Mixed Finite Element Formulation Research Articles

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.

Numerical Investigation of High-Order Solid Finite Elements for Anisotropic Finite Strain Problems

In this paper, a hierarchic high-order three-dimensional finite element formulation is studied for hyperelastic and anisotropic elastoplastic problems at finite strains. The element formulation allows for anisotropic ansatz spaces supporting efficient discretizations of beam-, plate-, and shell-like structures. Several benchmark examples are investigated and the results of the high-order formulation are compared to analytic solutions and different mixed finite element formulations. Special emphasis will be placed on locking effects, robustness with respect to high aspect ratios and element distortion as well as anisotropies related to the material model. Furthermore, the interplay between the chosen ansatz space for the displacement field and mapping function in the context of geometrically nonlinear problems are studied.

A gradient-enhanced formulation for thermoviscoplastic metals accounting for ductile damage

In this work, we extend a gradient-enhancement formulation for ductile damage analysis to thermoviscoplastic metals. The original contribution of the study is the nonlocal extension of our previously developed thermoviscoplastic Gurson's model accounting for plastic work heat generation, thermal diffusion and void shearing mechanism. The gradient enrichment is applied to the porosity evolution equation by adding a nonlocal length scale parameter together with the Laplacian of the void volume fraction, following a void diffusion law. A mixed finite element formulation is used to discretize the system with an implicit time integration and a monolithic solution of the linearized system at every time step. Three ductile fracture problems involving plates under plane-strain conditions and tensile loading are simulated, in order to compare the local and nonlocal formulations. Results show that the nonlocal term delays the damage evolution and somewhat alleviates mesh sensitivity in terms of the stress-strain response and the porosity distribution along specific lines. Nevertheless, the most notable effect of the nonlocal formulation is the regularization (or smoothing) of the porosity field, reducing its gradients and mesh dependence, and leading to more consistent porosity contours. The choice of the nonlocal parameter is found to be dependent of the strain rate employed. • A gradient-enhanced thermoviscoplastic ductile damage formulation is proposed. • Coupling of Gurson's porous plasticity, void shearing mechanism, thermoviscoplasticity and thermal diffusion. • Three structural problems involving plates under plain strain assumption are numerically analyzed in detail. • The model is suitable to alleviate mesh dependence and to smooth the porosity field.

Soil-structure interaction with time-dependent behaviour of both concrete and soil

As the consideration of soil-structure interaction is increasingly being incorporated into design practice, there is a need for solutions that consider the most relevant aspects of the behaviour of both the concrete and the soil, some of which are time-dependent. The present work introduces a model that suits both the structure and the foundation – including the supporting soil – allowing the coupled analysis of hardening on cure, creep, shrinkage and cracking of concrete, and consolidation of soil. Both the structure and the foundation are modeled as one-dimensional finite elements. The time-dependent behaviour of the concrete and soil is modeled using Kelvin chains. For the structural elements, a mixed finite element formulation is used. Validation tests were conducted in the proposed modeling, comparing its results with available experimental and analytical data. The study of a reinforced concrete continuous beam supported by foundations on consolidating clay considering the time-dependent behaviour of the materials showed considerable changes in the effects of the soil-structure interaction.

Port-Hamiltonian FE models for filaments

In this article, we present the port-Hamiltonian representation, the structure preserving discretization and the resulting finite-dimensional state space model of one-dimensional filaments based on a mixed finite element formulation. Due to the fact that the equations of motion of a filamentous body are based on the theory of geometrically nonlinear mechanical systems, the port-Hamiltonian formulation is expressed by means of its co-energy (effort) variables. The resulting port-Hamiltonian state space model features a quadratic Hamiltonian and the nonlinearity is reflected in the state dependence of its interconnection matrix. Numerical experiments generated with FEniCS illustrate the properties of the resulting finite element models.

A new first-order mixed beam element for static bending analysis of functionally graded graphene oxide powder-reinforced composite beams

In this paper, a new first-order mixed beam element is established and applied to investigate the static bending of the functionally graded graphene oxide powder-reinforced composite beams. The proposed beam element is developed via a mixed finite element formulation based on first-order shear deformation theory. The new element consists of two nodes and three degrees of freedom per node. In addition, the present beam element uses linear shape functions. The proposed beam element is free of shear locking without using selective or reduced integration. The comparison study demonstrates that the present beam element can predict exact solutions with very few elements, so, the computation cost is reduced. Then the proposed beam element is used to analyze the static bending of functionally graded graphene oxide powder-reinforced composite beams. A comprehensive parameter study is carried out to demonstrate the effects of some parameters such as the boundary conditions, graphene oxide powder weight fraction, graphene distribution patterns, graphene oxide powder dimensions, and the slender ratio of the beams.

