Showing papers on "Discretization published in 1970"

Thermomechanical analysis of viscoelastic solids

Abstract: SUMMARY This paper is concerned with the development of a computational algorithm for the solution of the uncoupled, quasi-static boundary value problem for a linear viscoelastic solid undergoing thermal and mechanical deformation. The method evolves from a finite element discretization of a stationary value problem, leading to the solution of a system of linear integral equations determining the motion of the solid. An illustrative example is included.

Incremental elastoplastic analysis and quadratic optimization

Abstract: The paper discusses the incremental boundary value problem for elastoplastic workhardening continua, allowing for distributed dislocations. A pair of “dual” extremum theorems reduces the problem to the optimization of convex quadratic forms subject to linear inequalities and equations: the first theorem takes as variables stress and plastic multiplier rates, the latter velocities and plastic multiplier rates. The conclusions reached are specialized to elastic perfectly plastic (nonhardening) cases. The problem is discussed both in the traditional terms of continuum mechanics and in matrix notation on the basis of finite element discretization, using some quadratic programming concepts. Finally a comparison is made with the classical incremental minimum principles of plasticity (Prager-Hodge, Greenberg), which are deduced from the present theorems in a form generalized to the distributed dislocations.

A minimum principle for incremental elastoplasticity with non-associated flow laws

Abstract: E lastoplastic constitutive laws with non-associated flow laws and work-hardening or non-hardening or work-softening behaviour are assumed. The incremental boundary-value problem is shown to be amenable to the inequality-constrained minimization of a quadratic functional of the plastic multiplier field. A weak sufficient condition for uniqueness of solution and overall stability is formulated. Application methods founded on finite-element discretization are indicated for solving threedimensional problems.

Stability of a spinning body containing elastic parts via Liapunov's direct method

Abstract: A general and rigorous method for the stability analysis of a spinning body which is part rigid and part elastic is presented. The motion of the body is described by a "hybrid" system of equations, namely a system consisting of both ordinary and partial differential equations. The stability analysis is based on the Liapunov direct method and it works directly with the hybrid set of equations in contrast with the common practice of replacing the partial differential equations with ordinary differential equations by means of modal truncation or spatial discretization. This new approach permits a more rigorous analysis which not only produces sharper stability criteria but also avoids questions as to the effect of the truncation and discretization processes on the results. The general formulation can be used to test the stability of orbiting satellites with various types of elastic members and should also be applicable to the stability analysis of hybrid systems encountered in other areas.

A gradient computational procedure for the solution of large problems arising from the finite element discretization method

Abstract: A computational procedure based on gradient iterative techniques is proposed for the solution of large problems to which the finite element method is applicable. In linear problems the procedure can be used either for solving the set of algebraic equations or for the complete inversion of the matrix of coefficients. Special attention is focused on the practical aspects of the procedure concerning its realization on the digital computer.

Application Of The Fast Multipole Method ToThe 3-D BEM Analysis Of Electron Guns

Abstract: Although BEM enjoys the boundary only discretization, the computational work and memory requirements become prohibitive for large-scale 3-D problems due to its dense matrix formulation. The computation of n-body problems, such as in charged particle simulation, has a similar inherent difficulty. In this paper, we use the Fast Multipole Method to overcome these difficulties and apply it to the 3-D analysis of electron guns, which involves both solving BEM and n-body problems.

The Method Of Fundamental Solutions For Time-Dependent Problems

Abstract: We show how a number of second order time-dependent partial differential equations can be solved numerically by reformulating them in terms of inhomogeneous modified Helmholtz equations. Using recently derived analytic particular solutions for these equations, the resulting boundary value problems can be reduced to solving homogeneous Helmholtz equations which can be done effectively using the MFS. The resulting algorithms are efficient in that they require neither domain nor boundary discretization.

An Evaluation Of Boundary Element Schemes For Convection-diffusion Problems

Abstract: This paper presents a numerical evaluation of the performance of three different boundary element schemes for solving steady-state convectiondiffusion problems with variable velocity fields. Two of these schemes are based on the dual reciprocity BEM: the first uses the fundamental solution of Laplace's equation and treats the whole convective part through the DRM; the other decomposes the velocity field into an average and a perturbation, and uses the fundamental solution of the convectiondiffusion equation for constant velocity. In this case, only the perturbation is treated using a dual reciprocity approximation. The third scheme also decomposes the velocity field into an average and a perturbation, but the effects of the perturbation velocity are included through domain discretization. A comparison of the performance of the three schemes is presented for two different problems.

