Mixed Finite Element Formulation Research Articles

Hybrid-Mixed Low-Order Finite Elements for Geometrically Exact Shell Models: Overview and Comparison

We critically review and compare fifteen mixed and mixed-hybrid nonlinear shell finite element formulations with 4 nodes in order to identify those that are closest to the “optimal” one. We consider formulations that are based either on Assumed Natural Strain concept, Enhanced Assumed Strain concept, Hellinger–Reissner functional or Hu–Washizu functional, and those that effectively combine several of the mentioned approaches. Most of the formulations are state-of-the-art, but some are also presented for the first time. We show that the flexibility of mixed-hybrid formulations requires careful consideration of theoretical and numerical aspects in the process of design of a high-performance element. We make extensive numerical experiments in order to assess convergence properties, mesh distortion sensitivity, and computational speed (that is associated with the ability to achieve large steps during the solution search) of the considered formulations. We show that the application of the idea to combine mixed-hybrid formulations of the Hellinger–Reissner and Hu–Washizu type with the Assumed Natural Strain and/or the Enhanced Assumed Strain concepts generates the formulations that perform excellent. We also identify the formulations that are currently closest to the “optimal” one.

Misorientation and grain boundary orientation dependent grain boundary response in polycrystalline plasticity

This paper studies the evolution of intergranular localization and stress concentration in three dimensional micron sized specimens through the Gurtin grain boundary model (J Mech Phys Solids 56:640–662, 2008) incorporated into a three dimensional higher-order strain gradient crystal plasticity framework (Yalcinkaya et al. in Int J Solids Struct 49:2625–2636, 2012). The study addresses continuum scale dislocation-grain boundary interactions where the effect of crystal orientation mismatch and grain boundary orientation are taken into account through the grain boundary model in polycrystalline metallic specimens. Due to the higher-order nature of the model, a mixed finite element formulation is used to discretize the problem in which both displacements and plastic slips are considered as primary variables. For the treatment of grain boundaries within the solution algorithm, an interface element is formulated and implemented together with the bulk plasticity model. The capabilities of the framework is demonstrated through 3D polycrystalline examples considering grain boundary conditions, grain boundary strength, the orientation distribution and the specimen size. A detailed grain boundary condition and stress concentration analysis is presented. The advantages and the disadvantages of the model is discussed in detail through numerical examples.

Mixed finite element formulation for the nonlinear analysis of buckling behavior of truss under effects of temperature variation

This paper proposes a novel approach to consider the effects of temperature variability in the nonlinear finite element analysis of buckling behavior of truss system subjected to initial mechanical load. Generally, using displacement-based finite formulation the procedure for solving geometrically nonlinear problem of the truss system subjected to temperature load is required to implement the temperature deformation constraint depending on the incremental element length to the master stiffness equation. For escaping the mathematical difficulties of treating the additional constraint of temperature deformation this research proposes to formulate this problem based on mixed finite element formulation. The balanced mixed equation of truss considering temperature loading effect is constructed based on the principle virtual work with establishing the mixed matrix of truss element, which consists the element temperature deformation, considering large displacement. For investigating the effects of temperature variation to the nonlinear buckling behavior of truss this research develops the incremental-iterative algorithm for solving the nonlinear equation with incremental displacement and temperature by using Newton-Raphson technique. Based on proposed algorithm the calculation program is written for numerical investing the effects of temperature variation the buckling behavior of plan truss.

A robust and efficient iterative strategy for nonlinear analysis of structures subjected to buckling

A robust and efficient iterative strategy for nonlinear analysis of structures subjected to buckling

Mixed Finite Element Method for Geometrically Nonlinear Buckling Analysis of Truss with Member Length Imperfection

