Finite Number Of Grid Points Research Articles

A Note on the Stochastic Dominance Test Statistics

Davidson and Duclos (DD, 2000) develop the stochastic dominance statistics, T_j(x)(j=1,2,3), to test the hypothesis on statistically significant differences between any two cumulative density functions F and G for assets Y and Z, respectively. The DD test compares distributions at only a finite number of grid points and, under the null of equality of F and G, Davidson and Duclos show that T_j is asymptotically distributed as the studentized maximum modulus (SMM) distribution. Various studies examine the choice of grid points. One may argue that too many grids will violate the independence assumption required by the SMM distribution. We note that the limiting distributions of T_j(x) are not independent even for two points partition. More importantly, the limiting correlations would depend on the null distribution F=G. To overcome the difficulty, in this paper we improve the DD statistics by deriving the limiting process of the DD test statistic T_j(x) for the situations in which assets Y and Z which could be dependent and independent, and hence, the modified DD test can be performed by max_x|T_j(x)| and to take care of the dependency of the partitions. In addition, we propose a Bootstrap method to decide the critical point of the modified DD test.

Three-dimensional non-free-parameter lattice-Boltzmann model and its application to inviscid compressible flows

In this Letter, a three-dimensional (3D) lattice-Boltzmann model is presented following the non-free-parameter lattice-Boltzmann method of Qu et al. [K. Qu, C. Shu, Y.T. Chew, Phys. Rev. E 75 (2007) 036706]. A simple function, which satisfies the zeroth- through third-order moments of the Maxwellian distribution function, is introduced to replace the Maxwellian distribution function as the continuous equilibrium distribution function in 3D space. The function is then discretized to discrete-velocity directions via a 25-point Lagrangian interpolation polynomial. To simulate compressible flows with shock waves, an implicit–explicit finite-difference scheme based on the total variation diminishing flux limitation is adopted to solve the discrete Boltzmann–BGK equation in order to capture the shock waves in compressible flows with a finite number of grid points. The model is validated by its application to some typical inviscid compressible flows ranging from 1D to 3D, and the numerical results are found to be in excellent agreement with the analytical solutions and/or other numerical results.

Multipoint Explicit Differencing (MED) for Time Integrations of the Wave Equation

Abstract For time integrations of the wave equation, it is desirable to use a scheme that is stable over a wide range of the Courant number. Implicit schemes are examples of such schemes, but they do that job at the expense of global calculation, which becomes an increasingly serious burden as the horizontal resolution becomes higher while covering a large horizontal domain. If what an implicit scheme does from the point of view of explicit differencing is looked at, it is a multipoint scheme that requires information at all grid points in space. Physically this is an overly demanding requirement because wave propagation in the real atmosphere has a finite speed. The purpose of this study is to seek the feasibility of constructing an explicit scheme that does essentially the same job as an implicit scheme with a finite number of grid points in space. In this paper, a space-centered trapezoidal implicit scheme is used as the target scheme as an example. It is shown that an explicit space-centered scheme with forward time differencing using an infinite number of grid points in space can be made equivalent to the trapezoidal implicit scheme. To avoid global calculation, a truncated version of the scheme is then introduced that only uses a finite number of grid points while maintaining stability. This approach of constructing a stable explicit scheme is called multipoint explicit differencing (MED). It is shown that the coefficients in an MED scheme can be numerically determined by single-time-step integrations of the target scheme. With this procedure, it is rather straightforward to construct an MED scheme for an arbitrarily shaped grid and/or boundaries. In an MED scheme, the number of grid points necessary to maintain stability and, therefore, the CPU time needed for each time step increase as the Courant number increases. Because of this overhead, the MED scheme with a large time step can be more efficient than a usual explicit scheme with a smaller time step only for complex multilevel models with detailed physics. The efficiency of an MED scheme also depends on how the advantage of parallel computing is taken.

Optimization of natural frequencies of bidirectional functionally graded beams

We propose a methodology to optimize the natural frequencies of functionally graded structures by tailoring their material distribution. The element-free Galerkin method is used to analyze the two-dimensional steady-state free and forced vibration of functionally graded beams. To optimize the material composition, the spatial distribution of volume fractions of the material constituents is defined using piecewise bicubic interpolation of volume fraction values that are specified at a finite number of grid points. Subsequently, we use a real-coded genetic algorithm to optimize the volume fraction distribution for three model problems. In the first problem, we seek material distributions that maximize each of the first three natural frequencies of a functionally graded beam. The goal of the second model problem is to minimize the mass of a functionally graded beam while constraining its natural frequencies to lie outside certain prescribed frequency bands. The last problem aims to minimize the mass of a functionally graded beam by simultaneously optimizing its thickness and material distribution such that the fundamental frequency is greater than a prescribed value.

