Mixed Finite Element Formulation Research Articles

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.

Penalty function method for geometrically nonlinear buckling analysis of imperfect truss with multi-freedom constraints based on mixed FEM

This paper is concerned with the approach to implementing the Penalty function method for imposing multi-freedom constraints in geometrically nonlinear analysis of imperfect trusses based on mixed finite element formulation. Using a finite element model based on displacement formulation, it is required to incorporate both the dependent boundary relations and initial length imperfection to the nonlinear master stiffness system of equations for solving the geometrically nonlinear problem of imperfect truss with multi-freedom constraints. For decreasing the mathematical complexion of the incorporating process, the author proposes a novel mixed finite truss element considering initial imperfection, used in building the model for solving the geometrically nonlinear problem of truss with multi-freedom constraints. The modified nonlinear stiffness equation is constructed by employing the penalty function method to convert a constrained problem into an unconstrained problem by extremizing the augmented energy function established based on the proposed mixed finite element formulation. For solving the nonlinear equilibrium equation of imperfect trusses with multi-freedom constraints, the incremental equilibrium equation is constructed, and the incremental-iterative algorithm for calculation is established utilizing the arc-length method, used for writing the calculation program for investigating geometrically nonlinear behavior of imperfect truss with multi-freedom constraints. The results of the numerical test show the influence of initial imperfection and choosing weight values on the equilibrium path of the truss.

AN ANISOTROPIC EXTENSION FOR A THERMOVISCOPLASTIC GTN DUCTILE DAMAGE MODEL

In this work, a thermoviscoplastic GTN ductile damage model is extended to account for anisotropic plastic behavior. The novel feature of the proposed model is the enrichment of a GTN model, with the Hill's yield criterion accounting for hardening, softening, strain-rate effect, plastic work heating, thermal diffusion, and void shearing mechanism. The anisotropy is introduced by changing the equivalent stress definition, which provides greater flexibility to the model. The resultant modifications in the yield criterion and in the damage evolution equations are described. Two numerical problems involving plates subjected to tension loading that leads to ductile damage and eventual fracture are presented and discussed in order to validate the present approach. A mixed finite element formulation is adopted together with the fully damaged zone technique to represent the ductile fracture. Results obtained through computational simulations highlight the effect of the anisotropic parameters on mechanical behavior in terms of the global stress-strain curve and the local evolution of plastic strains, porosity and temperature. We conclude that the proposed anisotropic model is simple to implement yet provides more flexibility in capturing complex ductile fracture processes.

Stress Analysis of Laminated HSDT Beams Considering Bending Extension Coupling

This study demonstrates a mixed finite element formulation procedure for the bending and stress analyses of laminated composite beams. The finite element method is based on the Hellinger-Reissner variational principle, while the beam assumptions are based on the Higher Order Shear Deformation Theory (HSDT). Reddy’s shear function is employed for the beam theory where the beam is discretized by two-noded linear elements. The displacements and stress resultants are obtained directly at the nodes according to the proposed mixed formulation. The validation of current study is performed by comparison and convergence analyzes for various lamination cases under different boundary conditions.

A mixed-finite element approach for full-field residual stress estimation

In this paper, a full-field residual stress identification technique has been presented via a mixed finite element approach from limited measured detail. Being an inverse problem, complete identification of residual stresses from incomplete measurement is always suffered by inherent ill-posedness. To control this issue, the identification problem has been formulated as an optimization problem via a Tikhonov type cost functional. The first part of this cost functional is associated with an energy norm of mismatch between predicted and measured residual stress. The second part represents the regularization term. A Lagrangian is then considered to satisfy the equilibrium constraint over the region of interest. The incorporation of Lagrangian with aforementioned cost functional provides a form which is very similar to the standard Hellinger-Reissner (HR) functional except the discrepancy term between the data. The minimization of this formulation leads to the solution of two-field mixed variational form associated with residual stress and Lagrangian variable. To show the efficacy of the proposed procedure a two-dimensional welded specimen is considered where the weld bead is assumed to be spatially varying in all direction for residual stress identification. Line scan details are also presented to examine the deviation between the actual and predicted residual stress in the heat affected zone and at the region away from the weld bead. The results indicate that the mixed finite element formulation has the capability to provide the complete information of residual stress despite of having very few information of measurement.

