Showing papers on "Discretization published in 1992"

Wavelet analysis and synthesis of fractional Brownian motion

Abstract: Fractional Brownian motion (FBM) offers a convenient modeling for nonstationary stochastic processes with long-term dependencies and 1/f-type spectral behavior over wide ranges of frequencies. Statistical self-similarity is an essential feature of FBM and makes natural the use of wavelets for both its analysis and its synthesis. A detailed second-order analysis is carried out for wavelet coefficients of FBM. It reveals a stationary structure at each scale and a power-law behavior of the coefficients' variance from which the fractal dimension of FBM can be estimated. Conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed, consequences of discretization are considered, and some connections between the wavelet point of view and previous approaches based on length measurements (analysis) or dyadic interpolation (synthesis) are briefly pointed out. >

Gradient-dependent plasticity: formulation and algorithmic aspects

Abstract: A plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure (hardening parameter), but also on the Laplacian thereof. The consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity. To properly solve the set of non-linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements. For appropriate boundary conditions this formulation can also be derived from a variational principle. Accordingly, the theory is complete

A viscosity solutions approach to shape-from-shading

Abstract: The problem of recovering a Lambertian surface from a single two-dimensional image may be written as a first-order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions. A new approach based on Hamilton–Jacobi–Bellman equations and viscosity solutions theories enables one to study non-uniqueness phenomenon and thus to characterize the surface among the various solutions.A consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle, and numerical results are presented.

ChiMerge: discretization of numeric attributes

Abstract: Many classification algorithms require that the training data contain only discrete attributes To use such an algorithm when there are numeric attributes, all numeric values must first be converted into discrete values--a process called discretization This paper describes ChiMerge, a general, robust algorithm that uses the χ2 statistic to discretize (quantize) numeric attributes

Resolution of Resistivity Tomography Inferred from Numerical SIMULATION1

Abstract: Some factors affecting the resolution and accuracy of resistivity tomography are examined using numerical simulation. The inversion method used is based on smoothness-constrained least-squares and finite-element methods. An appropriate block discretization is obtained by dividing the target region into square blocks of size equal to half the minimum electrode spacing. While the effect of the damping factor on the resolution is significant, the resolution is not very sensitive to Gaussian noise as long as the damping factor is properly chosen, according to the noise level. The issue of choosing an optimum electrode array should be considered at the planning stage of a survey. When the instrumental accuracy is high, the dipole-dipole array is more suitable for resolving complex structures than the pole-pole array. The pole-dipole array gives somewhat less resolution than the dipole-dipole array but yields greater signal strength; thus, the pole-dipole array may be a good compromise between resolution and signal strength. The effect of an inhomogeneity located outside the target region may be very small if block discretization is done so as to represent the resistivity variations in both the target and outside regions.

Discrete approximations to optimal trajectories using direct transcription and nonlinear programming

Abstract: A recently developed method for solving optimal trajectory problems uses a piecewise-polynomial representation of the state and control variables, enforces the equations of motion via a collocation procedure, and thus approximates the original calculus-of-variations problem with a nonlinear-programming problem, which is solved numerically. This paper identifies this method as a direct transcription method and proceeds to investigate the relationship between the original optimal-control problem and the nonlinear-programming problem. The discretized adjoint equation of the collocation method is found to have deficient accuracy, and an alternate scheme which discretizes the equations of motion using an explicit Runge-Kutta parallel-shooting approach is developed. Both methods are applied to finite-thrust spacecraft trajectory problems, including a low-thrust escape spiral, a three-burn rendezvous, and a low-thrust transfer to the moon.

