Mixed Finite Element Formulation Research Articles

Mixed Finite Element Formulation in Non-linear Geometrical Analysis of Space Trusses and Application to Trusses with Member Length Imperfection

Abstract Space trusses usually have a significant number of elements. As a result, some elements inevitably have imperfections. In this study, the authors proposed a mixed finite element method for non-linear geometrical analysis of space trusses. The post-buckling behavior of space trusses with length imperfections was predicted using a code list written in the programming software. The numerical results indicate that when the length imperfection approaches zero, the results converge to those obtained in studies on standard trusses. A second approach based on the displacement model is also proposed, in which the Lagrange multiplier method is used to deal with the member length imperfection. The results show that the two approaches proposed by the authors are effective, in which the mix formulation based on the finite element model is superior to that based on the displacement model. These results verify the accuracy of the method. Based on the proposed method, the effects of imperfections in the element layers on the truss were studied. The results show that the imperfections in the top two element layers have the maximum influence on the limit load value, whereas the remaining layers have a negligible influence. The combination of the imperfect lengths of these two layers of elements can produce results in which the limit load value is much larger than in the case of a perfect system. Based on the obtained results, it is possible to provide a solution for improving the overall stability of truss structures by creating reasonable imperfections.

Dynamic large strain formulation for nematic liquid crystal elastomers

AbstractLiquid crystal elastomers (LCEs) are a class of materials which exhibit an anisotropic behavior in their nematic state due to the main orientation of their rod-like molecules called mesogens. The reorientation of mesogens leads to the well-known actuation properties of LCEs, i.e. exceptionally large deformations as a consequence of particular external stimuli, such as temperature increase. Another key feature of nematic LCEs is the capability to undergo deformation by constant stresses while being stretched in a direction perpendicular to the orientation of mesogens. During this plateau stage, the mesogens rotate towards the stretching direction. Such characteristic is defined as semisoft elastic response of nematic LCEs. We aim at modeling the semisoft behavior in a dynamic finite element method based on a variational-based mixed finite element formulation. The reorientation process of the rigid mesogens relative to the continuum rotation is introduced by micropolar drilling degrees of freedom. Responsible for the above-mentioned characteristics is an appropriate free energy function. Starting from an isothermal free energy function based on the small strain theory, we aim to widen it into the framework of large strains by identifying tensor invariants. In this work, we analyze the isothermal influence of the tensor invariants on the mechanical response of the finite element formulation and show that its space-time discretization preserves mechanical balance laws in the discrete setting.

Error estimates for a mixed finite element method for the Maxwell’s transmission eigenvalue problem

In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell’s transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the derivatives of eigenvalues.

Hydromechanical coupling in unsaturated clayey rocks with double porosity based on a multiscale homogenization procedure

Shales and clay rocks are porous media with multiscale microstructures used in many engineering applications. Intact clay rocks often exhibit a bimodal pore size distribution in which the nanopores are related to the interlayer spacing in the clay platelet and the micropores are related to the interparticle pores between clay particles. The double-porosity microstructure has significant implications for the hydromechanical behavior of these rocks especially in unsaturated conditions. In this study, we develop a double-porosity hydromechanical framework incorporating a homogenization scheme to simulate fluid flow and elastic deformation in anisotropic clay rocks. The homogenization scheme provides an enriched description of the elastic behavior of clay rocks during changes in the degree of saturation by bridging the nano, micro, and macroscale characterizations. We also model the nanopores and micropores as two independent pore networks with distinct fluid flow mechanisms and permeabilities. The proposed framework is cast into a three-field mixed finite element formulation for solving fully coupled flow and deformation problems. Numerical examples of free and confined swelling in Opalinus clay samples are presented to demonstrate the impacts of a multiscale microstructure on wetting processes in clay rocks.

On the necessity of the inf-sup condition for a mixed finite element formulation

Abstract We study a nonstandard mixed formulation of the Poisson problem, sometimes known as dual mixed formulation. For reasons related to the equilibration of the flux, we use finite elements that are conforming in $\textbf{H}(\operatorname{\textrm{div}};\varOmega )$ for the approximation of the gradients, even if the formulation would allow for discontinuous finite elements. The scheme is not uniformly inf-sup stable, but we can show existence and uniqueness of the solution, as well as optimal error estimates for the gradient variable when suitable regularity assumptions are made. Several additional remarks complete the paper, shedding some light on the sources of instability for mixed formulations.

State of art for hybrid mixed finite element formulation in non-linear analysis of structures

Since the late 80’s the Structural Analysis Research Group of the Instituto Superior Técnico (IST) has been involved in the development of non-conventional finite element formulations in order to overcome some of the limitations associated with the use of the CFE method and to develop high performance numerical tools for the analysis of structural engineering problems. Several alternative models for the linear and non-linear structural analysis have been developed using hybrid and mixed models techniques. These works are summarized in this paper, in which their past and future applications of this formulation in non-linear analysis of structures are fully detailed.

