scholarly journals Treat All Integrals as Volume Integrals: A Unified, Parallel, Grid-Based Method for Evaluation of Volume, Surface, and Path Integrals on Implicitly Defined Domains

2018 ◽  
Vol 18 (2) ◽  
Author(s):  
Mete Yurtoglu ◽  
Molly Carton ◽  
Duane Storti
Keyword(s):  
Convergence Rates ◽  
Path Integrals ◽  
Computing Time ◽  
Region Of Interest ◽  
Uniform Grid ◽  
Grid Refinement ◽  
Specific Domain ◽  
Stencil Computation ◽  
Grid Points ◽  
Grid Based

We present a unified method for numerical evaluation of volume, surface, and path integrals of smooth, bounded functions on implicitly defined bounded domains. The method avoids both the stochastic nature (and slow convergence) of Monte Carlo methods and problem-specific domain decompositions required by most traditional numerical integration techniques. Our approach operates on a uniform grid over an axis-aligned box containing the region of interest, so we refer to it as a grid-based method. All grid-based integrals are computed as a sum of contributions from a stencil computation on the grid points. Each class of integrals (path, surface, or volume) involves a different stencil formulation, but grid-based integrals of a given class can be evaluated by applying the same stencil on the same set of grid points; only the data on the grid points changes. When functions are defined over the continuous domain so that grid refinement is possible, grid-based integration is supported by a convergence proof based on wavelet analysis. Given the foundation of function values on a uniform grid, grid-based integration methods apply directly to data produced by volumetric imaging (including computed tomography and magnetic resonance), direct numerical simulation of fluid flow, or any other method that produces data corresponding to values of a function sampled on a regular grid. Every step of a grid-based integral computation (including evaluating a function on a grid, application of stencils on a grid, and reduction of the contributions from the grid points to a single sum) is well suited for parallelization. We present results from a parallelized CUDA implementation of grid-based integrals that faithfully reproduces the output of a serial implementation but with significant reductions in computing time. We also present example grid-based integral results to quantify convergence rates associated with grid refinement and dependence of the convergence rate on the specific choice of difference stencil (corresponding to a particular genus of Daubechies wavelet).

Direct simulations of turbulent flow in a pipe rotating about its axis

1997 ◽  
Vol 343 ◽  
pp. 43-72 ◽  
Author(s):  
P. ORLANDI ◽  
M. FATICA
Keyword(s):  
Reynolds Number ◽  
Drag Reduction ◽  
Radial Direction ◽  
Uniform Grid ◽  
Grid Refinement ◽  
Streamwise Vorticity ◽  
Vorticity Field ◽  
Vortical Structures ◽  
The Mean ◽  
Grid Points

Flow in a circular pipe rotating about its axis, at low Reynolds number, is investigated. The simulation is performed by a finite difference scheme, second-order accurate in space and in time. A non-uniform grid in the radial direction yields accurate solutions with a reasonable number of grid points. The numerical method has been tested for the non-rotating pipe in the limit ν→0 to prove the energy conservation properties. In the viscous case a grid refinement check has been performed and some conclusions about drag reduction have been reached. The mean and turbulent quantities have been compared with the numerical and experimental non-rotating pipe data of Eggels et al. (1994a, b). When the pipe rotates, a degree of drag reduction is achieved in the numerical simulations just as in the experiments. Through the visualization of the vorticity field the drag reduction has been related to the modification of the vortical structures near the wall. A comparison between the vorticity in the non-rotating and in the high rotation case has shown a spiral motion leading to the transport of streamwise vorticity far from the wall.

Numerical Uncertainty in Heat Conduction Calculations for a Tube Immersed in a Fluidized Bed

1995 ◽  
Vol 209 (5) ◽  
pp. 323-336 ◽  
Author(s):  
A I Karamavruc ◽  
N N Clark ◽  
I Celik
Keyword(s):  
Heat Flux ◽  
Heat Conduction ◽  
Horizontal Tube ◽  
Numerical Code ◽  
Uniform Grid ◽  
Grid Refinement ◽  
Transfer Coefficients ◽  
Parabolic Profile ◽  
Grid Convergence ◽  
Spatial Domains

