Analytical Solutions of Saint Venant Equations Decomposed in Frequency Domain

2004 ◽  
Vol 20 (3) ◽  
pp. 187-197
Author(s):  
W. H. Chung ◽  
Y. L. Kang
Keyword(s):  
Analytical Solution ◽  
Finite Difference Method ◽  
Frequency Domain ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Single Equation ◽  
Response Functions ◽  
Generalized Model ◽  
Saint Venant Equations ◽  
Flow Domain

AbstractThe Saint Venant equations are often merged into a single equation for being easily solvable. By doing so, the most general form of the single equation is formulated in this study if all terms are preserved. As a result, the generalized model (GM) results and contains several unexpected nonlinear terms that may impose a great limitation on model analyses. In order to identify these redundant terms, this paper discusses the employment of the linearized Saint Venant equations (LSVE) governing subcritical flow in prismatic channels. The LSVE is solved by a new procedure that separates, in the Laplace frequency domain, the governing equation of water depth from that of flow velocity and thus enables us to consider two independent equations rather than two coupled ones. This allows us to obtain analytical solutions in a much easier way. Comparisons of the response functions of LSVE and the linearized generalized model (LGM) show that the two equations provide identical solutions if the redundant terms embedded in LGM are neglected. It then follows that the response function of LGM can be utilized as a replacement for solving the analytical solution of LSVE that is valid for prismatic channels of any shape. Validity of the analytical solution is verified by repeatedly comparing with the corresponding numerical solutions of finite difference method or Crump's algorithm [1], depending on whether the flow domain is finite or semi-infinite. It is clearly demonstrated in this paper that LSVE serves as an excellent substitution for LGM whose variants have been employed for quite a few years.

A Study of Photo-Thermoelastic Wave in Semiconductor Materials With Spherical Holes Using Analytical-numerical Methods

2021 ◽  
Author(s):  
Faris S. Alzahrani ◽  
Ibrahim Abbas
Keyword(s):  
Finite Difference ◽  
Analytical Solution ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Solutions ◽  
Governing Equations ◽  
Thermoelastic Wave ◽  
Spherical Cavities ◽  
Implicit Finite Difference ◽  
Implicit Finite Difference Method

Abstract Analytical and numerical solutions are two basic tools in the study of photothermal interaction problems in semiconductor medium. In this paper, we compare the analytical solutions with the numerical solutions for thermal interaction in semiconductor mediums containing spherical cavities. The governing equations are given in the domain of Laplace transforms and the eigenvalues approaches are used to obtained the analytical solution. The numerical solutions are obtained by applying the implicit finite difference method (IFDM). A comparison between the numerical solutions and analytical solution are presented. It is found that the implicit finite difference method (IFDM) is applicable, simple and efficient for such problems.

Verification of ANSYS and Matlab Conduction Results Using Analytical Solutions

2020 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
Giampaolo D'Alessandro ◽  
James V. Beck
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Early Time ◽  
The Body ◽  
Numerical Code ◽  
Transient Temperature ◽  
Dimensionless Time ◽  
Time Step ◽  
First Case