Existence and convergence theorems for the boundary layer equations based on the line method

Abstract: : A constructive existence proof for the boundary layer equations u(u sub x) + v(u sub y) = nu(u sub yy) + U(U prime). The v. Mises transform and the longitudinal line method (discretization of the space variable) are used. (Author)

Data Assimilation Schemes For Non-linear Shallow Water Flow Models

Abstract: In theory K aim an filters can be used to solve many on-line data assimilation problems. However, for models resulting from the discretization of partial differential equations the number of state variables is usually very large, leading to a huge computational burden. Therefore approximation of the Kalman filter equations is in general necessary. In this paper two new algorithms are proposed, that extend the idea of the Reduced Rank Square Root filter [15] for use with non-linear models. The algorithms are based on a low rank approximation of the error covariance matrix and use a square root representation of the error covariance. For both algorithms the tangent linear model is not needed. The first algorithm proposed is accurate up to first order terms, which is comparable to the extended Kalman filter. The second, at the cost of twice the number of computations, is second order accurate, which may be important for strongly nonlinear models. Several experiments were performed on a model of the southern part of the North Sea to measure the performance of both algorithms. Both algorithms perform well when the the number of modes, i.e. the rank of the approximation, is set to 30. This corresponds to a computation time of approximately 30 model runs for the first order algorithm and 60 for the second order algorithm.

FEFLOW: A Finite Element Code For Simulating Groundwater Flow, Heat Transfer And Solute Transport

Abstract: The FEFLOW code is a finite element program package developed at VTT Energy to model flow, solute transport and heat transfer in coupled and non-coupled, steady-state and transient situations, as well as in deterministic and stochastic modes. The code offers a novel finite element technique to model groundwater phenomena in fractured crystalline rock. Linear and bi-quadratic one-, twoand threedimensional finite elements can be used for describing engineered and natural bedrock structures. One of the solute transport models implemented in the package is capable of taking into account matrix diffusion as well. Highly convective cases are handled with different kinds of upwind schemes. The system of linear algebraic equations emerging from the standard Galerkin approximation can be solved with a direct frontal solver, as well as with an array of iterative solvers partly from the NSPCG package. The nonlinear algebraic equations resulting from coupled cases are solved with the Picard iterative approach with options for relaxation. The discretization of time is based on a simple finite difference scheme. For each result quantity to be determined, the code offers a wide selection of nodal boundary conditions including prescribed values, sources, sinks and/or fluxes. These may be constant or a function of time. Hydraulic properties of the bedrock features may also be constant or vary with depth. Besides the finite element analysis code the FEFLOW package comprises several programs to compute derived quantities (like flow paths and flow rates) and to facilitate generic modelling tasks. The code has been tested in a series of test cases, and verified in the international HYDROCOIN project. Main application areas of the FEFLOW package have been site investigations and safety analyses undertaken by the Finnish power company Teollisuuden Voima/Posiva Oy operating two nuclear power plants. It has also been employed to simulate various hydraulic disturbances and solute transport phenomena in the Aspo Hard Rock Laboratory, Sweden. Transactions on Ecology and the Environment vol 10, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541

A discrete element stress and displacement analysis of elastoplasticplates

Abstract: A displacement method of analysis for elastoplastie plates is presented with particular emphasis on accurate stress results. The Hencky-Nadai stress-strain law is assumed and a discretized potential energy function is formed using bicubic Hermite displacement functions. To achieve interelement continuous stress fields, bicubic spline constraints are introduced that produce interelement curvature continuity. The planar variation in material properties caused by plastic strains is approximated using additional nodes that uniformly subdivide the element planform into four quadrants. Integration of the strain energy through the plate thickness is accomplished using gaussian quadrature. Solutions of the nonlinear discretized equilibrium equations are obtained by energy minimization using the conjugate gradient algorithm. Results in the elastic range indicate the displacements converge with the fourth power of the grid size and the stresses converge with the grid size squared. These convergence rates were obtained with both bicubic Hermite and spline displacement functions. Elastoplastic results are presented that compare well with both deformation and incremental theory solutions. Stress results are also presented that demonstrate an elastic compressibility effect in elastoplastie plates.

Structural vibration isolation by rows of piles

Abstract: Three-dimensional passive structural vibration isolation by rows of piles is studied numerically by the frequency domain boundary element method. The source of soil vibration is assumed to be a vertical force harmonically varying with time. The piles and the soil material behaviour are assumed to be linear elastic or viscoelastic. Coupling between the soil and piles is accomplished through equilibrium and compatibility at their interfaces. Both continuous and discontinuous quadratic quadrilateral elements are employed and advanced direct numerical integration schemes are used for the treatment of the various singular integrals. The full-space dynamic fundamental solution is used and this requires a discretization of not only the substructure interfaces but also a finite portion of the free soil surface around the vibration isolation system. Symmetry and antisymmetry considerations reduce the complexity of the problem considerably. The above methodology is tested for accuracy by solving a problem of active vibration isolation by trenches for which there exist numerical solutions and then applied to the problem of structural vibration isolation by rows of piles and compared with an existing approximate analytical solution.