Improving the accuracy of discretisations of the vector transport equation on the lowest-order quadrilateral Raviart-Thomas finite elements

Within finite element models of fluids, vector-valued fields such as velocity or momentum variables are commonly discretised using the Raviart-Thomas elements. However, when using the lowest-order quadrilateral Raviart-Thomas elements, standard finite element discretisations of the vector transport equation typically have a low order of spatial accuracy. This paper describes two schemes that improve the accuracy of transporting such vector-valued fields on two-dimensional curved manifolds.The first scheme that is presented reconstructs the transported field in a higher-order function space, where the transport equation is then solved. The second scheme applies a mixed finite element formulation to the vector transport equation, simultaneously solving for the transported field and its vorticity. An approach to stabilising this mixed vector-vorticity formulation is presented that uses a Streamline Upwind Petrov-Galerkin (SUPG) method. These schemes are then demonstrated, along with their accuracy properties, through some numerical tests. Two new test cases are used to assess the transport of vector-valued fields on curved manifolds, solving the vector transport equation in isolation. The improvement of the schemes is also shown through two standard test cases for rotating shallow-water models.

A new mixed finite-element method for H2 elliptic problems

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed.

A mixed polygonal finite element formulation for nearly-incompressible finite elasticity

In this work, we present a mixed formulation based on the scaled boundary parameterization for the analysis of nearly-incompressible hyperelastic materials at finite strains. The formulation leads to a polygonal element with an arbitrary number of edges. The element domain is parameterized in both the boundary and scaling directions and necessitates the definition of a scaling center. Interpolation functions are introduced in both directions, which allow for an application to physically and geometrically nonlinear problems. This is an extension to the original scaled boundary finite element method (SBFEM), in which an analytical solution in the scaling direction is employed. Whereas the SBFEM approach is restricted to star-convex element geometries, the present approach allows for non-star-convex polygonal element shapes. A mixed displacement-pressure formulation is proposed to alleviate volumetric locking, which typically occurs in case of nearly-incompressible material behavior. Several numerical examples demonstrate that the proposed approach is applicable to arbitrary polygonal meshes, such as Voronoi meshes. The results demonstrate an excellent behavior in terms of computational efficiency compared to standard quadrilateral elements and to the virtual element method (VEM). The ability of the present formulation to alleviate volumetric locking is shown and it is demonstrated that a higher accuracy can be reached on Voronoi meshes compared to quadrilateral discretizations.

Objectivity in quasi-brittle structural failure via adaptive formulation and mesh refinement

In this work, the combined implementation of Adaptive Formulation Refinement (AFR) and Adaptive Mesh Refinement (AMR) strategies is proposed for the efficient and mesh objective evaluation of quasi-brittle fracture.A critical issue when computing localized structural failure is the spurious mesh bias dependence of the crack trajectories and collapse mechanisms that ensues when adopting the standard displacement-based finite element (FE) formulation. This problem is resolved by the introduction of the mixed strain/displacement FE formulation, which allows for local convergence, producing mesh objective results.On the one hand, the proposed AFR approach allows to adaptively switch between the standard and mixed FE formulations during the numerical simulation depending on the level of accuracy required in different regions of the domain. Within this framework, the more accurate mixed formulation is activated only in the areas where the crack onset and evolution take place, while the standard formulation is adopted for the rest of the structure.On the other hand, the adopted octree-based AMR strategy allows to adaptively refine only certain areas of the domain, also in function of the level of accuracy required. Using this methodology, it is possible to start the simulation with an initially relatively coarse mesh and adaptively introduce several levels of mesh refinement only in the regions of interest, where cracks appear and propagate. This approach guarantees the necessary mesh resolution to capture the fracture phenomenon accurately while further increasing the computational efficiency of the simulation. The subsequent adaptive coarsening of the FE mesh is also allowed.Structural failure and fracture propagation are reproduced via a local isotropic damage model. For this reason, a refinement/coarsening criterion defined in function of the equivalent effective stress is conveniently adopted with both AFR and AMR strategies.To assess the capabilities of the proposed method, a set of numerical simulations featuring benchmark problems and experiments is included. Results show the mesh objectivity and cost-efficiency of the approach combining AFR and AMR in fracture problems. The proposed model is able to reproduce experimental crack paths, collapse mechanisms and force–displacement curves with accuracy and without spurious mesh bias dependence.The paper includes a comparative assessment of the performance of the proposed method and the standard FE formulation exclusively, showing the superior performance of the AFR methodology and its mesh objectivity. Convergence of the computed results when introducing different levels of AMR is also verified. Computational efficiency of the AFR and AMR techniques is specifically assessed. The combination of AFR and AMR approaches allows to provide accurate and mesh objective results in quasi-brittle fracture problems with very significant savings in computational cost.