Abstract A two-dimensional transient thermal conduction problem is examined and numerical solutions to the problem generated by ANSYS and Matlab, employing the finite element method, are compared against an analytical solution. Various different grid densities and time-step combinations are used in the numerical solutions, including some as recommended by default in the ANSYS software, including coarse, medium and fine spatial grids. The transient temperature solutions from the analytical and numerical schemes are compared at four specific locations on the body and time-dependent error curves are generated for each point. Additionally, tabular values of each solution are presented for a more detailed comparison. The errors found in the numerical solutions by comparing them directly with the analytical solution vary depending primarily on the time step size used. The errors are much larger if calculated using the analytical solution at a given time as a basis of the comparison between the two solutions as opposed to using the steady-state temperature as a basis. The largest errors appear in the early time steps of the problem, which is typically the regime wherein the largest errors occur in mathematical solutions to transient conduction problems. Conversely, errors at larger values of dimensionless time are extremely small and the numerical solutions agree within one tenth of one percent of the analytical solutions at even the worst locations. In addition to difficulties during the early time values of the problem, temperatures calculated on convective boundaries or prescribed-heat-flux boundaries are locations generating larger-magnitude errors. Corners are particularly difficult locations and require finer gridding and finer time steps in order to generate a very precise solution from a numerical code. These regions are compared, using several grid densities, against the analytical solutions. The analytical solutions are, in turn, intrinsically verified to eight significant digits by comparing similar analytical solutions against one another at very small values of dimensionless time. The solution developed using the Matlab differential equation solver was found to have errors of a similar magnitude to those generated using ANSYS. Two different test cases are examined for the various numerical solutions using the selected grid densities. The first case involves steady heating on a portion of one surface for a long duration, up to a dimensionless time of 30. The second test case involves constant heating for a dimensionless time of one, immediately followed by an insulated condition on that same surface for another duration of one dimensionless time unit. Although the errors at large times were extremely small, the errors found within the short duration test were more significant.

scholarly journals Corrigendum

1985 ◽  
Vol 34 (3) ◽  
pp. 481-483
Author(s):  
T. G. Forbes ◽  
E. R. Priest ◽  
A. W. Hood
Keyword(s):  
Analytical Solution ◽  
Numerical Solution ◽  
Current Sheet ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Linear Solution ◽  
Significant Change ◽  
Parameter Values ◽  
A Current

Numerical solutions were obtained by Forbes, Priest & Hood (1982) for the resistive decay of a current sheet in an MHD fluid. To check the accuracy of the numerical solutions, a linear, analytical solution was also deived for the regime where diffusion is dominant. In a subsequent reinvestigation of this problem an error in the linear, analytical solution has been discovered. For the parameter values used in the numerical solution this error is too small (≲ 2%) to produce any significant change in the previous test comparison between the numerical and analytical solutions. However, for parameter values much different from those used in the numerical solution, the error in the linear solution can be significant.

THEORETICAL ANALYTICAL METHOD IN PROBLEM SOLUTION OF LIQUID VACUUM FREEZING IN QUIET STAT

2016 ◽  
Vol 2016 (3) ◽  
pp. 123-128
Author(s):  
Игорь Лобанов ◽  
Igor Lobanov
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Numerical Solutions ◽  
Stationary Process ◽  
Problem Solution ◽  
Quiet State ◽  
Stationary Vacuum ◽  
Vacuum Freezing

A generalized closed analytical solution of the problem of a quasi-stationary process in liquid vacuum freezing in a quiet state with regard to the thickness of the frosting layer ξ whereas heretofore numerical solutions of this problem occurred. The advantage of the analytical solutions obtained of the problem of a quasi-stationary vacuum freezing of moisture in a finedispersion state over existing numerical ones consists in the identification of an immanent tie between defining and determined parameters regarding a thickness of the frosting layer ξ. It is also possible to use them directly at the computation without resorting to the help of computers.

About Verification of Correct Wavelet-Based Approach to Local Static Analysis of Bernoulli Beam

2014 ◽  
Vol 580-583 ◽  
pp. 3013-3016 ◽  
Author(s):  
Pavel A. Akimov ◽  
Mojtaba Aslami
Keyword(s):  
Finite Difference ◽  
Analytical Solution ◽  
Finite Difference Method ◽  
Static Analysis ◽  
Analytical Solutions ◽  
High Accuracy ◽  
Sample Problem ◽  
Bernoulli Beam ◽  
Numerical Result ◽  
High Level

This paper is devoted to verification of correct and efficient wavelet-based method of local static analysis of Bernoulli beam on elastic foundation, proposed by authors. Corresponding results for sample problem have been compared with analytical solutions. Comparison shows that the localization of the problem with reducing its size by proposed method provide high-accuracy results for desired regions of the structure even in high level of reduction in wavelet coefficients. It should be noted that, the wavelet analysis can exactly decompose problem and the sources of the observed error between the analytical solution and numerical result, is based mainly on applied method for solving (in this case, finite difference method), and can be improved.

