Mixed Finite Element Formulation Research Articles

Ductile damage modelling of heterogeneous materials using a two-scale computational approach

The paper presents a new two-scale methodology for modelling of damage in ductile heterogeneous materials employing homogenization approach over the two microstructural models. The first microstructural model comprises elastoplastic behaviour without softening, while only microstructural softening is homogenized in the second model. The scale transition of the state variables is based on the first-order computational homogenization scheme. The microstructural damage response is governed by the nonlocal variable embedded into the gradient elastoplastic formulation using the implicit localization approach. The averaging computation over the microstructural softening zones is driven by the damage variable and the local equivalent strain. Therein discretization is performed by using the mixed finite element formulation including the nonlocal equivalent plastic strain interpolation, while the standard displacement based finite elements, employing the simple constitutive relation with damage variable, are used at the macrolevel. The capability of modelling material damage responses at both the micro and macroscale is demonstrated by the benchmark examples.

A two-dimensional adaptive remeshing method for solving melting and solidification problems with convection

Melting and solidification problems in presence of natural convection are known to require high computational resources. Very fine meshes are essential to accurately determine the flow structure and interface position between the different phases. In this work, a time-dependent adaptive remeshing method based on an efficient error estimator is presented. The proposed method greatly reduces the number of mesh elements while maintaining and even enhancing the efficiency of the simulations. A variant of the enthalpy-porosity formulation is used where the different thermo-physical properties between the solid–liquid phases are easily taken into account. A second order mixed finite element formulation for both space and time is employed for solving the momentum and energy equations. The efficiency of the proposed methodology is established by comparing solutions on very fine meshes with those obtained on adapted meshes and with existing experimental and numerical results found in the literature.

Accuracy of a coupled mixed and Galerkin finite element approximation for poroelasticity

In this paper, we consider a combined mixed finite element and continuous Galerkin finite element formulation for a coupled flow and geomechanics model. We use the lowest order Raviart–Thomas finite elements for the spatial approximation of the flow variables and continuous piecewise linear finite elements for the deformation variable. This numerical approach appears to be a common choice in the existing reservoir engineering simulators. We focus on deriving error estimates in a discrete-in-time setting. Previous a priori error estimates described in the literature e.g. \[2], \[19], which are optimal, show first order convergence in space with respect to the $L^2$-norm for the pressure and for the average fluid velocity and also first order convergence in space with respect to the $H^1$-norm for the displacement. Here we prove one extra order of convergence for the displacement approximation with respect to the $L^2$-norm. We also demonstrate that, by including a post-processing step in the scheme, the order of convergence of the approximation of pressure can be improved. Even though this result is critical for deriving the $L^2$-norm error estimates for the approximation of the deformation variable, surprisingly the corresponding gain of one convergence order holds independently of including or not the post-processing step in the method.

Mixed beam formulation with cross-section warping for dynamic analysis of thin-walled structures

This paper presents the formulation of a three-dimensional beam finite element (FE) that accounts for cross-section warping and dynamic inertia effects. The model is the extension of an existing mixed formulation, originally developed for the static analysis of thin-walled beams, to the case of dynamic loading conditions.Four independent fields are considered to derive the element governing equations, i.e. material rigid displacements, strains and stresses and an additional displacement field, describing the out-of-plane warping displacement of the beam cross-sections. The latter is independently interpolated in the element volume by including additional degrees of freedom (DOF) to the nodal translations and rotations classically considered in beam formulations.To obtain a consistent form of the element mass matrix, the cross-section displacement shape functions are computed, relating the generalized cross-section displacement fields to the element nodal variables. In mixed FE formulations, these are not assigned a priori, as in displacement-based approaches, but are derived on the basis of material stiffness and element geometry, together with compatibility conditions. Thus, the Unit Load method is applied to deduce the expressions of the shape functions consistent with the force-based approach, assuming the simply-supported beam as reference element configuration.As opposed to the original FE model, the additional warping DOFs are not condensed-out with the definition of the element quantities but are treated as additional global unknowns. This permits a correct description of the inertia effects and ensures continuity of the warping displacement fields between adjacent FEs.Correlation studies are presented to validate the proposed model and investigate the effects of cross-section warping on the dynamic behavior of thin-walled structures. For selected specimens, the studies compare solutions obtained adopting the proposed beam element with those resulting from shell or brick FE models. Modal decompositions and time-history analyses are conducted, assuming both linear elastic and nonlinear constitutive behavior for the latter.

Incorporating variable viscosity in vorticity-based formulations for Brinkman equations

In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity, and pressure with non-constant viscosity. The analysis is performed by the classical Babuška–Brezzi theory, and we state that any inf–sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete space of arbitrary order for the vorticity. We establish optimal a priori error estimates, which are further confirmed through computational examples.

