Showing papers in "Applied Mechanics and Engineering in 1987"
TL;DR: In this paper, a fully three-dimensional finite-strain viscoelastic model is developed, characterized by general anisotropic response, uncoupled bulk and deviatoric response over any range of deformations, general relaxation functions, and recovery of finite elasticity for very fast or very slow processes; in particular, classical models of rubber elasticity (e.g. Mooney-Rivlin).
Abstract: A fully three-dimensional finite-strain viscoelastic model is developed, characterized by: (i) general anisotropic response, (ii) uncoupled bulk and deviatoric response over any range of deformations, (iii) general relaxation functions, and (iv) recovery of finite elasticity for very fast or very slow processes; in particular, classical models of rubber elasticity (e.g. Mooney-Rivlin). Continuum damage mechanics is employed to develop a simple isotropic damage mechanism, which incorporates softening behavior under deformation, and leads to progressive degradation of the storage modulus in a cyclic test with increasing amplitude (Mullins' effect). A numerical integration procedure is proposed which trivially satisfies objectivity and bypasses the use of midpoint configurations. The resulting algorithm can be exactly linearized in closed form, and leads to symmetric tangent moduli. Quasi-incompressible response is accounted for within the context of a three-field variational formulation of the Hu-Washizu type.
911 citations
TL;DR: In this paper, a bifurcation analysis is used to determine the geometry of the localized deformation modes, and suitably defined shape functions are added to the element interpolation which closely reproduce the localized modes.
Abstract: A method is proposed which aims at enhancing the performance of general classes of elements in problems involving strain localization. The method exploits information concerning the process of localization which is readily available at the element level. A bifurcation analysis is used to determine the geometry of the localized deformation modes. When the onset of localization is detected, suitably defined shape functions are added to the element interpolation which closely reproduce the localized modes. The extra degrees of freedom representing the amplitudes of these modes are eliminated by static condensation. The proposed methodology can be applied to 2-D and 3-D problems involving arbitrary rate-independent material behavior. Numerical examples demonstrate the ability of the method to resolve the geometry of localized failure modes to the highest resolution allowed by the mesh.
613 citations
TL;DR: Symmetric finite element formulations are proposed for the primitive-variables form of the Stokes equations and shown to be convergent for any combination of pressure and velocity interpolations as mentioned in this paper.
Abstract: Symmetric finite element formulations are proposed for the primitive-variables form of the Stokes equations and shown to be convergent for any combination of pressure and velocity interpolations. Various boundary conditions, such as pressure, are accommodated.
415 citations
TL;DR: Several examples involving shock-shock interactions and the impact of shocks on structures demonstrate the performance of the adaptive finite element scheme for transient problems, indicating that considerable savings in CPU time and storage can be realized even for strongly unsteady flows.
Abstract: An adaptive finite element scheme for transient problems is presented. The classic h-enrichment / coarsening is employed in conjunction with a triangular finite element discretization in two dimensions. A mesh change is performed every n timesteps, depending on the Courant number employed and the number of ‘protective layers’ added ahead of the refined region. In order to simplify the refinement/ coarsening logic and to be as fast as possible, only one level of refinement/coarsening is allowed per mesh change. A high degree of vectorizability has been achieved on the CRAY XMP 12 at NRL. Several examples involving shock-shock interactions and the impact of shocks on structures demonstrate the performance of the method, indicating that considerable savings in CPU time and storage can be realized even for strongly unsteady flows.
345 citations
TL;DR: In this article, an SUPG-type finite element method for linear symmetric multidimensional advective-diffusive systems is described and analyzed, and optimal and near optimal error estimates are obtained for the complete range of ADD behavior.
Abstract: An SUPG-type finite element method for linear symmetric multidimensional advective-diffusive systems is described and analyzed. Optimal and near optimal error estimates are obtained for the complete range of advective-diffusive behavior.
323 citations
TL;DR: Some nonstandard finite element spaces are introduced, which, though based on the usual square bilinear elements, permit local mesh refinement and an “equivalent estimator” for the H 1 finite element error is developed.