This paper focuses on the numerical method for the geometrically nonlinear buckling analysis of truss with initial member length imperfection. The solution of nonlinear buckling problem of truss with imperfection using a displacement-based finite element is dependent on the imperfection implemented. Generally, the operation of incorporating the initial imperfection to the master stiffness equation develops by master-slave elimination method, penalty augmentation method or Lagrange multiplier adjunction methods. Obviously, the initial imperfection considerably increases the difficulty in finite element formulation nonlinear buckling problem. This research proposes a novel approach to formulate the nonlinear buckling problem of truss with imperfection using mixed finite element method. The mixed balanced equation of truss is formulated using the principle of stationary potential energy. The paper presents novel mixed finite truss element, including initial member length imperfection, considering large displacement. Using the arc length technique, the research develops a new incremental-iterative algorithm for solving the nonlinear buckling problem of truss with initial imperfection in different cases of model formulation, including displacement-based finite element and mixed finite element formulation. The numerical test is presented to investigate the equilibrium path for plan truss with initial member length imperfection. The calculation results in solving problem formulated in both displacement and mixed finite model are converged showing the efficiency and reliability of the proposed method.

Geometrically nonlinear buckling analysis of truss under mechanical and thermal load based on mixed finite element formulation

This paper is concerned with the numerical method for the geometrically nonlinear buckling analysis of truss under mechanical load and thermal load due to constant temperature change. The solution of a geometrically nonlinear problem of the system under thermal load based on a displacement-based finite formulation requires the implementation of thermal deformation constraint depending on the incremental element length. The additional constraint of thermal expansion considerably increases the difficulty in finite element formulation and solving method of the nonlinear buckling problem. This research proposes a novel approach to formulate the nonlinear buckling problem of truss under thermal load based on mixed finite formulation. The mixed balanced equation of thermal loading truss considering large displacement is formulated using the principle of virtual work. Based on the formulated equation, the research develops an incremental-iterative algorithm and calculation program for solving the nonlinear buckling problem of thermal loading truss using the arc length technique. The numerical test is presented to investigate the effect of thermal load to buckling behaviour of plan truss.

Mixed finite element formulation based on the discontinuous stress approximation

The paper is devoted to the development of the mixed form of finite element method for the calculation of constructions and structures. We consider the approach based on the approximation of discontinuous stress fields and the use of the penalty function method to satisfy the equilibrium equations. It is shown that the continuity of both normal and tangential stresses only on the adjacent sides of the finite elements contributes to the expansion of the class of statically admissible stress fields. At the same time, the consistent approximation is provided, both of the main part of the functional of additional energy, and its penalty part. Using the given approach, solutions of some problems were obtained and analyzed, in particular, the calculation of the cantilever beam-wall. The results obtained were compared with the decisions on the FEM in displacements and stresses, as well as with the exact solutions. It is shown that solutions based on the approximation of discontinuous stress fields in mixed form FEM converge to exact values from below and are more flexible in comparison with the solutions obtained by FEM in displacements. In addition, the accuracy of determining stresses is higher here.

Hierarchical basis in $H^{div}$ space for a mixed finite element formulation of the Darcy problem

Hierarchical basis in $H^{div}$ space for a mixed finite element formulation of the Darcy problem

Forced vibration analysis of composite-geometrically exact elliptical cone helices via mixed FEM

In this pioneering study, the cross-sectional warping included transient response and normal/shear stress components of composite elliptical and elliptical cone helices over exact axis geometry are investigated using a mixed FEM. The transient analysis is performed using the Newmark time integration algorithm with or without the amplitude decay factor. The constitutive equations of composite curved rods are derived from three-dimensional elasticity theory. A displacement-type finite element formulation computing the warping-included torsional rigidity is incorporated with the mixed finite element formulation. The curvatures and displacement-type finite elements are used to estimate the normal and shear stress distributions on the respective cross-sections. The maximum normal/shear stresses of a composite straight beam are compared with the literature. An excellent agreement is obtained for the results of an exact elliptical cone helix under dynamic loads compared to the results of 3D solid finite elements. During the implementation of the time integration scheme, the first and second time derivatives of forces and moments are preserved, and their time histories are discussed. Finally, the influences of helix geometry, lamination, and the ratios of material constants on the transient response besides the stresses are investigated. All the numerical examples in this paper are original for the literature.