Direct simulation of fluid particle motions

Abstract: Continuum models of two-phase flows of solids and liquids use constitutive assumptions to close the equations. A more fundamental approach is a “molecular dynamic” simulation of flowing “big” particles based on reliable macroscopic equations for both solid and liquid. We developed a package that simulates the unsteady two-dimensional solid-liquid two-phase flows using the Navier-Stokes equations for the liquid and Newton's equations of motion for the solid particles. The Navier-Stokes equations are solved using a finite-element formulation and Newton's equations of motion are solved using an explicit-implicit scheme. We show that the simplest fully explicit scheme to update the particle motion using Newton's equations is unstable. To correct this instability we propose and implement and Explicit-Implicit Scheme in which, at each time step, the positions of the particles are updated explicitly, the computational domain is remeshed, the solution at the previous time is mapped onto the new mesh, and finally the nonlinear Navier-Stokes equation and the implicitly discretized Newton's equations for particle velocities are solved on the new mesh iteratively. The numerical simulation reveals the effect of vortex shedding on the motion of the cylinders and reproduces the drafting, kissing, and tumbling scenario which is the dominant rearrangement mechanism in two-phase flow of solids and liquids in beds of spheres which are constrained to move in only two dimensions.

Simulating the grassfire transform using an active contour model

Abstract: A method for shape description of planar objects that integrates both region and boundary features is presented. The method is an implementation of a 2D dynamic grassfire that relies on a distance surface on which elastic contours minimize an energy function. The method is based on an active contour model. Numerous implementation aspects of the shape description method were optimized. A Euclidean metric was used for optimal accuracy, and the active contour model permits bypassing some of the discretization limitations inherent in using a digital grid. Noise filtering was performed on the basis of both contour feature measures and region measures, that is, curvature extremum significance and ridge support, respectively, to obtain robust shape descriptors. Other improvements and variations of the algorithmic implementation are proposed. >

The Laplace Transform Galerkin Technique for large-scale simulation of mass transport in discretely fractured porous formations

Abstract: The ability to simulate contaminant migration in large-scale porous formations containing a complex network of discrete fractures is limited by traditional modeling approaches. One primary reason is because of vastly different transport time scales in different regions due to rapid advection along the discrete fractures and slow but persistent diffusion in the porous matrix. In addition to time-related complexities, standard numerical methods require a fine spatial discretization in the porous matrix to represent sharp concentration gradients at the interface between the fractures and the matrix. In order to circumvent these difficulties, the Laplace transform Galerkin method is extended for application to discretely fractured media with emphasis on large-scale modeling capabilities. The technique avoids time stepping and permits the use of a relatively coarse grid without compromising accuracy because the Laplace domain solution is relatively smooth and devoid of discontinuities even in advection-dominated problems. Further computational efficiency for large-grid problems is achieved by employing a preconditioned, ORTHOMIN-accelerated iterative solver. A unique feature of the method is that each of the several needed p space solutions are independent, thus making the scheme highly parallel. Other features include the accommodation of advective-dispersive transport in the porous matrix and the straightforward inclusion of dual-porosity theory to represent matrix diffusion in regions where microfractures exist below the modeling scale. An example problem involving contaminant transport through an aquitard into an underlying aquifer leads to the conclusion that deep, essentially undetectable fractures in clayey aquitards can greatly compromise the quality of groundwater in the impacted aquifer.

Efficient rounding of approximate designs

Abstract: SUMMARY Discretization methods to round an approximate design into an exact design for a given sample size n are discussed. There is a unique method, called efficient rounding, which has the smallest loss of efficiency under a wide family of optimality criteria. The efficient rounding method is a multiplier method of apportionment which otherwise is known as the method of John Quincy Adams or the method of smallest divisors.

Accuracy of operator splitting for advection‐dispersion‐reaction problems

Abstract: An operator-splitting approach is often used for the numerical solution of advection-dispersion-reaction problems. Operationally, this approach advances the solution over a single time step in two stages, one involving the solution of the nonreactive advection-dispersion equation and the other the solution of the reaction equations. The first stage is usually solved with a finite difference, finite element, or related technique, while the second stage is normally solved with an ordinary differential equation integrator. The only generally published guidelines on numerical accuracy suggest that the discretization errors associated with each stage must be small in order to achieve high accuracy of the overall solution. However, in this note we demonstrate that there is an inherent mass balance error present in the operator-splitting algorithm for problems involving continuous mass influx boundary conditions. The mass balance error does not exist for instantaneous mass input problems. These conclusions are based upon analysis of a simple first-order decay problem for which each stage of the calculation can be performed analytically (i.e., without discretization error). For this linear decay problem we find that the product of the first-order decay coefficient (k) times Δt must be less than approximately 0.1 in order for the mass balance error to be less than 5%. We also present a variant of the normal operator-splitting algorithm in which the order of solving the advection-dispersion and reaction operators is reversed at each time step. This modification reduces the mass balance error by more than a factor of 10 for a wide range of kΔt values.