Abstract: This paper is the first in a series of two in which we discuss some theoretical and practical aspects of a feedback finite element method for solving systems of linear second-order elliptic partial differential equations (with particular interest in classical linear elasticity). In this first part we introduce some nonstandard finite element spaces, which, though based on the usual square bilinear elements, permit local mesh refinement. The algebraic structure of these spaces and their approximation properties are analyzed. An “equivalent estimator” for the H 1 finite element error is developed. In the second paper we shall discuss the asymptotic properties of the estimator and computational experience.
280 citations
TL;DR: The advantages and difficulties of using various composite grid schemes are reviewed and a trend in computational aerodynamics has been toward the use of composite grids.
Abstract: In finite difference flow field simulations the use of a single well-ordered body-conforming curvilinear mesh can lead to efficient solution procedures. However, it is generally impractical to build a single grid of this type for complex three-dimensional aircraft configurations. As a result, a trend in computational aerodynamics has been toward the use of composite grids. Composite grids use more than one grid to mesh an overall configuration with each individual subgrid of the system patched or overset together. Because each individual subgrid in the system is well ordered, the overall grid is suitable for efficient finite difference solution using vectorized or multitasking computers. Some of the advantages and difficulties of using various composite grid schemes are reviewed in this paper.
224 citations
TL;DR: In this article, a vectorized implementation of the EBE preconditioned conjugate gradients (PCG) algorithm is presented in the context of a nonlinear stress analysis code nike 3 d.
Abstract: The major costs of large implicit finite element calculations, particularly in three dimensions, arise from computing solutions to systems of linear equations. Direct methods, i.e., those based upon Gaussian elimination, can easily require prohibitively large amounts of both CPU time and storage, even on current supercomputers. Iterative procedures avoiding the formation and factorization of a global system of equations can circumvent these difficulties. The element-by-element (EBE) preconditioned conjugate gradients (PCG) algorithm is presented in the context of a vectorized implementation within the production nonlinear stress analysis code nike 3 d . Due to continued confusion as to the ease of vectorizing finite element procedures, we include examples of the main EBE subroutines in their entirety. The concept of a fractal dimension of a finite element mesh is introduced, and proves useful in characterizing the efficiency of this iterative algorithm with respect to a variable band, active column direct method. Sample calculations on a Cray X-MP/48 with solid-state storage device (SSD) illustrate the economy and range of applicability of EBE/PCG. Asymptotic cost formulae derived for two linear problems underscore differences between the direct and iterative algorithms for large problems and lead to predictions of problem size limitations imposed by the computing environment.
192 citations
TL;DR: In this article, an extension of the dual reciprocity boundary element method (DRBEM) to nonlinear diffusion problems is presented, in which thermal conductivity, specific heat, and density coefficients are all functions of temperature.
Abstract: This paper presents an extension of the dual reciprocity boundary element method (DRBEM) to deal with nonlinear diffusion problems in which thermal conductivity, specific heat, and density coefficients are all functions of temperature. The DRBEM, recently applied to the solution of problems governed by parabolic and hyperbolic equations, consists in the transformation of the differential equation into an integral equation involving boundary integrals only, the solution of which is achieved by employing a standard boundary element discretization coupled with a two-level finite difference time integration scheme. Contrary to previous formulations for the diffusion equation, the dual reciprocity BEM utilizes the well-known fundamental solution to Laplace's equation, which is space-dependent only. This avoids complex time integrations that normally appear in formulations employing time-dependent fundamental solutions, and permits accurate numerical solutions to be obtained in an efficient way. For nonlinear problems, the integral of conductivity is introduced as a new variable to obtain a linear diffusion equation in the Kirchhoff transform space. This equation involves a modified time variable which is itself a function of position. The problem is solved in an iterative way by using an efficient Newton-Raphson technique which is shown to be rapidly convergent.
182 citations
TL;DR: A review of adaptive grid generation is presented with an emphasis on the basic concepts and the interrelationship between the various methods, developed in a multifaceted progressive sense with enough detail so as to instill an operative spirit for the methods.
Abstract: The fundamental principles of adaptive grid generation for the numerical analysis of physical phenomena described by systems of partial differential equations are examined in an analytical review. Topics addressed include weight functions, equidistribution in one dimension, the specification of coefficients in the linear weight, the attraction to a given grid on a curve, evolutionary forces, and metric notation. Consideration is given to curve-by-curve methods, finite-volume methods, variational methods, and temporal aspects.