A novel functional and mixed finite element analysis of functionally graded micro-beams based on modified couple stress theory

Static bending, free vibration and buckling analyses of functionally graded (FG) Timoshenko micro-beams based on modified couple stress theory are presented under the framework of a new functional and mixed finite element method (MFEM). The new functional has been constituted for FG Timoshenko micro-beams through a scientific procedure based on the Gâteaux differential. Detailed derivation of mixed finite element formulation which allows C0 type continuous shape functions is given. The results of MFEM formulation for static bending, fundamental vibration frequency and critical buckling load of a straight FG micro-beam under various boundary conditions are obtained and compared with the existing literature. It is shown that, with simple C0 type shape functions it is possible to effectively capture variations of dimensionless fundamental frequencies and critical buckling load of moderately thick FG micro-beams for various boundary conditions and material parameters by completely avoiding the shear locking. Also FG tapered micro-beams are analyzed as a demonstration of versatility. Another distinctive feature of the new formulation is its capability of identification of the separate contributions of the coupled and the classical parts of the section moment. The new functional and the mixed finite element formulation are the original contributions of this study.

A fictitious domain approach for a mixed finite element method solving the two-phase Stokes problem with surface tension forces

In this article we study a mixed finite element formulation for solving the Stokes problem with general surface forces that induce a jump of the normal trace of the stress tensor, on an interface that splits the domain into two subdomains. Equality of velocities is assumed at the interface. The interface conditions are taken into account with multipliers. A suitable Lagrangian functional is introduced for deriving a weak formulation of the problem. A specificity of this work is the consideration of the interface with a fictitious domain approach. The latter is inspired by the XFEM approach in the sense that cut-off functions are used, but it is simpler to implement since no enrichment by singular functions is provided. In that context, getting convergence for the dual variables defined on the interface is non-trivial. For that purpose, an augmented Lagrangian technique stabilizes the convergence of the multipliers, which is important because their value would determine the dynamics of the interface in an unsteady framework. Theoretical analysis is provided, where we show that a discrete inf-sup condition, independent of the mesh size, is satisfied for the stabilized formulation. This guarantees optimal convergence rates, that we observe with numerical tests. The capacity of the method is demonstrated with robustness tests, and with an unsteady model tested for deformations of the interface that correspond to ellipsoidal shapes in dimension 2.

A conforming primal–dual mixed formulation for the 2D multiscale porous media flow problem

In this paper, a new primal–dual mixed finite-element method is introduced, aimed to model multiscale problems with several geometric subregions in the domain of interest. In each of these regions, porous media fluid flow takes place, but governed by physical parameters at a different scale; in addition, a fluid exchange through contact interfaces occurs between neighboring regions. The well-posedness of the primal–dual mixed finite-element formulation on bounded simply connected polygonal domains of the plane is presented. Next, the convergence of the discrete solution to the exact solution of the problem is discussed, together with the convergence rate analysis. Finally, the numerical examples illustrate the method’s capabilities to handle multiscale problems and interface discontinuities as well as experimental rates of convergence.

A Mixed Finite Element Method for Nearly Incompressible Multiple-Network Poroelasticity

In this paper, we present and analyze a new mixed finite element formulation of a general family of quasi-static multiple-network poroelasticity (MPET) equations. The MPET equations describe flow a...

On the use of local maximum entropy approximants for Cahn–Hilliard phase-field models in 2D domains and on surfaces

We apply a local maximum entropy (LME) approximation scheme to the fourth order phase-field model for the traditional Cahn–Hilliard theory. The discretization of the Cahn–Hilliard equation by Galerkin method requires at least C1-continuous basis functions. This requirement can be fulfilled by using LME shape functions which are C∞-continuous. In this case, the primal variational formulations of the fourth-order partial differential equation are well defined and integrable. Hence, there is no need to split the fourth-order partial differential equation into two second-order partial differential equations; this splitting scheme is a common practice in mixed finite element formulations with C0-continuous Lagrange shape functions. Furthermore, we use a general and simple numerical method such as statistical manifold learning technique that allows dealing with general point set surfaces avoiding a global parameterization, which can be applied to tackle surfaces of complex geometry and topology, and to solve Cahn–Hilliard equation on general surfaces. Finally, the flexibility and robustness of the presented methodology is demonstrated for several representative numerical examples.

A mixed extended finite element formulation for the simulation of cracks in nearly incompressible materials

AbstractA low order mixed extended finite element for the simulation of cracks in nearly incompressible materials is presented. The finite element is based on a classical Q1P0 formulation in 3D including enrichment functions and corresponding degrees of freedom to account for the discontinuity in the displacement field. In addition, in the mixed formulation the volumetric stress field is enriched as well. The additional degrees of freedom can be eliminated on element level. The superior convergence behaviour of the presented formulation especially for nearly incompressible material behaviour is demonstrated by means of examples from linear elastic fracture mechanics.

