Localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for nonlocal diffusion problems

Spectral/pseudo-spectral methods based on high order polynomials have been successfully used for solving partial differential and integral equations. In this paper, we will present the use of a localized radial basis functions-based pseudo-spectral method (LRBF-PSM) for solving 2D nonlocal problems with radial nonlocal kernels. The basic idea of the LRBF-PSM is to construct a set of orthogonal functions by RBFs on each overlapping sub-domain from which the global solution can be obtained by extending the approximation on each sub-domain to the entire domain. Numerical implementation indicates that the proposed LRBF-PSM is simple to use, efficient and robust to solve various nonlocal problems.     

Conforming spectral methods for Poisson problems in cuboidal domains

A Chebyshev collocation strategy is introduced for the subdivision of cuboids into cuboidal subdomains (elements). These elements are conforming, which means that the approximation to the solution isC ⁰ continuous at all points across their interfaces.     

Conforming Chebyshev spectral methods for Poisson problems in rectangular domains

Four and six-element conforming domain decomposition techniques are developed for Chebyshev spectral collocation methods for Poisson problems in rectangular domains. The applicability of the methods is demonstrated on standard test problems.Parts of this work were performed while the author was at the Mathematics Department, Southern Methodist University, Dallas, Texas 75275.     

A spectral method for bonds

We present an spectral numerical method for the numerical valuation of bonds with embedded options. We use a CIR model for the short-term interest rate. The method is based on a Galerkin formulation of the partial differential equation for the value of the bond, discretized by means of orthogonal Laguerre polynomials. The method is shown to be very efficient, with a high precision for the type of problems treated here and is easy to use with more general models with nonconstant coefficients. As a consequence, it can be a possible alternative to other approaches employed in practice, specially when a calibration of the parameters of the model is needed to match the observed market data.     

A practical assessment of spectral accuracy for hyperbolic problems with discontinuities

Numerical experiments are performed to compare the accuracy obtained when physical and transform space filters are used to smooth the oscillations in Fourier collocation approximations to discontinuous solutions of a linear wave equation. High-order accuracy can be obtained away from a discontinuity but the order is strongly filter dependent. Polynomial order accuracy is demonstrated when smooth high-order Fourier filters are used. Spectral accuracy is obtained with the physical space filter of Gottlieb and Tadmor.     

A distance template for octree traversal in CPU-based volume ray casting

Several optimization techniques have been proposed to improve the speed of direct volume rendering. A hierarchical representation formed by an octree is a data structure to skip over transparent regions while requiring little preprocessing and data storage. However, in order to skip over an octant estimated to be transparent (a transparent octant), the distance from a boundary to another boundary of the octant should be calculated. Because the distance computation is expensive, we propose a precomputed data structure, the distance template, which stores direction and distance values from one boundary voxel on a face to all the boundary voxels on the remaining five faces. In the rendering step, if a ray reaches a transparent octant, it leaps over the octant by referring to the stored distance value.     

A greedy algorithm for feedrate planning of CNC machines along curved tool paths with confined jerk

In this paper, the problem of optimal feedrate planning along a curved tool path for 3-axis CNC machines with the acceleration and jerk limits for each axis and the tangential velocity bound is addressed. It is proved that the optimal feedrate planning must be “Bang–Bang” or “Bang–Bang-Singular” control, that is, at least one of the axes reaches its acceleration or jerk bound, or the tangential velocity reaches its bound throughout the motion. As a consequence, the optimal parametric velocity can be expressed as a piecewise analytic function of the curve parameter u. The explicit formula for the velocity function when a jerk reaches its bound is given by solving a second-order differential equation. Under a “greedy rule”, an algorithm for optimal jerk confined feedrate planning is presented. Experiment results show that the new algorithm can be used to reduce the machining vibration and improve the machining quality.     

A comparative study of turbulence models performance for separating flow in a planar asymmetric diffuser

This paper presents a computational study of the separated flow in a planar asymmetric diffuser. The steady RANS equations for turbulent incompressible fluid flow and six turbulence closures are used in the present study. The commercial software code, FLUENT 6.3.26, was used for solving the set of governing equations using various turbulence models. Five of the used turbulence models are available directly in the code while the v²–f turbulence model was implemented via user defined scalars (UDS) and user defined functions (UDF). A series of computational analysis is performed to assess the performance of turbulence models at different grid density. The results show that the standard k–ω, SST k–ω and v²–f models clearly performed better than other models when an adverse pressure gradient was present. The RSM model shows an acceptable agreement with the velocity and turbulent kinetic energy profiles but it failed to predict the location of separation and attachment points. The standard k–ε and the low-Re k–ε delivered very poor results.     