174 citations
TL;DR: In this paper, the authors presented some weak formulations for an eigenvalue problem in electromagnetism, in particular, mixed and penalty formulations, and some numerical results based on the mixed finite element method.
Abstract: In this paper, we will present some weak formulations (variational formulations) for an eigenvalue problem in electromagnetism. In particular, mixed and penalty formulations are discussed, and some numerical results based on the mixed finite element method are given.
TL;DR: In this article, a general three-dimensional elliptic grid generation system is discussed, which includes automatic evaluation of control functions from an algebraic grid or interpolated from boundary point distributions.
Abstract: A general three-dimensional elliptic grid generation system is discussed. This system includes automatic evaluation of control functions from an algebraic grid or interpolated from boundary point distributions. Also included is iterative adjustment of control functions for boundary orthogonality. The solution is by point SOR iteration using a field of optimum acceleration parameters. This system is incorporated in a code that can treat any physical configuration by using a multiblock structure with complete continuity across interfaces.
TL;DR: In this paper, an accurate numerical algorithm for the integration of constitutive equations under both large deformations and/or large rotations is presented, based on the choice of the unrotated configuration as the frame of reference for all constitutive equation.
Abstract: An accurate numerical algorithm for the integration of constitutive equations under both large deformations and/or large rotations is presented. The algorithm is based on the choice of the unrotated configuration as the frame of reference for all constitutive equations. The algorithm does not entail excessive computational time or memory expense. The accuracy of the method is demonstrated by several numerical examples.
TL;DR: In this article, the elastoplastic boundary value problem in terms of rates, formulated using integral equations of linear elasticity, is shown to be amenable, through discretizations, to a linear complementarity problem endowed with a symmetric matrix.
Abstract: The elastoplastic boundary value problem in terms of rates, formulated using integral equations of linear elasticity, is shown to be amenable, through discretizations, to a linear complementarity problem endowed with a symmetric matrix (in contrast to the traditional boundary element formulation). This symmetrization is achieved by suitably choosing the fundamental solutions and formulating the integral equations for boundary quantities and stresses, and by enforcing in a Galerkin weighted-residual sense (instead of collocation) the boundary integral equations and the plastic constitutive laws over domain cells.
TL;DR: In this paper, a recently proposed smeared crack model which properly handles nonorthogonal cracks is further elaborated, and it is proved that the model obeys the principle of material frame-indifference.
Abstract: A recently proposed smeared crack model which properly handles nonorthogonal cracks is further elaborated. It is proved that the model obeys the principle of material frame-indifference and some comments on possible stress-strain laws within a smeared crack are made. Algorithms are presented or indicated for the combination of plasticity and possibly multiple crack formation, for the combination of visco-elasticity and cracking, and for the combination of cracking and temperature-dependent material properties and phenomena like thermal dilatation and shrinkage.
TL;DR: In this article, an equation for the properties of the following fluids: air, liquid water, water vapor, carbon dioxide, Freon-12, engine oil, and mercury.
Abstract: Equations are developed for property data of seven common fluids. These equations are useful in computer applications where it is desirable to determine properties without using tables or requiring the user to input data during the execution of programs. An equation is found that accurately approximates the variation of the fluid property data with temperature and is in a simple and convenient form. Equations are presented for the properties of the following fluids: air, liquid water, water vapor, carbon dioxide, Freon-12, engine oil, and mercury. The properties presented in equation form are: density, dynamic viscosity, constant pressure specific heat, thermal conductivity, and Prandtl number. Properties are presented for wide ranges of temperature and at atmospheric pressure.
TL;DR: The present paper explores the use of approximations based upon the value of the constraint and its derivative at two points based on suggestions and tests for randomly generated rational constraint functions.
Abstract: The use of constraint approximations is recognized as a primary means of achieving computational efficiency in structural optimization Existing approximation methods are based upon the value of the constraint function and its derivatives at a single point The present paper explores the use of approximations based upon the value of the constraint and its derivative at two points Several candidate approximations are suggested and tested for randomly generated rational constraint functions* Several of the approximations prove to be superior to the single point approximations
TL;DR: In this article, a new Kirchhoff triangular plate-bending element with nine degrees of freedom was derived based on the free formulation of Bergan and Nygard, which satisfies the conditions of invariance and the patch test.
