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.

scholarly journals The surface layer integral method for the modelling of the radial gravity gradient component over sea surface

2019 ◽  
Vol 9 (1) ◽  
pp. 127-132
Author(s):  
D. Zhao ◽  
Z. Gong ◽  
J. Feng
Keyword(s):  
Surface Layer ◽  
Integral Method ◽  
Integral Formula ◽  
Radial Component ◽  
Second Order ◽  
Sea Surface ◽  
Gravity Potential ◽  
Gravity Gradient ◽  
Disturbing Potential ◽  
Gradient Component

Abstract For the modelling and determination of the Earth’s external gravity potential as well as its second-order radial derivatives in the space near sea surface, the surface layer integral method was discussed in the paper. The reasons for the applicability of the method over sea surface were discussed. From the original integral formula of disturbing potential based on the surface layer method, the expression of the radial component of the gravity gradient tensor was derived. Furthermore, an identity relation was introduced to modify the formula in order to reduce the singularity problem. Numerical experiments carried out over the marine area of China show that, the modi-fied surface layer integral method effectively improves the accuracy and reliability of the calculation of the second-order radial gradient component of the disturbing potential near sea surface.

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.

scholarly journals Compact Schemes for the Spatial Discretization of Linear Elliptic Partial Differential Equations

2020 ◽  
Vol 9 (4) ◽  
pp. 1529-1535
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Elliptic Partial Differential Equations ◽  
Uniform Grid ◽  
Test Problems ◽  
Computational Domain ◽  
Compact Schemes ◽  
True Error ◽  
Partial Differential ◽  
Grid Line

A system of compact schemes used, to approximate the partial derivative 2 2 1 f x   and 2 2 2 f x   of Linear Elliptic Partial Differential Equations (LEPDE) ,on the non-boundary nodes, located along a particular horizontal grid line for 2 2 1 f x   and along a particular vertical grid line for 2 2 2 f x   of a two-dimensional structured Cartesian uniform grid. The aim of the numerical experiment is to demonstrate the higher order spatial accuracy and better rate of convergence of the solution, produced using the developed compact scheme. Further, these solutions are compared with the same, produced using the conventional 2 nd order scheme. The comparison is made, in terms of the discrete l l 2 &  norms, of the true error. The true error is defined as, the difference between the computed numerical and the available exact solution, of the chosen test problems. It is computed on every non-boundary node bounded in the computational domain.

scholarly journals Collocation polynomial neural forms and domain fragmentation for solving initial value problems

2021 ◽  
Author(s):  
Toni Schneidereit ◽  
Michael Breuß
Keyword(s):  
Neural Network ◽  
Differential Equations ◽  
Initial Value Problems ◽  
Feedforward Neural Networks ◽  
Computational Domain ◽  
Initial Value ◽  
Trial Solution ◽  
Grid Points ◽  
Set Up ◽  
Network Approaches

AbstractSeveral neural network approaches for solving differential equations employ trial solutions with a feedforward neural network. There are different means to incorporate the trial solution in the construction, for instance, one may include them directly in the cost function. Used within the corresponding neural network, the trial solutions define the so-called neural form. Such neural forms represent general, flexible tools by which one may solve various differential equations. In this article, we consider time-dependent initial value problems, which require to set up the neural form framework adequately. The neural forms presented up to now in the literature for such a setting can be considered as first-order polynomials. In this work, we propose to extend the polynomial order of the neural forms. The novel collocation-type construction includes several feedforward neural networks, one for each order. Additionally, we propose the fragmentation of the computational domain into subdomains. The neural forms are solved on each subdomain, whereas the interfacing grid points overlap in order to provide initial values over the whole fragmentation. We illustrate in experiments that the combination of collocation neural forms of higher order and the domain fragmentation allows to solve initial value problems over large domains with high accuracy and reliability.

scholarly journals Resistive Instabilities in a Two-Dimensional MHD Turbulent Flow

1990 ◽  
Vol 140 ◽  
pp. 357-358
Author(s):  
H. Politano ◽  
P. L. Sulem ◽  
A. Pouquet
Keyword(s):  
Turbulent Flow ◽  
Dynamic Range ◽  
Initial Conditions ◽  
Magnetic Energy ◽  
Uniform Grid ◽  
Two Dimensional ◽  
Fourier Decomposition ◽  
Time Stepping ◽  
Grid Points ◽  
Incompressible Mhd

