Showing papers in "Applied Mechanics and Engineering in 1990"

Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods

Abstract: This paper discusses the homogenization method to determine the effective average elastic constants of linear elasticity of general composite materials by considering their microstructure. After giving a brief theory of the homogenization method, a finite element approximation is introduced with convergence study and corresponding error estimate. Applying these, computer programs PREMAT and POSTMAT are developed for preprocessing and postprocessing of material characterization of composite materials. Using these programs, the homogenized elastic constants for macroscopic stress analysis are obtained for typical composite materials to show their capability. Finally, the adaptive finite element method is introduced to improve the accuracy of the finite element approximation.

On a stress resultant geometrically exact shell model. Part III: computational aspects of the nonlinear theory

Abstract: Computational aspects of a geometrically exact stress resultant model presented in Part I of this work are considered in detail. In particular, by exploiting the underlying geometric structure of the model, a configuration update procedure for the director (rotation) field is developed which is singularity free and exact regardless the magnitude of the director (rotation) increment. Our mixed finite element interpolation for the membrane, shear and bending fields presented in PartII of this work are extended to the finite deformation case. The exact linearization of the discrete form of the equilibrium equations is derived in closed form. The formulation is then illustrated by a comprehensive set of numerical experiments which include bifurcation and post-buckling response, we well as comparisons with closed form solutions and experimental results.

Finite deformation constitutive equations and a time integrated procedure for isotropic hyperelastic—viscoplastic solids

Abstract: Constitute equations for finite deformation, isotropic, elastic-viscoplastic solids are formulated. The concept of a multiplicative decomposition of the deformation gradient into an elastic and a plastic part is used. The constitutive equation for stress is a hyperelastic relation in terms of the logarithmic elastic strain. Since the material is assumed to be isotropic in every local configuration determined by the plastic part of deformation gradient, the internal variables are necessarily scalars. We use a single scalar as an internal variable to represent the isotropic resistance to plastic flow offered by the internal state of the material. The constitutive equation for stress is often expressed in a rate form, and for metals it is common to approximate this rate equation, under the assumption of infinitesimal elastic strains, to arrive at a hypoelastic equation for the stress. Here, we do not express the stress constitutive equation in a rate form, nor do we make this approximative assumption. For the total form of the stress equation we present a new implicit procedure for updating the stress and other relevant variables. Also, the principle of virtual work is linearized to obtain a consistent, closed-from elasto-viscoplastic tangent operator (the ‘Jacobian’) for use in solving for global balance of linear momentum in implicit, two-point, deformation driven finite element algorithms. The time integration algorithm is implemented in the finite element program ABAQUS. To check the accuracy and stability of the algorithm, some representative problems involving large, pure elastic and combined elastic-plastic deformations are solved.

On a stress resultant geometrically exact shell model. Part IV: variable thickness shells with through-the-thickness stretching

Abstract: This paper in concerned with the extension of the shell theory and numerical analysis presented in Part I, II and III to include finite thickness stretch and initial variable thickness. These effects play a significant role in problems involving finite membrane strains, contact, concentrated surface loads and delamination (in composite shells). We show that a direct numerical implementation of the standard single extensible director shell model circumvents the need for rotational updates, but exhibits numerical ill-conditioning in the thin shell limit. A modified formulation obtained via a multiplicative split of the director field into an extensible and inextensible part is presented, which involves only a trivial modification of the weak form of the equilibrium equations considered in Part III, and leads to a perfectly well-conditioned formulation in the thin-shell limit. In sharp contrast with previous attempts in the context of the degenerated solid approach, the thickness stretch is an independent field, not a dependent variable updated iteratively via the plane stress condition. With regard to numerical implementation, an exact update procedure which automatically ensures that the thickness stretch remains positive is presented. For the present theory, standard displacement models would exhibit ‘locking’ in the incompressible limit as a result of the essentially three-dimensional character of the constitutive equations. A mixed formulation is described which circumvents this difficulty. Numerical examples are presented that illustrate the effects of the thickness stretch, the performance of the proposed mixed interpolation, and the well-conditioned response exhibited by the present approach in the thin-shell (inextensible director) limit.