Abstract: A new Kirchhoff triangular plate-bending element with nine degrees of freedom is derived. The assumed shape functions are nonconforming and consist of a complete quadratic expansion complemented by three cubic modes that are energy-orthogonal to the quadratic expansion. The derivation is based on the free formulation of Bergan and Nygard, which satisfies the conditions of invariance and the patch test. An explicit expression is obtained for the inverse transformation matrix required in this formulation, which makes the stiffness computations extremely efficient. The aspect ratio sensitivity of the element under cylindrical bending modes can be lessened by scaling the higher-order stiffness through coefficients determined by a superlinear patch test technique. Numerical experiments indicate that the new element outperforms previously derived 9-dof triangular elements based on displacement modes.
TL;DR: In this article, a rate-independent constitutive theory for the behavior of concrete in the inelastic range is proposed, which is based on damage mechanics concepts previously applied to rock materials by the author.
Abstract: A rate-independent constitutive theory for the behaviour of concrete in the inelastic range is proposed. It is based on damage mechanics concepts previously applied to rock materials by the author. The inelasticity is provided by two basic damage mechanisms, namely, shear damage and hydrostatic tension damage. A scalar damage parameter is used to represent the degradation of the elastic properties. In addition to the damage mechanisms, a plasticity yield surface is included to bound the model in the hydrostatic compression sense. A simple calibration procedure is presented and the model is shown to predict reasonable behaviour for a variety of monotonic and reversed loading paths. The model is readily implemented in finite element codes and a number of representative boundary value problems is solved. Suggestions for future developments conclude the paper.
TL;DR: In this article, two C 0 curved beam elements based on the hybrid-mixed formulation are studied in the form of membrane-shear locking, mesh convergence, and stress predictions.
Abstract: Two C 0 curved beam elements based on the hybrid-mixed formulation are studied in the form of membrane-shear locking, mesh convergence, and stress predictions. At the element level, both the displacement and stress fields are approximated separately. The stress parameters are then eliminated from the stationary condition of the Hellinger-Reissner variational principle so that the standard stiffness equations are obtained. The stress functions are chosen from two important considerations: (i) kinematic deformation modes must be avoided, and (ii) the constraint index counting of the element, when applied to limiting cases, must be equal to or greater than one. Based on these considerations, two curved beam elements are derived by including the effect of shear deformation and with linear and quadratic displacement fields. The elements are found to be lock-free for thin-walled beams. Several numerical examples are given to demonstrate the performance of the two curved elements.
TL;DR: In this paper, a numerical method based on a modified full implicit continuous Eulerian (FICE) scheme and projected normal characteristic boundary conditions was developed for simulating MHD flows which undergo a long process of evolution.
Abstract: A numerical method has been developed based on a modified full implicit continuous Eulerian (FICE) scheme and projected normal characteristic boundary conditions for simulating MHD flows which undergo a long process of evolution. An astrophysical flow is chosen for illustration of this procedure, and numerical tests are made to verify the computational stability and physically realistic solution. Three computational tests have been accomplished; they are tests of solving methods, characteristic boundary condition, and time steps. The tests show that the program from the modified FICE scheme with proper boundary conditions and time steps can be made numerically stable for a time long enough to obtain physically plausible solutions.
TL;DR: In this article, a special treatment of transverse shear components in C0 bilinear and biquadratic plate elements is presented, which allows the full integration of the element stiffness matrix, thus avoiding the development of kinematic modes.
Abstract: In this paper we present a special treatment of transverse shear components in C0 bilinear and biquadratic plate elements that excludes the occurrence of locking completely and simultaneously allows the full integration of the element stiffness matrix, thus avoiding the development of kinematic modes. The suggested approach rests upon the standard isoparametric description of the kinematic field and only modifies the shear-strain energy without any change of the bending energy. The first step in constructing modified shear strain polynomials consists of expressing the Cartesian components of transverse shear in terms of the normalized coordinates by means of the usual isoparametric transformation. This process brings to light two distinct polynomials expressed in the normalized coordinates which are subsequently modified with the view of eliminating those coefficients which would lead to spurious constraining in the Kirchhoff limit of vanishing shear strains. Numerical examples are presented that illustrate the performance of the proposed procedure.
TL;DR: In this article, a multilevel optimization scheme for large laminated composite structures is proposed, and a suitable element/lower-level optimization scheme using a multicriteria objective function is developed.