A numerical code has been evaluated with regard to numerical uncertainties involved in calculating heat flux through the wall of a horizontal tube in a bubbling bed of sand. The two-dimensional unsteady heat conduction equation is solved numerically with a non-linearly varying temperature boundary condition prescribed according to measurements. The finite difference method used is an implicit method with a second-order accurate discretization scheme both in temporal and spatial domains. Previous literature dealing with numerical calculations in heat conduction usually reports any detailed study about numerical errors. In the present analysis, a rigorous grid dependence test is applied, and it is shown that the results, in particular heat flux, are very sensitive to the grid size and distribution. Therefore, to achieve better grid convergence when heat flux is sought, the discretization error in the heat flux rather than in the temperature calculations should be considered. This should be done even in cases where temperature is the primary unknown, because it is usually the derivative of temperature which is of any physical importance. The errors are also strongly dependent on the number of iterations which need to be increased as the grid is refined. The present application showed that a non-uniform grid refinement throughout the calculation domain gives a more efficient (less expensive) solution than uniform grid refinement. Furthermore, for calculation of the temperature gradient at the wall, a parabolic profile assumption gives a faster grid convergence compared to a linear profile assumption. The present study shows that the previously published results concerning calculated heat transfer coefficients should be interpreted with caution, unless the authors have provided some measure of grid dependency of their results.

The WiMap

2013 ◽  
pp. 31-41
Author(s):  
Junjun Xu ◽  
Haiyong Luo ◽  
Fang Zhao ◽  
Rui Tao ◽  
Yiming Lin ◽  
...  
Keyword(s):  
Region Of Interest ◽  
Signal Strength ◽  
Gaussian Mixture ◽  
Threshold Control ◽  
Localization Accuracy ◽  
Signal Distribution ◽  
Localization System ◽  
Design And Implementation ◽  
The Internet Of Things ◽  
Grid Based

As positioning technology is an important foundation of the Internet of Things, a dynamic indoor WLAN localization system is proposed in this paper. This paper mainly concentrates on the design and implementation of the WiMap-a dynamic indoor WLAN localization system, which employs grid-based localization method using RSS (received signal strength). To achieve high localization accuracy and low computational complexity, Gaussian mixture model is applied to approximate the signal distribution and a ROI (region of interest) is defined to limit the search region. The authors also discuss other techniques like AP selection and threshold control, which affects the localization accuracy. The experimental results indicate that an accuracy of 3m with 73.8% probability can be obtained in WiMap. Moreover, the running time is reduced greatly with limited ROI method.

scholarly journals Approximation of a Complex Geometric Domain in Polar Coordinates

2019 ◽  
Vol 38 ◽  
pp. 105-118
Author(s):  
Gour Chandra Paul ◽  
Md Masum Murshed ◽  
Md Mamunur Rasid ◽  
Md Morshed Bin Shiraj
Keyword(s):  
Finite Difference ◽  
Coastal Region ◽  
Region Of Interest ◽  
Tangential Direction ◽  
Polar Coordinates ◽  
Computational Grids ◽  
The Matrix ◽  
Grid Points ◽  
Step Algorithm ◽  
Circular Grid

In this study, a complex geometric domain having a colour picture is approximated through a stair- step representation of the coastal and island boundaries to make it suitable for implementing finite difference method in solving shallow water equations (SWEs) in polar coordinates. As a complex domain, we choose the coastal region of Bangladesh situated at the northern tip of the Bay of Bengal (BOB). To cover the whole coastal region, the pole is selected at the point in the  plane assuming it on the mean sea level (MSL). Along the tangential direction, 265 uniformly distributive straight lines are considered through the pole and 959 circular grid lines centered at  are drawn towards the radial direction covering up to  latitude in the BOB. Firstly, a matrix with 960´265 computational grids is constructed from the colour information of the picture. By representing the grids with suitable notations, a proper stair-step algorithm is employed to the matrix obtained with the 960´265 grids to approximate the coastal and island boundaries to the nearest finite difference grid lines using an Arakawa C-grid system. The whole procedure is done with our developed MATLAB program. The grids representing the coastal stations are also identified closely in the obtained approximated domain. Such a type of presentation of the coastal geometry of the region of interest is found to incorporate its complexities properly with minimum computational grid points. GANIT J. Bangladesh Math. Soc.Vol. 38 (2018) 105-118

Multiple reflection and transmission phases in complex layered media using a multistage fast marching method

Geophysics ◽  
2004 ◽  
Vol 69 (5) ◽  
pp. 1338-1350 ◽  
Author(s):  
N. Rawlinson ◽  
M. Sambridge
Keyword(s):  
Layered Media ◽  
Computational Domain ◽  
Grid Refinement ◽  
Fast Marching Method ◽  
Fast Marching ◽  
Reflection And Transmission ◽  
Local Grid Refinement ◽  
Local Grid ◽  
Marching Method ◽  
Grid Based

