Showing papers on "Discretization published in 1972"

Recursive computational procedure for two-dimensional stock cutting

Abstract: A recursive algorithm is implemented to give high computational speeds in the solution of a cutting-stock problem. Optimal edge-to-edge cutting is shown to be achieved more easily by recursive programming than by conventional methods. The technique features preliminary discretization, which lowers the memory requirements in the computational procedure. A comparison is made between this recursive algorithm and two iterative algorithms previously given by Gilmore-Gomory. The limitations of the algorithms are discussed and some numerical results given.

Condition of finite element matrices generated from nonuniform meshes.

Abstract: N a previous Note1 it has been shown (see also Refs. 2 and 3) that the spectral condition number Cn(K) of the global (stiffness) matrix K arising from a uniform mesh of finite elements (or of finite differences) discretization can be expressed by Cn(K) = cNes2m where 2m is the order of the differential equation and c a coefficient independent of Nes, the number of elements per side, but dependent on the order of the interpolation polynomials inside the element. This condition is "natural", since it is inherently associated with the approximation of the continuous problem by the discrete (algebraic) one. Nonuniform meshes of finite elements introduce many additional factors which may adversely affect the condition of the system. It is the purpose of this Note to describe a technique for establishing bounds on the condition number for irregular meshes of finite elements. With these bounds it becomes possible to study the effect of element geometry, the order of interpolation functions and other intrinsic and discretization parameters on Cn(K) and to isolate the factors that may lead to ill-conditioning. The matrix K is termed ill-conditioned when \Q~sCn(K) = 1, where s denotes the number of decimals in the computer. The bounds on Cn(K) are expressed in terms of the extremal eigenvalues of the element matrices. Since the element matrices are of restricted size, derivation of the bounds on Cn(K) as a function of the discretization parameters become straightforward for any problem and any element. Particular attention is focused on the possibility of improving the condition of the matrix by scaling. Bounds on the Extremal Eigenvalues

Seepage Analysis of Earth Banks under Drawdown

Abstract: The problem of transient unconfined seepage under drawdown in riverbanks and dams is solved by using a finite element procedure. An iterative procedure is employed to compute movements of the free surface caused by time wise fluctuations in the external water levels. The finite element solutions are compared with laboratory experiments on a parallel-plate viscous flow model and field observations at a section along the Mississippi River. Correlation between the numerical solutions and observations is found to be good. Consideration is given to such special requirements of numerical techniques as discretization of infinite media and various possible flow situations at discretized end boundaries. An effort is made to obtain the numerical formulations and the computer codes that yield acceptable accuracy with economy. Some projections for use of the method for design analysis are presented.

Applications of finite elements in space and time

Abstract: The paper initiates a major extension of the matrix displacement method concerning the analysis of dynamic phenomena in the presence of material and geometric non-linearities. In particular, elasto-plastic behaviour as well as large displacements are taken into account. An iterative procedure of solution of the nonlinear matrix equations is discussed. The application of the theory is described in detail in two examples. The first considers the simple static problem of a rectangular flat strip in a tensile test. The iterative calculation may be carried out for deformations as large as required and shows clearly the necking effect. More ambitious is the second example which demonstrates the non-linear dynamic theory on a cyclindrial deformable billet under the impact by a heavy rigid body. The momentum of the weight and the property of the billet are such that the latter will undergo large plastic deformations. If so required, it is straightforward to incorporate damping and also allow for friction forces on the contacts. The direct applicability of the technique to forging problems is evident. The solution of the dynamic phenomenon is accomplished by extending the discretisation of space also into time. In particular, the inertia forces are taken to vary over a finite time element as a third order polynomial. Exceptional accuracy is achieved by this method.

Numerical simulation of turbulent flows

Abstract: Numerical simulation of turbulent flow is of practical importance in geophysics and is useful as a test of turbulence theory. From a basic scale analysis of turbulent flow it is shown that complete simulation is only practicable for two-dimensional or marginally turbulent three-dimensional flows. Some examples of such simulations are described. A more useful approach for practical applications is to directly simulate the larger scales of motion and only consider the small unresolved scales with respect to their gross statistical interactions with the larger scales. A variable eddy viscosity model has proved useful for this purpose, although better approximations can be devised. Methods of discretization and numerical integration are discussed, and it is proposed that Galerkin methods are preferred when feasible in order to avoid truncation and aliasing errors.

A quadratic programming approach to the impulsive loading analysis of rigid plastic structures

Abstract: This paper discusses the dynamic problem of rigid plastic structures subjected to impulsive loading. A couple of “dual” extremum theorems reduces the problem to the optimization of convex quadratic functions subject to linear inequalities and equations: the first theorem takes as variables stress and accelerations, the second accelerations and plastic multiplier rates. The problem is discussed in matrix notation on the basis of finite element discretization of the structure and piecewise linear approximation of the yield surfaces, using some quadratic programming concepts. The procedure is illustrated by a simple numerical example.

