The moment redistribution effect of the C0 structural elements formulation

A new formulation was proposed recently for the removal of the shear and membrane locking mechanisms from the C⁰ structural elements. The performance of the new formulation was shown to be excellent in many cases of beam, plate and shell element applications, completely eliminating all locking problems. However, this formulation has its own problems. The potential introduction of softening effects (yielding softer models) and a rotational zero energy mode describe the problematic behaviour of the new formulation in cases of C⁰ plate and shell element applications. Analysis of this behaviour reveals some interesting aspects of the classical finite element formulation and allows for a better insight into the overall behaviour of the C⁰ structural elements. As a result of the present analysis, a modification of new formulation, remedying its problematic behaviour, will appear soon.     

Three-dimensional solid-to-beam transition elements for structural dynamics analysis

Complex structural components such as those encountered in many industrial applications may generally be considered as being composed of shell- or beam-like portions linked to three-dimensional solid continua. When discretized into finite elements, these structures present geometrical and mathematical difficulties at the connections between the different element types since the nodal degrees of freedom allocated to the solid, shell and beam elements are incompatible with each other. The development of specific and reliable transition finite elements is, thus, of outstanding practical importance. This paper presents efficient C⁰ compatible transition elements with a variable number of nodes for modelling solid to beam junctions. Based upon the standard isoparametric solid and beam formulations, the current approach includes the properties of both solids and beams, verifies the basic continuity, smoothness and completeness criteria inherent in the finite element convergence requirements, and avoids the shear locking phenomenon typical of C⁰ elements by using a strain-projection method. Several numerical examples which compare this formulation to analytical and experimental solutions are provided in order to show the applicability and efficiency of this approach.     

Dual-mixed hp finite element methods using first-order stress functions and rotations

 A two-field dual-mixed variational formulation of three-dimensional elasticity in terms of the non-symmetric stress tensor and the skew-symmetric rotation tensor is considered in this paper. The translational equilibrium equations are satisfied a priori by introducing the tensor of first-order stress functions. It is pointed out that the use of six properly chosen first-order stress function components leads to a (three-dimensional) weak formulation which is analogous to the displacement-pressure formulation of elasticity and the velocity-pressure formulation of Stokes flow. Selection of stable mixed hp finite element spaces is based on this analogy. Basic issues of constructing curvilinear dual-mixed p finite elements with higher-order stress approximation and continuous surface tractions are discussed in the two-dimensional case where the number of independent variables reduces to three, namely two components of a first-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dual-mixed hp finite elements is presented and compared to displacement-based hp finite elements when the Poisson's ratio converges to the incompressible limit of 0.5. It is shown that the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, both for h- and p-extensions. Received 22 October 1999     

An <Emphasis Type="Italic">rp</Emphasis>-adaptive finite element method for the deformation theory of plasticity

In this paper, we present an rp-discretization strategy for physically non-linear problems based on a high order finite element formulation. In order to achieve convergence, the p-version leaves the mesh unchanged and increases the polynomial degree of the shape functions locally or globally, whereas the r-method moves nodes and edges of an existing FE-mesh. The basic idea of our rp-version approach is to adjust the finite element mesh to the shape of the elastic–plastic interface in order to take into account the loss of regularity which arises along the curve of the plastic front. Numerical examples will demonstrate that this approach leads to an exponential rate of convergence and highly accurate results.     

Formulation of reference surface element and its applications in dynamic analysis of delaminated composite beams

This paper presents a new finite element formulation, referred to as reference surface element (RSE) model, for numerical prediction of dynamic behaviour of delaminated composite beams and plates using the finite element method. The RSE formulation can be readily incorporated into all elements based on the Timoshenko beam theory and the Reissner–Mindlin plate theory taking into account the transverse shear deformations. The ‘free model' and ‘constrained model' for dynamic analysis of delaminated composite beams and/or plates have been unified in this RSE formulation. The RSE formulation has been applied to an existing 2-node Timoshenko beam element taking into account the transverse shear deformations and the bending–extension coupling. Frequencies and vibration mode shapes are determined through solving an eigenvalue problem. Numerical results show that the present RSE model is reliable and practical when used to predict frequencies and mode shapes of delaminated composite beams. The RSE formulation has also been used to investigate the effects of the number, size and interfacial loci of delaminations on frequencies and mode shapes of composite beams.     

