Showing papers on "Discretization published in 1982"
Book•
01 Jan 1982
TL;DR: In this paper, the Finite Element Method is used to derive a system equation from a set of finite element vectors and matrices and then to solve the problem of finding the solution.
Abstract: 1. Overview of the Finite Element Method, 2. Discretization of the Domain, 3. Interpolation Models, 4. Higher Order and Isoparametric Elements, 5. Derivation of Element Matrices and Vectors, 6. Assembly of Element Matrices and Vectors and Derivation of System Equations, 7. Numerical Solution of Finite Element Equations, 8. Basic Equations and Solution Procedure, 9. Analysis of Trusses, Beams and Frames, 10. Analysis of Plates, 11. Analysis of Three-Dimensional Problems, 12. Dynamic Analysis, 13. Formulation and Solution Procedure, 14. One-Dimensional Problems, 15. Two-Dimensional Problems, 16. Three-Dimensional Problems, 17. Basic Equations of Fluid Mechanics, 18. Inviscid and Incompressible Flows, 19. Viscous and Non-Newtonian Flows, 20. Solution of Quasi-Harmonic Equations, 21. Solution of Helmhotz Equation, 22. Solution of Reynolds Equation, Appendix-A Green Greass Theorem.
1,247 citations
TL;DR: Second-order error estimates are proven for spatial discretization, using conforming or nonconforming elements, and indicate a fluid-like behavior of the approximations, even in the case of large data, so long as the solution remains regular.
Abstract: This is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary Navier–Stokes equations. Second-order error estimates are proven for spatial discretization, using conforming or nonconforming elements. The results indicate a fluid-like behavior of the approximations, even in the case of large data, so long as the solution remains regular. The analysis is based on sharp a priori estimates for the solution, particularly reflecting its behavior as$t \to 0$ and as $t \to \infty $. It is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the Navier–Stokes equations, as it depends on nonlocal compatibility conditions for the data. The results which are presented here are independent of such compatibility conditions, which cannot be verified in practice.
784 citations
TL;DR: An error bound is given that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation) and for thePDE in Lagrangian form.
Abstract: This paper deals with an algorithm for the solution of diffusion and/or convection equations where we mixed the method of characteristics and the finite element method. Globally it looks like one does one step of transport plus one step of diffusion (or projection) but the mathematics show that it is also an implicit time discretization of thePDE in Lagrangian form. We give an error bound (h+Δt+h×h/Δt in the interesting case) that holds also for the Navier-Stokes equations even when the Reynolds number is infinite (Euler equation).
697 citations
TL;DR: In this paper, an explicit closed-form solution for the Green functions (displacements due to unit loads) corresponding to dynamic loads acting on (or within) layered strata is presented.
Abstract: This paper presents an explicit, closed-form solution for the Green functions (displacements due to unit loads) corresponding to dynamic loads acting on (or within) layered strata. These functions embody all the essential mechanical properties of the medium and can be used to derive solutions to problems of elastodynamics, such as scattering of waves by rigid inclusions, soil-structure interaction, seismic sources, etc. The solution is based on a discretization of the medium in the direction of layering, which results in a formulation yielding algebraic expressions whose integral transforms can readily be evaluated. The advantages of the procedure are: (a) the speed and accuracy with which the functions can be evaluated (no numerical integration necessary); (b) the potential application to problems of elastodynamics solved by the boundary integral method; and (c) the possibility of comparing and verifying numerical integral solutions implemented in computer codes.
243 citations
TL;DR: In this article, a model of an inextensible elastic rod with equal principal stiffnesses is presented, which permits large deflections and finite rotations and accounts for tension variation along its length.
Abstract: A new three-dimensional finite element model of an inextensible elastic rod with equal principal stiffnesses is presented. The model permits large deflections and finite rotations and accounts for tension variation along its length. Its use in static analysis is described and a time integration method for dynamic analysis is developed. Accuracy of the spatial discretization and stability of the time integration method are demonstrated by comparison of numerical results with exact solutions for certain nonlinear problems.
212 citations
Book•
01 Jan 1982
209 citations
TL;DR: A theoretical analysis indicated that this warping can be interpreted as the manifestation of the frequency dependent phase velocity of waves in the discrete system.
196 citations
TL;DR: A novel integrodifferential approach to steady-state skin effect problems is applied to two-dimensional geometries using traditional triangular finite elements.
Abstract: A novel integrodifferential approach to steady-state skin effect problems is applied to two-dimensional geometries using traditional triangular finite elements. The forcing function in the new formulation is the total measurable current in the conductors. An explanation of why the new approach works is included. The application of Galerkin's criterion to the integrodifferential equation is outlined and it is shown how to proceed with the finite element discretization. The methods used in the computation of performance measuring quantities useful to designers are described in detail. Simple examples illustrate the method. Results are validated by comparison to known solutions.