Optimization-based decision support for scenario analysis in electronic sourcing markets with volume discounts

E-Sourcing software has become an integral part of electronic commerce. Beyond the use of single-lot auction formats, there has been an emerging interest in using e-sourcing software for complex negotiations. Procurement markets typically exhibit scale economies leading to various types of volume discounts which are in wide-spread use in practice. The analysis of bids in such negotiations typically leads to computationally hard optimization problems. Scenario analysis describes a process, in which procurement managers compute different award allocations as a result of different allocation constraints and parameters that they put in place. This paper discusses an optimization model and computational methods which allow for effective scenario analysis with allocation problems in the presence of different types of discount policies and allocation constraints. The model reduces the number of parameter settings to explore considerably. The models are such that they can often not be solved exactly for realistic problem sizes in practically acceptable time frames. Therefore, we provide results of numerical experiments using exact algorithms and heuristics to solve the problem. We find that RINS and Variable Neighborhood Search can be effectively used in traditional branch-and-cut algorithms for this problem. Overall, new computational approaches allow procurement managers to evaluate offers even in markets with a complex set of volume discounts and multiple allocation constraints.     

A heterogeneous computing system for coupling 3D endomicroscopy with volume rendering in real-time image visualization

Modern microscopic volumetric imaging processes lack capturing flexibility and are inconvenient to operate. Additionally, the quality of acquired data could not be assessed immediately during imaging due to the lack of a coherent real-time visualization system. Thus, to eliminate the requisition of close user supervision while providing real-time 3D visualization alongside imaging, we propose and describe an innovative approach to integrate imaging and visualization into a single pipeline called an online incrementally accumulated rendering system. This system is composed of an electronic controller for progressive acquisition, a memory allocator for memory isolation, an efficient memory organization scheme, a compositing scheme to render accumulated datasets, and accumulative frame buffers for displaying non-conflicting outputs. We implement this design using a laser scanning confocal endomicroscope, interfaced with an FPGA prototyping board through a custom hardware circuit. Empirical results from practical implementations deployed in a cancer research center are presented in this paper.     

A Spectral Method for the Time Evolution in Parabolic Problems

Numerical time propagation of linear parabolic problems is commonly performed by Taylor expansion based schemes, such as Runge–Kutta. However, explicit schemes of this type impose a stringent stability restriction on the time step when the space discretization matrix is poorly conditioned. Thus the computational work required for integration over a long and fixed time interval is controlled by stability rather than by accuracy of the scheme. We develop an improved time evolution scheme based on a new Chebyshev series expansion for solving time-dependent inhomogeneous parabolic initial-boundary value problems in which the stability condition is relaxed. Spectral accuracy of the time evolution scheme is achieved. Additionally, the approximation derived here can be useful for solving quasi-linear parabolic evolution problems by exponential time differencing methods     

A Dynamic Multilevel Model for the Simulation of the Small Structures in Homogeneous Isotropic Turbulence

In turbulence simulations, the small scales of motion, even if they carry only a very small percentage of the whole kinetic energy, must be taken into account in order to accurately reproduce the statistical properties of the flows. This induces strong computational restrictions. In an attempt to understand and model the nonlinear interaction between the small and large scales, a dynamic multilevel procedure is proposed and applied to homogeneous turbulence. As in large eddy simulation, filtering operators are used to separate the different scales of the velocity field. In classical models (Smagorinsky), only the large scale equation is resolved. A different approach is proposed here. Indeed, by analyzing the nonlinear interaction term in the large scale equation, we show that they locally have a very small contribution to the whole dynamic of the flow. We then propose to treat them less accurately. Specific treatments for these terms are achieved by a space and time adaptative procedure; the cut-off value (filter width) which defines the scale separation varies as time evolves. Simulations at Re in the range of 60 to 150 have been performed until statistical steady states are reached, i.e. over long time period. Comparisons with direct simulations (DNS) show that this numerical modeling provides an efficient resolution of the nonlinear interaction term. The multilevel algorithm is shown to be stable; the corresponding simulated flows reach a statistically steady state very close to the DNS ones. The shape of the energy spectrum functions as well as the characteristic statistical properties of the velocity and its derivatives are accurately recovered.     

A fourth order finite volume scheme for turbulent flow simulations in cylindrical domains

A finite volume scheme which is based on fourth order accurate central differences in the spatial directions and on a hybrid explicit/semi-implicit time stepping scheme was developed to solve the incompressible Navier-Stokes equations on cylindrical staggered grids. This includes a new fourth order accurate discretization of the velocity at the singularity of the cylindrical coordinate system and a new stability condition. The new method was applied in the direct numerical simulations (DNS) of the fully developed non-swirling turbulent flow through straight pipes with circular cross-section for the Reynolds number Re_τ = 360 based on the friction velocity u_τ and the pipe diameter. The obtained results are expressed in terms of statistical moments of the velocity components and are presented in comparison with those obtained with a second order accurate scheme and by measurements. It is shown that the fourth order spatial discretization leads to improved higher order statistical moments, while the first and the second order moments are more or less insensitive to the spatial discretization order.     

