Three-field Variational Formulation Research Articles

Three-field mixed hp-finite element method for the solution of the Guyer–Krumhansl heat conduction model

On the basis of numerous experimental studies the Guyer–Krumhansl heat conductivity model can be considered as one of the most promising theoretical models to simulate the low temperature processes and the thermal behavior of such complex structures as materials with inhomogeneities and interfaces, as well as porous materials. However, the classical h-version finite element methods do not provide convergent and accurate results for the solution of the Guyer–Krumhansl heat conductivity model. In recent paper, a new three-field variational formulation is derived treating the temperature, the heat flow and its current density as independent variables. Both the temperature- and the heat flow boundary condition are weakly imposed, i.e., built in the variational form.Based on this variational background, a new, hp-version mixed finite element method is constructed. The h- and p-convergence behaviors of the temperature and the heat flow are analyzed on the transient region for two representative model problems: (i) a rapid heating process with exponentially changing rate and (ii) a ramp-type heating process. The relative and absolute errors are measured in maximum norm. From the computational experiments it follows that the mixed hp-finite element method gives reliable, robust (uniformly stable) results not only for the h- but also for the p-approximation in the case of both the temperature and the heat flow.

A constitutive framework for rocks undergoing solid dissolution

We formulate a time-dependent damage theory for rocks subjected to mechanical deformation and solid dissolution. The constitutive description is inspired by the transition state theory, which states that the rate of dissolution is a function of the reactive surface area measured through the crack density in the volume. We use a gradient-enhanced damage framework in which damage depends on the deformation of the material as well as on the amount of solid mass dissolved over time. The gradient-enhanced formulation is characterized by a three-field variational formulation with the solid displacement, nonlocal equivalent strain, and nonlocal rate of solid dissolution as the basic state variables. Traditionally, time-independent damage theories have only allowed damage to increase with increasing external load. In the proposed framework, the degree of damage may increase due to solid dissolution even when the external load is held fixed. In this way, solid dissolution is viewed as a process that is responsible for bringing about rate-dependent effects such as creep and stress-relaxation, which are two common features of geomaterial behavior.

A Three-Field Variational Formulation for a Frictional Contact Problem with Prescribed Normal Stress

In the present work, we address a nonlinear boundary value problem that models frictional contact with prescribed normal stress between a deformable body and a foundation. The body is nonlinearly elastic, the constitutive law being a subdifferential inclusion. We deliver a three-field variational formulation by means of a new variational approach governed by the theory of bipotentials combined with a Lagrange-multipliers technique. In this new approach, the unknown of the mechanical model is a triple consisting of the displacement field, a Lagrange multiplier related to the friction force and the Cauchy stress tensor. We obtain existence, uniqueness, boundedness and convergence results.

A posteriori error estimates by weakly symmetric stress reconstruction for the Biot problem

A posteriori error estimates are constructed for the three-field variational formulation of the Biot problem involving the displacements, the total pressure and the fluid pressure. The discretization under focus is the H1(Ω)-conforming Taylor–Hood finite element combination, consisting of polynomial degrees k+1 for the displacements and the fluid pressure and k for the total pressure. An a posteriori error estimator is derived on the basis of H(div)-conforming reconstructions of the stress and flux approximations. The symmetry of the reconstructed stress is allowed to be satisfied only weakly. The reconstructions can be performed locally on a set of vertex patches and lead to a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. Particular emphasis is given to nearly incompressible materials and the error estimates hold uniformly in the incompressible limit. Numerical results on the L-shaped domain confirm the theory and the suitable use of the error estimator in adaptive strategies.

Coupling finite elements and auxiliary sources

We consider second-order scalar elliptic boundary value problems on unbounded domains, which model, for instance, electrostatic fields. We propose a discretization that relies on a Trefftz approximation by multipole auxiliary sources in some parts of the domain and on standard mesh-based primal Lagrangian finite elements in other parts. Several approaches are developed and, based on variational saddle point theory, rigorously analyzed to couple both discretizations across the common interface:1. Least-squares-based coupling using techniques from PDE-constrained optimization.2. Coupling through Dirichlet-to-Neumann operators.3. Three-field variational formulation in the spirit of mortar finite element methods.We compare these approaches in a series of numerical experiments.

Damage-based fracture with electro-magnetic coupling

A coupled elastic and electro-magnetic analysis is proposed including finite displacements and damage-based fracture. Piezo-electric terms are considered and resulting partial differential equations include a non-classical wave equation due to the specific constitutive law. The resulting wave equation is constrained and, in contrast with the traditional solutions of the decoupled classical electro-magnetic wave equations, the constraint is directly included in the analysis. The absence of free current density allows the expression of the magnetic field rate as a function of the electric field and therefore, under specific circumstances, removal of the corresponding magnetic degrees-of-freedom. A Lagrange multiplier field is introduced to exactly enforce the divergence constraint, forming a three-field variational formulation (required to include the wave constraint). No vector-potential is required or mentioned, eliminating the need for gauges. The classical boundary conditions of electromagnetism are specialized and a boundary condition involving the electric field is obtained. The spatial discretization makes use of mixed bubble-based (of the MINI type) finite elements with displacement, electric field and Lagrange multiplier degrees-of-freedom. Three verification examples are presented with very good qualitative conclusions and mesh-independence.