Effect of elastocapillarity on the swelling kinetics of hydrogels

The swelling behavior of hydrogels, involving coupled diffusion and large deformations, makes them ideal for biomedical applications such as micro- and nano-scale drug delivery systems. Understanding the transient swelling or drying behavior of hydrogels at relevant length-scales will provide insight for the development of specialized and controlled drug release. At sub-millimeter length-scales, surface stresses have been shown to significantly influence material behavior for soft polymers and hydrogels, but very little is known about the influence of surface stresses on the swelling kinetics of hydrogels. In this paper, we present a non-linear theory and mixed finite element formulation that takes into account mass transport, large deformations, and elastocapillary effects for hydrogels. Focusing on hydrogel micro-spheres, we provide a comparison of swelling kinetics between the presented non-linear theory and an analytical solution using linear poroelasticity that incorporates surface stresses. Our results demonstrate that when the surface free energy is constant per unit current area (fluid-like) and the elastocapillary length-scale is on the order of the size of the hydrogel micro-sphere, the transient response equilibrates approximately an order of magnitude faster in time compared to the case without surface effects irrespective of swelling or drying. This difference in equilibration time suggests the interplay between competing processes of solvent diffusion, large deformations, and surface effects. Furthermore, we demonstrate that a Neo-Hookean type surface free energy can result in an even faster equilibration as compared to a fluid-like surface free energy. Lastly, our finite element implementation predicts the transient response of complex shapes and constrained structures.

Efficient solvers for hybridized three-field mixed finite element coupled poromechanics

We consider a mixed hybrid finite element formulation for coupled poromechanics. A stabilization strategy based on a macro-element approach is advanced to eliminate the spurious pressure modes appearing in undrained/incompressible conditions. The efficient solution of the stabilized mixed hybrid block system is addressed by developing a class of block triangular preconditioners based on a Schur-complement approximation strategy. Robustness, computational efficiency and scalability of the proposed approach are theoretically discussed and tested using challenging benchmark problems on massively parallel architectures.

A mixed extended finite element for the simulation of cracks and heterogeneities in nearly incompressible materials and metal plasticity

This paper describes a mixed extended finite element (XFEM) formulation that can handle nearly incompressible material behaviour for fracture mechanics problems as well as for heterogeneous materials under small and finite elastic and elastoplastic deformations without showing locking behaviour. Typical applications are the simulation of cracks and their propagation in rubber materials, microstructure simulations of rubber and simulations of heterogeneities in J2 and crystal plasticity. The developed finite element technique is an extension of the well established mixed Q1P0 formulation. Herein, enrichments as they are commonly used within the XFEM are considered to capture discontinuities within the displacement and/or strain field. The volumetric stress is discontinuous across a crack face or an interface between two materials. For that reason the piecewise constant ansatz used in the Q1P0 formulation for the volumetric stress needs to be enriched with suitable enrichment functions as well. Therefore, the developed mixed extended finite element formulation is named XQ1XP0. With several examples it is shown that the new formulation does not show locking effects even for almost incompressible materials and for significant plastic deformations and that it shows superior convergence behaviour compared to a standard formulation.

A phase field model for elastic-gradient-plastic solids undergoing hydrogen embrittlement

We present a gradient-based theoretical framework for predicting hydrogen assisted fracture in elastic-plastic solids. The novelty of the model lies in the combination of: (i) stress-assisted diffusion of solute species, (ii) strain gradient plasticity, and (iii) a hydrogen-sensitive phase field fracture formulation, inspired by first principles calculations. The theoretical model is numerically implemented using a mixed finite element formulation and several boundary value problems are addressed to gain physical insight and showcase model predictions. The results reveal the critical role of plastic strain gradients in rationalising decohesion-based arguments and capturing the transition to brittle fracture observed in hydrogen-rich environments. Large crack tip stresses are predicted, which in turn raise the hydrogen concentration and reduce the fracture energy. The computation of the steady state fracture toughness as a function of the cohesive strength shows that cleavage fracture can be predicted in otherwise ductile metals using sensible values for the material parameters and the hydrogen concentration. In addition, we compute crack growth resistance curves in a wide variety of scenarios and demonstrate that the model can appropriately capture the sensitivity to: the plastic length scales, the fracture length scale, the loading rate and the hydrogen concentration. Model predictions are also compared with fracture experiments on a modern ultra-high strength steel, AerMet100. A promising agreement is observed with experimental measurements of threshold stress intensity factor Kth over a wide range of applied potentials.

