Mixed Finite Element Formulation Research Articles

Mixed displacement–pressure formulations and suitable finite elements for multimaterial problems with compressible and incompressible models

Multimaterial problems in linear and nonlinear elasticity are some of the least explored using mixed finite element formulations with higher-order elements. The fundamental issue in adapting the mixed displacement–pressure formulations with linear and higher-order continuous elements for the pressure field is their inability to capture pressure and stress jumps across material interfaces. In this paper, for the first time in literature, we perform comprehensive studies of multimaterial problems in elasticity consisting of compressible and incompressible material models using the mixed displacement–pressure formulation to assess the performance of different element types in accurately resolving pressure fields within the domains and pressure jumps across material interfaces. In particular, inf–sup stable displacement–pressure combinations with element-wise discontinuous pressure for triangular and tetrahedral elements are considered and their performance is assessed along with the Q1/P0 element and Taylor–Hood elements using several numerical examples. The results show that Taylor–Hood elements fail to capture the stress jumps due to the continuity of DOFs across elements, the Crouzeix–Raviart (P2b/P1dc) element yields substantially poor pressure fields despite a significant increase in pressure degrees of freedom and that the P3/P1dc element produces superior quality results fields when compared with the P2b/P1dc element.

Macroscopic modeling of flexoelectricity-driven remanent polarization in piezoceramics

This paper presents a variational modeling framework for investigating the flexoelectricity-driven evolution of remanent polarization in piezoceramics. In small-scale electromechanical systems, strain gradients can exhibit polarization in dielectric materials via the direct flexoelectric effect. In ferroelectrics, it is reasonable to expect that a sufficiently large magnitude of the flexoelectricity leads to a switching of the domain structure and thus the material becomes remanently polarized. It is interesting to note that this means that poling would be able to occur in the absence of any external electrical source. This provides the motivation to gain a better understanding of this effect for a possible technical use in e.g. sensor applications. For this purpose, a macroscopic model is presented that couples flexoelectricity and ferroelectric domain switching processes. By embedding the model in the variational framework of the generalized standard materials (GSM), a minimum-type potential structure and thus a stable and efficient numerical treatment is obtained. A mixed finite element formulation based on the Helmholtz free energy is introduced to solve the higher-order flexoelectric boundary value problem. In order to realistically predict the flexoelectric material behavior, the model response is adapted to experimental results in literature obtained for a piezoceramic in a bending test. By simulations based on the adapted model the evolution of the flexoelectricity-driven remanent polarization in the vicinity of a notch is shown.

Discretisations of mixed-dimensional Thermo-Hydro-Mechanical models preserving energy estimates

In this study, we explore mixed-dimensional Thermo-Hydro-Mechanical (THM) models in fractured porous media accounting for Coulomb frictional contact at matrix fracture interfaces. The simulation of such models plays an important role in many applications such as hydraulic stimulation in deep geothermal systems and assessing induced seismic risks in CO2 storage. We first extend to the mixed-dimensional framework the thermodynamically consistent THM models derived in [19] based on first and second principles of thermodynamics. Two formulations of the energy equation will be considered based either on energy conservation or on the entropy balance, assuming a vanishing thermo-poro-elastic dissipation. Our focus is on space time discretisations preserving energy estimates for both types of formulations and for a general single phase fluid thermodynamical model. This is achieved by a Finite Volume discretisation of the non-isothermal flow based on coercive fluxes and a tailored discretisation of the non-conservative convective terms. It is combined with a mixed Finite Element formulation of the contact-mechanical model with face-wise constant Lagrange multipliers accounting for the surface tractions, which preserves the dissipative properties of the contact terms. The discretisations of both THM formulations are investigated and compared in terms of convergence, accuracy and robustness on 2D test cases. It includes a Discrete Fracture Matrix model with a convection dominated thermal regime, and either a weakly compressible liquid or a highly compressible gas thermodynamical model.

Modeling the mechanosensitive collective migration of cells on the surface and the interior of morphing soft tissues.