Direct numerical simulations of decaying two-dimensional incompressible MHD flows at Reynolds numbers of several thousands are reported here, using resolutions of 10242 collocation points on a uniform grid. Spatial derivation is performed using Fourier decomposition, assuming periodic boundary conditions, and nonlinear terms are computed in configuration space. The time-stepping scheme is semi-implicit, Crank–Nicolson and third-order Runge–Kutta. The magnetic Prandtl number is equal to unity in all runs. Both deterministic and random initial conditions are used, concentrated in the large scales, with quasi–equipartition between kinetic and magnetic energy. The dynamic range in amplitude of the fields is 107, ensuring well–resolved current and vorticity sheets, over roughly 20 grid points. This leaves sufficient space for tearing instabilities to develop, embedded in a turbulent flow.

THE INVESTIGATION OF THERMOHYDRODYNAMIC CHARACTERISTIC OF SINGLE AXIAL GROOVE JOURNAL BEARINGS WITH FINITE LENGTH BY USING CFD TECHNIQUES

2013 ◽  
Vol 10 (05) ◽  
pp. 1350031 ◽  
Author(s):  
ALIREZA ARAB SOLGHAR ◽  
S. A. GANDJALIKHAN NASSAB
Keyword(s):  
Stokes Equations ◽  
Control Volume ◽  
Three Dimensional ◽  
Theoretical Data ◽  
Simple Algorithm ◽  
Journal Bearings ◽  
Rate Equations ◽  
Navier Stokes ◽  
Computational Domain ◽  
Oil Flow

The three-dimensional steady state thermohydrodynamic (THD) analysis of an axial grooved oil journal bearing is obtained theoretically. Navier–Stokes equations are solved simultaneously along with turbulent kinetic energy and its dissipation rate equations coupled with the energy equation in the lubricant flow and the heat conduction equation in the bush. The AKN low-Re κ–ε turbulence model is used to simulate the mean turbulent flow field. Considering the complexity of the physical geometry, conformal mapping is used to generate an orthogonal grid and the governing equations are transformed into the computational domain. Discretized forms of the transformed equations are obtained by the control volume method and solved by the SIMPLE algorithm. The numerical results of this analysis can be used to investigate the pressure distribution, volumetric oil flow rate and the loci of shaft in the journal bearings. To validate the computational results, comparison with the experimental and theoretical data of other investigators is made, and reasonable agreement is found.

An optimal frequency-domain finite-difference operator with a flexible stencil and its application in discontinuous-grid modeling

Geophysics ◽  
2021 ◽  
pp. 1-41
Author(s):  
Na Fan ◽  
Xiao-Bi Xie ◽  
Lian-Feng Zhao ◽  
Xin-Gong Tang ◽  
Zhen-Xing Yao
Keyword(s):  
Finite Difference ◽  
Frequency Domain ◽  
Waveform Inversion ◽  
Uniform Grid ◽  
Optimal Method ◽  
Velocity Models ◽  
Full Waveform ◽  
Rotationally Symmetric ◽  
Grid Points ◽  
Coarse To Fine

We develop an optimal method to determine expansion parameters for flexible stencils in 2D scalar wave finite-difference frequency-domain (FDFD) simulation. The proposed stencil only requires the involved grid points to be paired and rotationally symmetric around the central point. We apply this method to the transition zone in discontinuous-grid modeling, where the key issue is designing particular FDFD stencils to correctly propagate the wavefield passing through the discontinuous interface. The proposed method can work in FDFD discontinuous-grid with arbitrary integer coarse-to-fine gird spacing ratios. Numerical examples are presented to demonstrate how to apply this optimal method for the discontinuous-grid FDFD schemes with spacing ratios 3 and 5. The synthetic wavefields are highly consistent to those calculated using the conventional dense uniform grid, while the memory requirement and computational costs are greatly reduced. For velocity models with large contrasts, the proposed discontinuous-grid FDFD method can significantly improve the computational efficiency in forward modeling, imaging and full waveform inversion.