A new approach to the discretization of continuous-time controllers

Abstract: A new method for discretizing continuous-time controllers is derived, using principles of controller approximation. It focuses on the closed-loop use of the discrete-time controller. The resulting approximation criterion is a measure for stability of the control system, provides an upper bound on the sampling time for which stability can be guaranteed, and because it is based on continuous-time controller approximation it indicates the cost of discretization in terms of performance degradation. The discrete-time controller is obtained through minimization of this criterion, which can be performed with standard software used in H/sub infinity / controller design. >

Fast capacitance extraction of general three-dimensional structures

Abstract: K. Nabors and J. White (1991) presented a boundary-element-based algorithm for computing the capacitance of three-dimensional m-conductor structures whose computational complexity grows nearly as mn, where n is the number of elements used to discretize the conductor surfaces. In that algorithm, a generalized conjugate residual iterative technique is used to solve the n*n linear system arising from the discretization, and a multipole algorithm is used to compute the iterates. Several improvements to that algorithm are described which make the approach applicable and computationally efficient for almost any geometry of conductors in a homogeneous dielectric. Results using these techniques in a program which computes the capacitance of general 3D structures are presented to demonstrate that the new algorithm is nearly as accurate as the more standard direct factorization approach, and is more than two orders of magnitude faster for large examples. >

Multilevel Schwarz methods

Abstract: We consider the solution of the algebraic system of equations which result from the discretization of second order elliptic equations. A class of multilevel algorithms are studied using the additive Schwarz framework. We establish that the condition number of the iteration operators are bounded independent of mesh sizes and the number of levels. This is an improvement on Dryja and Widlund's result on a multilevel additive Schwarz algorithm, as well as Bramble, Pasciak and Xu's result on the BPX algorithm. Some multiplicative variants of the multilevel methods are also considered. We establish that the energy norms of the corresponding iteration operators are bounded by a constant less than one, which is independent of the number of levels. For a proper ordering, the iteration operators correspond to the error propagation operators of certain V-cycle multigrid methods, using Gauss-Seidel and damped Jacobi methods as smoothers, respectively.

A Class of Semi-Lagrangian Approximations for Fluids

Abstract: This paper discusses a class of finite-difference approximations to the evolution equations of fluid dynamics These approximations derive from elementary properties of differential forms Values of a fluid variable ψ at any two points of a space-time continuum are related through the integral of the space-time gradient of ψ along an arbitrary contour connecting these two points (Stokes' theorem) Noting that spatial and temporal components of the gradient are related through the fluid equations, and selecting the contour composed of a parcel trajectory and an appropriate residual, leads to the integral form of the fluid equations, which is particularly convenient for finite-difference approximations In these equations, the inertial and forcing terms are separated such that forces are integrated along a parcel trajectory (the Lagrangian aspect), whereas advection of the variable is evaluated along the residual contour (the Eulerian aspect) The virtue of this method is an extreme simplicity of t

Error estimates with smooth and nonsmooth data for a finite element method for the Cahn-Hilliard equation

Abstract: A finite element method for the Cahn-Hilliard equation (a semilinear parabolic equation of fourth order) is analyzed, both in a spatially semidiscrete case and in a completely discrete case based on the backward Euler method. Error bounds of optimal order over a finite time interval are obtained for solutions with smooth and nonsmooth initial data. A detailed study of the regularity of the exact solution is included. The analysis is based on local Lipschitz conditions for the nonlinearity with respect to Sobolev norms, and the existence of a Ljapunov functional for the exact and the discretized equations is essential. A result concerning the convergence of the attractor of the corresponding approximate nonlinear semigroup (upper semicontinuity with respect to the discretization parameters) is obtained as a simple application of the nonsmooth data error estimate.