Stability analysis of finite difference schemes for inertial oscillations in ocean general circulation models

Abstract: Numerical time finite difference schemes in widely used ocean general circulation models are systematically examined to ensure the correct and accurate discretization of the Coriolis terms. Two groups of numerical schemes are categorized. One group is suitable for simulating an inertial wave in the ocean with the necessary condition for stability being F=fAt

Invited Paper Three-dimensional Numerical Codes For Simulating Groundwater Contamination: FLOW3D, Flow In Saturated And Unsaturated Porous Media

Abstract: In this paper we describe the numerical code for a three-dimensional groundwater flow model. The model is developed for the case of variably saturated porous media, applicable to both the unsaturated (soil) zone and the saturated (groundwater) zone. The governing equation is nonlinear, and is linearized using either Picard or Newton iteration. The large sparse systems of linear equations generated by the finite element discretization are solved using efficient preconditioned conjugate gradient-like methods. Tetrahedral elements and linear basis functions are used for the discretization in space, and a weighted finite difference formula is used for the discretization in time. The code handles: temporally and spatially variable boundary conditions, including seepage faces and evaporation/precipitation inputs; heterogeneous material properties and hydraulic characteristics, including saturated conductivities, porosities, and storage coefficients; and various expressions to describe the moisture content-pressure head and relative conductivity-pressure head relationships. The model solves for nodal pressure heads, and uses these values to compute the water saturations and velocities over the flow domain. The water saturation and velocity values can be used as input to the LEA3D and NONLEA3D transport codes, which are described in a companion paper.

Algorithm 378: discretized Newton-like method for solving a system of simultaneous nonlinear equations [C5]

Abstract: comment Functional procedure nielin, of the integer type, solves a system of simultaneous nonlinear algebraic or trans-cendental equations. Let us consider a given system of n equations with n variables: A kth approximation of the solution of the system (1) is supposed to be given: Yo (~) = (y~), y~), ..-, y~)). (2) If for every i, Ifi(Yo(~))[ < e, (3) where e > 0 is a given number, then the approximation (2) is considered as a solution of the system (1), otherwise a further approximation is calculated. Let h (k) > 0 be given and construct the n new points: For every function of the system (1) a new interpolating polynomial of the first order is constructed on the points (2) and (4) such that: w,(y~k)) = f, A solution of the linear system: wi(y~ , y2 , \"'', Y,) = O, i = 1, 2, \"', n, (6) is used as the (kT1)-th successive approximation. The special choice of the interpolation points (2) and (4) assures existence and uniqueness of the interpolating poly-nomials wi (5). Namely, the kth approximation has for the ith function the form: n w,(~> (Y) = /,(r2)) + ~ g~(yj-y~(~)' j, (7) i-1 where g~) = (f,(y]k)) _ f,(y~))/h(k). The solution of the system (6) where w~ is given by (7) can be written in the form (see [2]): y(~+l) = y~k) _ (1/a(k))z~) X h Ck), i = 1, 2, ..., n, (9) where z (~) = (z~ k), z~ (k), ..., z(= k)) is a solution of the following linear system: n ~f~(y(k))

Solution Of One-phase Solid-liquid Phase ChangeEnthalpy Equation By The Dual ReciprocityBoundary Element Method

Abstract: This paper presents a solution to the entalpy equation corresponding to solid-liquid phase change systems by the dual reciprocity boundary element method. The physical model is based on the one-phase averaged formulation of incompressible distinct or continuous phase-change materials with temperature-dependent thermal conductivities and specific heats of the solid and liquid phases. The boundary-domain integral equation is structured by Green's function of the Laplace equation and by the dual reciprocity boundary representation of the domain integrals. Discretization in a two-dimensional cartesian frame is based on straight line boundary elements with constant space and linear time shape functions, and on global so-called first order radial interpolation functions. The convergence of the method with respect to discretization, Peclet and Stefan number is investigated by comparing the quasi-one-dimensional numerical solution with the one-dimensional exact solution.

Iterative Solutions Of Boundary IntegralEquations

Abstract: Collocation discretisation of boundary integral equations leads to fully populated complex valued non-hermitian boundary element equations. In this paper we study the efficient solution of these linear systems by various iterative methods based on the splitting of the discrete operators. In particular the boundary element solution of the Burton and Miller formulation for the exterior Helmholtz equation is considered where the hypersingular operator, the derivative of the double layer Helmholtz potential, is present. The choice of the coupling parameter in the formulation and the splitting of the operator are shown to play an important role in the convergence of the iterative methods.