Multi-objective optimization of functionally graded materials with temperature-dependent material properties

We propose a methodology for the multi-objective optimization of material distribution of functionally graded materials with temperature-dependent material properties for steady thermomechanical processes. The proposed approach focuses on metal/ceramic and metal/metal functionally graded materials, which offer great promise in high temperature and high heat flux applications. The two-dimensional quasi-static heat conduction and thermoelasticity problems are analyzed using the element-free Galerkin method. The formulation takes into account the change in material properties due to temperature. The spatial distribution of the volume fraction is obtained by piecewise bicubic interpolation of volume fractions defined at a finite number of grid points. The effective material properties at a point in the domain are estimated from the local volume fractions of the material constituents using the Mori–Tanaka and self-consistent homogenization schemes while the effective yield stress for metal/metal functionally graded materials is determined using either the Hashin–Shtrikman lower estimate or the Hashin–Shtrikman upper bound. The volume fraction distribution is optimized using an elitist, non-dominated sorting multi-objective genetic algorithm. We use the genetic algorithm and element-free Galerkin analyses to optimize the material composition for two model problems. In the first problem, we simultaneously minimize the mass and minimize the peak effective stress of a zirconia/titanium alloy component subjected to a specified temperature load. The goal of the second model problem is to simultaneously minimize the mass and maximize the factor of safety of a two-dimensional tungsten/copper alloy FGM under an intense heat flux, subject to constraints on the factor of safety and maximum temperature experienced by the constituent phases.

Two-dimensional optimization of material composition of functionally graded materials using meshless analyses and a genetic algorithm

We propose a methodology for the two-dimensional simulation and optimization of material distribution of functionally graded materials for thermomechanical processes. The proposed approach focuses on metal/ceramic functionally graded materials, which offer great promise in applications where the operating conditions are severe. The two-dimensional quasi-static heat conduction and thermoelasticity problems are analyzed using the element-free Galerkin method. The spatial distribution of ceramic volume fraction is obtained by piecewise bicubic interpolation of volume fractions defined at a finite number of grid points. The effective material properties at a point in the domain are estimated from the local volume fractions of the material constituents using the Mori–Tanaka and self-consistent homogenization schemes. The volume fraction distribution is optimized using a real-coded genetic algorithm. The element-free Galerkin code is validated by comparing the results with an exact thermoelasticity solution for a simply supported plate. Subsequently, we use the genetic algorithm and element-free Galerkin analyses to optimize the material composition for two model problems. In the first problem, we minimize the peak residual stress when the functionally graded component is cooled from a high fabrication temperature. The goal of the second model problem is to minimize the mass of a graded component with constraints on the peak effective stress and maximum temperature experienced by the metal.

Optimal reflection-free complex absorbing potentials for quantum propagation of wave packets

The conditions for optimal reflection-free complex-absorbing potentials (CAPs) are discussed. It is shown that the CAPs as derived from the smooth-exterior-scaling transformation of the Hamiltonian,[J. Phys. B. 31, 1431 (1998)], serve as optimal reflection-free CAPs (RF-CAPs) in wave-packet propagation calculations of open systems. The initial wave packet, can be located in the interaction region (as in half collision experiments) where the CAPs have vanished or in the asymptote where the absorbing potential non zero. As we show the optimal CAPs can be introduced also in the region where the physical potential has not vanished. The un-avoided reflections due to the use of finite number of grid points (or basis functions) are discussed. A simple way to reduce the "edge grid" reflection effect is described.

Minimax filters for microphone arrays

Conventionally, minimax spatial/spectral filters for microphone arrays are designed by first discretizing the spatial and frequency domains into a finite number of grid points and then performing the optimization over these points. The drawback with this approach is that the response of the spatial/spectral filters in between the grid points can be poor. More recently, an approach that performs the minimax design over the continuum of points in the decision space has appeared in the literature. In this brief, we describe an approach to solving this continuous decision space design problem that is numerically more elegant and efficient. The effectiveness of the new method is illustrated by a numerical example.