A continuum framework for coupled solid deformation–fluid flow through anisotropic elastoplastic porous media

We present a continuum framework for coupled solid deformation–fluid flow in anisotropic elastoplastic porous media. A thermodynamic formulation of the coupled processes gives rise to an anisotropic Biot tensor that is a function of drained elastic tangent moduli tensor of the solid skeleton and the intrinsic bulk modulus of the solid constituent. Two effective stress measures emerge from the formulation, namely, σ′, which is energy-conjugate to the elastic strain, and σ′′, which is energy-conjugate to the plastic strain. For the special case of transverse isotropy that is commonly encountered in natural rocks, the Biot tensor can be expressed in terms of its normal and tangential components to the bedding plane, along with a microstructure tensor. Apart from its thermodynamic consistency, an advantage of this new formulation is that standard mixed finite element formulation can be employed to discretize the domain and solve initial boundary-value problems. We conduct plane strain simulations of coupled solid deformation–fluid flow in a transversely isotropic porous medium to demonstrate the impacts of material anisotropy, stress history, and the Biot tensor on the system response.

Accurate iteration-free mixed-stabilised formulation for laminar incompressible Navier–Stokes: Applications to fluid–structure interaction

Stabilised mixed velocity–pressure formulations are one of the widely-used finite element schemes for computing the numerical solutions of laminar incompressible Navier–Stokes. In these formulations, the Newton–Raphson scheme is employed to solve the nonlinearity in the convection term. One fundamental issue with this approach is the computational cost incurred in the Newton–Raphson iterations at every load/time step. In this paper, we present an iteration-free mixed finite element formulation for incompressible Navier–Stokes that preserves second-order temporal accuracy of the generalised-alpha and related schemes for both velocity and pressure fields. First, we demonstrate the second-order temporal accuracy using numerical convergence studies for an example with a manufactured solution. Later, we assess the accuracy and the computational benefits of the proposed scheme by studying the benchmark example of flow past a fixed circular cylinder. Towards showcasing the applicability of the proposed technique in a wider context, the inf–sup stable P2–P1 pair for the formulation without stabilisation is also considered. Finally, the resulting benefits of using the proposed scheme for fluid–structure interaction problems are illustrated using two benchmark examples in fluid-flexible structure interaction.

Nearly-constrained transversely isotropic linear elasticity: energetically consistent anisotropic deformation modes for mixed finite element formulations

Strong anisotropies and/or near-incompressibility properties introduce internal constraints in material deformation. Numerical simulations comprising such a constrained behaviour show an overstiff structural response, referred to as element locking. Implementations based on mixed variational methods can heal locking but available solutions in the state-of-the-art are still non-optimal for anisotropic materials. This paper addresses this issue, by proposing a novel decomposition of anisotropic deformation modes on the basis of kinematic and energy requirements. Theoretical results exploit the Walpole’s formalism. The proposed kinematic split allows to introduce a new class of variational principles, referred to as energetically decoupled, for nearly-constrained transversely isotropic materials in linear elasticity. Low-order mixed finite element models are thus derived for treating near-inextensibility and/or near-incompressibility. Two-dimensional benchmark tests reproducing pure-bending and Cook’s membrane problems are conducted. Numerical results show that the accuracy of energetically decoupled formulations is high and robust with respect to variations of material properties, while the accuracy of non-energetically decoupled formulations is more sensitive.

Implicit time discretization schemes for mixed least-squares finite element formulations

This work is an extension of the ideas in Averweg et al. (2019) with the focus on a detailed investigation of implicit time discretization schemes to model instationary fluid flow, based on the incompressible Navier–Stokes equations, and linear elastodynamic structural behavior. The variational approaches for fluid and solid mechanics are based on a mixed least-squares finite element method. The L2-norm minimization of the residuals of the constructed first-order systems of the governing differential equations is based on two-field stress–velocity (SV) functionals. For the time discretization of the SV-fluid formulation, four different types of implicit integration schemes are investigated, namely the Houbolt method, the Crank–Nicolson method and two explicit, singly diagonally implicit Runge–Kutta methods (ESDIRK). The SV-formulation for the solid is discretized applying the Houbolt method. The presented time integration schemes are validated investigating an unsteady fluid flow and an elastodynamic structural benchmark. Since both (fluid and solid) SV formulations are discretized using conforming finite element spaces in H(div) and H1, respectively, the inherent fulfillment of coupling conditions, when modeling fluid–structure interaction problems, is given a priori. Therefore, the applicability is also examined by two simplified FSI problems for small deformations, in order to represent the main characteristics of the presented approach.