A Potential Field Method for Path Planning Into Cavities Defined by B-Splines

1998 ◽  
Author(s):  
Alley C. Butler ◽  
Steven R. LeClair
Keyword(s):  
Path Planning ◽  
Potential Function ◽  
Potential Field ◽  
Robot Manipulator ◽  
Field Method ◽  
Robot Path Planning ◽  
B Splines ◽  
Field Approach ◽  
Potential Field Method ◽  
Robot Path

Abstract This paper describes a potential field approach to robot path planning into cavities defined by B-splines. It discusses existing methods, and describes the development of a Voronoi tree using intersecting hodographs. The Voronoi tree is similar to the Voronoi diagram in that it is maximally distant from all objects in the robot’s environment, but the Voronoi tree is developed for cavities modeled by splines. By adding a potential field to the robot manipulator, a robot path can be found by seeking the manipulator position with the minimum value of the potential function. This position as defined by the potential field is also maximally distant from cavity walls. Results with the potential field method are discussed and conclusions are drawn.

Validation of a 1D Transient Simulation Model of a Multistage Axial Compressor

2015 ◽  
Author(s):  
Reema Kundu ◽  
J. V. R. Prasad ◽  
Yedidia Neumeier
Keyword(s):  
Axial Compressor ◽  
Rotating Stall ◽  
Second Order ◽  
Working Fluid ◽  
Uniform Grid ◽  
Computational Domain ◽  
Source Terms ◽  
One Dimensional ◽  
The Impact ◽  
Multistage Axial Compressor

An unsteady one-dimensional dynamic model has been developed at Georgia Tech to investigate the impact of stage characteristics as well as load distribution on the compression and expansion waves that develop prior to a surge event in a multistage axial compressor. In the developed model, each of the blade rows is replaced by a duct of varying cross-sectional area with force and work source terms. The source terms model the force and energy imparted by a blade row to the working fluid. The modeling assumes the flow to be inviscid, unsteady, compressible and axisymmetric. While rotating stall cannot be explicitly modeled in a 1D mean-line method, the effect of rotating stall can be captured by a judicious choice of source terms that reflects the loss of pumping capability of a stage. Conservation of mass, momentum and energy are applied to an elemental control volume resulting in one-dimensional quasi-linear Euler system of equations. A non-uniform grid and the second-order central difference Kurganov-Tadmor (KT) scheme are used to discretize the one-dimensional computational domain. The resulting ODEs are solved with an explicit second order Runge-Kutta solver. A throttle schedule is used to introduce perturbations at a selected operating condition in order to study flow oscillations that can lead to a stall event. The current study is aimed at validation of the developed flow solver using an industrial compressor database. Further, the current study is aimed at understanding the interaction between the stages with regards to pressure oscillations leading to stall.

Drag Coefficients of Viscous Spheres at Intermediate and High Reynolds Numbers

2001 ◽  
Vol 123 (4) ◽  
pp. 841-849 ◽  
Author(s):  
Zhi-Gang Feng ◽  
Efstathios E. Michaelides
Keyword(s):  
Drag Coefficient ◽  
Viscosity Ratio ◽  
Stokes Equations ◽  
Hydrodynamic Force ◽  
Density Ratio ◽  
Reynolds Numbers ◽  
Navier Stokes ◽  
Uniform Grid ◽  
Computational Domain ◽  
High Reynolds

A finite-difference scheme is used to solve the Navier-Stokes equations for the steady flow inside and outside viscous spheres in a fluid of different properties. Hence, the hydrodynamic force and the steady-state drag coefficient of the spheres are obtained. The Reynolds numbers of the computations range between 0.5 and 1000 and the viscosity ratio ranges between 0 (inviscid bubble) and infinity (solid particle). Unlike the numerical schemes previously implemented in similar studies (uniform grid in a stretched coordinate system) the present method introduces a two-layer concept for the computational domain outside the sphere. The first layer is a very thin one [ORe−1/2] and is positioned at the interface of the sphere. The second layer is based on an exponential function and covers the rest of the domain. The need for such a double-layered domain arises from the observation that at intermediate and large Reynolds numbers a very thin boundary layer appears at the fluid-fluid interface. The computations yield the friction and the form drag of the sphere. It is found that with the present scheme, one is able to obtain results for the drag coefficient up to 1000 with relatively low computational power. It is also observed that both the Reynolds number and the viscosity ratio play a major role on the value of the hydrodynamic force and the drag coefficient. The results show that, if all other conditions are the same, there is a negligible effect of the density ratio on the drag coefficient of viscous spheres.

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