Showing papers on "Finite element method published in 1990"

Stochastic Finite Elements: A Spectral Approach

Abstract: Representation of stochastic processes stochastic finite element method - response representation stochastic finite element method - response statistics numerical examples.

Approximation scheme with applications to computational fluid-dynamics-- i surface approximations and partial derivative estimates

Abstract: A~traet--We present a powerful, enhanced multiquadrics (MQ) scheme developed for spatial approximations. MQ is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions. It is continuously differentiable and integrable and is capable of representing functions with steep gradients with very high accuracy. Monotonicity and convexity are observed properties as a result of such high accuracy. Numerical results show that our modified MQ scheme is an excellent method not only for very accurate interpolation, but also for partial derivative estimates. MQ is applied to a higher order arbitrary Lagrangian-Eulerian (ALE) rezoning. In the second paper of this series, MQ is applied to parabolic, hyperbolic and elliptic partial differential equations. The parabolic problem uses an implicit time-marching scheme whereas the hyperbolic problem uses an explicit time-marching scheme. We show that MQ is also exceptionally accurate and efficient. The theory of Madych and Nelson shows that the MQ interpolant belongs to a space of functions which minimizes a semi-norm and gives credence to our results. 1. BACKGROUND The study of arbitrarily shaped curves, surfaces and bodies having arbitrary data orderings has immediate application to computational fluid-dynamics. The governing equations not only include source terms but gradients, divergences and Laplacians. In addition, many physical processes occur over a wide range of length scales. To obtain quantitatively accurate approximations of the physics, quantitatively accurate estimates of the spatial variations of such variables are required. In two and three dimensions, the range of such quantitatively accurate problems possible on current multiprocessing super computers using standard finite difference or finite element codes is limited. The question is whether there exist alternative techniques or combinations of techniques which can broaden the scope of problems to be solved by permitting steep gradients to be modelled using fewer data points. Toward that goal, our study consists of two parts. The first part will investigate a new numerical technique of curve, surface and body approximations of exceptional accuracy over an arbitrary data arrangement. The second part of this study will use such techniques to improve parabolic, hyperbolic or elliptic partial differential equations. We will demonstrate that the study of function approximations has a definite advantage to computational methods for partial differential equations. One very important use of computers is the simulation of multidimensional spatial processes. In this paper, we assumed that some finite physical quantity, F, is piecewise continuous in some finite domain. In many applications, F is known only at a finite number of locations, {xk: k = 1, 2 ..... N} where xk = x~ for a univariate problem, and Xk = (x~,yk .... )X for the multivariate problem. From a finite amount of information regarding F, we seek the best approximation which can not only supply accurate estimates of F at arbitrary locations on the domain, but will also provide accurate estimates of the partial derivatives and definite integrals of F anywhere on the domain. The domain of F will consist of points, {xk }, of arbitrary ordering and sub-clustering. A rectangular grid is a very special case of a data ordering. Let us assume that an interpolation function, f, approximates F in the sense that

The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case

Abstract: In this paper we study the two-dimensional version of the Runge- Kutta Local Projection Discontinuous Galerkin (RKDG) methods, already de- fined and analyzed in the one-dimensional case. These schemes are defined on general triangulations. They can easily handle the boundary conditions, verify maximum principles, and are formally uniformly high-order accurate. Prelimi- nary numerical results showing the performance of the schemes on a variety of initial-boundary value problems are shown.

A class of mixed assumed strain methods and the method of incompatible modes

Abstract: A three-field mixed formulation in terms of displacements, stresses and an enhanced strain field is presented which encompasses, as a particular case, the classical method of incompatible modes. Within this frame-work, incompatible elements arise as particular ‘compatible’ mixed approximations of the enhanced strain field. The conditions that the stress interpolation contain piece-wise constant functions and be L2-ortho-gonal to the enhanced strain interpolation, ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation. The preceding conditions are formulated in a form particularly convenient for element design. As an illustration of the methodology three new elements are developed and shown to exhibit good performance: a plane 3D elastic/plastic QUAD, an axisymmetric element and a thick plate bending QUAD. The formulation described herein is suitable for non-linear analysis.

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.

Finite-element approximations of the nonstationary Navier-Stokes problem. Part IV: error estimates for second-order time discretization

Abstract: This paper provides an error analysis for the Crank–Nicolson method of time discretization applied to spatially discrete Galerkin approximations of the nonstationary Navier–Stokes equations. Second-order error estimates are proven locally in time under realistic assumptions about the regularity of the solution. For approximations of an exponentially stable solution, these local error estimates are extended uniformly in time as ${\text{t}} \to \infty $.