Mixed Finite-Element Formulation Based on Refined Sinusoidal Model for Buckling of Layered Beams

In the last 3 decades, sinusoidal theory has been increasingly utilized to research mechanical behaviors of layered composite and sandwich structures. Nevertheless, the existing sinusoidal model will encounter trouble in precisely yielding the buckling loads of layered structures composed of layers with different material properties. Thus, a refined sinusoidal model is offered for the buckling analysis of composite and sandwich structures that can simulate the zigzag effect of the in-plane displacement and meet the free conditions of transverse shear stresses on the surfaces. In the light of the elegant sinusoidal model, a three-node beam element has been constructed to work out the discrete eigenvalue equation coming from the stability behavior. Making use of a mixed variational theorem from the literature, the finite-element formulation can meet beforehand the continuous conditions of transverse stress at the interfaces. The three-dimensional finite element method (3D-FEM) results are utilized to evaluate the precision and efficiency of the proposed approach through a numerical example. The proposed finite-element formulation can produce satisfactory results with lower calculational cost, and some interesting conclusions are presented. However, the results for stability of layered structures made of plies with different material properties will be overestimated by utilizing the models violating the continuous prerequisite of interlaminar stresses.

Structural size effect: Experimental, theoretical and accurate computational assessment

In this paper, experimental evidence, theoretical predictions and the finite element modelling of the structural size effect in cracking problems of quasi-brittle materials are discussed and assessed against each other. The fracture process is modelled through the crack band approach, using an isotropic damage constitutive law. The correct dissipation of the fracture energy, essential for modelling the phenomenon with precision, is introduced. An enhanced accuracy mixed finite element formulation is used to ensure mesh bias independent results.Several experimental campaigns where size effect is investigated are numerically reproduced in 2D and in 3D to assess the feasibility and the performance of the method. For this, mode I and mixed mode I and II fracture situations are considered in notched and unnotched beams.The correlation of the experimental results with the numerical simulations shows the capacity of the mixed FE formulation to reproduce crack paths, force-displacement curves and collapse mechanisms with precision for a wide range of structural sizes. The enhanced accuracy FE formulation eliminates the spurious mesh dependency that is characteristic of standard FE simulations.In addition, the model is able to follow Bazant’s size effect law with precision. Results confirm that the energy release rate in the progressing fracture is the fundamental cause of size effect in quasi-brittle materials. This is additionally verified in a study of the relative influence of statistical and energetic size effect. Computations show that the essential requirements to suitably simulate the phenomenon are (1) a fracture model ensuring the correct energy dissipation at the crack and (2) a method guaranteeing mesh objective results.

A solid–shell formulation based on the assumed natural inhomogeneous strains for modeling the viscoelastic response of electro-active polymers

In this paper, an advanced low-order solid–shell formulation is presented for modeling electro-active polymers (EAPs). This advanced finite element is of great importance due to the fact that EAPs actuators are typically designed as shell-like formations, in which the application of standard finite element formulation will lead to various locking pathologies (e.g. shear locking, trapezoidal locking, volumetric locking, etc.). Thus, for alleviating the various locking pathologies, both the assumed natural inhomogeneous strains (ANIS) and the enhanced assumed strain (EAS) methods are adopted for modifying the strain measure. Within the modified kinematics, a strain energy function that accounts for the elastic and the viscoelastic response as well as the electromechanical coupling is adopted. The developed formulation is implemented in the finite-element software Abaqus for further numerical applications, in which the developed ANIS solid–shell is compared with the classical assumed natural strains solid–shell and the mixed finite element formulation.