Mixed approximation of nonlinear acoustic equations: Well-posedness and a priori error analysis

Accurate simulation of nonlinear acoustic waves is essential for the continued development of a wide range of (high-intensity) focused ultrasound applications. In this article, we explore mixed finite element formulations of classical strongly damped quasilinear models of ultrasonic wave propagation, the Kuznetsov and Westervelt equations. Such formulations allow simultaneous retrieval of the acoustic particle velocity and either the pressure or acoustic velocity potential, thus characterizing the complete ultrasonic field at once. Using non-standard energy analysis and a fixed-point technique, we establish sufficient conditions for the well-posedness, stability, and optimal a priori errors in the energy norm for the semi-discrete equations. For the Westervelt equation, we also determine the conditions under which the error bounds can be made uniform with respect to the involved dissipation parameter. Additionally, we discuss convergence in the maximum error norm of the involved scalar quantities. Finally, computer experiments for Raviart–Thomas (RT) and Brezzi–Douglas–Marini (BDM) elements are performed to confirm the theoretical findings.

Modified third order shear-deformation model for anisotropic glulam beams via three-field mixed finite-element formulation

ABSTRACT Glulam timber structural linear elements often show low span-to-depth ratios. Moreover, timber is intrinsically orthotropic with very low shear moduli. These two circumstances justify the substantial contribution of shear strain to the overall deformation of linear elements. Since design specifications for timber beams commonly adopt 1D models, two such elastic models are proposed in this paper for static plane behavior, where, instead of fully orthotropic, glulam is assumed to be transversely isotropic. These models provide a good compromise between accuracy and simplicity for design proposes. A 3-field mixed formulation is generalized in order to produce two new 1D finite elements suitable for modeling transversely isotropic elastic glulam beams. To assess the accuracy of these new finite elements the results they produce are compared with those determined with a plane stress transversely isotropic 2D finite element. The included examples show that the proposed combination of transverse isotropy with a third order shear deformation model provides a suitable background to the 1D modeling of glulam beams with improved accuracy when compared to the models conventionally used in design. The paper is concluded with a parametric analysis which identifies the cases where the improved models should be used instead of simpler ones.

A C0 continuous mixed FE formulation for bending of laminated composite plates based on unified HSDT

AbstractThis study proposes a unified C0 continuous mixed finite element (MFE) formulation for the accurate and efficient prediction of stress components in laminated composite plates relying on Higher Order Shear Deformation Theory (HSDT). This unified form of the MFE accepts any convenient function for the representation of transverse shear deformation. The Hellinger–Reissner variational principle is employed for the derivation of MFE equations within a two‐field formulation involving stress resultants along with kinematical variables. Thus, the displacement and stress resultant fields are obtained directly from the global solution of the system of equations. In this manner, the in‐plane stress components are calculated over constitutive relations at the nodes without any need for error‐prone spatial derivatives. Furthermore, the independent interpolation of kinematic and stress resultant type variables allows the numerical solution to overcome the shear‐locking problem and ensure C0 continuity requirement. Numerical examples include convergence and comparison tests of predicted displacements and stress components under various boundary conditions and material configurations. Various test cases are considered for both the thin and thick plates subjected to sinusoidal and uniformly distributed loads. It is demonstrated that the proposed MFE formulation can capture stress components with high accuracy while being computationally efficient.

Saddle point least squares discretization for convection-diffusion

We consider a model convection-diffusion problem and present our recent analysis and numerical results regarding mixed finite element formulation and discretization in the singular perturbed case when the convection term dominates the problem. Using the concepts of optimal norm and saddle point reformulation, we found new error estimates for the case of uniform meshes. We compare the standard linear Galerkin discretization to a saddle point least square discretization that uses quadratic test functions, and explain the non-physical oscillations of the discrete solutions. We also relate a known upwinding Petrov–Galerkin method and the stream-line diffusion discretization method, by emphasizing the resulting linear systems and by comparing appropriate error norms. The results can be extended to the multidimensional case in order to find efficient approximations for more general singular perturbed problems including convection dominated models.

