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Showing papers in "International Journal for Numerical Methods in Engineering in 1992"


Journal ArticleDOI
TL;DR: In this article, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes, which has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems.
Abstract: This is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation. In this paper, a general recovery technique is developed for determining the derivatives (stresses) of the finite element solutions at nodes. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of widely used linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements. In particular, an O(h4) convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time. The performance of the proposed technique is compared with the widely used smoothing procedure of global L2 projection and other methods. It is found that the derivatives recovered at interelement nodes, by using L2 projection, are also superconvergent for linear elements but not for quadratic elements. Numerical experiments on the convergence of the recovered solutions in the energy norm are also presented. Higher rates of convergence are again observed. The results presented in this part of the paper indicate clearly that a new, powerful and economical process is now available which should supersede the currently used post-processing procedures applied in most codes.

1,993 citations


Journal ArticleDOI
TL;DR: In this paper, the authors derived a theorem showing the dependence of the effectivity index for the Zienkiewicz-Zhu error estimator on the convergence rate of the recovered solution.
Abstract: In this second part of the paper, the issue of a posteriori error estimation is discussed. In particular, we derive a theorem showing the dependence of the effectivity index for the Zienkiewicz–Zhu error estimator on the convergence rate of the recovered solution. This shows that with superconvergent recovery the effectivity index tends asymptotically to unity. The superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the Zienkiewicz–Zhu error estimator to demonstrate accurate estimation of the exact error attainable. Numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient (stress) error estimation. Several examples of the performance of the error estimator in adaptive mesh refinement are also presented.

1,106 citations


Journal ArticleDOI
TL;DR: In this paper, the yield strength not only depends on an equivalent plastic strain measure (hardening parameter), but also on the Laplacian thereof, and the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity.
Abstract: A plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure (hardening parameter), but also on the Laplacian thereof. The consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity. To properly solve the set of non-linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements. For appropriate boundary conditions this formulation can also be derived from a variational principle. Accordingly, the theory is complete

924 citations


Journal ArticleDOI
TL;DR: In this paper, a class of assumed strain mixed finite element methods for fully nonlinear problems in solid mechanics is presented which, when restricted to geometrically linear problems, encompasses the classical method of incompatible modes as a particular case.
Abstract: A class of ‘assumed strain’ mixed finite element methods for fully non-linear problems in solid mechanics is presented which, when restricted to geometrically linear problems, encompasses the classical method of incompatible modes as a particular case. The method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part, formulated in the context of a three-field variational formulation. The resulting class of mixed methods provides a possible extension to the non-linear regime of well-known incompatible mode formulations. In addition, this class of methods includes non-linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements. The good performance of the proposed methodology is illustrated in a number of simulations including 2-D, 3-D and axisymmetric finite deformation problems in elasticity and elastoplasticity. Remarkably, these methods appear to be specially well suited for problems involving localization of the deformation, as illustrated in several numerical examples.

763 citations


Journal ArticleDOI
TL;DR: In this article, a two-dimensional dual boundary element method for linear elastic crack problems is presented. But the authors focus on the effective numerical implementation of the method, and they do not address the problem of collocation at crack tips, crack kinks and crack-edge corners.
Abstract: SUMMARY The present paper is concerned with the effective numerical implementation of the two-dimensional dual boundary element method, for linear elastic crack problems. The dual equations of the method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation on the other, general mixed-mode crack problems can be solved with a single-region formulation. Both crack surfaces are discretized with discontinuous quadratic boundary elements; this strategy not only automatically satisfies the necessary conditions for the existence of the finite-part integrals, which occur naturally, but also circumvents the problem of collocation at crack tips, crack kinks and crack-edge corners. Examples of geometries with edge, and embedded crack are analysed with the present method. Highly accurate results are obtained, when the stress intensity factor is evaluated with the J-integral technique. The accuracy and efficiency of the implementation described herein make this formulation ideal for the study of crack growth problems under mixed-mode conditions.

656 citations


Journal ArticleDOI
TL;DR: In this article, a solution strategy to find the shape and topology of structures that maximize a natural frequency is presented, based on a homogenization method and the representation of the shape of the structure as a material property.
Abstract: A solution strategy to find the shape and topology of structures that maximize a natural frequency is presented. The methodology is based on a homogenization method and the representation of the shape of the structure as a material property. The problem is formulated as a reinforcement problem in which a given structure is reinforced using a prescribed amount of material. Two dimensional, plane elasticity problems are considered. Examples are presented for illustration.