Cellular contractility, migration, and extracellular matrix (ECM) mechanics are critical for a wide range of biological processes including embryonic development, wound healing, tissue morphogenesis, and regeneration. Even though the distinct response of cells near the tissue periphery has been previously observed in cell-laden microtissues, including faster kinetics and more prominent cell-ECM interactions, there are currently no models that can fully combine coupled surface and bulk mechanics and kinetics to recapitulate the morphogenic response of these constructs. Mailand et al. (Biophys J 117(5):975-986, 2019) had shown the importance of active elastocapillarity in cell-laden microtissues, but modeling the distinct mechanosensitive migration of cells on the periphery and the interior of highly deforming tissues has not been possible thus far, especially in the presence of active elastocapillary effects. This paper presents a framework for understanding the interplay between cellular contractility, migration, and ECM mechanics in dynamically morphing soft tissues accounting for distinct cellular responses in the bulk and the surface of tissues. The major novelty of this approach is that it enables modeling the distinct migratory and contractile response of cells residing on the tissue surface and the bulk, where concurrently the morphing soft tissues undergo large deformations driven by cell contractility. Additionally, the simulation results capture the changes in shape and cell concentration for wounded and intact microtissues, enabling the interpretation of experimental data. The numerical procedure that accounts for mechanosensitive stress generation, large deformations, diffusive migration in the bulk and a distinct mechanism for diffusive migration on deforming surfaces is inspired from recent work on bulk and surface poroelasticity of hydrogels involving elastocapillary effects, but in this work, a two-field weak form is proposed and is able to alleviate numerical instabilities that were observed in the original method that utilized a three-field mixed finite element formulation.

Modeling and simulation of electrochemical and surface diffusion effects in filamentary cation-based resistive memory devices

Cation-based (or electrochemical) resistive memory devices are gaining increasing interest in neuromorphic applications due to their capability to emulate the dynamic behavior of biological neurons and synapses. The utilization of such devices in neuromorphic systems necessitates a reliable physical model for the resistance switching mechanism, which is based on the formation and dissolution of a conductive filament in a thin dielectric layer, sandwiched between two metal electrodes. We propose a comprehensive model to simulate the evolution of the filament geometry under the effect of both surface diffusion caused by curvature gradient and electromechanical stress, and mass injection due to electrodeposition of cations. The model has been implemented in a C++ platform using a level-set approach based on a mixed finite element formulation, enriched by a mesh adaptation strategy to accurately and efficiently track the evolution of the filament shape. The numerical scheme is initially validated on various benchmark case studies. We then simulate the growth and self-dissolution of the filamentary geometry, incorporating an electrical model allowing a comparison with conventional cation-based memories. The simulations showcase filament formation under varying applied voltages and filament dissolution under different initial resistances.

Finite element approximation of stabilized mixed models in finite strain hyperelasticity involving displacements and stresses and/or pressure—An overview of alternatives

AbstractThis paper presents mixed finite element formulations to approximate the hyperelasticity problem using as unknowns the displacements and either stresses or pressure or both. These mixed formulations require either finite element spaces for the unknowns that satisfy the proper inf‐sup conditions to guarantee stability or to employ stabilized finite element formulations that provide freedom for the choice of the interpolating spaces. The latter approach is followed in this work, using the Variational Multiscale concept to derive these formulations. Regarding the tackling of the geometry, we consider both infinitesimal and finite strain problems, considering for the latter both an updated Lagrangian and a total Lagrangian description of the governing equations. The combination of the different geometrical descriptions and the mixed formulations employed provides a good number of alternatives that are all reviewed in this paper.

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.

Topology optimization of incompressible structures subject to fluid–structure interaction

In this work, an algorithm for topology optimization of incompressible structures is proposed, in both small and finite strain assumptions and in which the loads come from the interaction with a surrounding fluid. The algorithm considers a classical block-iterative scheme, in which the solid and the fluid mechanics problems are solved sequentially to simulate the interaction between them. Several stabilized mixed finite element formulations based on the Variational Multi-Scale approach are considered to be capable of tackling the incompressible limit for the numerical approximation of the solid. The fluid is considered as an incompressible Newtonian fluid flow which is combined with an Arbitrary-Lagrangian Eulerian formulation to account for the moving part of the domain. Several numerical examples are presented and discussed to assess the robustness of the proposed algorithm and its applicability to the topology optimization of incompressible elastic solids subjected to Newtonian incompressible fluid loads.

Dynamic large strain formulation for nematic liquid crystal elastomers

Liquid 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.

Analysis of aWeak Galerkin Mixed Formulation for Maxwell’s Equations

In this paper we introduce and analyse a mixed weak Galekin finite element method for the Maxwell equations in the primary electric field-Lagrange multiplier. Our weak Galerkin method is equipped with stable finite elements composed of habitual polynomials of degree k for the electric field and polynomials of degree k + 1 for the Lagrange multiplier. Optimal order error estimations for the proposed weak Galerkin mixed finite element formulation are demonstrated and are confirmed numerically on a two dimensional bounded domain.

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.