Space-time finite element methods for second-order hyperbolic equations

Abstract: Space-time finite element methods are presented to accurately solve elastodynamics problems that include sharp gradients due to propagating waves. The new methodology involves finite element discretization of the time domain as well as the usual finite element discretization of the spatial domain. Linear stabilizing mechanisms are included which do not degrade the accuracy of the space-time finite element formulation. Nonlinear discontinuity-capturing operators are used which result in more accurate capturing of steep fronts in transient solutions while maintaining the high-order accuracy of the underlying linear algorithm in smooth regions. The space-time finite element method possesses a firm mathematical foundation in that stability and convergence of the method have been proved. In addition, the formulation has been extended to structural dynamics problems and can be extended to higher-order hyperbolic systems.

A single surface contact algorithm for the post-buckling analysis of shell structures

Abstract: In some of our applications we are interested in how a structure behaves after it buckles. When a structure collapses completely, a single surface may buckle enough that it comes into contact with itself. The traditional approach of defining master and slave contact surfaces will not work because we do not know a priori how to partition the surface of the structure. This paper presents a contact algorithm that requires only a single surface definition for its input.

Mixed finite element methods for elliptic problems

Abstract: This paper treats the basic ideas of mixed finite element methods at an introductory level. Although the viewpoint presented is that of a mathematician, the paper is aimed at practitioners and the mathematical prerequisites are kept to a minimum. A classification of variational principles and of the corresponding weak formulations and Galerkin methods—displacement, equilibrium and mixed—is given and illustrated through four significant examples. The advantages and disadvantages of mixed methods are discussed. The concepts of convergence, approximability and stability and their interrelations are developed, and a resume is given of the stability theory which governs the performance of mixed methods. The paper concludes with a survey of techniques that have been developed for the construction of stable mixed methods and numerous examples of such methods.

Cam-Clay plasticity, part I: implicit integration of elastoplastic constitutive relations

Abstract: Two stress integration algorithms based on implicit calculation of plastic strain are implemented and tested for the modified Cam-Clay plasticity model. The integration rules fall under the category of return mapping algorithms in which the return directions are computed by closest point projection for associative flow rule and by central return mapping for non-associative flow rule applied to the Cam-Clay ellipsoids. Stress updates take place at the Gauss points upon enforcement of the consistency condition in which the appropriate consistency parameters are determined iteratively on the scalar level. Numerical examples with geotechnical applications, which include an analysis of foundation bearing capacity and an investigation of deformations in vertical cuts, are discussed to demonstrate the global accuracy and stability of the numerical solution. The relationships among various return mapping schemes are discussed in the context of both associative and non-associative flow rule formulations.

The p - and h-p version of the finite element method, an overview

Abstract: We survey the advances in the p- and the h-p versions of the finite element method. An up-to-date list of references related to these methods is provided.

A discourse on the stability conditions for mixed finite element formulations

Abstract: We discuss the general mathematical conditions for solvability, stability and optimal error bounds of mixed finite element discretizations. Our objective is to present these conditions with relatively simple arguments. We present the conditions for solvability and stability by considering the general coefficient matrix of mixed finite element discretizations, and then deduce the conditions for optimal error bounds for the distance between the finite element solutions and the exact solution of the mathematical problem. To exemplify our presentation we consider the solutions of various example problems. Finally, we also present a numerical test that is useful to identify numerically whether, for the solution of the general Stokes flow problem, a given finite element discretization satisfies the stability and optimal error bound conditions.

Dynamic strain localization in elasto-(visco-)plastic solids, Part 1. General formulation and one-dimensional examples

Abstract: Viscoplasticity is introduced as a procedure to regularize the elasto-plastic solid, especially for those situations in which the underlying inviscid material exhibits instabilities which preclude further analysis of initial-value problems. The procedure is general, and therefore has the advantage of allowing the regularization of any inviscid elastic-plastic material. Rate-dependency is shown to naturally introduce a length-scale into the dynamical initial-value problem. Furthermore, the width of the localized zones in which high strain gradients prevail and strain accumulations take place, is shown to be proportional to the characteristic length c η, which is the distance the elastic wave travels in the characteristic time η. Viscosity can thus be viewed either as a regularization parameter (computational point of view), or as a substructural/micromechanical parameter to be determined from observed shear-band widths (physical point of view). Finally, from a computational point of view, the proposed approach is shown to have striking advantages: (1) the wave speeds always remain real (even in the softening regime) and are set by the elastic moduli; (2) the elasto-(visco-)plastic constitutive equations are amenable to unconditionally stable integration; (3) the resulting well-posedness of the dynamical initial-value problem guarantees stable and convergent solutions with mesh refinements. The initial-value problems reported in this first part are essentially one-dimensional. They are used because they offer the simplest possible context to illustrate both the physical and computational significance of the proposed viscoplastic regularization procedure. The methods used in multi-dimensional analysis and examples will be reported in Part 2.