465 citations


Journal ArticleDOI
TL;DR: In this paper, a new class of time-stepping algorithms for strongly coupled thermomechanical problems is presented which retains the computational convenience of traditional staggered algorithms without upsetting the unconditional stability property characteristic of fully implicit (monolithic) schemes.
Abstract: A new class of time-stepping algorithms for strongly coupled thermomechanical problems is presented which retains the computational convenience of traditional staggered algorithms without upsetting the unconditional stability property characteristic of fully implicit (monolithic) schemes.11 The proposed schemes are fractional step methods associated with a two phase operator split of the full non-linear system of thermoelasticity into an adiabatic elastodynamics phase, followed by a heat conduction phase at fixed configuration. This operator split is shown to inherit the contractivity property of the full problem of evolution, thus leading to unconditionally B-stable product formula algorithms. In sharp contrast with this stability result, traditional staggered algorithms based on an isothermal mechanical phase followed by a heat conduction phase with an effective heat source are shown to lead, at best, to conditionally stable schemes. It is further shown that the actual implementation of these two classes of schemes is essentially identical. The numerical simulations presented include both dynamic and quasi-static problems and are shown to closely replicate the stability estimates and non-linear stability results derived for these two classes of staggered schemes.

294 citations



Journal ArticleDOI
TL;DR: The problem of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix has been studied in the literature as mentioned in this paper.
Abstract: The problem consists of determining the location and size of a circular inclusion in a finite matrix and the elastic material properties of the inclusion and the matrix.

235 citations


Journal ArticleDOI
TL;DR: In this paper, the friction forces are assumed to follow the Coulomb law, with a slip criterion treated in the context of a standard return mapping algorithm Consistent linearization of the field equations is performed which leads to a fully implicit scheme with non-symmetric tangent stiffness which preserves asymptotic quadratic convergence of the Newton-Raphson method.
Abstract: The friction forces are assumed to follow the Coulomb law, with a slip criterion treated in the context of a standard return mapping algorithm Consistent linearization of the field equations is performed which leads to a fully implicit scheme with non-symmetric tangent stiffness which preserves asymptotic quadratic convergence of the Newton-Raphson method

230 citations


Journal ArticleDOI
TL;DR: In this paper, a study on shear correction factors is presented as well as typical applications, such as a multilayered plate, a symmetrical sandwich and a structure formed by two plates glued to each other.
Abstract: Multilayered plate and shell finite elements usually have a constant shear distribution across the thickness. This causes a decrease of accuracy, especially for sandwich structures. The problem is overcome by using shear correction factors which are defined by energy considerations. In this paper a study on shear correction factors is presented as well as typical applications. These factors are defined by studying a crossply laminate without midplane symmetry in cylindrical bending. It is possible to define one factor for a given cross section or several ones, one factor per ply. The two ways are equivalent. Special cases are studied: a multilayered plate, a symmetrical sandwich and a structure formed by two plates glued to each other. The importance of shear correction factors is illustrated in the case of plate and shell applications; the results are obtained by using three-dimensional degenerated shell finite elements.

Journal ArticleDOI
TL;DR: In this article, a "symmetric" boundary element method based on a weighted residual Galerkin approach for elastoplastic analysis is revisited and its computer implementation for two-dimensional homogeneous problems is described.
Abstract: A “symmetric” boundary element method based on a weighted residual Galerkin approach for elastoplastic analysis is revisited and its computer implementation for two-dimensional homogeneous problems is described.

Journal ArticleDOI
TL;DR: In this paper, the time-discontinuous Galerkin and least-squares methods for structural dynamics were used to prove a general convergence theorem in a norm stronger than the energy norm.
Abstract: SUMMARY Time finite element methods are developed for the equations of structural dynamics. The approach employs the time-discontinuous Galerkin method and incorporates stabilizing terms having least-squares form. These enable a general convergence theorem to be proved in a norm stronger than the energy norm. Results are presented from finite difference analyses of the time-discontinuous Galerkin and least-squares methods with various temporal interpolations and commonly used finite difference methods for structural dynamics. These results show that, for particular interpolations, the time finite element method exhibits improved accuracy and stability.

Journal ArticleDOI
TL;DR: In this paper, a non-linear shell theory, including transverse shear strains, with exact description of the kinematical fields is developed, and the strain measures are derived via the polar decomposition theorem allowing for an explicit use of a three parametric rotation tensor.
Abstract: A non-linear shell theory, including transverse shear strains, with exact description of the kinematical fields is developed. The strain measures are derived via the polar decomposition theorem allowing for an explicit use of a three parametric rotation tensor. Thus in-plane rotations, also called drilling degrees of freedom, are included in a natural way. Various alternatives of the theory are derived. For a special version of the theory, with altogether six kinematical fields, different mixed variational principles are given. A hybrid finite element formulation, which does not exhibit locking phenomena, is developed. Numerical examples of shell deformation at finite rotations, with excellent element performance, are presented. Comparison with results reported in the literature demonstrates the features of the theory as well as the proposed finite element formulation.