Drill Temperature Distributions by Numerical Solutions

1971 ◽  
Vol 93 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
U. K. Saxena ◽  
M. F. DeVries ◽  
S. M. Wu
Keyword(s):  
Experimental Data ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Three Dimensional ◽  
Temperature Profiles ◽  
Temperature Distributions ◽  
Drill Cutting ◽  
Drill Point

The backward finite-difference method is used to determine three-dimensional drill temperature distributions. The geometry of the drill was described by (1) approximating the drill as a one-quarter cone and (2) sectioning a true drill point and measuring its profiles. The three-dimensional temperature distributions provided both drill cutting edge and drill flank temperature profiles which were close to prior experimental data and showed improvement over the previous analytical solutions.

scholarly journals Soluções variacionais e numéricas da Equação de Schrödinger 1D submetida ao potencial de Pöschl-Teller

2020 ◽  
Vol 1 (48) ◽  
pp. 156
Author(s):  
Lucas Carvalho Pereira ◽  
João Vítor Batista Ferreira ◽  
Valter Aragão do Nascimento
Keyword(s):  
Field Theory ◽  
Quantum Mechanics ◽  
Graduate Students ◽  
Analytical Solution ◽  
Finite Difference Method ◽  
Numerical Solutions ◽  
Topological Defects ◽  
Variational Solutions ◽  
The Finite Difference Method ◽  
Bose Einstein

<p>This paper presents the numerical and variational solutions of the 1D Schrödinger Equation submitted to the Pöschl-Teller potential. The methods used were the Variational Method and the Finite Difference Method. They were presented in a didactic and detailed way with the purpose of instructing both undergraduate and graduate students, about the applicability and effectiveness of the aforementioned methods. We use the Pöschl-Teller potential due to the fact that it is little explored in the books of Quantum Mechanics used in undergraduation courses and also because of its diverse applications, such as in Bose-Einstein condensates, waveguides, topological defects in field theory and so on. We conclude this paper comparing the variational and numerical solutions with the analytical solution and present the advantages of each method.</p>

scholarly journals Long wave generation and coastal amplification due to propagating atmospheric pressure disturbances

2021 ◽  
Vol 106 (2) ◽  
pp. 1195-1221
Author(s):  
Gozde Guney Dogan ◽  
Efim Pelinovsky ◽  
Andrey Zaytsev ◽  
Ayse Duha Metin ◽  
Gulizar Ozyurt Tarakcioglu ◽  
...  
Keyword(s):  
Numerical Model ◽  
Analytical Solution ◽  
Shallow Water ◽  
Analytical Solutions ◽  
Atmospheric Pressure ◽  
Numerical Solutions ◽  
Ocean Waves ◽  
Tsunami Waves ◽  
Long Wave ◽  
Dance Suite

AbstractMeteotsunamis are long waves generated by displacement of a water body due to atmospheric pressure disturbances that have similar spatial and temporal characteristics to landslide tsunamis. NAMI DANCE that solves the nonlinear shallow water equations is a widely used numerical model to simulate tsunami waves generated by seismic origin. Several validation studies showed that it is highly capable of representing the generation, propagation and nearshore amplification processes of tsunami waves, including inundation at complex topography and basin resonance. The new module of NAMI DANCE that uses the atmospheric pressure and wind forcing as the other inputs to simulate meteotsunami events is developed. In this paper, the analytical solution for the generation of ocean waves due to the propagating atmospheric pressure disturbance is obtained. The new version of the code called NAMI DANCE SUITE is validated by comparing its results with those from analytical solutions on the flat bathymetry. It is also shown that the governing equations for long wave generation by atmospheric pressure disturbances in narrow bays and channels can be written similar to the 1D case studied for tsunami generation and how it is integrated into the numerical model. The analytical solution of the linear shallow water model is defined, and results are compared with numerical solutions. A rectangular shaped flat bathymetry is used as the test domain to model the generation and propagation of ocean waves and the development of Proudman resonance due to moving atmospheric pressure disturbances. The simulation results with different ratios of pressure speed to ocean wave speed (Froude numbers) considering sub-critical, critical and super-critical conditions are presented. Fairly well agreements between analytical solutions and numerical solutions are obtained. Additionally, basins with triangular (lateral) and stepwise shelf (longitudinal) cross sections on different slopes are tested. The amplitudes of generated waves at different time steps in each simulation are presented with discussions considering the channel characteristics. These simulations present the capability of NAMI DANCE SUITE to model the effects of bathymetric conditions such as shelf slope and local bathymetry on wave amplification due to moving atmospheric pressure disturbances.