An energy–momentum couple stress formula for variational-based macroscopic modelings of roving-matrix composites in dynamics

In dynamical systems, fiber-reinforced materials with fiber rovings in a cured matrix material, woven or manufactured by the technology of tailored fiber placement (TFP), for instance, have gained great significance for engineers. Therefore, many finite element simulations are currently in progress during the design of composite parts. But, standard finite element formulations do not take into account dimensions and the direction-dependent flexure and twist stiffness of the rovings. In order to avoid computationally costly mesoscopic finite element models with a discretization of roving-matrix interior boundary interfaces, a macroscopic material model can be derived by introducing micropolar rotation degrees of freedom. Micropolar rotation degrees of freedom allow for a separate constitutive description of the direction-dependent stiffness and inertia of the rovings in the cured matrix material. This continuum model represents an extended continuum formulation with an extended set of balance laws. In its computational setting, the discrete preservation of the complete set of balance laws can be realized by a mixed principle of virtual power in connection with energy–momentum tension stress and couple stress formulas. The guaranteed algorithmic energy–momentum consistency then allows an one-off virtual parameter identification of the macroscopic material constants with the mesoscopic model based on the error of the balance functions. Therefore, a mixed finite element formulation with independent micropolar rotation degrees of freedom is recently published, which are able to simulate dynamical problems by taking into account flexure and twist of the rovings. In this previously published paper, the constitutive laws are at most quadratic such that energy–momentum time integrations are possible without a special couple stress approximation. If a nonlinear strain energy function for flexure and twist of the rovings is considered, a new special couple stress approximation is necessary. Therefore, the present paper completes this mixed finite element formulation with a new couple stress approximation and shows its application by means of that numerical experiments with unidirectional rovings, which are designed for the virtual parameter identification. These fully dynamical examples with large translations, rotations and deformations demonstrate the flexure and twist stiffness of the rovings with roving curvature–twist strain energy functions of Kauderer-type by parameter studies. The complete set of discrete balance laws are satisfied independent of boundary conditions in order to allow the upcoming virtual parameter identification.

A mixed finite element formulation for energy‐momentum time integrations of composites with fiber bending stiffness

AbstractWe aim at dynamic simulations of 3D‐fiber‐reinforced materials in light‐weight structures, which are reinforced by fiber rovings comprised of thousands of filaments. Each roving possesses a separate bending stiffness, which has a large influence on the material behaviour on the macro scale. For the description of this material behavior, we extend a hyperelastic anisotropic Cauchy continuum formulation, which models only the macro‐scopic behaviour of a reinforcement by filaments, by a higher‐order gradient of the deformation mapping as an independent field. The new mixed finite element formulation is based on well‐known mixed finite elements for anisotropic polyconvex material formulations, but include new additional mixed fields concernd with the fiber bending stiffness. As we aim at dynamic long‐term simulations, we apply higher‐order accurate time integrators. Here, we include the new mixed finite element formulation in a variational‐based time finite element method. As numerical examples serve a simple cantilever beam. Here, we first analyze the influence of the fiber bending stiffness.

Modeling of fiber‐bending stiffness in fiber‐reinforced composites with a dynamic mixed finite element method based on the principle of virtual power

AbstractInvestigating the bending stiffness of fibers in fiber‐reinforced composites for rotor‐dynamical systems which are subjected to dynamical loads are essential in the development of system design. The proposed numerical modeling approach of fiber‐reinforced composites uses a multi‐field mixed finite element formulation based on a dynamic variational approach to perform long‐term dynamic simulations with CPU‐time efficient and increased accurate numerical solutions. To model the bending stiffness of fibers accurately, we extend a Cauchy continuum with higher‐order gradients of the deformation mapping as an independent field in the functional formulation. This extended continuum also takes into account the higher‐order energy contributions including the fiber curvature along with popular proven approaches that efficiently avoid the numerical locking effect of the fibers. The effect of the proposed approach is demonstrated through transient dynamic analysis on a cantilever beam with a hyperelastic, transversely isotropic, polyconvex material behavior. The beam is subjected to bending load with a strong dependence of the overall stiffness on the fiber orientation. The material and conservation properties are analyzed.

A Mixed Acoustic Fluid Finite Element in Boundary Representation for Fluid‐Structure Interaction

AbstractThis contribution focuses on a mixed finite element formulation for an acoustic fluid in boundary representation to model fluid‐structure interaction problems. In fact, acoustic fluids are characterized by their lack of dynamic viscosity and undergo displacements through a pure pressure wave propagation. Since many fluids, for instance water, show nearly‐incompressible behavior, volumetric locking occurs in the classical finite element formulation. This drawback is herein evaded by a two‐field, or mixed, finite element formulation. To facilitate the contact condition at the fluid‐structure interface, the acoustic fluid is expressed by structural field variables, which are nodal displacements and internal pressures. Both the displacements and internal pressures are parameterized by a Scaled‐Boundary (SB) approach, in which an interpolation in the scaling direction is employed. From this parametrization, the element mass and stiffness matrices of the fluid can be formulated. Due to the special parametrization of the element boundary, pressure continuity is required between scaled sections in the element interior, although pressure continuity may be interrupted between neighboring Scaled‐Boundary elements. Since the lack of shear viscosity leads to unwanted zero‐energy modes concerned with the fluid's vorticity, this vorticity is penalized on a sectional level. As a result, these modes do not show up in the frequency spectrum of the dynamic eigenvalue problem any longer. The natural frequencies and mode shapes in this spectrum deliver accurate results, even for very coarse meshes.