Journal ArticleDOI
TL;DR: The paper describes an object oriented implementation of the finite element method, and illustrates the advantages of the approach.
Abstract: The finite element method is by its nature very modular. Object oriented programming enables full advantage to be taken of this modularity. This makes for safer and easier programming, and extending or modifying object oriented programs is very straightforward. The paper describes an object oriented implementation of the finite element method, and illustrates the advantages of the approach.

Journal ArticleDOI
TL;DR: In this article, a thermomechanically coupled contact element is presented which accounts for the real microscopic shape of the surfaces, the microscopic mechanism of force transmission and heat exchange, and the macroscopic related stiffnesses are calculated and continuously updated taking into account changes in significant parameters.
Abstract: The solution of contact problems involves great numerical efforts to satisfy non-penetration conditions. The search for numerical efficiency hence has limited the modelling of the real physical interface behaviour. Up to now mainly simple laws, usually formulated using constant coefficients, have been available to study contact problems in uncoupled from. Here a thermomechanically coupled contact element is presented which accounts for the real microscopic shape of the surfaces, the microscopic mechanism of force transmission and heat exchange. The contact element geometrical behaviour has been put together with experimental and theoretical well founded micro-mechanical and micro-thermal laws adapted to Finite Element Method (FEM) necessities. Based on these laws the macroscopic related stiffnesses are calculated and continuously updated taking into account changes in significant parameters. The linearization of the set of equations has been obtained using a consistent technique which implies computational efficiency.

Journal ArticleDOI
TL;DR: The tabu search method with random moves to solve approximately minimum weight design problems of non-convex optimization problems of continuous variables is introduced and outperforms the random search method and composite genetic algorithm.
Abstract: Optimum engineering design problems are usually formulated as non-convex optimization problems of continuous variables. Because of the absence of convexity structure, they can have multiple minima, and global optimization becomes difficult. Traditional methods of optimization, such as penalty methods, can often be trapped at a local optimum. The tabu search method with random moves to solve approximately these problems is introduced. Its reliability and efficiency are examined with the help of standard test functions. By the analysis of the implementations, it is seen that this method is easy to use, and no derivative information is necessary. It outperforms the random search method and composite genetic algorithm. In particular, it is applied to minimum weight design examples of a three-bar truss, coil springs, a Z-section and a channel section. For the channel section, the optimal design using the tabu search method with random moves saved 26.14 per cent over the weight of the SUMT method.

Journal ArticleDOI
TL;DR: In this paper, a theory of rubber-like membrane shells undergoing large elastic deformations is derived and the stresses are deduced from Ogden's material law, which is formulated in terms of the principal values of the right stretch tensor.
Abstract: A theory of rubberlike membrane shells undergoing large elastic deformations is derived. The stresses are deduced from Ogden's material law, which is formulated in terms of the principal values of the right stretch tensor. Incompressibility is fulfilled exactly using the plane stress constraint. Furthermore, a finite element formulation of the membrane theory is given. The use of the tangential stiffness matrix, derived analytically, provides a quadratically convergent solution process. Several numerical examples show the robustness of the developed finite element.

Journal ArticleDOI
TL;DR: In this article, the authors examined the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method, the treatment of finite rotations through an arbitrary parametrization of the rotation group, the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis.
Abstract: The underlying theory is statically and geometrically exact, and it naturally includes small strain and finite strain problems of thin as well as thick shells. This paper examines the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method, the treatment of finite rotations through an arbitrary parametrization of the rotation group, the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis.

Journal ArticleDOI
TL;DR: In this article, a triangular element based on the Reissner/Mindlin plate theory is presented, which has three nodes and three d.o.f. per node only.
Abstract: In this paper the formulation of a new triangular element based on the Reissner/Mindlin plate theory is presented. The element has three nodes and three d.o.f. per node only. It is based on constant bending modes plus incompatible energy orthogonal higher order bending modes. The transverse shear effects are represented using the moment equilibrium and the constitutive equations. Discrete (collocation) shear constraints are considered on each side to relate the kinematical and the independent shear strains. The element has a proper rank, is completely locking free, passes all constant patch-tests exactly. The detailed numerical evaluation shows that the element, called DST-BK, is a robust and high-performance element for thick and thin plates.

