A variational finite element formulation for dielectric waveguides in terms of transverse magnetic fields

A spurious-mode-free variational formulation for the finite-element analysis of anisotropic, inhomogeneous dielectric waveguides is derived and demonstrated with examples. Apart from avoiding the occurrence of spurious modes, this formulation has numerous advantages, including the ability to treat problems including significant amount of loss, the direct solution for the propagation constant at a given frequency, and the use of the most efficient representation of the problem, needing only two vector components. This is achieved without losing the sparsity of the resultant (canonical) matrix eigenvalue problem, which depends on the topology of the mesh used.<>     

A finite element formulation for creep analysis

As an alternative to the initial strain method, a variable stiffness method is presented for creep analysis. The method is developed by incorporating the change in stress state during a time interval in determining the creep strain increments concurrent with the change. It is shown by means of examples that this method provides solution stability for relatively large time intervals for which the initial strain method may fail to function properly.     

A scaled boundary finite element formulation for poroelasticity

This paper develops the scaled boundary finite element formulation for applications in coupled field problems, in particular, to poroelasticity. The salient feature of this formulation is that it can be applied over arbitrary polygons and/or quadtree decomposition, which is widely employed to traverse between small and large scales. Moreover, the formulation can treat singularities of any order. Within this framework, 2 sets of semianalytical, scaled boundary shape functions are used to interpolate the displacement and the pore fluid pressure. These shape functions are obtained from the solution of vector and scalar Laplacian, respectively, which are then used to discretise the unknown field variables similar to that of the finite element method. The resulting system of equations are similar in form as that obtained using standard procedures such as the finite element method and, hence, solved using the standard procedures. The formulation is validated using several numerical benchmarks to demonstrate its accuracy and convergence properties.     

A finite element displacement formulation for gradient elastoplasticity

We present a second gradient elastoplastic model for strain‐softening materials based entirely on a finite element displacement formulation. The stress increment is related to both the strain increment and its Laplacian. The displacement field is the only field needed to be discretized using a C¹ continuity element. The required higher‐order boundary conditions arise naturally from the displacement field. The model is developed to regularize the ill‐posedness caused by strain‐softening material behaviour. The gradient terms in the constitutive equations introduce an extra material parameter with dimensions of length allowing robust modelling of the post‐peak material behaviour leading to localization of deformation. Mesh insensitivity is demonstrated by modelling localization of deformation in biaxial tests. It is shown that both the thickness and inclination of the shear‐band zone are insensitive to the mesh directionality and refinement and agree with the expected theoretical and experimental values. Copyright © 2001 John Wiley & Sons, Ltd.     

ALE finite element formulation for ring rolling analysis

In this paper, a ring rolling process is analysed by the Arbitrary Lagrangian Eulerian (ALE) finite element method. Phenomena associated with the process, such as large deformations, elastoplastic material behaviour and the friction on the interface, are included in the analysis. Special modelling on driven, idle and guide rolls is given. Results which include the overall shape of the formed ring, the time histories of roll separating force and driving torque, the distribution of the normal pressure on the ring–roll interface as well as the distribution of effective stresses in the formed ring, are also presented.     

A mixed finite element formulation for plate problems

A mixed triangular finite element model has been developed for plate bending problems in which effects of shear deformation are included. Linear distribution for all variables is assumed and the matrix equation is obtained through Reissner's variational principle. In this model, interelement compatibility is completely satisfied whereas the governing equations within the element are satisfied ‘in the mean’. A detailed error analysis is made and convergence of the scheme is proved. Numerical examples of thin and moderately thick plates are presented.     

A thermomechanical interface element formulation for finite deformations

Interfacial layers are thermally and mechanically described in the presented approach. The combination of temperature evolution and mechanical loading influences significantly the deformation and thermal behavior. A consistent framework is derived from principle thermodynamical laws and balance equations. The approach is incorporated in the finite element analysis framework, wherein the unknown temperature- and displacement fields are obtained, e.g. by a Newton-type solution scheme. The derived finite element equations are linearized and a fully coupled interface element formulation is given with respect to thermomechanical residuals and stiffnesses. Bonds between the opening crack flanks are the main mechanisms of the delamination process. These bonds can be of different nature, depending on the bulk material. They are constitutively described in the presented approach in terms of transmission of tractions and heat. Numerical examples are shown in order to demonstrate the predictive capabilities of the thermomechanical interface element.     