A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equation

Abstract: A velocity pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations

Least-squares finite element method for fluid dynamics

Abstract: An overview is given of new developments of the least squares finite element method (LSFEM) in fluid dynamics. Special emphasis is placed on the universality of LSFEM; the symmetry and positiveness of the algebraic systems obtained from LSFEM; the accommodation of LSFEM to equal order interpolations for incompressible viscous flows; and the natural numerical dissipation of LSFEM for convective transport problems and high speed compressible flows. The performance of LSFEM is illustrated by numerical examples.

Stabilised bilinear—constant velocity—pressure finite elements for the conjugate gradient solution of the Stokes problem

Abstract: In this paper, a sufficient condition for the stability of low-order mixed finite element methods is introduced. To illustrate the possibilities, two stabilisation procedures for the popular Q 1 −P 0 mixed method are theoretically analysed. The effectiveness of these procedures in practice is assessed by comparing results with those obtained using a conventional penalty formulation, for a standard test problem. It is demonstrated that with appropriate stabilisation, efficient iterative solution techniques of conjugate gradient type can be applied directly to the discrete Stokes system.

Least-square finite elements for Stokes problem

Abstract: A least-squares method based on the first-order velocity-pressure-vorticity formulation for the Stokes problem is proposed. This method leads to a minimization problem rather than to a saddle-point problem. The choice of the combinations of elements is thus not subject to the Ladyzhenskaya-Babuska-Brezzi (LBB) condition. Numerical results are given for the optimal rate of convergence for equal-order interpolations.

A Legendre spectral element method for simulation of unsteady incompressible viscous free-surface flows

Abstract: A new Legendre spectral element method is presented for the solution of viscous incompressible free-surface flows. It is based on the following extensions of the fixed-domain spectral element method: use of the full viscous stress tensor for natural imposition of traction (surface tension) boundary conditions; use of arbitrary-Lagrangian-Eulerian methods for accurate representation of moving boundaries; and use of semi-implicit time-stepping procedures to partially decouple the free-surface evolution and interior Navier-Stokes equations. For purposes of analysis and clarity of presentation, attention is focused on the stability of falling films. Analysis of the spectrum of the linear stability problem (Orr-Sommerfeld equation) associated with film flow reveals physical effects that limit the stability of semi-implicit schemes and suggests optimal formulas for temporal discretization of the spectral element equations. Detailed results are presented for the spectral element simulation of the film flow problem.

Optimal error analysis of spectral methods with emphasis on non-constant coefficients and deformed geometries

Abstract: The numerical analysis of spectral methods when non-constant coefficients appear in the equation, either due to the original statement of the equations or to take into account the deformed geometry, is presented. Particular attention is devoted to the optimality of the discretization even for low values of the discretization parameter. The effect of some overintegration is also addressed, in order to possibly improve the accuracy of the discretization.

The spectral overlay on finite elements for problems with high gradients

Abstract: A methodology has been developed for enhancing the accuracy of finite element solutions of problems with high gradients. This is accomplished by superimposing the spectral approximation on subdomains which overlay the finite element mesh in regions where high gradients are indicated by the solution. The orientation and shape of these subdomains are quite arbitrary. Results are obtained for a viscoplastic bar which is stretched beyond its bifurcation point. The results show that with a modest increase in the number of spectral unknowns, it is possible to achieve high resolution of the strain field in the shear band.

Shock capturing by the spectral viscosity method

Abstract: A main disadvantage of using spectral methods for nonlinear conservation laws lies in the formation of Gibbs phenomenon, once spontaneous shock discontinuities appear in the solution. The global nature of spectral methods then pollutes the unstable Gibbs oscillations over all the computational domain, and the lack of entropy dissipation prevents convergences in these cases. In this paper, we discuss the spectral viscosity method, which is based on high frequency-dependent vanishing viscosity regularization of the classical spectral methods. We show that this method enforces the convergence of nonlinear spectral approximations without sacrificing their overall spectral accuracy.