181 citations
01 Feb 1982
TL;DR: Onedant as mentioned in this paper solves the one-dimensional multigroup transport equation in plane, cylindrical, spherical, and two-angle plane geometries, subject to vacuum, reflective, periodic, white, albedo and inhomogeneous boundary flux conditions.
Abstract: ONEDANT is designed for the CDC-7600, but the program has been implemented and run on the IBM-370/190 and CRAY-I computers. ONEDANT solves the one-dimensional multigroup transport equation in plane, cylindrical, spherical, and two-angle plane geometries. Both regular and adjoint, inhomogeneous and homogeneous (k/sub eff/ and eigenvalue search) problems subject to vacuum, reflective, periodic, white, albedo, or inhomogeneous boundary flux conditions are solved. General anisotropic scattering is allowed and anisotropic inhomogeneous sources are permitted. ONEDANT numerically solves the one-dimensional, multigroup form of the neutral-particle, steady-state form of the Boltzmann transport equation. The discrete-ordinates approximation is used for treating the angular variation of the particle distribution and the diamond-difference scheme is used for phase space discretization. Negative fluxes are eliminated by a local set-to-zero-and-correct algorithm. A standard inner (within-group) iteration, outer (energy-group-dependent source) iteration technique is used. Both inner and outer iterations are accelerated using the diffusion synthetic acceleration method. (WHK)
160 citations
TL;DR: In this paper, a finite element method for solving shallow water flow problems is presented, where the standard Galerkin method is employed for spatial discretization, and the numerical integration scheme for the time variation is the explicit two step scheme, which was originated by the authors and their co-workers.
Abstract: A finite element method for solving shallow water flow problems is presented. The standard Galerkin method is employed for spatial discretization. The numerical integration scheme for the time variation is the explicit two step scheme, which was originated by the authors and their co-workers. However, the original scheme has been improved to remove the erroneous artifical damping effect. Since the improved scheme employs a combination of lumped and unlumped coefficients, the scheme is referred to as a selective lumping scheme. Stability conditions and accuracy are investigated by considering several numerical examples. The method has been applied to the tidal flow in Osaka Bay and Yatsushiro Bay.
122 citations
TL;DR: In this paper, the use of different discretizations for the isotropic and deviatoric parts of the strain tensor is advocated for preventing collapse loads overestimation by finite element or finite difference procedures when the material discretization does not allow locally incompressible plastic flow: for example, constant strain triangles or tetrahedra give unacceptable results.
Abstract: Collapse loads are overestimated by finite-element or finite-difference procedures when the material discretization does not allow locally incompressible plastic flow: for example, constant-strain triangles or tetrahedra give unacceptable results The use of different discretizations for the isotropic and deviatoric parts of the strain tensor is advocated for preventing this problem The technique can be easily implemented in existing two- and three-dimensional computer programs Its effectiveness is demonstrated by several examples
TL;DR: The resulting approximations of the velocity are shown to have optimal rate of convergence in L2 under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space.
Abstract: We consider approximating the solution of the initial and boundary value problem for the Navier-Stokes equations in bounded twoand three-dimensional domains using a nonstandard Galerkin (finite element) method for the space discretization and the third order accurate, three-step backward differentiation method (coupled with extrapolation for the nonlinear terms) for the time stepping. The resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions; all these linear systems have the same matrix. The resulting approximations of the velocity are shown to have optimal rate of convergence in L2 under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space.
TL;DR: In this article, a detailed description of an atmospheric boundary layer model capable of simulating the diurnal cycles of wind, temperature and humidity is given, which includes a formulation of various physical processes (radiative effects, variation of soil surface temperature, etc.).
Abstract: We give a detailed description of an atmospheric boundary layer model capable of simulating the diurnal cycles of wind, temperature and humidity. The model includes a formulation of various physical processes (radiative effects, variation of soil surface temperature and humidity, etc.) and uses a first-order closure for turbulent fluxes that relies upon a time-dependent equation for turbulent kinetic energy and on a mixing length governed by a relaxation process. The exchange processes taking place in the surface layer are dealt with in a separate micrometeorological module. The one-dimensional model uses a Galerkin technique based on linear finite elements, variable resolution in the vertical, and a time discretization of the Crank-Nicholson type. A simulation test based on day 33 of the Wangara Australian experiment indicates that the model, despite its relative simplicity, gives realistic results that compare favorably with those from higher order models while taking much less space and time o...