On a New Method of Quasi-static and Dynamic Analysis of Viscoelastic Plate on Elastic Foundation

In the present work, an alternative solution technique based on mixed finite element (MFE) formulation in the Laplace-Carson domain is proposed for quasi-static and dynamic analyses of viscoelastic plate (VEP) resting on an elastic foundation (EF). This work contributed a numerical solution to the problem of a viscoelastic Kirchhoff plate supported on a Winkler foundation. VEP-EF interaction problems are taken into account under different wave-type loadings. The viscoelastic material behavior of the plate is modeled by the Zener rheological solid model. A four-nodded linear isoparametric element containing sixteen degrees of freedom is used to model the VEP. Developed functional in the Laplace-Carson domain based on the Gâteaux differential method is transformed to the real time domain by utilizing the Dubner and Abate (D&A) inverse Laplace transform technique (ILTT). To evaluate the applicability of the results, five numerical samples are considered. Further analyzes are performed on different wave type loadings to offer a new perspective on the time-dependent behavior of VEP on EF.

On the stability of mixed polygonal finite element formulations in nonlinear analysis

AbstractThis article discusses the accuracy and stability of the pressure field in nonlinear mixed displacement‐pressure finite element formulations in solid mechanics. We focus on two‐dimensional mixed polygonal finite element formulations with linear displacement and constant pressure approximations in particular. The inf‐sup stability of these formulations is assessed and compared with classical mixed finite element formulations. An analytical proof is presented, which concludes that the occurrence of spurious pressure modes depends on the chosen meshing strategy. It is shown that these spurious modes are successfully suppressed on any Voronoi mesh in both linear elasticity and nonlinear hyperelasticity without the need for any kind of stabilization. Several linear and nonlinear nearly‐incompressible examples with different discretization strategies and boundary conditions are considered to validate the analytical proof. A mixed polygonal finite element formulation based on the scaled boundary parameterization is used to approximate the field variables, however, the derivations presented herein hold for any lowest‐order mixed polygonal finite element formulation. The nonexistence of checkerboard modes on linear elastic Voronoi discretizations is shown graphically. By evaluating the incremental pressure in each Newton–Raphson iteration, the stabilization effect of the Voronoi discretization is demonstrated for nonlinear problems. In addition, the analytical proof is validated by the numerical (generalized) inf‐sup test.

BELFEM: a special purpose finite element code for the magnetodynamic modeling of high-temperature superconducting tapes

Predicting the performance and reliability of high-temperature superconducting (HTS) cables and magnets is a critical component of their research and development process. Novel mixed finite element formulations, particularly the h -φ-formulation with thin-shell simplification, present promising opportunities for more efficient simulations of larger geometries. To make these new methods accessible in a flexible tool, we are developing the Berkeley Lab Finite Element Framework (BELFEM). This paper provides an overview of the relevant formulations, discusses the current state of the art, and discusses the main aspects of the BELFEM code structure. We validate a first 2D thin-shell implementation in BELFEM against selected benchmarks computed in COMSOL Multiphysics and compare the performance of our code with a comparable formulation in GetDP. We also outline the next steps in the development process, paving the way for more advanced and robust modeling capabilities.

A selectively reduced degree basis for efficient mixed nonlinear isogeometric beam formulations with extensible directors

The effect of higher order continuity in the solution field by using NURBS basis function in isogeometric analysis (IGA) is investigated for an efficient mixed finite element formulation for elastostatic beams. It is based on the Hu–Washizu variational principle considering geometrical and material nonlinearities. Here we present a reduced degree of basis functions for the additional fields of the stress resultants and strains of the beam, which are allowed to be discontinuous across elements. This approach turns out to significantly improve the computational efficiency and the accuracy of the results. We consider a beam formulation with extensible directors, where cross-sectional strains are enriched to avoid Poisson locking by an enhanced assumed strain method. In numerical examples, we show the superior per degree-of-freedom accuracy of IGA over conventional finite element analysis, due to the higher order continuity in the displacement field. We further verify the efficient rotational coupling between beams, as well as the path-independence of the results.

A mixed finite element method for nonlinear radiation–conduction equations in optically thick anisotropic media

We propose a new mixed finite element formulation for solving radiation–conduction heat transfer in optically thick anisotropic media. At this optical regime, the integro-differential equations for radiative transfer can be replaced by the simplified PN approximations using an asymptotic analysis. The conductivity is assumed to be nonlinear depending on the temperature along with anisotropic absorption and scattering depending on both the direction and location variables. The simplified PN approximations are enhanced by considering a diffusion tensor capable of describing anisotropic radiative heat transfer. In the present study, we investigate the performance of the unified and mixed formulations combining cubic P3, quadratic P2, and linear P1 finite elements to approximate the temperature in the simplified P3 model. To demonstrate the performance of the proposed methodology, three-dimensional examples of nonlinear radiation–conduction equations in optically thick anisotropic media are presented. The obtained numerical results demonstrate the accuracy and efficiency of the proposed mixed finite element formulation over the conventional unified finite element formulation to accurately solve the simplified P3 equations in anisotropic media.