Introduction to finite element vibration analysis

Abstract: 1 Formulation of the equations of motion 2 Element energy functions 3 Introduction to the finite element displacement method 4 In-plane vibration of plates 5 Vibration of solids 6 Flexural vibration of plates 7 Vibration of stiffened plates and folded plate structures 8 Vibration of shells 9 Vibration of laminated plates and shells 10 Hierarchical finite element method 11 Analysis of free vibration 12 Forced response 13 Forced response II 14 Computer analysis technique

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.

Paving: A new approach to automated quadrilateral mesh generation

Abstract: This paper presents a new mesh generation technique, paving, which meshes arbitrary 2-D geometries with an all-quadrilateral mesh. Paving allows varying element size distributions on the boundary as well as the interior of a region. The generated mesh is well formed (i.e. nearly square elements, elements perpendicular to boundaries, etc.) and geometrically pleasing (i.e. mesh contours tend to follow geometric contours of the boundary). In this paper we describe the theory behind this algorithmic/heuristic technique, evaluate the performance of the approach and present examples of automatically generated meshes.

On the theory of semi‐implicit projection methods for viscous incompressible flow and its implementation via a finite element method that also introduces a nearly consistent mass matrix. Part 1: Theory

Abstract: Ever since the expansion of the finite element method (FEM) into unsteady fluid mechanics, the «consistent mass matrix» has been a relevant issue. Applied to the time-dependent incompressible Navier-Stokes equations, it virtually demands the use of implicit time integration methods in which full «velocity-pressure coupling» is also inherent. We re-introduce the consistent mass matrix into some semi-implicit projection methods in such a way that the cost advantage of lumped mass and the accuracy advantage of consistent mass are simultaneously realized

Simulation of piezoelectric devices by two- and three-dimensional finite elements

Abstract: A method for the analysis of piezoelectric media based on finite-element calculations is presented in which the fundamental electroelastic equations governing piezoelectric media are solved numerically The results obtained by this finite-element calculation scheme agree with theoretical and experimental data given in the literature The method is applied to the vibrational analysis of piezoelectric sensors and actuators with arbitrary structure Natural frequencies with related eigenmodes of those devices as well as their responses to various time-dependent mechanical or electrical excitations are computed The theoretically calculated mode shapes of piezoelectric transducers and their electrical impedances agree quantitatively with interferometric and electric measurements The simulations are used to optimize piezoelectric devices such as ultrasonic transducers for medical imaging The method also provides deeper insight into the physical mechanisms of acoustic wave propagation in piezoelectric media >

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.

Fixed grid techniques for phase change problems: A review

Abstract: SUMMARY The aim of this paper is to categorize the major fixed grid formulations and solution methods for conduction controlled phase change problems. Using a two phase model of. a solid/liquid phase change, the basic enthalpy equation is derived. Starting from this equation, a number of alternative formulations are obtained. All the formulations are reduced to a standard form. From this standard form, finite element and finite volume discretizations are developed. These discretizations are used as the basis for a number of fixed grid numerical solution techniques for solidification phase change systems. In particular, various apparent capacity and source based enthalpy methods are explored.

The quantum transmitting boundary method

Abstract: A numerical algorithm for the solution of the two‐dimensional effective mass Schrodinger equation for current‐carrying states is developed. Boundary conditions appropriate for such states are developed and a solution algorithm constructed that is based on the finite element method. The utility of the technique is illustrated by solving problems relevant to submicron semiconductor quantum device structures.

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.

A test problem for outflow boundary conditions—flow over a backward-facing step

Abstract: A numerical solution for steady incompressible flow over a two-dimensional backward-facing step is developed using a Galerkin-based finite element method. The Reynolds number for the simulations is 800. Computations are performed on an extended channel length to minimize the effect of the outflow boundary on the upstream recirculation zones. A thorough mesh refinement study is performed to validate the results. Extensive profile data at several channel locations are provided to allow future testing and evaluation of outflow boundary conditions.

Introduction to Finite Elements in Engineering

Abstract: Fundamental concepts matrix algebra and Gaussian elimination one-dimensional problems trusses two-dimensional problems using constant strain triangles axisymmetric solids subjected to axisymmetric loading two-dimensional isoparametric elements and numerical integration beams and frames three-dimensional problems in stress analysis scalar field problems dynamic considerations preprocessing and postprocessing.

Rotordynamics prediction in engineering

Abstract: Part 1 Characteristics of rotor elements: disc shaft bearings and seals mass unbalance. Part 2 Simple models - basic phenomena: determination of the model symmetric rotor dissymmetric rotor instability damped rotors. Part 3 Rotor equations - solutions of equations: finite elements of rotor elements equations for monorotors equations for coaxial multirotors solution of equations computer program. Part 4 Towards industrial applications: pseudo-modal method influence of modeling natural frequencies - bearing stiffness diagram transmissibility dual rotor. Part 5 Industrial applications: steam compressor high pressure centrifugal compressor low pressure centrifugal compressor centrifugal compressor fitted with active magnetic bearings power gas turbine industrial steam turbine textile machine spindle jet engine. Part 6 Transient motions: equations and solutions speed of rotation law behaviour of a simple rotor experimental and theoretical results for a three disc rotor behaviour of an industrial rotor. Part 7 Torsion: determination from statics of the minimum shaft radius branched systems in torsion analysis of a branched system in torsion industrial electric motor-centrifugal compressor unit. Part 8 Miscellaneous topics: dissymmetric shaft equations with periodic coefficients axial torque influence.