A new mixed finite element formulation for reorientation in liquid crystalline elastomers

Liquid crystalline elastomers (LCE) belong to the class of soft materials and are capable of large deformations induced by temperature changes and ultraviolet irradiation. These materials are under investigation as actuator materials in light-weight structures. In order to numerically simulate LCE materials by using a space–time finite element method, a continuum model is necessary which combines a thermo-viscoelastic material formulation with a dissipative reorientation process of mesogens. Mesogens are rigid and rod-shaped molecules linked with the polymer chains of the elastomer. The dissipative reorientation of the mesogens can be described by a combination of an independent field of orientation vectors with unit length and micropolar rotational degrees of freedom. The orientation vector field follow from a partial differential equation and possesses initial–boundary conditions. The unit length of the orientation vector field for any time is guaranteed by a local rotation with the micropolar rotational degrees of freedom. The dissipative reorientation process is introduced by a local time evolution equation with respect to the micropolar rotational degrees of freedom. This is analogous to thermo-viscoelasticity and can be variationally-based formulated. In this paper, we present this new variational-based reorientation finite element formulation in a dynamical framework. We arrive at an energy–momentum consistent mixed finite element formulation with a Galerkin-based space–time integration.

Dynamic Versus Quasi-Static Analysis of Crack Propagation in Soft Materials

Abstract Cracks usually propagate dynamically that makes them so dangerous. However, most crack simulations are based on quasi-static analyses because they are simpler than the dynamic ones. Is it correct to use quasi-static analyses instead of the dynamic ones? Will the quasi-static and dynamic simulations provide similar results? We try to answer these questions in the present work. We compare results of quasi-static and dynamic simulations of crack propagation in aneurysm material. We use the material-sink (MS) approach, which is based on the notion of the diffused bond breakage. The latter feature implies a local loss of material and, consequently, decrease of mass density, which, in its turn, means that both stiffness and inertia go down in the damaged zone. The cancellation of inertia is an important feature of the MS approach in contrast to more formal regularization theories as phase field, gradient damage, and other nonlocal formulations. The MS approach is implemented within commercial finite-element software abaqus. A reduced mixed finite-element formulation is adopted to circumvent the volumetric locking and an implicit staggered solution algorithm is developed via the user-defined element subroutine UEL. Considered examples show that the onset of crack instability under static loads is followed by the dynamic rather than quasi-static crack propagation. Moreover, dynamic and quasi-static simulations, generally, provide different results.

Adaptive stabilized mixed formulation for phase field fracture modeling of nearly incompressible finite elasticity

The recently proposed mixed displacement-pressure-phase field formulation characterized by damage-mitigating incompressibility constraints excels in fracture modeling of nearly incompressible hyperelastic materials. However, such formulations generally call for mixed finite element (FE) allocations that satisfy the inf-sup condition, denying the popular lower equal-order linear FE pairs, like linear polynomial (P1) interpolation for all fields. To overcome this drawback, we develop a stabilized mixed formulation (P1/P1S) by resorting to a pressure projection technique. The lure of this formulation lies in its simplicity and versatility, allowing low-order P1 interpolation for all field variables. With this trait in mind, an efficient adaptive meshing strategy is designed, thereby speeding up computation by more than 15 times. For better coping with the adaptive scenarios involving crack nucleation, we also propose a novel energy-based mesh refinement criterion. The numerical treatment of the stabilized mixed FE formulation is elaborated, as well as the core operations of the adaptive meshing technique. The accuracy, efficiency, and robustness of the proposed formulation are validated by several representative numerical examples. Remarkably, a mixed I/II fracture test thereof is in good agreement with the experiment and thus promises to be a new benchmark.

Robust A Posteriori Error Analysis for Rotation-Based Formulations of the Elasticity/Poroelasticity Coupling

We develop the a posteriori error analysis of three mixed finite element formulations for rotation-based equations in elasticity, poroelasticity, and interfacial elasticity-poroelasticity. The discretizations use $H^1$-conforming finite elements of degree $k+1$ for displacement and fluid pressure, and discontinuous piecewise polynomials of degree $k$ for rotation vector, total pressure, and elastic pressure. Residual-based estimators are constructed, and upper and lower bounds (up to data oscillations) for all global estimators are rigorously derived. The methods are all robust with respect to the model parameters (in particular, the Lamé constants); they are valid in 2D and 3D, and also for arbitrary polynomial degree $k\geq 0$. The error behavior predicted by the theoretical analysis is then demonstrated numerically on a set of computational examples including different geometries on which we perform adaptive mesh refinement guided by the a posteriori error estimators.

A mixed finite element formulation for elastoplasticity

AbstractThe present article, provides a novel mixed finite element formulation for elastoplasticity which is suitable for the discretization of thin‐walled structures using highly anisotropic volume elements. We present a thermodynamically consistent framework for the modeling of elastoplastic stress response, where dissipative effects are considered through a dissipation function instead of explicit flow rules. Along these lines, a mixed incremental principle, in which stresses are included as independent unknowns, is derived. We propose to employtangential‐displacement normal‐normal‐stress(TDNNS) elements for a Galerkin discretization of the underlying variational problem. These elements were originally developed for linear elasticity, where it could be shown that they do not suffer from shear locking if the element's aspect ratio deteriorates. Excellent computational performance and accuracy of the proposed method are demonstrated in several benchmark problems.