Zur stetigen Abhängigkeit des Extremalwertes eines konvexen Optimierungsproblems von einer stetigen Änderung des Problems

Abstract: In dieser Arbeit wird das Problem behandelt, ein konvexes Funktional unter konkaven Nebenbedingungen auf einer konvexen Teilmenge eines linearen Raumes zum Minimum zu machen. Es wird gezeigt, das sich der gesuchte Extremalwert des Funktionals unter geeigneten Voraussetzungen stetig andert, wenn die im Problem vorgegebenen Grosen stetig abgeandert werden. Die Ergebnisse werden angewandt auf Approximationsprobleme in normierten Raumen und Optimierungsaufgaben mit unendlich vielen Nebenbedingungen, bei denen die stetige Anderung des Problems durch Diskretisierung (Ubergang zu Halbnormen oder endlich vielen Nebenbedingungen) durchgefuhrt wird. This paper is concerned with the minimization of a convex functional on a convex subset of a linear space subject to concave constraints. It is shown that, under suitable conditions the extreme value of the functional changes continuously if the data of the given problem are changed continuously. The results will be applied to approximation problems in normed linear spaces and to optimization problems, with infinitely many constraints where the continuous change of the problem is performed by discretization (use of seminorms or restriction to finitely many constraints).

Distributed optimal control in a nuclear reactor

Abstract: The linear deterministic mathematical model of a nuclear reactor system, which can bo described by parabolic partial differential equations, and the problems associated with this model arc discussed in this paper. Also studied is the derivation of a maximum principle for this distributed nuclear reactor system and the problems associated with approximating this distributed system by discretization into a lumped system. Examples concerning the regulator and the servomechanism problems of a nuclear reactor system are presented.

Optimal smoothing for continuous-time systems with multiple time delays

Abstract: Equations for the smoothed state estimate and for the error covariances of a continuous-time system with multiple time delays, based on observations involving time delays, are derived through a combination of discretization, state augmentation, and subsequent dediscretization procedures.

Über die näherungsweise Lösung nichtlinearer Integralgleichungen

Abstract: In this paper we study the iterative solvability of nonlinear systems of equations which arise from the discretization of Hammerstein integral equations. It is shown that, for a large class of equations satisfying monotonicity assumptions, it is possible to solve these systems by means of a linearly convergent iteration method. Moreover, for general monotone operators on a Hilbert space a globally convergent variant of Newton's method is given. Finally it is shown that this method effectively can be applied in a natural way to the systems of equations under consideration.

Computation of Best Monotone Approximations

Abstract: A numerical procedure to compute the best uniform approximation to a given continuous function by algebraic polynomials with nonnegative rth derivative is presented and analyzed. The method is based on discretization and linear programming. Several numerical experiments are discussed.

Finite element solution for energy conservation using a highly stable explicit integration algorithm

Abstract: Theoretical derivation of a finite element solution algorithm for the transient energy conservation equation in multidimensional, stationary multi-media continua with irregular solution domain closure is considered. The complete finite element matrix forms for arbitrarily irregular discretizations are established, using natural coordinate function representations. The algorithm is embodied into a user-oriented computer program (COMOC) which obtains transient temperature distributions at the node points of the finite element discretization using a highly stable explicit integration procedure with automatic error control features. The finite element algorithm is shown to posses convergence with discretization for a transient sample problem. The condensed form for the specific heat element matrix is shown to be preferable to the consistent form. Computed results for diverse problems illustrate the versatility of COMOC, and easily prepared output subroutines are shown to allow quick engineering assessment of solution behavior.

Limit Design in the Absense of a Given Layout: A Finite Element, Zero-One Programming Approach∗

Abstract: Limit design in three dimensions is discussed and formulated as a constrained minimax problem in kinematic and geometric variables. A finite element discretization is proposed which, combined with piecewise linearization of the yield surfaces, reduces the minimum weight design to a pair of dual problems in linear mixed zero one programming. The relevant duality theory is shown to be useful for the theoretical frame of the mechanical problem. Various ways of reducing the number of variables and constraints are pointed out, in order to make available algorithms economically applicable to practical situations.

COMOC: Thermal analysis variant. User's manual

Abstract: The Thermal Analysis Variant of the COMOC (computational continuum mechanics) computer system solves problems involving transient heat conduction and convection in stationary continua spanning arbitrarily irregular two-dimensional and axisymmetric solution domains. COMOC is based upon a finite element solution algorithm for the energy equation, and solves for the transient nodal temperature distribution using a highly stable and automatic explicit integration procedure. COMOC is extensively user-oriented, requires minimal input, and no a priori knowledge concerning the stability character of the differential equation system. It can readily output computed data in user-specified format fields, that geometrically resemble the solution domain discretization (for rapid engineering evaluation). Complete information is provided for applying COMOC to a specific problem.