Multiscale methods for model order reduction of non-linear multiphase flow problems

Numerical simulations for flow and transport in subsurface porous media often prove computationally prohibitive due to property data availability at multiple spatial scales that can vary by orders of magnitude. A number of model order reduction approaches are available in the existing literature that alleviate this issue by approximating the solution at a coarse scale preserving fine scale features. We attempt to present a comparison between two such model order reduction techniques, namely (1) adaptive numerical homogenization and (2) generalized multiscale basis functions. We rely upon a non-linear, multi-phase, black-oil model formulation, commonly encountered in the oil and gas industry, as the basis for comparing the aforementioned two approaches. An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow, and transport problems. A numerical benchmark is setup using fine scale property information from the 10th SPE comparative project dataset for the purpose of comparing accuracies of these two schemes. An adaptive criterion is employed by both the schemes for local enrichment that allows us to preserve solution accuracy compared to the fine scale benchmark problem. The numerical results indicate that both schemes are able to adequately capture the fine scale features of the model problem at hand.

A mixed finite element formulation for compressible finite hyperelasticity with two fibre family reinforcement

A mixed finite element formulation for compressible finite hyperelasticity, possibly with strongly deformation dependent stiffening fibre reinforcement, is developed and verified. Reinforcement by two, mechanically equivalent, in general oblique, fibre families is considered. In the provided numerical experiments, residual based error estimation is essential for a simple h-adaptive mesh refinement strategy. It is shown to be effective to resolve stress concentrations, also in combination with stress oscillations (checker boarding) in the presence of strong anisotropy especially when mixed element constructs with enriched polynomial order are used. The novel mixed generalised (Lagrangian) metric based formulation is illustrated using a simple but fully coupled material model. The ground substance is the common so-called compressible neo-Hooke model and the standard reinforcement part is augmented by second order anisotropic invariants. The theoretical formulation admits approaching the inextensible limit seamlessly, by-passing the use of a separate decoupled material model. The Hu–Washizu implementation used here relies on a certain extensibility for elimination of the auxiliary variables, approximated in L2 at the element level. The condensation yields a purely H1-based displacement formulation at the global level.

Simulation of magnetised microstructure evolution based on a micromagnetics-inspired FE framework: application to magnetic shape memory behaviour

Microstructure evolution in magnetic materials is typically a non-local effect, in the sense that the behaviour at a material point depends on the magnetostatic energy stored within the demagnetisation field in the entire domain. To account for this, we propose a finite element framework in which the internal state variables parameterising the magnetic and crystallographic microstructure are treated as global fields, optimising a global potential. Contrary to conventional micromagnetics, however, the microscale is not spatially resolved and exchange energy terms are neglected in this approach. The influence of microstructure evolution is rather incorporated in an effective manner, which allows the computation of meso- and macroscale problems. This approach necessitates the development and implementation of novel mixed finite element formulations. It further requires the enforcement of inequality constraints at the global level. To handle the latter, we employ Fischer–Burmeister complementarity functions and introduce the associated Lagrange multipliers as additional nodal degrees-of-freedom. As a particular application of this general methodology, a recently established energy-relaxation-based model for magnetic shape memory behaviour is implemented and tested. Special cases—including ellipsoidal specimen geometries—are used to verify the magnetisation and field-induced strain responses obtained from finite element simulations by comparison to calculations based on the demagnetisation factor concept.

A reduced mixed finite-element formulation for modeling the viscoelastic response of electro-active polymers at finite deformation

In this work, a parameter identification procedure has been held for characterizing the widely used dielectric elastomer VHB. The calibration procedure has been performed using various experimental data found in previous works including uniaxial and multiaxial tests. Unlike the uniaxial tests, the multiaxial tests yield inhomogeneous deformation fields and, therefore, the finite-element method is adopted to obtain numerical solutions for the multiaxial tests. Here, a numerical scheme has been developed using the reduced mixed finite-element formulation, which eliminates the possible volumetric locking in electro-active polymers and enhances the computational efficiency as the static condensation is circumvented. The objective function, which calculates the discrepancy between the results obtained from the computational model and the measured experimental data, has been formulated taking into account the different types of experiments. The material parameters have been extracted when the objective function reached a minimum value, which is obtained by applying an iterative procedure using the fminsearch in Matlab. Within the extracted parameters and the developed finite-element formulation, number of simulations showing the efficiency of the computational model have been performed.