Primal and mixed finite element formulations for the relaxed micromorphic model

The classical Cauchy continuum theory is suitable to model highly homogeneous materials. However, many materials, such as porous media or metamaterials, exhibit a pronounced microstructure. As a result, the classical continuum theory cannot capture their mechanical behaviour without fully resolving the underlying microstructure. In terms of finite element computations, this can be done by modelling the entire body, including every interior cell. The relaxed micromorphic continuum offers an alternative method by instead enriching the kinematics of the mathematical model. The theory introduces a microdistortion field, encompassing nine extra degrees of freedom for each material point. The corresponding elastic energy functional contains the gradient of the displacement field, the microdistortion field and its Curl (the micro-dislocation). Therefore, the natural spaces of the fields are [H1]3 for the displacement and [H(curl)]3 for the microdistortion, leading to unusual finite element formulations. In this work we describe the construction of appropriate finite elements using Nédélec and Raviart–Thomas subspaces, encompassing solutions to the orientation problem and the discrete consistent coupling condition. Further, we explore the numerical behaviour of the relaxed micromorphic model for both a primal and a mixed formulation. The focus of our benchmarks lies in the influence of the characteristic length Lc and the correlation to the classical Cauchy continuum theory.

Mixed Finite Element Formulation for Navier-Stokes Equations for Magnetic Effects on Biomagnetic Fluid in a Rectangular Channel.

The article presents the mixed finite element formulation for examining the biomagnetic fluid dynamics as governed by the Navier–Stokes equation, coupled with energy and magnetic expressions. Both ferrohydrodynamics and magnetohydrodynamics describe the additional magnetic effects. For model discretization, the Galerkin weighted residual method was performed. Departing from a good agreement with existing findings, a biomagnetic flow (blood) in a straight rectangular conduit was then simulated in the presence of a spatially changing magnetic distribution. By virtue of negligible spatial variation influence from the magnetic field, the effects of Lorentz force were not presently considered. It was further found that the model accurately exhibits the formation and distribution of vortices, temperature, and skin friction located adjacent to and remotely from the source of magnetic load following a rise in the magnetic intensity.

A p‐adaptive, implicit‐explicit mixed finite element method for diffusion‐reaction problems

AbstractA new class of implicit‐explicit (IMEX) methods combined with a ‐adaptive mixed finite element formulation is proposed to simulate the diffusion of reacting species. Hierarchical polynomial functions are used to construct a conforming base for the flux vectors, and a non‐conforming base for the mass concentration of the species. The mixed formulation captures the distinct nonlinearities associated with the flux constitutive equations and the reaction terms. The IMEX method conveniently treats these two sources of nonlinearity implicitly and explicitly, respectively, within a single time‐stepping framework. A reliable a posteriori error estimate is proposed and analyzed. A ‐adaptive algorithm based on the proposed a posteriori error estimate is also constructed. The combination of the proposed residual‐based a posteriori error estimate and hierarchical finite element spaces allows for the formulation of an efficient ‐adaptive algorithm. A series of numerical examples demonstrate the performance of the approach for problems involving travelling waves, and possessing discontinuities and singularities. The flexibility of the formulation is illustrated via selected applications in pattern formation and electrophysiology.

A compatible mixed finite element method for large deformation analysis of two‐dimensional compressible solids in spatial configuration

AbstractIn this article, a new mixed finite element formulation is presented for the analysis of two‐dimensional compressible solids in finite strain regime. A three‐field Hu–Washizu functional, with displacement, displacement gradient and stress tensor considered as independent fields, is utilized to develop the formulation in spatial configuration. Certain constraints are imposed on displacement gradient and stress tensor so that they satisfy the required continuity conditions across the boundary of elements. From theoretical standpoint, simplex elements are best suited for the application of continuity constraints. The techniques that are proposed to implement the constraints facilitate their automatic imposition and, hence, they can be regarded as an important feature of the work. Since the exterior calculus provides the basis for the developments presented herein, the relevant topics are discussed within the context of the work. Various technical aspects of the formulation are described in detail. These aspects help to illuminate the mathematical formulation that might seem vague in the first place and, more importantly, they help to provide an efficient implementation for ensuing developments. The performance of the mixed finite element method is studied through benchmark numerical examples and it is compared with other similar elements. It is shown that the element has excellent convergence properties and it is numerically stable, especially for problems where classical first order elements demonstrate stiff or unstable behavior.