The Importance of Analytic Solutions for Verification: Application to the Secondary Magnetic Flux of a Conductive Disk

2007 ◽  
Vol 44 (4) ◽  
pp. 342-357 ◽  
Author(s):  
Nicolaos J. Siakavellas
Keyword(s):  
Graduate Students ◽  
Magnetic Flux ◽  
Numerical Solution ◽  
Analytic Solutions ◽  
Independent Method ◽  
Time Varying ◽  
Sample Problem ◽  
Single Integral ◽  
Improper Integrals ◽  
Quantities Of Interest

The method and value of analytical methods and the importance of analytic solutions for verification must be emphasised to both under- and post-graduate students. So, they must be able to: (i) derive by their own expressions for the quantities of interest, and (ii) test their numerical results by an independent method. Hence, they must be motivated to try to solve any problem analytically, before proceeding to a numerical solution. Towards this aim, a sample problem is proposed and an analytic expression is derived for the secondary magnetic flux induced in a conductive disk by a time-varying magnetic field. This expression, involving elliptic and improper integrals, is finally reduced into a single integral, depending on the current distribution on the disk. Next, an analytical solution for a simplified version of the problem is obtained, which may be used for verification of the numerical solution of the more complex and general problem.

Optimal Performance of Ships Under Combined Power and Sail

1982 ◽  
Vol 26 (03) ◽  
pp. 209-218
Author(s):  
John S. Letcher
Keyword(s):  
Numerical Solution ◽  
Economic Model ◽  
Analytic Solutions ◽  
Simultaneous Equations ◽  
Simultaneous Optimization ◽  
Fixed Cost ◽  
University Of Michigan ◽  
Cost Rate ◽  
Optimal Setting ◽  
Set Up

A simplified hydrodynamic and economic model is developed to describe the operation of a ship equipped with both sails and engine. In the range of light-to-moderate winds in which use of the engine is likely to be economical, the vessel is described by a characteristic speed, a characteristic fixed-cost rate, and five dimensionless parameters (four hydrodynamic, one economic). The model includes simultaneous optimization of three control variables: sail lift, throttle setting, and course angle; optimal setting of variable draft devices can be included optionally. Although no analytic solutions are attained, the simultaneous equations expressing minimization of cost per mile made good are set up, and a general algorithm is given for numerical solution of these problems. As an illustrative example, numerical values are worked out for the 30,000-dwt square-rigged bulk cargo ship from the 1975 University of Michigan study.

An Evaluation of Material Weight and Contained Fluid Volume for Eccentric Intersecting Cylinders of Arbitrary Size

1989 ◽  
Vol 111 (3) ◽  
pp. 342-347
Author(s):  
Y. J. Chao ◽  
M. A. Sutton
Keyword(s):  
Vessel Diameter ◽  
Fluid Volume ◽  
Wide Range ◽  
Single Integral ◽  
Material Volume ◽  
Thin Walled ◽  
Special Case ◽  
Range Of Values ◽  
Engineering Personnel ◽  
Quantities Of Interest

Engineering personnel in industries which use pressurized containment vessels having attached nozzles are required not only to design portions of the lifting mechanism, but also to estimate the fluid volume which the vessel and nozzles will contain; most designers use simplified formulas for computing the quantities of interest. Typically, these formulas are valid approximations when the nozzle diameter is much smaller than the vessel diameter. The enclosed work develops three single-integral expressions which can be programmed and numerically integrated to obtain accurate estimates for both the material volume and also the containment volume present in a pair of eccentrically, or concentrically, intersecting thin-walled cylinders of arbitrary diameters. A table of such values is presented for a wide range of values of the standard nozzle pipe diameter and vessel diameter, for the special case of a concentric nozzle. In addition, an example is presented which compares the numerically integrated values for both the material volume and the containment volume to simplified upper and lower-bound estimates.

A semi-infinite hydraulic fracture with leak-off driven by a power-law fluid

2017 ◽  
Vol 837 ◽  
pp. 210-229 ◽  
Author(s):  
E. V. Dontsov ◽  
O. Kresse
Keyword(s):  
Numerical Solution ◽  
Power Law ◽  
Hydraulic Fracture ◽  
Closed Form Solution ◽  
Analytic Solutions ◽  
Form Solution ◽  
Rapid Evaluation ◽  
Power Law Fluid ◽  
Parametric Space ◽  
Fluid Rheology

This study investigates the problem of a semi-infinite hydraulic fracture that propagates steadily in a permeable formation. The fracturing fluid rheology is assumed to follow a power-law behaviour, while the leak-off is modelled by Carter’s model. A non-singular formulation is employed to effectively analyse the problem and to construct a numerical solution. The problem under consideration features three limiting analytic solutions that are associated with dominance of either toughness, leak-off or viscosity. Transitions between all the limiting cases are analysed and the boundaries of applicability of all these limiting solutions are quantified. These bounds allow us to determine the regions in the parametric space, in which these limiting solutions can be used. The problem of a semi-infinite fracture, which is considered in this study, provides the solution for the tip region of a hydraulic fracture and can be used in hydraulic fracturing simulators to facilitate solving the moving fracture boundary problem. To cater for such applications, for which rapid evaluation of the solution is necessary, the last part of this paper constructs an approximate closed form solution for the problem and evaluates its accuracy against the numerical solution inside the parametric space.

Analytical Solutions for a Spinning Rigid Body Subject to Time-Varying Body-Fixed Torques, Part I: Constant Axial Torque

1993 ◽  
Vol 60 (4) ◽  
pp. 970-975 ◽  
Author(s):  
J. M. Longuski ◽  
P. Tsiotras
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
Axisymmetric Body ◽  
Attitude Motion ◽  
Time Varying ◽  
Torque Vector ◽  
Constant Torque ◽  
Symmetric Rigid Body ◽  
Body Subject

Analytic solutions are derived for the general attitude motion of a near-symmetric rigid body subject to time-varying torques in terms of certain integrals. A methodology is presented for evaluating these integrals in closed form. We consider the case of constant torque about the spin axis and of transverse torques expressed in terms of polynomial functions of time. For an axisymmetric body with constant axial torque, the resulting solutions of Euler’s equations of motion are exact. The analytic solutions for the Eulerian angles are approximate owing to a small angle assumption, but these apply to a wide variety of practical problems. The case when all three components of the external torque vector vary simultaneously with time is much more difficult and is treated in Part II.

scholarly journals Analytic-Numerical Solution of Random Parabolic Models: A Mean Square Fourier Transform Approach

2018 ◽  
Vol 23 (1) ◽  
pp. 79-100 ◽  
Author(s):  
Maria-Consuelo Casaban ◽  
Juan-Carlos Cortes ◽  
Lucas Jodar
Keyword(s):  
Stochastic Process ◽  
Fourier Transform ◽  
Standard Deviation ◽  
Stochastic Processes ◽  
Numerical Solution ◽  
Integral Form ◽  
Initial Condition ◽  
Mean Square ◽  
Improper Integrals ◽  
Gauss Hermite Quadrature

This paper deals with the construction of mean square analytic-numerical solution of parabolic partial differential problems where both initial condition and coefficients are stochastic processes. By using a random Fourier transform, an inf- nite integral form of the solution stochastic process is firstly obtained. Afterwards, explicit expressions for the expectation and standard deviation of the solution are obtained. Since these expressions depend upon random improper integrals, which are not computable in an exact manner, random Gauss-Hermite quadrature formulae are introduced throughout an illustrative example.

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