01 Feb 1982
TL;DR: In this article, an updated user's manual for DYNA2D, an explicit two-dimensional axisymmetric and plane strain finite element code for analyzing the large deformation dynamic and hydrodynamic response of inelastic solids is provided.
Abstract: This revised report provides an updated user's manual for DYNA2D, an explicit two-dimensional axisymmetric and plane strain finite element code for analyzing the large deformation dynamic and hydrodynamic response of inelastic solids A contact-impact algorithm permits gaps and sliding along material interfaces By a specialization of this algorithm, such interfaces can be rigidly tied to admit variable zoning without the need of transition regions Spatial discretization is achieved by the use of 4-node solid elements, and the equations-of motion are integrated by the central difference method An interactive rezoner eliminates the need to terminate the calculation when the mesh becomes too distorted Rather, the mesh can be rezoned and the calculation continued The command structure for the rezoner is described and illustrated by an example
TL;DR: In this paper, the authors show that the scaled stability region of a method, satisfying some reasonable conditions, cannot be properly contained in the scaled stabilizer region of another method, for general nonlinear ordinary differential systems, for systems obtained from parabolic problems, and for hyperbolic problems.
Abstract: Stability regions of explicit "linear" time discretization methods for solving initial value problems are treated. If an integration method needsm function evaluations per time step, then we scale the stability region by dividing bym. We show that the scaled stability region of a method, satisfying some reasonable conditions, cannot be properly contained in the scaled stability region of another method. Bounds for the size of the stability regions for three different purposes are then given: for "general" nonlinear ordinary differential systems, for systems obtained from parabolic problems and for systems obtained from hyperbolic problems. We also show how these bounds can be approached by high order methods.
TL;DR: In this paper, a numerical formulation of a static method in limit analysis based on discretization in finite elements, is proposed, which leads to a separable programming problem if the medium obeys the von Mises yield criterion, or to a linear problem if it is a Tresca-Coulomb yield criterion.
TL;DR: A unified framework for calculating the order of the error for a class of finite-difference approximations to the monoenergetic linear transport equation in slab geometry is developed in this article.
Abstract: A unified framework is developed for calculating the order of the error for a class of finite-difference approximations to the monoenergetic linear transport equation in slab geometry. In particular, the global discretization errors for the step characteristic, diamond, and linear discontinuous methods are shown to be of order two, while those for the linear moments and linear characteristic methods are of order three, and that for the quadratic method is of order four. A superconvergence result is obtained for the three linear methods, in the sense that the cell-averaged flux approximations are shown to converge at one order higher than the global errors.
TL;DR: In this paper, the theoretical and numerical treatment of dynamic unilateral problems is discussed, where the governing equations are formulated as an equivalent variational inequality expressing D' Alembert's principle in its inequality form.
Abstract: The present paper deals with the theoretical and numerical treatment of dynamic unilateral problems. The governing equations are formulated as an equivalent variational inequality expressing D' Alembert's principle in its inequality form. The discretization with respect to time and space leads to a static nonlinear programming problem which is solved by an appropriate algorithm. Some properties of dynamic unilateral problems are outlined and the influence of several parameters on the solution is investigated by means of numerical examples.
TL;DR: In this article, a method to obtain boundary conditions for the wave equation in one dimension, fitting to the discretization scheme and stable, was presented, and error estimates on the reflected part were given.
Abstract: When computing a partial differential equation, it is often necessary to introduce artificial boundaries. Here we explain a systematic method to obtain boundary conditions for the wave equation in one dimension, fitting to the discretization scheme and stable. Moreover, we give error estimates on the reflected part.
TL;DR: Three methods of digital simulation of partially coherent imagery are presented and compared and it is shown that when the imaging is of narrow point spread function, the modal expansion method is very efficient, especially for relatively high coherence.
Abstract: Three methods of digital simulation of partially coherent imagery are presented and compared. The first method is a direct discretization of imaging equations. In the second, the computations are performed in the Fourier domain. The third method is based on a modal expansion of the imaging as an incoherent sum of a number of coherent modes; this allows full utilization of FFT algorithms. It is shown that when the imaging is of narrow point spread function, the modal expansion method is very efficient, especially for relatively high coherence. Examples of 1-D and 2-D images are shown.
TL;DR: It is found that irregularity sets in when two homoclinic structures are present and, in this case, many and continual homocinic crossings occur throughout the irregular time series.
01 Jan 1982
TL;DR: In this paper, the sensitivity analysis is treated as an integrated part of a unified approach to eigenvalue analysis of elastic solids, and the gradient functions, the dependence of slenderness and the inherent problem of local optima are obtained.