Finite element computation of sloshing modes in containers with elastic baffle plates

A finite element method to approximate the vibration modes of a plate in contact with an incompressible fluid is analysed in this paper. The effect of the fluid is taken into account by means of an added mass formulation, discretized by standard piecewise linear tetrahedral finite elements. Gravity waves on the free surface of the liquid are considered in the model. The plate is modelled by Reissner–Mindlin equations discretized by MITC3 locking‐free elements. Implementation issues are discussed and numerical experiments are presented. In particular, the method is compared with analytical approximations and with an experimental study which has been recently reported. Copyright © 2002 John Wiley & Sons, Ltd.     

Dual‐grid‐based tree/cotree decomposition of higher‐order interpolatory H(∇∧, Ω) basis

This work extends the zeroth‐order tree/cotree (TC) decomposition method into higher order (HO) interpolatory elements and develops the constraints operator required for the elimination of spurious solutions for general HO spectral basis. Earlier methods explicitly enforce the divergence condition that requires a mixed finite element (FE) formulation with both H¹ and H(?∧) expansions and involves repeated solutions of the Poisson equation. A recent approach, which avoids the mixed formulation and the Poisson problem, uses TC decomposition of edge DoF over the primal graph and construction of integration and gradient matrices. The approach is easily applied to HO hierarchical elements but becomes quite complex for HO spectral elements. In the presence of internal DoF, it is difficult to utilize the primal graph for an explicit decomposition of the spectral DoF. In contrast, this work utilizes the dual grid, resulting in an explicit decomposition of DoF and construction of constraint equations from a fixed element matrix. Thus, mixed formulation and the Poisson problems are avoided while eliminating the need for evaluation of integration and gradient matrices. The proposed constraints matrix is element‐geometry independent and possesses an explicit sparsity formulation reducing the need for dynamic memory allocation. Numerical examples are included for verification. Copyright © 2010 John Wiley & Sons, Ltd.     

A methodology for computing nonlinear fracture parameters for a bulging crack in a pressurised aircraft fuselage

A computational methodology for obtaining nonlinear fracture parameters which account for the effects of plasticity at the tips of a bulging crack in a pressurised aircraft fuselage is developed. The methodology involves a hierarchical three stage analysis (global, intermediate, and local) of the cracked fuselage, with the crack incorporated into the model at each stage. The global analysis is performed using a linear elastic shell finite element model in which the stiffeners are treated as beam elements. The geometrically nonlinear nature of the bulging phenomenon is emulated in the intermediate analysis using a geometrically nonlinear shell finite element model. The local analysis is a three-dimensional solid finite element model of the cracked skin using a hypoelastic-plastic rate formulation. Kinematic boundary conditions for each stage are obtained from the preceding stage in the hierarchy using a general mesh independent mechanism. The T ^*integral, which accounts for both large deformations and plasticity, is taken to be the fracture parameter characterising the severity of the conditions at the crack tip, and is evaluated from the local analysis using the Equivalent Domain Integral (EDI) method. The implementation of the EDI technique for finite deformations in shell space is also outlined. The methodology is applied to a number of example problems for which correction factors relating the nonlinear T ^* values to those obtained from a linear elastic stiffened shell analysis are computed. The issue of flapping is addressed by investigating the behaviour of the longitudinal stress parallel to the crack for various cases.This research is supported under a grant to the Center of Excellence for Computational Modeling of Aircraft Structures at Georgia Institute of Technology, from the Federal Aviation Administration. The authors also thank Dr. R. Singh, Mr. V Nagaswamy, Mr. L. Wang for many helpful discussions. The first author also wishes to express his heartfelt thanks to S. for helping him through some turbulent times of this life     

On consistent discretization of inertia terms for C0 structural elements with modified stiffness matrices

In the existing finite element calculations of dynamic problems using C⁰ structural elements, the inertia terms are evaluated without any reference to the modifications such as reduced integration, projections etc., typically needed in the discretization of the stiffness terms. A different discretization of inertia is discussed here. It is based on the following two observations. First, as shown in this work (at least for the beam problems), the modified stiffness matrix for a given C⁰ element can be obtained by standard, unmodified approach, in which degrees of freedom remain unchanged, but the shape functions are different. Those modified functions are of higher order and define the translational field within the element in terms of both translational and rotational parameters. Second, if standard consistent approach to the formulation of dynamic problems is to be followed, approximation of the displacement field used in the unmodified evaluation of the stiffness terms should also be used in discretization of the inertia terms. This implies that the modified higher-order functions should be employed when evaluating the element mass matrix for the C⁰ elements with modified stiffness matrices. As a consequence of this approach, consistency between formulation of the inertia and stiffness terms is restored. This leads to inertial coupling between rotational and translational degrees of freedom, which is absent in standard evaluation of inertia. It is demonstrated that this approach tends to improve accuracy of dynamic computations. © 1997 by John Wiley & Sons, Ltd.     