Assessment of advances in structural analysis for space shuttle

Abstract: The latest technology for structural analysis in relation to the design tasks that lie ahead for the space shuttle is reviewed. For shell-of-revolution structures, the analysis can be formulated as a one-dimensional problem which is readily solved by using finite-difference or numerical-integration techniques. For more general asymmetric shells, a two-dimensional formulation is required. However, the governing equations are readily formulated and are amenable to solution by finite-difference techniques. For a completely general structural arrangement, such as structural frameworks, recourse is usually made to discretized formulations using finite elements. Of course, the finite-element programs could be used for shell structures, but at a loss in accuracy and increase in computer time compared with the special purpose programs.

A nonlinear analysis of statistically loaded plane frames using a discrete element model

Abstract: A DISCRETE ELEMENT ANALYSIS WHICH CONSIDERS GEOMETRIC, MATERIAL, AND SUPPORT NONLINEARITIES OF STATICALLY LOADED PLANE FRAMES IS DEVELOPED. A COMPUTER PROGRAM HAS BEEN WRITTEN TO IMPLEMENT AND VERIFY THE ANALYSIS. FRAME GEOMETRY, LOADS, CROSS SECTIONS, AND SUPPORTS (NONLINEAR CONCENTRATED AND DISTRIBUTED SPRINGS) CAN BE SUFFICIENTLY GENERAL TO WORK PRACTICAL FRAME PROBLEMS. THE METHOD OF ANALYSIS IS BASED ON AN ITERATIVE PROCEDURE CALLED THE TANGENT STIFFNESS METHOD. UNBALANCED NODAL POINT FORCES ARE APPLIED TO A TEMPORARILY LINEAR STRUCTURE WHOSE POSITION DEPENDENT STIFFNESS MATRIX IS THE TANGENT STIFFNESS MATRIX OF THE STRUCTURE. THE FRAME MEMBERS ARE DIVIDED INTO A NUMBER OF DISCRETE ELEMENTS. LOAD-DISPLACEMENT EQUATIONS FOR AN INDIVIDUAL DISCRETE ELEMENT ARE DERIVED WHICH ARE VALID FOR LARGE DISPLACEMENTS. A NUMERICAL TECHNIQUE IS USED TO DETERMINE THE FORCE-DEFORMATION RESPONSE OF A CROSS SECTION WITH NONLINEAR STRESS-STRAIN CURVES. LOADS AND NONLINEAR SUPPORTS ARE INPUT IN NORMAL ENGINEERING TERMS AND CAN BE REFERENCED EITHER TO THE STRUCTURE OR TO THE MEMBER AXES. WHEN NECESSARY, THE LOADS AND NONLINEAR SUPPORTS ARE INTERNALLY TRANSFORMED TO MEMBER COORDINTES AND DISCRETIZED TO CONCENTRATED VALUES AT THE NODAL POINTS. CASTIGLIANO'S FIRST THEOREM IS APPLIED TO DEVELOP MATRIX EXPRESSIONS FOR THE STIFFNESS MATRIX OF A GENERAL DISCRETE ELEMENT AND THESE EXPRESSIONS ARE USED TO OBTAIN THE STIFFNESS MATRIX FOR THE SPECIFIC DISCRETE ELEMENT USED IN THE FRAME SOLUTIONS. A NUMBER OF PROBLEMS ARE WORKED AND COMPARED WITH EXISTING ANALYTICAL OR EXPERIMENTAL SOLUTIONS.

Computer modelling of a holographic process

Abstract: WE present algorithms for the synthesis and reconstruction of holograms and the results of a theoretical study of the effect the discretization step on the hologram has on the quality of the reconstructed image and synthesized holographic images of objects of complex form.

Fluid simulation with incompressible navier-stokes equation

Abstract: Fluid mechanics are usually modeled after the Navier-Stokes equation (NSE). To solve it numerically, we apply an Eulerian discretization: at each time step, uid variables (velocity and pressure) are computed on a xed grid of points in the plane, without taking in account the moviment of the particles. Through a staggered grid, we avoid decoupling of pressure and velocity variables in the discretized NSE, and also improve our estimates of partial derivatives. Chorin-Temam Projection Method is used to solve NS, dividing it in two steps:

Thermal elastoplastic structural analysis of non-metallic thermal protection systems

Abstract: An incremental theory and numerical procedure to analyze a three-dimensional thermoelastoplastic structure subjected to high temperature, surface heat flux, and volume heat supply as well as mechanical loadings are presented. Heat conduction equations and equilibrium equations are derived by assuming a specific form of incremental free energy, entropy, stresses and heat flux together with the first and second laws of thermodynamics, von Mises yield criteria and Prandtl-Reuss flow rule. The finite element discretization using the linear isotropic three-dimensional element for the space domain and a difference operator corresponding to a linear variation of temperature within a small time increment for the time domain lead to systematic solutions of temperature distribution and displacement and stress fields. Various boundary conditions such as insulated surfaces and convection through uninsulated surface can be easily treated. To demonstrate effectiveness of the present formulation a number of example problems are presented.