Abstract: A finite element discretization, combined with a powerful numerical eigenvalue procedure, has proved to be a unified approach to eigenvalue analysis of elastic solids. Treating the sensitivity analysis as an integrated part of this approach, one obtains gradients of the eigenvalues without any new eigenvalue analysis. This forms the necessary information for an optimal redesign which is formulated as a linear programming problem. By a sequence of optimal redesigns, one then obtains a solution to the problem of optimal design or a solution to an inverse eigenvalue problem. Taking as an example the vibration of Timoshenko beams, we focus on the gradient functions, on the dependence of slenderness, and on the inherent problem of local optima.
TL;DR: The treatment of general linear discretization methods for initial value problems is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general ‘method-free’ statements are again obtained.
Abstract: This paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems. Here the treatment is extended to cover also implicit schemes, and by placing the accuracy of the schemes into a more central position in the discussion general `method-free' statements are again obtained. More specialized results are additionally given for linear multistep methods and for the Taylor series method.
TL;DR: In this article, the position, size and surface temperature of circular holes inside a two-dimensional heat conductor are optimized to produce a minimum variation in surface temperature over a portion of the outer boundary.
Abstract: The position, size and surface temperature of circular holes inside a two-dimensional heat conductor are optimized to produce a minimum variation in surface temperature over a portion of the outer boundary. This problem, whic arises in thermal desing of moulds and dies, resembles those encountered in structural shape optimization because the internal geometry of the heat conductor depends on the design variables. In this paper, some of the traditional difficulties associated with shape optimization are overcome by analysing steady heat conduction with a special boundary integral method developed for two-dimensional regions with circular hole. This approach eliminates the need to regenerate a finite element mesh over the interior of the region each time the geometry is changed during the design process. It also increases the efficiency of the analysis by reducing the number of unknowns in the numerical discretization of the region. Since the objective function depends only on the boundary temperatures, there is no need to determine temperatures in the interior.
The analysis method is applied to two problems arising in optimal thermal design of compression moulds. These examples show that the number of holes choson for the design strongly affects their resulting optimal arrangement as well as the ultimate uniformity of the cavity surface temperature.
TL;DR: The paper deals with the discretization of the continuous Nash cascade using the principles of state space modelling and a discrete state variable model is derived whose state and input transition matrices are in a dual relationship.
01 Jan 1982
TL;DR: A continuum approximation is proposed for problems of flows in large and dense networks, where each point of the continuum is characterized by ‘capacity’ and/or ‘distance/cost’ which are convex sets in the tangent space.
Abstract: A continuum approximation is proposed for problems of flows in large and dense networks, where each point of the continuum is characterized by ‘capacity’ and/or ‘distance/cost’ which are convex sets in the tangent space. The problems of flows in continua corresponding to various problems of flows in networks are formulated in variational forms, of which mathematical properties are investigated and for which numerical algorithms are derived by means of the finite-element discretization technique. A practical procedure, based upon concepts from integral geometry, for constructing the continuum which approximates the original network is also proposed.
TL;DR: Estimates are given that require only weak smoothness assumptions on the initial data and estimates of the error made by the resulting semidiscrete approximations, and of their time derivatives, are given.
Abstract: Semidiscrete methods for approximating the solutions of initial boundary value problems for parabolic equations are studied. The construction of these semidiscrete methods is based upon the availability of several different Galerkin type approximation methods for the associated elliptic steadystate problem. The properties required of the spacial discretization methods are listed and estimates of the error made by the resulting semidiscrete approximations, and of their time derivatives, are given. In particular, estimates are given that require only weak smoothness assumptions on the initial data. Verifications of the required properties for various Galerkin type methods are provided.
TL;DR: In this paper, Galerkin's method with appropriate discretization in time is considered for approximating the solution of the nonlinear integro-differential equation u t (x, t) = ∝ 0 t a(t − τ) ∂ ∂x σ(u x (x), τ)) dτ + f(x, T), 0 An error estimate in a suitable norm will be derived for the difference u − uh between the exact solution u and the approximant uh.
TL;DR: In this paper, a numerical formulation of the kinematic limit analysis method based on finite element discretization is presented for axisymmetric cases, and the influence of this approximation is quantified for the triaxial test problem and the stability of circular excavations.
Abstract: This paper deals with a numerical formulation of the kinematic limit analysis method. This formulation, based on a finite element discretization, is available for axisymmetric cases. The linearization of the yield Tresca and Von Mises criteria leads to linear programming problems. The velocity fields described here are plastically admissible only on an average. The influence of this approximation is quantified for the triaxial test problem and the stability of circular excavations. (Author/TRRL)