Frame Element for Metallic Shear-Yielding Members under Cyclic Loading

Modeling the energy dissipation capacity of shear-yielding members is important in the evaluation of the seismic response of earthquake resistant structural systems. This paper presents the model of a frame element for the hysteretic behavior of these members. The model is based on a three-field variational formulation with independent displacement, stress, and strain fields. The displacement field is based on the Timoshenko beam theory. The nonlinear response of the element is derived from the section integration of the multiaxial material stress-strain relation at several control points along the element, thus accounting for the interaction between normal and shear stress and the spread of inelastic deformations in the member. With the derivation of the axial force-shear-flexure interaction of short members from the material response the proposed model is general, in contrast to existing concentrated plasticity models that require parameter calibration for different loading and support conditions. Furthermore, the model does not suffer from shear locking and does not require mesh refinement for the accurate representation of inelastic member deformations. Correlation studies of analytical results with available experimental data of the hysteretic behavior of shear-yielding members confirm the capabilities of the proposed model. Its computational efficiency makes it suitable for large scale simulations of the earthquake response of structures with shear-yielding members.

A mixed hybrid finite beam element with an interface to arbitrary three-dimensional material models

A theory for spatial beams and an associated mixed finite element formulation are presented. The basic kinematic assumption allows for finite rotations and transverse shear strains. The relation of the Green–Lagrangean strains to the beam strains is derived. A three-field variational formulation with independent displacements, rotations, stress resultants and beam strains is developed using the kinematic and static field equations and the constitutive equations. Higher order stress resultants are set to zero within the Euler–Lagrange equations. The formulation in this paper is restricted to rectangular cross sections. The interpolation of the beam strains consists of two parts. The first part is identical to the stress resultant interpolation. The second part describes cross section warping and allows for transverse normal strains. Within this approach an interface to arbitrary three-dimensional constitutive laws can easily be realized. Due to the mixed interpolation technique full Gauss integration in length direction of the beam element can be applied to obtain a locking free element formulation. The effectiveness of the method is demonstrated by means of several numerical examples. We present results for elasticity, inelasticity, and stability problems and evaluate the load carrying capacities of spatial beam structures. The essential feature of the new element is the robustness in nonlinear applications. It allows very large load steps in comparison to other element formulations.

A NEW COMPACTNESS RESULT FOR ELECTROMAGNETIC WAVES: APPLICATION TO THE TRANSMISSION PROBLEM BETWEEN DIELECTRICS AND METAMATERIALS

We consider the time-harmonic Maxwell equations, involving wave transmission between media with opposite sign dielectric and/or magnetic coefficients. We prove that, in the case of sign-shifting dielectric coefficients, the space of electric fields is compactly embedded in L2. We build a three-field variational formulation equivalent to Maxwell system for sign-shifting magnetic coefficients and show that, under some suitable conditions, the formulation fits into the coercive plus compact framework.

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.

Mixed and selective reduced integration procedures in large strain hyperelastic analysis of nearly incompressible solids

We compare a mixed method and a displacement method with selective reduced integration for the finite element analysis of finite, nearly incompressible, hyperelasticity. The Simo-Taylor three-field variational formulation is extended to a general description which includes volumetric-deviatoric coupling and can readily be specialized to particular cases, such as the strain energy functions due to Rivlin or Ogden. The displacement formulation is obtained as a special case. Emphasis is placed on the treatment of incompressibility and the avoidance of volumetric locking. Classical incompressible strain energy functions, extended to include the effect of compressibility at the nearly incompressible limit, result in a penalty type algorithm for the imposition of near incompressibility. This is accomplished through the decomposition of strain into distortional and volumetric components. Shear locking is prevented through the use of higher order elements. All integrals are evaluated in the current configuration. The two methods have been implemented into the finite element code MSC/NASTRAN i . It is shown that the methods,

On a fully three-dimensional finite-strain viscoelastic damage model: Formulation and computational aspects

A fully three-dimensional finite-strain viscoelastic model is developed, characterized by: (i) general anisotropic response, (ii) uncoupled bulk and deviatoric response over any range of deformations, (iii) general relaxation functions, and (iv) recovery of finite elasticity for very fast or very slow processes; in particular, classical models of rubber elasticity (e.g. Mooney-Rivlin). Continuum damage mechanics is employed to develop a simple isotropic damage mechanism, which incorporates softening behavior under deformation, and leads to progressive degradation of the storage modulus in a cyclic test with increasing amplitude (Mullins' effect). A numerical integration procedure is proposed which trivially satisfies objectivity and bypasses the use of midpoint configurations. The resulting algorithm can be exactly linearized in closed form, and leads to symmetric tangent moduli. Quasi-incompressible response is accounted for within the context of a three-field variational formulation of the Hu-Washizu type.