A new efficient mixed formulation for thin shell finite element models

A nine node shell element is developed by a new and more efficient mixed formulation. The new shell element formulation is based on the Hellinger–Reissner principle with independent strain and the concept of a degenerate solid shell. The new formulation is made more efficient in terms of computing time than the conventional mixed formulation by dividing the assumed strain fields into a lower order part and a higher order part. Numerical results demonstrate that the present nine node element is free of locking even for very thin plates and shells and is also kinematically stable. In fact the stiffness matrix associated with the higher order assumed strain plays the role of a stabilization matrix.     

A 9-node mixed shell element based on the Hu-Washizu principle

This paper proposes a new 9-node degenerated shell element based on a nonlinear mixed formulation. To avoid locking phenomena, we present a mixed formulation based on a three-field Hu-Washizu principle in which displacements, the Green strain tensors, and the second Piola-Kirchhoff stress tensors are independently assumed. In approximating strain and stress fields, covariant components of the strains and stresses measured in the element curvilinear coordinate system are interpolated by the common polynomial functions over an element. Parameter vectors of stress and strain interpolants are elementwise eliminated so that we may obtain an element stiffness matrix similar to that of the displacement model. This formulation is mathematically clear in the variational context, and can include geometrical and material nonlinearities without spoiling such clearness. Numerical results based on our approach are illustrated with satisfactory behavior of the element observed.     

A new drilling quadrilateral membrane element with high coarse-mesh accuracy using a modified Hu-Washizu principle

By utilizing a modified Hu-Washizu principle, a new mixed variational framework and a corresponding high-performing four-node membrane element with drilling degrees of freedom, named as GCMQ element, are proposed. In this work, the generalized conforming concept, which is originally proposed within a displacement-based formulation, is extended to a mixed formulation. The new element is able to handle higher-order displacement, strain, and stress distributions. The interpolations are complete up to second order for stress and strain. The enhanced strain field is optimized so that a complete cubic displacement field can be represented. For numerical integration, a five-point scheme is proposed to minimize computational cost. Compared to other four-node elements in existing literature, numerical examples show that the proposed element has a better performance regarding predictions of both displacements and internal forces, particularly with coarse meshes. The new element is also free from shear locking and volumetric locking. Due to the nature of the mixed framework, GCMQ can be directly used in elastoplastic applications.     

A novel augmented Lagrangian‐based formulation for upper‐bound limit analysis

This paper describes a novel upper‐bound formulation of limit analysis. This formulation is an innovative variant of an existing two‐field mixed formulation based on the augmented Lagrangian method also developed by the authors. A natural approach is used to describe the deformation of each finite element. Furthermore, and in contrast to the previous formulation, two independent field approximations are now both used to define the velocity field, defined globally and at element level. It is shown that this feature allows a governing system of uncoupled linear equations to be obtained. Some numerical examples in plane strain conditions are presented in order to illustrate the current model performance. In conclusion, the potential and advantages of this new approach are discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

On the enhancement of low‐order mixed finite element methods for the large deformation analysis of diffusion in solids

A stabilized scheme is developed for mixed finite element methods for strongly coupled diffusion problems in solids capable of large deformations. Enhanced assumed strain techniques are employed to cure spurious oscillation patterns of low‐order displacement/pressure mixed formulations in the incompressible limit for quadrilateral elements and brick elements. A study is presented that shows how hourglass instabilities resulting from geometrically nonlinear enhanced assumed strain methods have to be distinguished from pressure oscillation patterns due to the violation of the inf‐sup condition. Moreover, an element formulation is proposed that provides stable results with respect to both types of instabilities. Comparisons are drawn between material models for incompressible solids of Mooney–Rivlin type and models for standard diffusion in solids with incompressible matrices such as polymeric gels. Representative numerical examples underline the ability of the proposed element formulation to cure instabilities of low‐order mixed formulations. Copyright © 2015 John Wiley & Sons, Ltd.     

Mixed shell element for seven‐parameter formulation

A new mixed shell element is developed for a seven‐parameter formulation in this paper. The mixed shell element is constructed by assuming stress field and displacement field together. Assumed stress field and assumed displacement field can be combined by stress–strain relationship with Hu‐Washizu functional. The developed mixed shell element can provide more flexible stiffness than other commercial softwares. Additionally, seven‐parameter shell formulation is used instead of Reissner/Mindlin formulation, since it can provide the thickness change. Even though some commercial engineering software are not proper for very thick shell structure, the developed mixed shell element for seven‐parameter formulation can be used without distinction of thick shell and thin shell. An example of shell models with different thickness is provided with solid model. Static and modal analyses are also performed for verification. Copyright © 2005 John Wiley & Sons, Ltd.     

Mixed‐enhanced formulation for geometrically linear axisymmetric problems

A four‐noded quadrilateral axisymmetric formulation in the context of a mixed‐enhanced method is presented. The strain field is represented by two sets of element parameters, which results in enhanced performance and coarse mesh accuracy in bending dominated problems and locking‐free response in the near incompressible limit. The mixed fields presented are such that variational stress recovery is permissible. In addition, the formulation is cast such that the mixed parameters are obtained explicitly yielding finite element arrays with the proper rank using standard order quadrature. In this paper our attention is restricted to the area of geometrically linear problems in solid mechanics. Representative simulations show favourable performance of the formulation. Copyright © 2002 John Wiley & Sons, Ltd.     