C0 finite element discretization of Kirchhoff's equations of thin plate bending

An alternative formulation of Kirchhoff's equations is given which is amenable to a standard C⁰ finite element discretization. In this formulation, the potential energy of the plate is formulated entirely in terms of rotations, whereas the deflections are the outcome of a subsidiary problem. The nature of the resulting equations is such that C⁰ interpolation can be used on both rotations and deflections. In particular, general classes of triangular and quadrilateral isoparametric elements can be used in conjunction with the method. Unlike other finite element methods which are based on three-dimensional or Mindlin formulations, the present approach deals directly with Kirchhoff's equations of thin plate bending. Excellent accuracy is observed in standard numerical tests using both distorted and undistorted mesh patterns.     

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.     

Simple and efficient shear flexible two-node arch/beam and four-node cylindrical shell/plate finite elements

Several simple and accurate C° two-node arch/beam and four-node cylindrical shell/plate finite elements are presented in this paper. The formulation used here is based on the refined theory of thick cylindrical shells and the quasi-conforming element technique. Unlike most C° elements, the element stiffness matrix presented here is given explicitly. In spite of their simplicity, these C° finite elements posseses linear bending strains and are free from the deficiencies existing in curved C° elements such as shear and membrane locking, spurious kinematic modes and numerical ill-conditioning. These finite elements are valid not only for thick/thin beams and plates, but also for arches/straight beams and cylindrical shells/plates. Furthermore, these C° elements can automatically reduce to the corresponding C¹ beam and plate elements and give the C° beam element obtained by the reduced integration as a special case. Several numerical examples indicate that the simple two-node arch/beam and four-node cylindrical shell/plate elements given in this paper are superior to the existing C° elements with the same element degrees of freedom. Only the formulation of the rectangular cylindrical shell and plate element is presented in this paper. The formulation of an arbitrarily quadrilateral plate element will be presented in a follow-up paper³².     

A method for creating a class of triangular C1 finite elements

Finite elements providing a C¹ continuous interpolation are useful in the numerical solution of problems where the underlying partial differential equation is of fourth order, such as beam and plate bending and deformation of strain‐gradient‐dependent materials. Although a few C¹ elements have been presented in the literature, their development has largely been heuristic, rather than the result of a rational design to a predetermined set of desirable element properties. Therefore, a general procedure for developing C¹ elements with particular desired properties is still lacking. This paper presents a methodology by which C¹ elements, such as the TUBA 3 element proposed by Argyris et al., can be constructed. In this method (which, to the best of our knowledge, is the first one of its kind), a class of finite elements is first constructed by requiring a polynomial interpolation and prescribing the geometry, the location of the nodes and the possible types of nodal DOFs. A set of necessary conditions is then imposed to obtain appropriate interpolations. Generic procedures are presented, which determine whether a given potential member of the element class meets the necessary conditions. The behaviour of the resulting elements is checked numerically using a benchmark problem in strain‐gradient elasticity. Copyright © 2011 John Wiley & Sons, Ltd.     

ASSUMED STRESS C0 QUADRILATERAL/TRIANGULAR PLATE ELEMENTS BY INTERRELATED EDGE DISPLACEMENTS

This paper presents two simple quadrilateral C⁰ plate bending elements with explicit element stiffness matrix. The element formulation is based on assumed element stress fields and the interrelated transverse displacement and rotation along element boundaries. The interrelated edge displacements not only can result in higher-order displacements interpolations for higher accuracy element and overcome the shear locking in thin plate analysis encountered by C⁰ plate elements, but can also unify the four-noded quadrilateral element and its corresponding three-noded triangular element. The latter cannot be achieved by the assumed displacement formulation. The numerical examples demonstrate the accuracy and robustness of the present assumed stress C⁰ plate elements.     

A general degree semihybrid triangular compatible finite element formulation for Kirchhoff plates

We revisit compatible finite element formulations for Kirchhoff plates and propose a new general degree hybridized approach that strictly imposes C¹ continuity. These new elements are triangular and based on nodal polynomial approximation functions that only use displacement and rotation degrees of freedom for assembly, and thereby “nearly” impose C¹ continuity. This condition is then strictly enforced by adding appropriately chosen hybrid constraints and the corresponding Lagrange multipliers. Unlike all other existing approaches, this formulation allows for the definition of elements of arbitrary degree considering a single polynomial basis for each element, without using degrees of freedom associated with second-order derivatives. The convergence is compared with that of alternative approaches in terms of numbers of elements and degrees of freedom.     