Dynamic finite element formulation for Cosserat elastic plates

A displacement and rotation‐based dynamic finite element formulation for Cosserat plates is discussed in detail in this paper. Special attention is given to the validation of the element through adequate benchmarks and patch tests. The choice of the interpolation functions and the order of integration of the stiffness and the mass matrices are extensively argued. The possibility of local system deficiencies is explored, and a shear locking investigation specifically elaborated for Cosserat plates is carried out. It is shown how the present formulation has interesting computational properties as compared to a classical continuum‐based formulation and how it can provide suitable results despite the use of reduced integration. An example of application of the finite element is given, in which the natural frequencies of a masonry panel modelled by means of discrete elements are computed and compared with the finite elements solution. Copyright © 2014 John Wiley & Sons, Ltd.     

p-version least-squares finite element formulation for convection-diffusion problems

This paper presents a p-version least-squares finite element formulation of the convection-diffusion equation. The second-order differential equation describing convection-diffusion is reduced to a series of equivalent first-order differential equations for which the least-squares formulation is constructed using the same order of approximation for each of the dependent variables. The hierarchical approximation functions and the nodal variable operators are established by first constructing the one-dimensional hierarchical approximation functions of orders pξ and pη and the corresponding nodal variable operators in ξ and η-direction and then taking their products. Numerical results are presented and compared with analytical and numerical solutions for a two-dimensional test problem to demonstrate the accuracy and the convergence characteristics of the present formulation. The Gaussian quadrature rule used to calculate the numerical values of the element matrices, vectors as well as the error functional I(E), is established based on the highest degree of the polynomial in the integrands. It is demonstrated that this quadrature rule with the present p-version formulation produces excellent results for very low as well as extremely high Peclet numbers (10-10⁶) and, furthermore, the error functional I (sum of the integrals of E²) is a monotonically decreasing function of the number of degrees of freedom as the p-levels are increased for a fixed mesh. It is shown that exact integration with the h-version (linear and parabolic elements) produces inaccurate solutions at high Peclet numbers. Results are also presented using reduced integration. It is shown that the reduced integration with p-version produces accurate values of the primary variable even for relatively low p-levels but the error functional I (when calculated using the proper integration rule) has a much higher value (due to errors in the derivatives of the primary variable) and is a non-monotonic function of the degrees of freedom as p-levels are increased for a fixed mesh. Similar behaviour of the error functional I is also observed for the h-models using linear elements when reduced integration is used. Although the h-models using parabolic elements produce monotonic error functional behaviour as the number of degrees of freedom are increased, the error values are inferior to the p-version results using exact integration.     

A finite element formulation for thermoelastic damping analysis

We present a finite element formulation based on a weak form of the boundary value problem for fully coupled thermoelasticity. The thermoelastic damping is calculated from the irreversible flow of entropy due to the thermal fluxes that have originated from the volumetric strain variations. Within our weak formulation we define a dissipation function that can be integrated over an oscillation period to evaluate the thermoelastic damping. We show the physical meaning of this dissipation function in the framework of the well‐known Biot's variational principle of thermoelasticity. The coupled finite element equations are derived by considering harmonic small variations of displacement and temperature with respect to the thermodynamic equilibrium state. In the finite element formulation two elements are considered: the first is a new 8‐node thermoelastic element based on the Reissner–Mindlin plate theory, which can be used for modeling thin or moderately thick structures, while the second is a standard three‐dimensional 20‐node iso‐parametric thermoelastic element, which is suitable to model massive structures. For the 8‐node element the dissipation along the plate thickness has been taken into account by introducing a through‐the‐thickness dependence of the temperature shape function. With this assumption the unknowns and the computational effort are minimized. Comparisons with analytical results for thin beams are shown to illustrate the performances of those coupled‐field elements. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptivity in hybrid-Trefftz finite element formulation

With the advent of computer aided design (CAD), the development of fully or semi-automated procedures by which a solution warranting the specified accuracy can most efficiently be reached has become a necessity. It is now of utmost interest to investigate finite element (FE) formulations offering a simple form of adaptivity, easy to implement. The present study (which is a sequel to an earlier paper¹ on error estimation by the authors) identifies as a particularly promising approach of attaining this goal the p-method associated with the so-called hybrid-Trefftz FE model.² Based on a simple stress error estimator¹ and prior knowledge of the convergence rate, the solution warranting specified stress accuracy may be reached in a single trial by suitably respecifying just one parameter in the input data. The reported approach has successfully been implemented into the general purpose FE program SAFE³ and its high efficiency is illustrated in the paper through practical applications involving corner singularities and stress concentrations.     