Journal ArticleDOI
TL;DR: In this paper, a family of p-method plane elasticity elements is derived based on the hybrid Trefftz formulation, where exact solutions of the Lame-Navier equations are used for the intra-element displacement field together with an independent displacement frame function field along the element boundary.
Abstract: A family of p-method plane elasticity elements is derived based on the hybrid Trefftz formulation.1 Exact solutions of the Lame-Navier equations are used for the intra-element displacement field together with an independent displacement frame function field along the element boundary. The final unknowns are the parameters of the frame function field consisting of the usual degrees of freedom at corner nodes and an optional number of hierarchic degrees of freedom associated with the mid-side nodes. Since the element matrices do not involve integration over the element area, the elements have a polygonal contour with an optional number of curved sides. The quadrilateral element has the same external appearance as the conventional p-method plane elasticity element.2,3 But unlike in the conventional p-method approach, suitable special-purpose Trefftz functions are generally used to handle the singularity and/or stress concentration problems rather than a local mesh refinement. The practical efficiency of the new elements is assessed through a series of examples.

Journal ArticleDOI
TL;DR: In this paper, a non-linear time-stepping algorithm is proposed to preserve the total linear and angular momentum of a shell in the presence of a nonlinear dynamic response.
Abstract: In Parts I to V of the present work, the formulation and finite element implementation of a non-linear stress resultant shell model are considered in detail. This paper is concerned with the extension of these results to incorporate completely general non-linear dynamic response. Of special interest here is the dynamics of very flexible shells undergoing large overall motion which conserves the total linear and angular momentum and, for the Hamiltonian case, the total energy. A main goal of this paper is the design of non-linear time-stepping algorithms, and the construction of finite element interpolations, which preserve exactly these fundamental constants of motion. It is shown that only a very special class of algorithms, namely a formulation of the mid-point rule in conservation form, exactly preserves the total linear and angular momentum. For the Hamiltonian case, a somewhat surprising result is proved: regardless of the degree of non-linearity in the stored-energy function, a generalized mid-point rule algorithm always exists which exactly conserves energy The conservation properties of a time-stepping algorithm need not, and in general will not, be preserved by the spatial discretization. Precise conditions which ensure preservation of these conservation properties are derived. A number of numerical simulations are presented which illustrate the exact conservation properties of the proposed methodology.

Journal ArticleDOI
TL;DR: In this paper, the authors describe a technique whereby this facet-shell formulation is extended to handle geometric non-linearity by means of a co-rotational procedure, which is increment-independent with both the internal force vector and tangent stiffness matrix being derived from the total strain measures in a consistent manner.
Abstract: The facet-shell formulation involves the combination of the constant-strain membrane triangle with a constant-curvature bending triangle. The paper describes a technique whereby this facet-formulation is extended to handle geometric non-linearity by means of a co-rotational procedure. Emphasis is placed on the derivation of a technique that is increment-independent with both the internal force vector and tangent stiffness matrix being derived from the «total strain measures» in a «consistent manner».

Journal ArticleDOI
TL;DR: In this paper, a general, well-structured and efficient method is advanced for the solution of-a large class of dynamic interaction problems including a non-linear dynamic system running at a prescribed time-dependent speed on a linear track or guideway.
Abstract: A general, well-structured and efficient method is advanced for the solution of-a large class of dynamic interaction problems including a non-linear dynamic system running at a prescribed time-dependent speed on a linear track or guideway. The method uses an extended state-space vector approach in conjunction with a complex modal superposition. It allows for the analysis of structures containing both physical and modal components. The physical components studied here are vehicles modelled as linear or non-linear discrete mass–spring–damper systems. The modal component studied is a linear continuous model of a track structure containing beam elements which can be generally damped and which can be embedded in a three-parameter damped Winkler-type foundation. The complex modal parameters of the track structure are solved for. Algebraic equations are established which impose constraints on the transverse forces and accelerations at the interfaces between the moving dynamic systems and the track. An irregularity function modelling a given non-straight profile of the non-loaded track or a non-circular periphery of the wheels is also accounted for. Loss of contact and recovered contact between a vehicle and the track can be treated. The system of coupled first-order differential equations governing the motion of the vehicles and the track and the set of algebraic constraint equations are together compactly expressed in one unified matrix format. A time-variant initial-value problem is thereby formulated such that its solution can be found in a straightforward way by use of standard time-stepping methods implemented in existing subroutine libraries. Examples for verification and application of the proposed method are given. The present study should be of particular value in railway engineering.