Abstract: A multilevel optimization scheme for large laminated composite structures is proposed, and a suitable element/lower-level optimization scheme using a multicriteria objective function is developed. The objective function combines a weight function and a strain energy change function into a utility function which is minimized and in which the relative importance of each part is reflected by weighting coefficients. Minimizing the change in strain energy ensures load path continuity in the overall structure when switching between upper and lower levels of optimization, and so decouples the problems at the two levels. Continuous lamina thickness and ply-angle variation is used to minimize the objective function while satisfying strain, buckling, and gauge constraints. Numerical applications are given to illustrate the effect of the weighting coefficients in the objective function on the final result, and to demonstrate the algorithm's effectiveness as a pure weight minimization routine.
TL;DR: An adaptive refinement algorithm is developed and implemented for the SUPG method applied to a linear time-dependent advection problem in two space dimensions which compares the adaptive solution and the solution obtained using a uniform mesh.
Abstract: An adaptive refinement algorithm is developed and implemented for the SUPG method applied to a linear time-dependent advection problem in two space dimensions. Accuracy comparisons and timing results are presented which compare the adaptive solution and the solution obtained using a uniform mesh. The comparisons are made for the well-known rotating cone benchmark problem.
TL;DR: In this paper, a new mixed Petrov-Galerkin method is presented for the Timoshenko beam problem, allowing new combinations of interpolation, in particular, equal-order stress and displacement fields.
Abstract: A new mixed Petrov-Galerkin method is presented for the Timoshenko beam problem. The method has enhanced stability compared to the Galerkin formulation, allowing new combinations of interpolation, in particular, equal-order stress and displacement fields. The methodology is easily generalizable for multi-dimensional Hellinger-Reissner systems.
TL;DR: The multisurface method and the two-boundary technique are described as univariate procedures that can be applied within the context of transfinite interpolation.
Abstract: Algebraic grid generation is the direct expression of a physical coordinate system as a function of a uniform grid in a rectangular computational coordinate system. Algebraic grid generation is based on mathematical interpolation and is presented in general terms of multivariate transfinite interpolation. The multisurface method and the two-boundary technique are described as univariate procedures that can be applied within the context of transfinite interpolation. A technique for grid clustering is described. Problems that are commonly encountered in three-dimensional grid generation are discussed and approaches for dealing with complex physical domains using multiple computational grid blocks are presented.
TL;DR: The experience with the 32-node CalTech Hypercube multiprocessor for solving structural mechanics problems by the finite element method is reported and the essential features of the employed concurrent algorithms and their implementation are illustrated.
Abstract: This paper reports our experience with the 32-node CalTech Hypercube multiprocessor for solving structural mechanics problems by the finite element method. We begin with an overview of the Hypercube that is pertinent to finite element computations, and the C-based system utilities used for message passing. We discuss the partitioning and mapping of the structure onto the Hypercube nodes. We illustrate the essential features of the employed concurrent algorithms and their implementation using a one-dimensional wave propagation problem and a two-dimensional stress analysis problem. Finally, we present some outstanding issues that need be addressed for the widespread use of parallel computers in engineering computations.
TL;DR: In this article, a limitation principle for finite elements based on the Hellinger-Reissner (two-field) variational formulation is presented in the context of the Hu-Washizu (three-field).
Abstract: In a pioneering paper Fraeijs de Veubeke introduced a limitation principle for finite elements based on the Hellinger-Reissner (two-field) variational formulation. In this paper, a similar principle is presented in the context of the Hu-Washizu (three-field) variational formulation.
TL;DR: In this article, a computer program has been developed to study the details of supersonic flows and its associated flame, and the analysis of a spatially developing and reacting mixing layer is presented, and conclusions are drawn regarding the structure of the evolving layer.
Abstract: A current research effort is underway at the NASA Langley Research Center to achieve a detailed understanding of important phenomena present when a supersonic flow undergoes a chemical reaction. A computer program has been developed to study the details of such flows. The program has been constructed to consider the multicomponent diffusion and convection of important species, the finite-rate reaction of these species, and the resulting interaction between the fluid mechanics and chemistry. Code results from the analysis of a spatially developing and reacting mixing layer are presented, and conclusions are drawn regarding the structure of the evolving layer and its associated flame.