Zur Lösung der BETHE-GOLDSTONE-Gleichung bei nicht-verschwindendem Gesamtimpuls I

1963 ◽  
Vol 18 (4) ◽  
pp. 531-538
Author(s):  
Dallas T. Hayes
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Nuclear Matter ◽  
Analytical Solutions ◽  
Projection Operator ◽  
Center Of Mass ◽  
Separable Potential ◽  
Localized Solutions ◽  
Non Local ◽  
Square Well

Localized solutions of the BETHE—GOLDSTONE equation for two nucleons in nuclear matter are examined as a function of the center-of-mass momentum (c. m. m.) of the two nucleons. The equation depends upon the c. m. m. as parameter due to the dependence upon the c. m. m. of the projection operator appearing in the equation. An analytical solution of the equation is obtained for a non-local but separable potential, whereby a numerical solution is also obtained. An approximate solution for small c. m. m. is calculated for a square-well potential. In the range of the approximation the two analytical solutions agree exactly.

Transient Two-Dimensional Heat Conduction Problem With Partial Heating Near Corners

2016 ◽  
Author(s):  
Robert L. McMasters ◽  
Filippo de Monte ◽  
James V. Beck ◽  
Donald E. Amos
Keyword(s):  
Heat Conduction ◽  
Numerical Integration ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Thermal Protection ◽  
Heat Conduction Problem ◽  
Separation Of Variables ◽  
Two Dimensional ◽  
Rectangular Region ◽  
Long Distance

This paper provides a solution for two-dimensional heating over a rectangular region on a homogeneous plate. It has application to verification of numerical conduction codes as well as direct application for heating and cooling of electronic equipment. Additionally, it can be applied as a direct solution for the inverse heat conduction problem, most notably used in thermal protection systems for re-entry vehicles. The solutions used in this work are generated using Green’s functions. Two approaches are used which provide solutions for either semi-infinite plates or finite plates with isothermal conditions which are located a long distance from the heating. The methods are both efficient numerically and have extreme accuracy, which can be used to provide additional solution verification. The solutions have components that are shown to have physical significance. The extremely precise nature of analytical solutions allows them to be used as prime standards for their respective transient conduction cases. This extreme precision also allows an accurate calculation of heat flux by finite differences between two points of very close proximity which would not be possible with numerical solutions. This is particularly useful near heated surfaces and near corners. Similarly, sensitivity coefficients for parameter estimation problems can be calculated with extreme precision using this same technique. Another contribution of these solutions is the insight that they can bring. Important dimensionless groups are identified and their influence can be more readily seen than with numerical results. For linear problems, basic heating elements on plates, for example, can be solved to aid in understanding more complex cases. Furthermore these basic solutions can be superimposed both in time and space to obtain solutions for numerous other problems. This paper provides an analytical two-dimensional, transient solution for heating over a rectangular region on a homogeneous square plate. Several methods are available for the solution of such problems. One of the most common is the separation of variables (SOV) method. In the standard implementation of the SOV method, convergence can be slow and accuracy lacking. Another method of generating a solution to this problem makes use of time-partitioning which can produce accurate results. However, numerical integration may be required in these cases, which, in some ways, negates the advantages offered by the analytical solutions. The method given herein requires no numerical integration; it also exhibits exponential series convergence and can provide excellent accuracy. The procedure involves the derivation of previously-unknown simpler forms for the summations, in some cases by virtue of the use of algebraic components. Also, a mathematical identity given in this paper can be used for a variety of related problems.

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