A linear discretization of the volume conductor boundary integral equation using analytically integrated elements (electrophysiology application)

Abstract: A method is presented to compute the potential distribution on the surface of a homogeneous isolated conductor of arbitrary shape. The method is based on an approximation of a boundary integral equation as a set of linear algebraic equations. The potential is described as a piecewise linear or quadratic function. The matrix elements of the discretized equation are expressed as analytical formulas. >

Fluid flow and heat transfer test problems for non‐orthogonal grids: Bench‐mark solutions

Abstract: Four problems of fluid flow and heat transfer were designed in which non-orthogonal, boundary-fitted grids were to be used. These are proposed to serve as test cases for testing new solution methods. This paper presents solutions of the test problems obtained by using a multigrid finite volume method with grids of up to 320 × 320 control volumes. Starting from zero fields, iterations were performed until the sum of the absolute residuals had fallen seven orders of magnitude, thus ensuring that the variable values did not change to six most significant digits. By comparing the solutions for successive grids at moderate Reynolds and Rayleigh numbers, the discretization errors were estimated to be lower than 0·1%. The results presented in this paper may thus serve for comparison purposes as bench-mark solutions.

Upwind Scheme for Solving the Euler Equations on Unstructured Tetrahedral Meshes

Abstract: An upwind scheme is presented for solving the three-dimensional Euler equations on unstructured tetrahedral meshes. Spatial discretization is accomplished by a cell-centered finite-volume formulation using flux-difference splitting. Higher-order differences are formed by a multidimensional linear reconstruction process. The solution gradients required for the higher-order differenes are computed by a novel approach that yields highly resolved solutions in regions of smooth flow while avoiding oscillations across shocks without explicitly applying a limiter. Solutions are advanced in time by a three-stage Runge-Kutta time-stepping scheme with convergence accelerated to steady state by local time stepping and implicit residual smoothing. Transonic solutions are presented for two meshes around the ONERA M6 wing and demonstrate substantial accuracy and insensitivity to mesh size.

An improved energy transport model including nonparabolicity and non-Maxwellian distribution effects

Abstract: An improved energy transport model for device simulation is derived from the zeroth and second moments of the Boltzmann transport equation (BTE) and from the presumed functional form of the even part of the carrier distribution in momentum space. Energy-band nonparabolicity and non-Maxwellian distribution effects are included to first order. The model is amenable to an efficient self-consistent discretization taking advantage of the similarity between current and energy flow equations. Numerical results for ballistic diodes and MOSFETs are presented. Typical spurious velocity overshoot spikes, obtained in conventional hydrodynamic (HD) simulations of ballistic diodes, are virtually eliminated. >

Adaptive h‐refinement on 3D unstructured grids for transient problems

Abstract: An adaptive finite element scheme for transient problems is presented. The classic h-enrichment/coarsening is employed in conjunction with a tetrahedral finite element discretization in three dimensions. A mesh change is performed every n time steps, 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 by pre-sorting the elements and then performing the refinement/coarsening groupwise according to the case at hand. Further reductions in CPU requirements arc realized by optimizing the identification and sorting of elements for refinement and deletion. The developed technology has been used extensively for shock-shock and shock-object interaction runs in a production mode. A typical example of this class of problems is given.

Direct discretization of planar div-curl problems

Abstract: A control volume method is proposed for planar div-curl systems. The method is independent of potential and least squares formulations, and works directly with the div-curl system. The novelty of the technique lies in its use of a single local vector field component and two control volumes rather than the other way around. A discrete vector field theory comes quite naturally from this idea and is developed. Error estimates are proved for the method, and other ramifications investigated.

Analysis of a finite element method for Maxwell's equations

Abstract: The use of finite elements to discretize the time dependent Maxwell equations on a bounded domain in three-dimensional space is analyzed Energy norm error estimates are provided when general finite element methods are used to discretize the equations in space In addition, it is shown that if some curl conforming elements due to Nedelec are used, error estimates may also be proved in the $L^2 $ norm