Two mixed finite element formulations for the weak imposition of the Neumann boundary conditions for the Darcy flow

Abstract We propose two different discrete formulations for the weak imposition of the Neumann boundary conditions of the Darcy flow. The Raviart–Thomas mixed finite element on both triangular and quadrilateral meshes is considered for both methods. One is a consistent discretization depending on a weighting parameter scaling as 𝒪(h −1), while the other is a penalty-type formulation obtained as the discretization of a perturbation of the original problem and relies on a parameter scaling as 𝒪(h −k−1), k being the order of the Raviart–Thomas space. We rigorously prove that both methods are stable and result in optimal convergent numerical schemes with respect to appropriate mesh-dependent norms, although the chosen norms do not scale as the usual L 2-norm. However, we are still able to recover the optimal a priori L 2-error estimates for the velocity field, respectively, for high-order and the lowest-order Raviart–Thomas discretizations, for the first and second numerical schemes. Finally, some numerical examples validating the theory are exhibited.

Dual length scale non-local model to represent damage and transport in porous media

Recent experiments and physical evidence show that fractured porous media feature cracks and fluid capillary networks at various scales. We present a multi-physics macro-scale model that can distinguish between the mechanics and transport interactions. The porous media is represented by a poroelastic domain incorporating non-local damage and non-local transport. The evolution of each of these processes is governed by a unique length scale and driving force, which allows for better flexibility in modeling hydraulic-deformation network systems. For consistency the governing equations of the non-local multi-physics problem are derived from thermodynamics principles. Hence, a four-field (u,P,σ̃eq,D,κ̃) mixed finite element formulation is developed. The non-linear system of equations is linearized and solved using Newton’s method and a backward Euler scheme is used to evolve the system in time, for which a consistent Jacobian matrix and residual vector are derived analytically. Two benchmark examples are investigated: hydraulic fracturing of rocks and soil consolidation. The numerical examples show the viability of this model, and how the variation of the two length scales and damage parameters can be used to describe different physical phenomena.

Energy based fracture initiation criterion for strain-crystallizing rubber-like materials with pre-existing cracks

Fracture prediction is indispensable for polymers, like rubbers, which have a broad range of applications mainly due to their high extensibility. The phenomenon known as strain-induced crystallization further contributes to the fracture toughness of certain rubbers. In this study, a criterion based on internal bond energy, incorporating the effects of crystallization, is proposed to predict fracture initiation in rubber-like materials with pre-existing cracks. First, a multi-scale mechanical model is developed for characterizing the behavior of rubber when subjected to both uniaxial and biaxial deformation states. At the microscale, both the amorphous and crystalline chain segments are modeled as elastic in order to consider the energy contribution by the molecular bond distortions. This internal energy is considered along with the entropic and crystalline free energy for each chain. In the chain model, the effects of loading condition and the relative orientation of a chain on its crystallinity are taken into account. At the macroscopic scale, an existing crystallinity distribution function is adapted and a mixed finite element formulation with an augmented Lagrangian multiplier is utilized to impose the incompressibility constraint. A non-affine maximal advance path constraint based homogenization model is utilized for bridging the two scales. Its potential to account for anisotropy in the stretched network compels the model to be preferable due to its physical significance, for the purpose of fracture modeling. The rigidity of the crystallites is accounted for by proposing a crystallite distortion energy in addition to the critical bond dissociation energy, for fracture initiation to occur. The model is validated by comparison with existing experimental results for both crystallizing and non-crystallizing rubbers. In addition to its potential to predict the material behavior when subjected to uniaxial and biaxial loading, the capability of the model to quantitatively estimate the effect of crystallization on fracture initiation is also verified.

Mixed finite element formulation for bending of laminated beams using the refined zigzag theory

This study presents a mixed finite element formulation for the stress analysis of laminated composite beams based on the refined zigzag theory. The Hellinger–Reissner variational principle is employed to obtain the first variation of the functional that is expressed in terms of displacements and stress resultants. Due to C0 continuity requirements of the formulation, linear shape functions are adopted to discretize the straight beam domain with two-noded finite elements. The proposed formulation is shear locking free from nature since it introduces displacement and stress resultant terms as independent field variables. A monolithic solution of the global finite element equations is preferred, hence the stress resultants are directly obtained from the solution of these equations. The in-plane strain measures of the beam are obtained directly at the nodes over the compliance matrix and stress resultants by avoiding error-prone spatial derivatives. Following, transverse shear stresses are calculated from the equilibrium equations at the post-processing level. This simple but effective finite element formulation is first verified and tested for convergence behavior. The robustness of the approach is shown through some examples and its accuracy in predicting the displacement and stress components is revealed.