Nonlinear analysis of reinforced concrete slabs using a quasi-3D mixed finite element formulation

This study presents an efficient quasi-3D mixed finite element (MFE) formulation for the materially nonlinear analysis of reinforced concrete (RC) slabs based on a refined layered global–local plate theory. The cross-section of the RC slab is divided into a series of concrete and steel layers. Each layer is treated as a plate structure and mechanical properties of the steel and concrete materials are individually applied to each layer. In addition to unknown variables of the displacement field, out-of-plane stress components are also assumed as independent field variables in the presented MFE formulation. It makes it possible to obtain the out-of-plane shear stresses at each node directly from constitutive equations with the enough accuracy. A four‐node rectangular element ensuring the C1‐continuity is used for discretizing the domain of the RC slab. The governing equations are derived based on a partially mixed-field variational principle. The accuracy and efficiency of the presented MFE have been investigated through some benchmark examples. Numerical tests show that the presented MFE yields results with sufficient accuracy at a low computational cost.

A comprehensive analysis of in-plane functionally graded plates using improved first-order mixed finite element model

In this study, the mechanical behaviors of the in-plane functionally graded plate (IFG) are comprehensively explored via a finite element (FE) model. The current FE model is developed base on improved first-order shear deformation theory (FSDT) in conjunction with the mixed FE formulation. The volume fraction as well as the material properties change smoothly along the width and length of the plates for six different patterns. A simple improved transverse shear stress is applied to modify its distribution across the thickness of the IFG plates. Hence, the transverse shear stress equals zeros on the lower and upper surfaces of the IFG plates. Besides, no shear correction factor is needed for calculation of shear strain energy. A four-node mixed plate element, IMQ4, is established via mixed FE formulation for mechanical analysis of the IFG plates. The accuracy and fast convergence rate are validated via some examples. A comprehensive parametric analysis is provided to demonstrate the roles of several parameters including the material gradient index, the side-to-thickness ratio, aspect ratio, and boundary conditions on the mechanical behaviors of the IFG plates.

Principle of virtual power and drilling degrees of freedom for dynamic modeling of the behavior of liquid crystal elastomer films

In this work, we aim to model the reorientation process of mesogens in nematic liquid crystal elastomers within the context of dynamics. We consider a continuum model with separate mappings for the deformation of the monolithic material and the orientation of the nematic director, where the latter describes the inclination of the mesogens. We achieve the inextensibility of the nematic director through the introduction of drilling degrees of freedom. We combine this approach with the application of the principle of virtual power and a mixed finite element formulation, in order to formulate distinct momentum and angular momentum balance laws for the two separate mappings. Furthermore, we include in our continuum model a volume load and a surface load associated only with the orientation mapping. We show in the presented three numerical examples that our formulation enables the fulfillment of all momentum and angular momentum balance laws.

Warping included mixed finite elements for bending and stresses of functionally graded exact curved beams

This study presents a warping included mixed finite element formulation for the static and stress analyses of functionally graded (FG) exact curved beams. Another distinctive feature of this study is to present the range of geometric and material parameters where the first-order shear deformation theory lacks necessary precision. The constitutive equations are derived from the 3D elasticity theory of an elastic continuum. The couple effects are considered in addition to the variations of FG material constituents through the transverse/axial directions. The mixed finite element formulation is enhanced by the cross-sectional warping deformations over a displacement-type finite element formulation. The two-noded curved mixed finite elements with 12 degrees of freedom at per node are derived over exact curvature and length. Satisfactory results are obtained for the static responses and stresses of axially FG-sandwich and transversely FG exact curved beams with power-law dependence compared to the 3D behavior of brick finite elements. Finally, the influences of ellipticity, width to thickness ratio, FG material constituents, and material gradient index on the static response of FG exact curved beams are investigated.

A simple and efficient lagrange multiplier based mixed finite element for gradient damage

A novel finite element formulation for gradient-regularized damage models is presented which allows for the robust, efficient, and mesh-independent simulation of damage phenomena in engineering and biological materials. The paper presents a Lagrange multiplier based mixed finite element formulation for finite strains. Thereby, no numerical stabilization or penalty parameters are required. On the other hand, no additional degrees of freedom appear for the Lagrange multiplier which is achieved through a suitable FE-interpolation scheme allowing for static condensation. In contrast to competitive approaches from the literature with similar efficiency, the proposed formulation does not require cross-element information and thus, a straightforward implementation using standard element routine interfaces is enabled. Numerical tests show mesh-independent solutions, robustness of the solution procedure for states of severe damage and under cyclic loading conditions. It is demonstrated that the computing time of the gradient damage calculations exceeds the one of purely elastic computations only by an insignificant amount. Furthermore, an improved convergence behavior compared to alternative approaches is shown.