A mixed finite element formulation of triphasic mechano‐electrochemical theory for charged,hydrated biological soft tissues

An equivalent new expression of the triphasic mechano‐electrochemical theory [9] is presented and a mixed finite element formulation is developed using the standard Galerkin weighted residual method. Solid displacement u ^s, modified electrochemical/chemical potentials ϵ^w, ϵ⁺and ϵ⁻ (with dimensions of concentration) for water, cation and anion are chosen as the four primary degrees of freedom (DOFs) and are independently interpolated. The modified Newton–Raphson iterative procedure is employed to handle the non‐linear terms. The resulting first‐order Ordinary Differential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme which is unconditionally stable. One‐dimensional (1‐D) linear isoparametric element is developed. The final algebraic equations form a non‐symmetric but sparse matrix system. With the current choice of primary DOFs, the formulation has the advantage of small amount of storage, and the jump conditions between elements and across the interface boundary are satisfied automatically. The finite element formulation has been used to investigate a 1‐D triphasic stress relaxation problem in the confined compression configuration and a 1‐D triphasic free swelling problem. The formulation accuracy and convergence for 1‐D cases are examined with independent finite difference methods. The FEM results are in excellent agreement with those obtained from the other methods. Copyright © 1999 John Wiley & Sons, Ltd.     

The C° structural finite elements reformulated

A new formulation was recently proposed by the present author aimed at removing the shear and membrane locking mechanisms from the C° structural elements. The performance achieved was shown to be excellent, completely eliminating all locking problems. In some cases of C° plate and shell element applications; however, the proposed formulation was shown to yield flexible (softer than expected) models. Analysis of this behaviour revealed the presence of an internal moment redistribution mechanism with the classical formulation. The absence of this mechanism from the new formulation was found to be responsible for the potential introduction of softening effects in the elastic finite element models. In the present paper, the internal moment redistribution effect is examined analytically and the key component responsible for its development is isolated. The new formulation, as originally proposed for the C° structural elements, is modified so that the internal moment redistribution mechanism is retained, yet, with all locking mechanisms being rejected. The proposed formulation has been subjected recently to extensive numerical investigation with excellent results.     

Refined h-adaptive finite element procedure for large deformation geotechnical problems

Adaptive finite element procedures automatically refine, coarsen, or relocate elements in a finite element mesh to obtain a solution with a specified accuracy. Although a significant amount of research has been devoted to adaptive finite element analysis, this method has not been widely applied to nonlinear geotechnical problems due to their complexity. In this paper, the h-adaptive finite element technique is employed to solve some complex geotechnical problems involving material nonlinearity and large deformations. The key components of h-adaptivity including robust mesh generation algorithms, error estimators and remapping procedures are discussed. This paper includes a brief literature review as well as formulation and implementation details of the h-adaptive technique. Finally, the method is used to solve some classical geotechnical problems and results are provided to illustrate the performance of the method.     

p-version least-squares finite element formulation for convection-diffusion problems

This paper presents a p-version least-squares finite element formulation of the convection-diffusion equation. The second-order differential equation describing convection-diffusion is reduced to a series of equivalent first-order differential equations for which the least-squares formulation is constructed using the same order of approximation for each of the dependent variables. The hierarchical approximation functions and the nodal variable operators are established by first constructing the one-dimensional hierarchical approximation functions of orders pξ and pη and the corresponding nodal variable operators in ξ and η-direction and then taking their products. Numerical results are presented and compared with analytical and numerical solutions for a two-dimensional test problem to demonstrate the accuracy and the convergence characteristics of the present formulation. The Gaussian quadrature rule used to calculate the numerical values of the element matrices, vectors as well as the error functional I(E), is established based on the highest degree of the polynomial in the integrands. It is demonstrated that this quadrature rule with the present p-version formulation produces excellent results for very low as well as extremely high Peclet numbers (10-10⁶) and, furthermore, the error functional I (sum of the integrals of E²) is a monotonically decreasing function of the number of degrees of freedom as the p-levels are increased for a fixed mesh. It is shown that exact integration with the h-version (linear and parabolic elements) produces inaccurate solutions at high Peclet numbers. Results are also presented using reduced integration. It is shown that the reduced integration with p-version produces accurate values of the primary variable even for relatively low p-levels but the error functional I (when calculated using the proper integration rule) has a much higher value (due to errors in the derivatives of the primary variable) and is a non-monotonic function of the degrees of freedom as p-levels are increased for a fixed mesh. Similar behaviour of the error functional I is also observed for the h-models using linear elements when reduced integration is used. Although the h-models using parabolic elements produce monotonic error functional behaviour as the number of degrees of freedom are increased, the error values are inferior to the p-version results using exact integration.