A Priori Estimates for Mixed Finite Element Methods for the Wave Equation

Abstract: This paper treats mixed methods for second order hyperbolic equations. The convergence of a mixed method continuous-time scheme for the hyperbolic problem is reduced to a question of convergence of the associated elliptic problem. Stability conditions are also derived for a conditionally stable explicit scheme. Numerical experiments are presented that verify the theoretical rates of convergence and compare two of the discrete schemes discussed.

Adaptive finite element methods for diffusion and convection problems

Abstract: We give a survey of recent results obtained together with K. Eriksson on adaptive h-methods for the basic linear partial differential equations of elliptic, parabolic and hyperbolic type. Our adaptive algorithms are based on a posteriori error estimates leading to reliable methods, and comparison with sharp a priori error estimates is made to prove efficiency of the procedures.

Predictor-corrector procedures for stress and free vibration analysis of multilayered composite plates and shells

Abstract: A study is made of two predictor-corrector procedures for the accurate determination of the global, as well as detailed, static and vibrational response characteristics of plates and shells. Both procedures use first-order shear deformation theory in the predictor phase, but differ in the elements of the computational model being adjusted in the corrector phase. The first procedure calculates a posteriori estimates of the composite correction factors and uses them to adjust the transverse shear stiffnesses of the plate (or shell). The second procedure calculates a posteriori the functional dependence of the displacement components on the thickness coordinate. The corrected quantities are then used in conjunction with the three-dimensional equations to obtain better estimates for the different response quantities. Extensive numerical results are presented showing the effects of variation in the geometric and lamination parameters for antisymmetrically laminated anisotropic plates, and simply supported multilayered orthotropic cylinders, on the accuracy of the linear static and free vibrational responses obtained by the predictor-corrector procedures. Comparison is also made with the solutions obtained by other computational models based on two-dimensional shear deformation theories. For each problem the standard of comparison is taken to be the analytic three-dimensional elasticity solution. The numerical examples clearly demonstrate the accuracy and effectiveness of the predictor-corrector procedures.

Iterative solvers by substructuring for the p -version finite element method

Abstract: We study a class of substructuring methods well-suited for iterative solution of large systems of linear equations arising from the p-version finite element method. The p-version offers a natural decomposition with every element treated as a substructure. We use the preconditioned conjugate gradient method with preconditioning constructed by a decomposition of the local function space on each element. We develop an elementary theory giving bounds on the condition numbers which do not depend on the number of elements if a sparse system with only few variables per element is solved in each iteration. This bound can be evaluated considering one element at a time and we compute such condition numbers numerically for various elements.

Dynamic strain localization in elasto-(visco-)plastic solids, Part 2. Plane strain examples

Abstract: Viscoplasticity is introduced as a procedure to regularize the elastic-plastic solid, especially for those situations in which the underlying inviscid material exhibits instabilities which preclude meaningful analysis of the initial-value problem. The procedure is general and therefore has the advantage of allowing the regularization of any inviscid elastic-plastic material. Rate dependence is shown to naturally introduce a length-scale that sets the width of the shear bands in which the deformations localize and high strain gradients prevail. Then, provided that the element size is appropriate for an adequate description of the shear band geometry, the numerical solutions are shown to be pertinent. Stable and convergent solutions with mesh refinements are obtained which are shown to be devoid of spurious mesh length-scale effects. The numerical framework adopted for this study is realistic and relevant to the solution of large scale nonlinear problems. An efficient explicit time stepping algorithm is used to advance the solution in time, and low-order finite elements with only one stress-point are used. An unconditionally stable stress-point algorithm is used to integrate the nonlinear elasto-(visco-) plastic constitutive equations. Therefore, the only numerical restriction of the proposed computational procedure stems from a time step size restriction which emanates from the explicit time integration of the equations of motion. However, since the wave speeds remain elastic, this restriction is trivially dealt with, resulting in a most efficient computational procedure.