A parallel adaptive method for simulating shock-induced combustion with detailed chemical kinetics in complex domains

An adaptive finite volume approach is presented to accurately simulate shock-induced combustion phenomena in gases, particularly detonation waves. The method uses a Cartesian mesh that is dynamically adapted to embedded geometries and flow features by using regular refinement patches. The discretisation is a reliable linearised Riemann solver for thermally perfect gas mixtures; detailed kinetics are considered in an operator splitting approach. Besides easily reproducible ignition problems, the capabilities of the method and its parallel implementation are quantified and demonstrated for fully resolved triple point structure investigations of Chapman–Jouguet detonations in low-pressure hydrogen–oxygen–argon mixtures in two and three space dimensions.     

A time-varying geometry modeling method for parts with deformation during machining process

Deformation due to residual stress is a significant issue during the machining of thin-walled parts with low rigidity. If there are multiple processes with deformation during machining, some process suitability issues will appear. On this occasion, the actual geometric state of the deformed workpiece is needed for process adjustment. However, it is still a challenge to obtain the complete geometry information of deformed workpiece accurately and efficiently. In order to address this issue, a time-varying geometry modeling method, combining cutting simulation and in-process measurement, is proposed in this paper. The deformed workpiece model can be reconstructed via transforming the deformed workpiece with only a small amount of the measurement points by superimposing material removal and workpiece deformation simulation according to a time sequence, which takes advantage of the proposed Curved Surface Mapping based Geometric Representation Model (CSMGRM). Machining experiment of a typical structural part has shown that the deformed geometry model of the whole workpiece can be reconstructed within the error of 0.05mm, which is less than one tenth of the finish machining allowance in general cases, and it is sufficient to meet the accuracy requirements for interference or overcut/undercut analysis and process adjustment.     

On the stability of the reduced basis method for Stokes equations in parametrized domains

We present an application of reduced basis method for Stokes equations in domains with affine parametric dependence. The essential components of the method are (i) the rapid convergence of global reduced basis approximations - Galerkin projection onto a space W_N spanned by solutions of the governing partial differential equation at N selected points in parameter space; (ii) the off-line/on-line computational procedures decoupling the generation and projection stages of the approximation process.The operation count for the on-line stage - in which, given a new parameter value, we calculate an output of interest - depends only on N (typically very small) and the parametric complexity of the problem; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. Particular attention is given (i) to the pressure treatment of incompressible Stokes problem; (ii) to find an equivalent inf-sup condition that guarantees stability of reduced basis solutions by enriching the reduced basis velocity approximation space with the solutions of a supremizer problem; (iii) to provide algebraic stability of the problem by reducing the condition number of reduced basis matrices using an orthonormalization procedure applied to basis functions; (iv) to reduce computational costs in order to allow real-time solution of parametrized problem.     

A method for high-order multipoint boundary value problems with Birkhoff-type conditions

In this paper an efficient numerical method for solving a class of multipoint boundary value problems with special boundary conditions of Birkhoff-type is presented. After a quick reference to Birkhoff-type interpolation polynomial which satisfies the particular conditions, and a result on the existence and uniqueness of solution of the given problem, an algorithm is introduced to find a polynomial that approximates the solution. It is a general collocation method. Then an a priori estimation of the error of this approximation is given. Finally, to show the efficiency and the applicability of the method, numerical results are presented. These numerical experiments provide favourable comparisons with the NDSolve command of Mathematica.     

A note on the satisfaction of the boundary conditions for Chebyshev collocation methods in rectangular domains

The way boundary conditions are imposed when applying Chebyshev collocation methods to Poisson and biharmonic-type problems in rectangular domains is investigated. It is shown that careful selection of the number of collocation points leads to a linear system ofn linearly independent equations inn unknowns.     

A trigonometrically-fitted method with two frequencies,one for the solution and another one for the derivative

In trigonometrically-fitted methods the determination of the parameter (usually known as the frequency) is a critical issue, as was shown in the article by H. Ramos and J. Vigo-Aguiar [Applied Mathematics Letters, 23 (2010) 1378–1381]. If the frequency estimation relies on the vanishing of the principal term of the local truncation error, then the first derivative is present in the formula for approximating the parameter. This requires the use of a procedure for approximating the first derivative. For this purpose we use another trigonometrically-fitted formula with a second parameter (different from the frequency of the principal method and also different from the frequency of the true solution). We describe how to approximate both parameters on each step and present different experiments concerning these questions. The numerical results indicate the good performance of the strategy.     

An approximation method for queues in series with blocking

An approximation method for the analysis of queues in series with exponential servers, Poisson arrivals and blocking, is presented. The analysis considers cells consisting of three nodes with revised arrival and service processes. A recursive scheme is developed for the calculation of the steady state probability distributions of the number of items/customers at each node, and also the joint steady state probability distributions of the number of items for triplets of nodes are provided. This approximation method seems to yield encouraging results when compared with those obtained by the other methods and simulation.