A least squares finite element formulation for elastodynamic problems

A residual finite element formulation is developed in this paper to solve elastodynamic problems in which body wave potentials are primary unknowns. The formulation is based on minimizing the square of the residuals of governing equations as well as all boundary conditions. Since the boundary conditions in terms of wave potentials are neither Dirichlet nor Neumann type it is difficult to construct a functional to satisfy all governing equations and boundary conditions following the variational principle designed for conventional finite element formulation. That is why the least squares technique is sought. All boundary conditions are included in the functional expression so that the satisfaction of any boundary condition does not become a requirement of the trial functions, but they should satisfy some continuity conditions across the interelement boundary to guarantee proper convergence. In this paper it is demonstrated that the technique works well for elastodynamic problems; however, it is equally applicable to any other field problem.     

An advanced finite element formulation for piezoelectric beam structures

Since many piezoelectric components are thin rod-like structures, a piezoelectric finite beam element can be utilized to analyse a wide range of piezoelectric devices effectively. The mechanical strains and the electric field are coupled by the constitutive relations. Finite element formulations using lower order functions to interpolate mechanical and electrical fields lead to unbalances within the numerical approximation. As a consequence incorrect computational results occur, especially for bending dominated problems. The present contribution proposes a concept to avoid these errors. Therefore, a mixed multi-field variational approach is introduced. The element employs the Timoshenko beam theory and considers strains throughout the width and the thickness enabling to directly use 3D constitutive relations. By means of several numerical examples it is shown that the element formulation allows to analyse piezoelectric beam structures for all typical load cases without parasitically affected results.     

A C0 finite element formulation for thin-walled beams

Timoshenko's and Vlasov's beam theories are combined to produce a C⁰ finite element formulation for arbitrary cross section thin-walled beams. Section properties are generated using a curvilinear co-ordinate system to describe the cross section dimensions. The element includes both shear and warping deformations caused by the bending moments and the bimoment. A Gauss quadrature order is employed which exactly integrates the bending and warping stiffness matrices and provides a reduced integration order for the shear stiffness matrices. Numerical results are presented for a channel section cantilever beam. The influence of shear deformation is investigated and the calculated results are shown to be in excellent agreement with the classical solutions.     

A novel finite element formulation for frictionless contact problems

This article advocates a new methodology for the finite element solution of contact problems involving bodies that may undergo finite motions and deformations. The analysis is based on a decomposition of the two-body contact problem into two simultaneous sub-problems, and results naturally in geometrically unbiased discretization of the contacting surfaces. A proposed two-dimensional contact element is specifically designed to unconditionally allow for exact transmission of constant normal traction through interacting surfaces.     

A finite element formulation for highly anisotropic incompressible elastic solids

A numerical approach is developed for the solution of problems of materials with extremely strong directions. Small deformations of a transversely isotropic linear elastic solid, reinforced by a single family of inextensible fibres, are considered. The kinematic constraint equations of incompressibility and inextensibility in the fibre direction lead to the appearance of an arbitrary hydrostatic pressure and an arbitrary tension stress in the constitutive equations. A Galerkin approach is used to discretize the virtual work and weak form of the constraint equations. Independent interpolation of the displacement, pressure and tension fields leads to a mixed system of equations, with characteristic zero-diagonal terms. The assumption of plane stress conditions in the plane of the fibres results in a simplified displacement-tension formulation, analogous to the primitive-variable formulation of Stokes flow. A mixed penalty approximation is then employed to solve for displacement and tension stress fields. Computations are carried out using a biquadratic displacement element with discontinuous bilinear tension stress interpolation. The formulation is used to solve a number of simple beam problems and the results compared to closed-form solutions.     

A scaled boundary finite element formulation for dynamic elastoplastic analysis

This study presents the development of the scaled boundary finite element method (SBFEM) to simulate elastoplastic stress wave propagation problems subjected to transient dynamic loadings. Material nonlinearity is considered by first reformulating the SBFEM to obtain an explicit form of shape functions for polygons with an arbitrary number of sides. The material constitutive matrix and the residual stress fields are then determined as analytical polynomial functions in the scaled boundary coordinates through a local least squares fit to evaluate the elastoplastic stiffness matrix and the residual load vector semianalytically. The treatment of the inertial force within the solution of the nonlinear system of equations is also presented within the SBFEM framework. The nonlinear equation system is solved using the unconditionally stable Newmark time integration algorithm. The proposed formulation is validated using several benchmark numerical examples.     

An alternative high-order finite element formulation for cylindrical field problems

Earlier formulations of the finite element approach to cylindrical (rz) field problems led initially to variational expressions containing a simple term in 1/r and consequent attempts to remove it by appropriate choice of interpolation functions. The present paper uses new interpolation functions which ensure that the field behaviour near the axis is correctly modelled. High-order finite elements up to order four are derived and tested on a special cylindrical geometry to confirm, in a practical case, the theoretical claims of improved rates of convergence in solving problems of engineering significance.