Spectral element—Fourier methods for incompressible turbulent flows

Abstract: A mixed spectral element-(Fourier) spectral method is proposed for solution of the incompressible Navier-Stokes equations in general, curvilinear domains. The formulation is appropriate for simulations of turbulent flows in complex geometries with only one homogeneous flow direction. The governing equations are written in a form suitable for both direct (DNS) and large-eddy (LES) simulations allowing a unified implementation. The method is based on skew-symmetric convective operators that induce minimal aliasing errors and fast Helmholtz solvers that employ efficient iterative algorithms (e.g. multigrid). Direct numerical simulations of channel flow verified that the proposed method can sustain turbulent fluctuations even at ‘marginal’ Reynolds numbers. The flexibility of the method to efficiently simulate complex-geometry flows is demonstrated through an example of transitional flow in a grooved channel and an example of transitional-turbulent flow over rough wall surfaces.

Compressible and incompressible flow; an algorithm for all seasons

Abstract: A semi-implicit algorithm is presented which allows the solution of both incompressible and compressible flows to be achieved in a single code. Both transient and steady state solutions are available and compressibility no longer dictates the time step limits. Difficulties of compressible flow with uniform interpolation which exist due to local incompressibility in low velocity regions are avoided. The transition to the fully explicit form at high Mach number flows can be accomplished automatically yielding a form slightly different from that of conventional procedures. The algorithm is available for shallow water equations where its advantages promise to be large. Several examples illustrate the paper.

An arbitrary condensing, noncondensing solution strategy for large scale, multi-zone boundary element analysis

Abstract: A multi-zone boundary element analysis (BEA) capability that includes substructuring and condensation in a completely general fashion is presented. This condensation procedure is shown to be an effective way to perform blocked matrix factorizations using a reduced amount of high speed computer memory, and an approach that largely removes the effect of the boundary element zone numbering scheme on the computational resources expended due to block fill-in. In iterative problems with changing configuration, the strategy of condensing (substructuring) the unchanging portion of an overall model, in an exact fashion, and subsequently iterating on the resulting reduced model, is shown to have the potential for extending the range of such iterative problems. The approach will also allow for the simultaneous condensation and subsequent expansion of multiple boundary element zones on computers with parallel processing facility. The overall algorithm is described that allows for the assembly and solution of boundary element zones connected in a quite general way that may also be arbitrarily either condensed or maintained at their original size. The approach thus allows for both condensed and uncondensed boundary element zones to consistently coexist in the same multi-zone problem. A consistent and general formulation for the treatment of the double values of traction components at boundary element zone corners is also presented. Sample problems are described to demonstrate the efficiency and usefulness of the resulting capability.

An error analysis of least-squares finite element method of velocity-pressure-vorticity formulation for Stokes problem

Abstract: A theoretical proof of the optimal rate of convergence for the least-squares method is developed for the Stokes problem based on the velocity-pressure-vorticity formula. The 2D Stokes problem is analyzed to define the product space and its inner product, and the a priori estimates are derived to give the finite-element approximation. The least-squares method is found to converge at the optimal rate for equal-order interpolation.

Lax-stability of fully discrete spectral methods via stability regions and pseudo-eigenvalues

Abstract: In many calculations, spectral discretization in space is coupled with a standard ordinary differential equation formula in time To analyze the stability of such a combination, one would like simply to test whether the eigenvalues of the spatial discretization operator (appropriately scaled by the time step k) lie in the stability region for the ode formula, but it is well known that this kind of analysis is in general invalid In the present paper we rehabilitate the use of stability regions by proving that a discrete linear multistep ‘method of lines’ approximation to a partial differential equation is Lax-stable, within a small algebraic factor, if and only if all of the e-pseudo-eigenvalues of the spatial discretization operator lie within O(e) of the stability region as e → 0 An e-pseudo-eigenvalue of a matrix A is any number that is an eigenvalue of some matrix A + E with ∥E ∥ ⩽ e; our arguments make use of resolvents and are closely related to the Kreiss matrix theorem As an application of our general result, we show that an explicit N-point Chebyshev collocation approximation of ut = −xux on [−1, 1] is Lax-stable if and only if the time step satisfies k = O(N−2), although eigenvalue analysis would suggest a much weaker restriction of the form k ⩽ CN−1