DEM: A new computational approach to sheet metal forming problems

Many current approaches to finite element modelling of large deformation elastic—plastic forming problems use a rate form of the virtual work (equilibrium) equations, and a finite element representation of the displacement components. Called the incremental method, this approach produces a three-field formulation in which displacements, stresses and effective strain are dependent variables. Next, the formulation is converted to a one-field displacement formulation by an algebraic time discretization which uses a low order explicit time-stepping procedure to integrate the equations. This approach does not produce approximations which satisfy the discrete equilibrium equations at all times and, moreover, the advantage of the single-field algebraic formulation is realized at the expense of very small time steps needed to produce stability and accuracy in the numerical calculations. This paper describes a variant of the mixed method in which all three field variables (displacements, stresses and effective strain) are given finite element representations. The discrete equilibrium equations then generate a nonlinear system of algebraic equations whose solutions represent a manifold, while the constitutive equations form a system of ordinary differential equations. A commercially available, variable time step/variable order code is then used to integrate this differential/algebraic system. When applied to the problem of hydrostatic bulging of a membrane, the new approach requires far less computer time than the incremental method.     

Formulation and numerical treatment of incompressibility constraints in large strain elastic–plastic analysis

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate‐independent finite strain analysis of solids undergoing large elastic‐isochoric plastic deformations. The formulation relies on the introduction of a mixed‐variant metric deformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure which can be additively decomposed into the exact isochoric (deviatoric) and volumetric (spheric) strain measures. This fact may be seen as the basic idea in the formulation of appropriate mixed finite elements which guarantee the accurate computation of isochoric strains. The mixed‐variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response whereas the plastic material behavior is assumed to be governed by a generalized J₂ yield criterion and rate‐independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on higher‐order Padé approximations. To be able to take into account the plastic incompressibility constraint a modified mixed variational principle is considered which leads to a quasi‐displacement finite element procedure. Finally, the numerical solution of finite strain elastic‐plastic problems is presented to demonstrate the efficiency and the accuracy of the algorithm. Copyright © 1999 John Wiley & Sons, Ltd.     

A systematic construction of B‐bar functions for linear and non‐linear mixed‐enhanced finite elements for plane elasticity problems

In a previous paper a modified Hu–Washizu variational formulation has been used to derive an accurate four node plane strain/stress finite element denoted QE2. For the mixed element QE2 two enhanced strain terms are used and the assumed stresses satisfy the equilibrium equations a priori for the linear elastic case. In this paper an alternative approach is discussed. The new formulation leads to the same accuracy for linear elastic problems as the QE2 element; however it turns out to be more efficient in numerical simulations, especially for large deformation problems. Using orthogonal stress and strain functions we derive B̄ functions which avoid numerical inversion of matrices. The B̄ ‐strain matrix is sparse and has the same structure as the strain matrix B obtained from a compatible displacement field. The implementation of the derived mixed element is basically the same as the one for a compatible displacement element. The only difference is that we have to compute a B̄ ‐strain matrix instead of the standard B ‐matrix. Accordingly, existing subroutines for a compatible displacement element can be easily changed to obtain the mixed‐enhanced finite element which yields a higher accuracy than the Q4 and QM6 elements. Copyright © 1999 John Wiley & Sons, Ltd.     

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

In this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.     

A WEAK–WEAK FORMULATION FOR LARGE DISPLACEMENTS BEAM STATICS: A FINITE VOLUMES APPROXIMATION

This paper addresses the problem of the numerical solution of beam statics undergoing large displacements. A kinematic analysis outlines the beam geometrical model through the definition of its Lagrangian co-ordinate and strain parameters. A definition of the stress parameters, a constitutive law and an expression for the strain energy of the beam are then provided under the hypothesis of small strain. The equations governing the beam equilibrium are introduced and their weak form is derived. These equations are then proved to be equivalent to the primal and mixed form of Principle of Virtual Work. The numerical approximation is introduced by applying the bidiscontinuous finite elements method on the linearized weak form. The weak–weak formulation is attained by using the lowest interpolation order both for test and trial functions on two staggered decompositions of the space domain. Some numerical examples prove the capability of present formulation in handling actual problems.     

Isogeometric analysis of shear bands

Numerical modeling of shear bands present several challenges, primarily due to strain softening, strong nonlinear multiphysics coupling, and steep solution gradients with fine solution features. In general it is not known a priori where a shear band will form or propagate, thus adaptive refinement is sometimes necessary to increase the resolution near the band. In this work we explore the use of isogeometric analysis for shear band problems by constructing and testing several combinations of NURBS elements for a mixed finite element shear band formulation. Owing to the higher order continuity of the NURBS basis, fine solution features such as shear bands can be resolved accurately and efficiently without adaptive refinement. The results are compared to a mixed element formulation with linear functions for displacement and temperature and Pian–Sumihara shape functions for stress. We find that an element based on high order NURBS functions for displacement, temperature and stress, combined with gauss point sampling of the plastic strain leads to attractive results in terms of rate of convergence, accuracy and cpu time. This element is implemented with a \(\overline{\hbox {B}}\) -bar strain projection method and is shown to be nearly locking free.     