A finite element/control volume approach to mold filling in anisotropic porous media

Abstract: Mold filling in anisotropic porous media is the governing phenomena in a number of composite manufacturing processes, such as resin transfer molding (RTM) and structural reaction injection molding (SRIM). In this paper we present a numerical simulation to predict the flow of a viscous fluid through a fiber network. The simulation is based on the finite element/control volume method. It can predict the movement of a free surface flow front in a thin shell mold geometry of arbitrary shape and with varying thickness. The flow through the fiber network is modeled using Darcy's law. Different permeabilities may be specified in the principal directions of the preform. The simulation permits the permeabilities to vary in magnitude and direction throughout the medium. Experiments were carried out to measure the characteristic permeabilities of fiber preforms. The results of the simulation are compared with experiments performed in a flat rectangular mold using a Newtonian fluid. A variety of preforms and processing conditions were used to verify the numerical model.

Finite Element Methods for Fluids

Abstract: Partial differential equations of fluid mechanics irrotational and weakly irrotational flows convention - diffusion phenomena the Stokes problem the Navier-Stokes equations Euler, compressible Navier-Stokes and the shallow water equations Appendix: a finite element program for fluids on the Macintosh

On the finite volume element method

Abstract: The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH 1 semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h 2). Results on the effects of numerical integration are also included.

Finite volume method for prediction of fluid flow in arbitrarily shaped domains with moving boundaries

Abstract: In this paper a method is presented that can be used for both the Lagrangian and the Eulerian solution of the Navier–Stokes equations in a domain of arbitrary shape, bounded by boundaries which move in any prescribed time-varying fashion. The method uses the integral form of the governing equations for an arbitrary moving control volume, with pressure and Cartesian velocity components as dependent variables. Care is taken to also satisfy the space conservation law, which ensures a fully conservative computational procedure. Fully implicit temporal differencing makes the method stable for any time step. A detailed description is provided for the discretization in two dimensions, with a collocated arrangement of variables. Central differences are used to evaluate both the convection and diffusion fluxes. The well known SIMPLE algorithm is employed for pressure–velocity coupling. The resulting algebraic equation systems are solved iteratively in a sequential manner. Results are presented for a flow in a channel with a moving indentation; they show favourable agreement with experimental observations.

Finite element methdos for calculation of steady, viscoelastic flow using constitutive equations with a Newtonian viscosity

Abstract: Adding a Newtonian solvent to most differential viscoelastic constitutive equations mathematically regularizes the coupled set formed by the momentum, continuity and constitutive equations. The momentum and continuity equations form an elliptic saddle point problem for velocity and pressure, and the constitutive equation is hyperbolic in stress. Three finite element algorithms are presented that exploit this elliptic behavior by expressing the momentum equation in different differential forms. The first is based on the viscous elliptic operator that arises naturally with the introduction of a Newtonian solvent viscosity; the second is based on the explicitly elliptic momentum equation formulation developed for the upper-convected Maxwell model; and the third is based on an elastic-viscous splitting of the momentum equation. Finite element discretizations are created by using Galerkin's method for the momentum and continuity equations and the streamline-upwind Petrov-Galerkin method for the components of the constitutive equation. In the latter two methods, additional interpolants are introduced so as to maintain continuous representations of velocity derivatives across element boundaries; this is a requirement if only those higher-order velocity terms which are explicitly elliptic are integrated by parts. Calculations for flow between eccentric cylinders and through a corrugated tube demonstrate the numerical stability and accuracy of each of the formulations. The robustness of each algorithm for calculation of flows at high Deborah numbers depends on the value of the ratio β ≡ ηS/gh0, where ηs is the solvent viscosity and η0 is the viscosity of the solution. The algorithm based on the elastic-viscous splitting is the most robust across the full range 0 ≤ β ≤ 1.

Computational Fluid Dynamics: An Introduction for Engineers

Abstract: 1. Introduction. 2. The simplest description of continuous fluid flow. 3. The finite difference method. 4. Diffusion problems. 5. Transport as a combination of addiction and diffusion. 6. Descriptions of unsteady flows. 7. Solution methods for unsteady free surface flows. 8. Equilibrium methods. 9. Computational fluid dynamics of turbulence. 10. An introduction to some other numerical methods. Index.