Generation of computational grids using optimization

A method for generation of computational grids using principles of nonlinear programming (optimization) is described. Grids are generated so that certain discrete measures of grid smoothness and orthogonality are max- imized using a fast iterative optimization procedure. The method can also be used to improve an existing grid ir- respective of the method used for its generation. If the original grid contains regions of overlap (nonpositive Jacobian), this method is capable of unraveling the grid and making it useful for computations. The iterative op- timization procedure is efficient due to the use of a conjugate direction method with exact line searching. Examples are given of the application of this technique to two- and three-dimensional computational grids. The extension of the method to generate solution adaptive grids is discussed. N the past few years, there has been a great deal of interest in the development of computational grid-generation techniques1 for discretizing complex regions for the numerical solution of partial differential equations. The commonly used grid-generation methods have achieved a high degree of sophistication and ease of use. However, for complex con- figurations, the existing methods do not always produce ac- ceptable grids for computations. Grid-quality deterioration is especially apparent in three-dimensional grid-generation methods and when two-dimensional grids are combined to form a three-dimensio nal grid.2'3 In addition to grid-quality problems, many methods suffer from computational ineffi- ciency. For these reasons, the authors have perceived a need for a grid-generatio n method that could be used to improve the quality of a given computational grid. In addition, the method should be capable of serving as a stand-alone grid- generation procedure, independent of other techniques. The basic concept behind this grid-generation method is to assume from the outset that the grid to be generated or op- timized consists of straight-line segments joining the nodal grid points. This is in contrast to methods that solve elliptic, parabolic, or hyperbolic partial differential equations or use conformal mapping. These methods are based on con- tinuous, not discrete, coordinate transformations and, therefore, suffer from discretization errors for any grid con- sisting of a finite number of grid points. The grid-generation technique described herein formulates discrete measures of grid smoothness and orthogonality at each grid point and, therefore, does not suffer from truncation errors on coarse grids. These measures of grid quality are then minimized us- ing a conjugate direction procedure with exact line searching. The resulting grids are optimal with respect to the particular smoothness and orthogonality measures chosen, and there are many possible choices for their exact forms. The next section describes the formulation of the method followed by some typical results of using this method to im- prove existing grids or generate new grids. Finally, exten- sions of the method for generating solution adaptive grids are discussed, and other areas of possible future research are pointed out. Analysis The formulation of the method will be illustrated for two- dimensional grids. The details of the method for the three- dimensional case are given in the Appendix. We will discuss the grid-generation method by first noting similarities with the Saltzman-Brackbill4 variational grid- generation method. In the variational method, two func- tionals are introduced that provide measures of grid smoothness

Computation of Steady Shocks by Second-Order Finite-Difference Schemes

The computational stability of steady shocks which satisfy the entropy condition is considered for the scalar conservation law \[ \frac {{\partial u}}{{\partial t}} + \frac {\partial }{{\partial x}}\left ( {\frac {1}{2}{u^2}} \right ) = 0.\] It is shown that the computation of the pure initial value problem by Lax-Wendroff type schemes approaches a steady state if the initial data satisfies a specified condition, and that this condition is always satisfied for the corresponding initial-boundary value problem with a finite number of grid points. The effect of machine accuracy on the influence of the boundaries on the error near the shock is also discussed.

Computation of steady shocks by second-order finite-difference schemes

The computational stability of steady shocks which satisfy the entropy condition is considered for the scalar conservation law \[ ∂ u ∂ t + ∂ ∂ x ( 1 2 u 2 ) = 0. \frac {{\partial u}}{{\partial t}} + \frac {\partial }{{\partial x}}\left ( {\frac {1}{2}{u^2}} \right ) = 0. \] It is shown that the computation of the pure initial value problem by Lax-Wendroff type schemes approaches a steady state if the initial data satisfies a specified condition, and that this condition is always satisfied for the corresponding initial-boundary value problem with a finite number of grid points. The effect of machine accuracy on the influence of the boundaries on the error near the shock is also discussed.

ÉTUDE DE L'ATTÉNUATION DE L'INFLUENCE DES ANOMALIES SUPERFICIELLES DANS LE CALCUL DE LA DÉRIVÉE SECONDE DE LA GRAVITÉ*)

ABSTRACTIn order to eliminate the effect of smoothing due to the use of a finite number of grid points, the second derivative is computed by integrating the product of g with a convenient continuous function, which yields the second derivative to the extent to which the first terms of the Taylor expansion of g represent its value correctly. By applying this method to the anomaly caused by an isolated mass, and to that caused by a homogeneous half plane, it is shown that, if the result obtained is interpreted as if it really were a second derivative, erroneous values for the depth and the mass are obtained. If the real depth of the mass is small, a too large apparent depth is obtained. In the case of a half plane the use of a system of grid points gives the same result. These considerations permit the rational choice of the method of computing the second derivative, such that the effects of too shallow mass irregularities are attenuated.