The Interaction Of Nonlinear Gravity Waves With Fixed And Floating Structures

Abstract: The paper presents the formulation of a numerical wave channel (NWC) for the investigation of the dynamics of free floating bodies in nonlinear gravity waves. A boundary element (BE) approach for two-dimensional configurations is introduced using cubic spline approximations for the free surface discretization and a double node concept for the modeling of contact points between structures and the fluid. The unknown time-dependent and nonlinear boundary conditions on the free surface are evaluated by a timestepping procedure. In addition, this initial value problem is applicable to the equations of motion of free floating bodies. In this case the right hand sides are the external forces, calculated by integrating the pressure distribution on the submerged surfaces at every time step. Here, the unknown time derivatives of the velocity potential of the fluid have to be derived e.g. by a finite difference scheme or, as proposed here, by a polynomial approximation. The advantages of this procedure are minimal discretization expenses for typical test configurations and a time domain solution, taking into account the fully nonlinear boundary conditions. Several applications of this approach are presented and discussed.

The Numerical Optimization of Distributed Parameter Systems by Gradient Methods

Abstract: : The numerical optimization of distributed parameter systems is considered. In particular the adaptation of the Davidon method, the conjugate gradient method, and the best step steepest descent method to distributed parameters is presented. The class of problems with quadratic cost functionals and linear dynamics is investigated. Penalty functions are used to render constrained problems amenable to these gradient techniques. Also considered is an analysis of the effects of discretization of continuous distributed parameter optimal control problems. Estimates of discretization error bounds are established and a measure of the suboptimality of the numerical solution is presented.

An initial value formulation of the CSDT method of solving partial differential equations

Abstract: Numerical methods of solving partial differential equations (PDEs) using analog or hybrid computers fall into three broad categories. Assuming, for concreteness, that one of the independent variables is time and the rest are spatial, the continuous-space and discrete-time (or CSDT) methods envisage to keep the space-like variable continuous and discretize the time-like variable. Similarly, the terms discrete-space and continuous time (DSCT) and discrete-space and discrete-time (DSDT) approximations are self-explanatory. For a one-space dimensional PDE, for instance, both the CSDT and DSCT approximations yield a set of ordinary differential equations while the DSDT approximations lead to a set of algebraic equations. Because of the inherent need to handle a continuous variable, both CSDT and DSCT approximations lend themselves well for computation on analog or hybrid computers. Indeed, several analog and hybrid computer implementations of all these three methods are currently in vogue each method claiming to be superior in some respect to the others. However, it was the CSDT method that showed great promise and produced little results. The purpose of this paper is to present another alternative to this problem.

A Note on the Generalized Inverses of Some Partitioned Matrices

Abstract: A method is developed for calculating the Moore–Penrose generalized inverse of certain singular partitioned (tensor product) matrices in terms of the generalized inverse of matrices of lower order. Such matrices may arise, for example, from the discretization of boundary value problems with von Neumann conditions.

Finite Elements For Shallow Water Equations:Stabilized Formulations And ComputationalAspects

Abstract: In this paper we present a space-time finite element formulation for problems governed by the shallow water equations. A linear time-discontinuous approximation is adopted, and linear three node triangles are used for the spatial discretization. Computational aspects are also discussed. It is shown that an edge-based data structure turns out to be a very fast and efficient way to solve the non-linear system of equations. Numerical examples are shown, illustrating the quality of the solution and the performance of the method.

Boundary Integral Formulation Of General Source- Based Method For Convective-diffusive Solid-liquid Phase Change Problems

Abstract: A new one-domain dual reciprocity boundary integral method technique for solving one-phase continuum formulation of the convective-diffusive energy equation as appears when treating energy transport in solid-liquid phase change systems is described. Laplace equation fundamental solution weighting, straight line geometry and constant field shape functions on the boundary, Crank-Nicolson time discretization and thin plate splines for transforming the domain integrals into a finite series of boundary integrals are employed. Iterations over the timestep are based on the Voller-Swaminathan scheme, upgraded to cope with the convective term. The technique could be applied to a wide range of solid-liquid phase change problems where finite volume or finite element solvers have been almost exclusively used in the past.

Dynamic Response Of 2-D ElastoplasticStructures By A BEM/FEM Scheme

Abstract: A combined boundary element / finite element method (BEM/FEM) in the time domain is developed and applied for solving two-dimensional dynamic elastoplastic problems. The BEM is used for the portion of the structural domain expected to remain elastic, while the FEM for the remaining expected to become elastoplastic during the deformation history. Thus, while an interior discretization is required for the FEM domain, only a boundary discretization is necessary for the BEM domain with obvious gains in efficiency. The two methods are coupled together at their interface through equilibrium and compatibility. The solution procedure is based on a step-by-step time integration algorithm of the Newmark type associated with an iterative scheme at every time step. The proposed method is illustrate by means of numerical examples, which also serve to demonstrate its merits.