Traditional grid‐based eikonal schemes for computing traveltimes are usually confined to obtaining first arrivals only. However, later arrivals can be numerous and of greater amplitude, making them a potentially valuable resource for practical applications such as seismic imaging. The aim of this paper is to introduce a grid‐based method for tracking multivalued wavefronts composed of any number of reflection and refraction branches in layered media. A finite‐difference eikonal solver known as the fast marching method (FMM) is used to propagate wavefronts from one interface to the next. By treating each layer that the wavefront enters as a separate computational domain, one obtains a refracted branch by reinitializing FMM in the adjacent layer and a reflected branch by reinitializing FMM in the incident layer. To improve accuracy, a local grid refinement scheme is used in the vicinity of the source where wavefront curvature is high. Several examples are presented which demonstrate the viability of the new method in highly complex layered media. Even in the presence of velocity variations as large as 8:1 and interfaces of high curvature, wavefronts composed of many reflection and transmission events are tracked rapidly and accurately. This is because the scheme retains the two desirable properties of a single‐stage FMM: computational speed and stability. Local grid refinement about the source also can increase accuracy by an order of magnitude with little increase in computational cost.

High-precision potential-field and gradient-component transformations and derivative computations using cubic B-splines

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. I35-I42 ◽  
Author(s):  
Bingzhu Wang ◽  
Edward S. Krebes ◽  
Dhananjay Ravat
Keyword(s):  
High Precision ◽  
Potential Field ◽  
Theoretical Data ◽  
Gravity Gradient ◽  
Uniform Grid ◽  
Computational Domain ◽  
B Splines ◽  
Direct Interpretation ◽  
Gradient Component ◽  
Grid Points

Potential-field and gradient-component transformations and derivative computations are necessary for many techniques of data enhancement, direct interpretation, and inversion. We advance new unified formulas for fast interpolation, differentiation, and integration and propose flexible high-precision algorithms to perform 3D and 2D potential-field- and gradient component transformations and derivative computations in the space domain using cubic B-splines. The spline-based algorithms are applicable to uniform or nonuniform rectangular grids for the 3D case and to regular or irregular grids for the 2D case. The fast Fourier transform (FFT) techniques require uniform grid spacing. The spline-based horizontal-derivative computations can be done at any point in the computational domain, whereas the FFT methods use only grid points. Comparisons between spline and FFT techniques through two gravity-gradient examples and one magnetic example show that results computed with the spline technique agree better with the exact theoretical data than do results computed with the FFT technique.

A fast genetic algorithm for solving architectural design optimization problems

2015 ◽  
Vol 29 (4) ◽  
pp. 457-469 ◽  
Author(s):  
Zhouzhou Su ◽  
Wei Yan
Keyword(s):  
Genetic Algorithms ◽  
Design Optimization ◽  
Domain Knowledge ◽  
Architectural Design ◽  
Optimization Problems ◽  
Computing Time ◽  
Divide And Conquer ◽  
Specific Domain ◽  
Building Performance Simulation ◽  
Simulation And Optimization

AbstractBuilding performance simulation and genetic algorithms are powerful techniques for helping designers make better design decisions in architectural design optimization. However, they are very time consuming and require a significant amount of computing power. More time is needed when two techniques work together. This has become the primary impediment in applying design optimization to real-world projects. This study focuses on reducing the computing time in genetic algorithms when building simulation techniques are involved. In this study, we combine two techniques (offline simulation and divide and conquer) to effectively improve the run time in these architectural design optimization problems, utilizing architecture-specific domain knowledge. The improved methods are evaluated with a case study of a nursing unit design to minimize the nurses’ travel distance and maximize daylighting performance in patient rooms. Results show the computing time can be saved significantly during the simulation and optimization process.

HPSO Algorithm of Actuator Placement for Vibration Suppression of Large Flexible Space Structure

2013 ◽  
Vol 401-403 ◽  
pp. 175-179
Author(s):  
Xiang Yi Zhou ◽  
Jin Zhang
Keyword(s):  
Convergence Rates ◽  
Vibration Suppression ◽  
Control Method ◽  
Computing Time ◽  
Space Structure ◽  
Hybrid Particle Swarm Optimization ◽  
Optimal Criterion ◽  
Active Vibration ◽  
Controllability And Observability ◽  
Actuator Placement

Optimal actuator placement is the key technology to be solved for active vibration suppression of large flexible space structure. According to the features of close mode and light damping, the optimal criterion derived from the controllability and observability of Grammian matrix is designed; Hybrid Particle Swarm Optimization (HPSO) algorithm is introduced to solve the problem in optimizing actuator placement, and the detail solving step is given. Compared with genetic algorithm (GA) in previous research, HPSO is better than GA in convergence rates and computing time. Based on the above optimal results, LQG/LTR control method is utilized when the large flexible structure under pulse and Gauss white noise excitation respectively. The numerical simulation results show that LQG/LTR, which has a better performance in suppressing structure vibration than LQG, can suppress the vibration of large flexible space structure and improve system robustness.

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