaviour is defined in the strain space and the effective stress turns out to be additively splitted in the actual stress and in the nonlocal counterpart of the relaxation stress related to damage phenomena. An important advantage of models with strain-based loading functions and explicit damage evolution laws is that the stress corresponding to a given strain can be evaluated directly without any need for solving a nonlinear system of equations. A mixed nonlocal variational formulation in the complete set of state variables is presented and is specialized to a mixed two-field variational formulation. Hence a finite element procedure for the analysis of the elastic model with nonlocal damage is established on the basis of the proposed two-field variational formulation. Two examples concerning a one-dimensional bar in simple tension and a two-dimensional notched plate are addressed. No mesh dependence or boundary effects are apparent.

An energy-momentum conserving algorithm for nonlinear transient analysis within the framework of hybrid elements

This work deals with the formulation and implementation of an energy-momentum conserving algorithm for conducting the nonlinear transient analysis of structures, within the framework of stress-based hybrid elements. Hybrid elements, which are based on a two-field variational formulation, are much less susceptible to locking than conventional displacement-based elements within the static framework. We show that this advantage carries over to the transient case, so that not only are the solutions obtained more accurate, but they are obtained in fewer iterations. We demonstrate the efficacy of the algorithm on a wide range of problems such as ones involving dynamic buckling, complicated three-dimensional motions, et cetera.

Multi-field variational formulations and related finite elements for piezoelectricshells

Smart structures technology characterized by structurally integrated sensors and actuatorshas recently expanded significantly especially as regards lightweight constructions inaeronautics and robotics, e.g. to allow vibration suppression and noise attenuation. In orderto be capable of solving these complex issues the finite element method as a wellestablished design tool has to be extended. This paper focuses on shallow sandwichcomposite shell structures with thin piezoelectric patches bonded to the surfaces. For theproper design of plate and shell structures with integrated piezoelectric materials, variousvariational formulations and corresponding finite elements are presented. The starting pointis the well known two-field variational formulation where the linear piezoelectric effect istaken into account so that the displacements and the electric potential serve asindependent variables. Here, the mostly assumed linear variation of the electricpotential through the thickness is assumed. Next, it is shown that a quadraticvariation of the electric potential through the thickness can be deduced directlyfrom the charge conservation condition. This quadratic variation of the electricpotential in the thickness direction is compared with the linear gradient of the firsttwo-field variational formulation. Moreover, in order to allow the implementation ofalternative formulations of the constitutive equations by switching of the independentvariables and nonlinear material behaviour, a three-field variational formulation ispresented in analogy to the Hu–Washizu principle. Adopting this variationalprinciple a hybrid finite element is derived where the dielectric displacement isformulated as an additional degree of freedom. This independent variable canbe condensed on the element level and does not enter the system of equations.For the first time all these different variational formulations are developed for aReissner–Mindlin shallow shell element formulation and compared with each other.

A two-field finite element formulation for elasticity coupled to damage

A two-field variational formulation is proposed for continuum damage mechanics problems. This formulation is applied to the numerical solution, via the finite element method, of initial boundary value problems arising in elasticity coupled to damage; the nodal variables are the displacement and damage fields. Simple examples are presented to demonstrate the advantages and limitations of this formulation.