Explicit synchronous partitioned algorithms for interface problems based on Lagrange multipliers

Traditional explicit partitioned schemes exchange boundary conditions between subdomains and can be related to iterative solution methods for the coupled problem. As a result, these schemes may require multiple subdomain solves, acceleration techniques, or optimized transmission conditions to achieve sufficient accuracy and/or stability. We present a new synchronous partitioned method derived from a well-posed mixed finite element formulation of the coupled problem. We transform the resulting Differential Algebraic Equation (DAE) to a Hessenberg index-1 form in which the algebraic equation defines the Lagrange multiplier as an implicit function of the states. Using this fact we eliminate the multiplier and reduce the DAE to a system of explicit ODEs for the states. Explicit time integration both discretizes this system in time and decouples its equations. As a result, the temporal accuracy and stability of our formulation are governed solely by the accuracy and stability of the explicit scheme employed and are not subject to additional stability considerations as in traditional partitioned schemes. We establish sufficient conditions for the formulation to be well-posed and prove that classical mortar finite elements on the interface are a stable choice for the Lagrange multiplier. We show that in this case the condition number of the Schur complement involved in the elimination of the multiplier is bounded by a constant. The paper concludes with numerical examples illustrating the approach for two different interface problems.

Micro–macro constitutive modeling and finite element analytical-based formulations for fibrous materials: A multiscale structural approach for crimped fibers

Materials with crimped fibers have special properties that can be effectively explored only when using a micro–macro perspective. In this framework, a novel constitutive model based on a multiscale structural rationale is introduced. Material micromechanics, depending on fiber straightening mechanisms, is described introducing a beam model which drives material model response. This rationale leads to a quasi-analytical formulation, coupling the advantages of purely-analytical and computational approaches. The proposed model is also proven to be polyconvex.Furthermore, a finite-element formulation is developed, enriched by a quasi-analytical core associated with the multiscale constitutive formulation. Different solution strategies are tested in order to optimize the numerical performances in terms of accuracy, robustness and cost. Moreover, a mixed finite element formulation based on a simplified-kinematics-for-anisotropy (SKA) is introduced. For the tested boundary value problems, the SKA-element is an optimal choice in terms of displacement and fiber stress convergence behavior, especially for coarse meshes.

The static and dynamic analyses of warping included composite exact conical helix by mixed FEM

The objective of this study is to investigate the combined influence of two important topics on the precision of static and dynamic analyses of non-circular composite helical bars, namely, exact helix geometry and the warping effect. Sometimes a conical helix over logarithmic spiral planar curve is formed by a degenerated plane curve. The most important goal of this study is to determine the range of the geometric parameters in which the degenerated plane curve lacks the precision necessary for the structural analysis of the conical helix compare to using an exact logarithmic spiral function. Another important topic on the precision of the results is the warping of non-circular composite sections. In this study, first, a parametric analysis is carried out in order to determine the maximum influence of warping on the torsional rigidity of non-circular sandwich/composite cross-sections. Then, some benchmark examples are employed to consider the combined influence of the two topics mentioned above. The analysis is performed over a curved mixed finite element formulation based on Timoshenko beam theory by considering the shear influence, rotary inertia and the warping included torsional rigidity. The curved element consists of two nodes and 24 degrees of freedom in total.

Morpho-elastic model of the tortuous tumour vessels

Solid tumours have the ability to assemble their own vascular network for optimizing their access to the vital nutrients. These new capillaries are morphologically different from normal physiological vessels. In particular, they have a much higher spatial tortuosity forcing an impaired flow within the peritumoral area. This is a major obstacle for the efficient delivery of antitumoral drugs. This work proposes a morpho-elastic model of the tumour vessels. A tumour capillary is considered as a growing hyperelastic tube that is spatially constrained by a linear elastic environment, representing the interstitial matter. We assume that the capillary is an incompressible neo-Hookean material, whose growth is modelled using a multiplicative decomposition of the deformation gradient. We study the morphological stability of the capillary by means of the method of incremental deformations superposed on finite strains, solving the corresponding incremental problem using the Stroh formulation and the impedance matrix method. The incompatible axial growth of the straight capillary is found to control the onset of a bifurcation towards a tortuous shape. The post-buckling morphology is studied using a mixed finite element formulation in the fully nonlinear regime. The proposed model highlights how the geometrical and the elastic properties of the capillary and the surrounding medium concur to trigger the loss of marginal stability of the straight capillary and the nonlinear development of its spatial tortuosity.

A mixed edge-based smoothed solid-shell finite element method (MES-FEM) for laminated shell structures

We in this study present a mixed solid-shell finite element formulation for laminated shell analysis. With the commonly adopted truncations applied to the strain components and the Jacobian determinant, a generalized stiffness matrix is established. Discrete shear gap (DSG) technique together with assumed natural strain (ANS) method is employed in order to alleviate shear locking and trapezoidal locking. Stiffness matrix formulation is analytically calculated based on the edge-based cell backgrounds, which result in simple and efficient computation. The proposed formulation does not require using cross-diagonal meshes, which can be an obstacle for several existing three-node degenerated shell elements. This aspect makes the present method much more appealing than others through examples tested. Numerical validations show that the present method performs well for both structured and unstructured triangular meshes.