Journal ArticleDOI
TL;DR: In this paper, a comparative analysis of two purely explicit and one semi-implicit finite element methods used for incompressible flow computation is presented, where the fundamental concepts and characteristics of the formulations and the solution methodology used are described in technical detail.
Abstract: A comparative investigation, based on a series of numerical tests, of two purely explicit and one semi-implicit finite element methods used for incompressible flow computation is presented. The ‘segregated’ approach is followed and the equations of motion are considered sequentially. The fundamental concepts and characteristics of the formulations and the solution methodology used are described in technical detail. Various modifications to Chorin's projection algorithm are investigated, particularly with respect to their effects on stability and accuracy. The stability of the semi-implicit method is shown to be less restrictive when compared to the explicit methods as the Reynolds number increases. At large time steps the artificial viscosity is also reduced and higher accuracy is obtained. The performance of the methods discussed in this paper is illustrated by the numerical solutions obtained for the cavity flow and flow past a rearward-facing step problems at high Reynolds numbers, and free convection flow problem at high Rayleigh numbers. It is shown that the semi-implicit method needs fewer iterations than the explicit methods, and the accuracy of the present methods is guaranteed by comparison with the existing methods.

Journal ArticleDOI
TL;DR: In this paper, the principles governing the formulation of hierarchic models for laminated composites are discussed and the essential features of the hierarchical models described herein are: (1) the exact solutions corresponding to the hierarchic sequence of models converge to the exact solution of the corresponding problem of elasticity for a fixed laminate thickness.
Abstract: The principles governing the formulation of hierarchic models for laminated composites are discussed. The essential features of the hierarchic models described herein are: (1) the exact solutions corresponding to the hierarchic sequence of models converge to the exact solution of the corresponding problem of elasticity for a fixed laminate thickness; and (2) the exact solution of each model converges to the same limit as the exact solution of the corresponding problem of elasticity with respect to the laminate thickness approaching zero. Hierarchic models make the computation of any engineering data possible to an arbitrary level of precision within the framework of the theory of elasticity.

Journal ArticleDOI
TL;DR: In this article, the necessary requirements for the good behaviour of shear constrained Reissner-Mindlin plate elements for thick and thin plate situations are re-interpreted and a simple explicit form of the substitute shear strain matrix is obtained.
Abstract: In this paper the necessary requirements for the good behaviour of shear constrained Reissner–Mindlin plate elements for thick and thin plate situations are re-interpreted and a simple explicit form of the substitute shear strain matrix is obtained. This extends the previous work of the authors presented in References 18 and 31. The general methodology is applied to the re-formulation of some well known quadrilateral plate elements and some new triangular and quadrilateral plate elements which show promising features. Some examples of the good behaviour of these elements are given.

Journal ArticleDOI
TL;DR: Non-symmetric versions of these algorithms are described and their performance is checked for BEM matrices, through various numerical experiments.
Abstract: The present paper is concerned with the application of iterative techniques to solve boundary element method (BEM) systems of equations. Initially, a brief description of some algorithms which have been employed for symmetric positive-definite matrices is given (finite element method matrices, for instance). Subsequently, non-symmetric versions of these algorithms are described and their performance is checked for BEM matrices, through various numerical experiments. Preconditioned algorithms were found to work quite well.

Journal ArticleDOI
TL;DR: In this paper, a finite element model and an associated algorithm are developed for the analysis of the thermomechanical coupling in the necking process of a bar in an uniaxial tension test.
Abstract: SUMMARY The necking process of a bar in an uniaxial tension test is influenced by the heat production due to inelastic deformations. Thus, for an analysis of this problem the thermomechanical coupling has to be considered. A finite element model and an associated algorithm are developed for this purpose. This computational tool allows the study of adiabatic processes as well as processes with heat flux. The analysis of the necking process of a perfectly cylindrical specimen shows that, in contrast to isothermal and adiabatic cases, no bifurcation occurs in the case where heat flux is considered. 1. INTRODUCTION A variety of experimental, analytical and numerical investigations have lead to a basic under- standing of the mechanisms pertaining to the diffuse necking process in a circular ductile bar tensile test. A number of papers have been published in which numerical simulations of this phenomenon within the framework of classical elastoplasticity can be found, see e.g. Chen6 or Needleman. Any advanced simulation of the necking problem should consider the fact that heat production due to dissipated mechanical work in plastic deformations will lead to temperature changes in the specimen when the test is performed at finite rates. In the context of a pure adiabatic process the associated bifurcation loads of cylindrical specimen has been investigated by Bruhns and Mielniczuk. Several numerical concepts for coupled thermomechanical processes have been developed in Argyris and Doltsinis2 and Argyris