Formulations in rates and increments for elastic-plastic analysis

Minimum principles in velocities, stress rates and plastic strain rates are extended in order to derive formulations for finite increments of displacement, stress and plastic strain fields defining complete numerical methods. Kinematical, statical and mixed principles are developed from a new variational formulation of the elastic-plastic work-hardening constitutive relation. The consequences of this time discretization are discussed independently of any discretization of the continuum. In particular, the incremental formulations derived from extended rate principles account for local elastic unloading and produce stress field approximations complying with equilibrium and plastic admissibility without any additional procedure, at least for piecewise linear yield functions. These properties are not fulfilled when the incremental analysis is based on direct discrete versions of classical rate principles. Finally, FEM approximations are formally introduced and the solution of the resulting finite dimensional quadratic optimization problem is considered.     

Nonlinear failure prediction of concrete composite columns by a mixed finite element formulation

This study focuses on developing a mixed frame finite element formulation of reinforced concrete and FRP composite columns in order to give more accuracy not only to predict the global behavior of the structural system but also to predict the local damage in the cross-section. A hypo-elastic constitutive law of concrete is presented under the basis of a three-dimensional stress state in order to model the compressive behavior of confined concrete wrapped with FRP jackets. To predict the nonlinear load path-dependent confinement model of FRP-confined concrete, the strength enhancement of concrete was determined by the failure surface of concrete in a tri-axial stress state, and its corresponding peak strain was computed by the strain-enhancement factor proposed in this study. The behavior of FRP jacket was modeled using the two-dimensional classical lamination theory. The flexural behavior of concrete and composite members was defined using a nonlinear fiber cross-sectional approach. The results obtained by developed mixed finite element formulation were verified with the experiments of concrete composite columns and also were compared with a displacement-based finite element formulation. It is shown that the proposed formulation gives e more accurate results in the global behavior of the column system as well as in the local damage in the column sections.     

Viscoelastic incremental formulation using creep and relaxation differential approaches

A new incremental formulation in the time domain for linear, non-ageing viscoelastic materials undergoing mechanical deformation is presented in this work. The formulation is derived from linear differential equations based on a discrete spectrum representation for the creep and relaxation tensors. The incremental constitutive equations are then obtained by finite difference integration. Thus the difficulty of retaining the stress and strain history in computer solutions is avoided. A complete general formulation of linear viscoelastic stress analysis is developed in terms of increments of strains and stresses in order to establish the constitutive stress–strain relationship. The presented method is validated using numerical simulations and reliable results are obtained.     

Bubble‐enhanced smoothed finite element formulation: a variational multi‐scale approach for volume‐constrained problems in two‐dimensional linear elasticity

This paper presents a bubble‐enhanced smoothed finite element formulation for the analysis of volume‐constrained problems in two‐dimensional linear elasticity. The new formulation is derived based on the variational multi‐scale approach in which unequal order displacement‐pressure pairs are used for the mixed finite element approximation and hierarchical bubble function is selected for the fine‐scale displacement approximation. An area‐weighted averaging scheme is employed for the two‐scale smoothed strain calculation under the framework of edge‐based smoothed FEM. The smoothed fine‐scale solution is shown to naturally contain the stress field jump of the smoothed coarse‐scale solution across the boundary of edge‐based smoothing domain and thus provides the possibility to stabilize the global solution for volume‐constrained problems. A global monolithic solution strategy is employed, and the fine‐scale solution is solved without the consideration of approximating the strong form of the fine‐scale equation. Several numerical examples are analyzed to demonstrate the accuracy of the present formulation. Copyright © 2014 John Wiley & Sons, Ltd.     

A hybrid‐mixed four‐node quadrilateral plate element based on sampling surfaces method for 3D stress analysis

The hybrid‐mixed assumed natural strain four‐node quadrilateral element using the sampling surfaces (SaS) technique is developed. The SaS formulation is based on choosing inside the plate body N not equally spaced SaS parallel to the middle surface in order to introduce the displacements of these surfaces as basic plate variables. Such choice of unknowns with the consequent use of Lagrange polynomials of degree N–1 in the thickness direction permits the presentation of the plate formulation in a very compact form. The SaS are located at Chebyshev polynomial nodes that allow one to minimize uniformly the error due to the Lagrange interpolation. To avoid shear locking and have no spurious zero energy modes, the assumed natural strain concept is employed. The developed hybrid‐mixed four‐node quadrilateral plate element passes patch tests and exhibits a superior performance in the case of coarse distorted mesh configurations. It can be useful for the 3D stress analysis of thin and thick plates because the SaS formulation gives the possibility to obtain solutions with a prescribed accuracy, which asymptotically approach the 3D exact solutions of elasticity as the number of SaS tends to infinity. Copyright © 2016 John Wiley & Sons, Ltd.