The Method of Rotated Solutions: A Highly Efficient Procedure for Code Verification

2020 ◽  
Author(s):  
Marc Horner
Keyword(s):  
Analytical Solution ◽  
Analytical Solutions ◽  
Source Code ◽  
Numerical Algorithms ◽  
Coordinate Transformations ◽  
Code Verification ◽  
Efficient Procedure ◽  
One Dimensional ◽  
Engineering Problems

Abstract Code verification provides mathematical evidence that the source code of a scientific computing software platform is free of bugs and that the numerical algorithms are consistent. The most stringent form of code verification requires the user to demonstrate agreement between the formal and observed orders of accuracy. The observed order is based on a determination of the discretization error, and therefore requires the existence of an analytical solution. One drawback of analytical solutions based on traditional engineering problems is that most derivatives are identically zero, which limits their scope during code verification. The Method of Rotated Solutions is introduced herein as a methodology that utilizes coordinate transformations to generate additional non-zero derivatives in the numerical and analytical solutions. These transformations extend the utility of even the simplest one-dimensional solutions to be able to perform more thorough evaluations. This paper outlines the rotated solutions methodology and provides an example that demonstrates and confirms the utility of this new technique.

Semi-analytical solution to one-dimensional consolidation of viscoelastic saturated soils subject to an inner point sink

2020 ◽  
Vol 37 (5) ◽  
pp. 1787-1804
Author(s):  
Peichao Li ◽  
Linzhong Li ◽  
Mengmeng Lu
Keyword(s):  
Analytical Solution ◽  
Pore Pressure ◽  
Analytical Solutions ◽  
Surface Settlement ◽  
Great Benefit ◽  
Constitutive Law ◽  
Saturated Soils ◽  
Content Type ◽  
One Dimensional ◽  
Consolidation Behavior

Purpose The purpose of this paper is to present a semi-analytical solution to one-dimensional (1D) consolidation induced by a constant inner point sink in viscoelastic saturated soils. Design/methodology/approach Based on the Kelvin–Voigt constitutive law and 1D consolidation equation of saturated soils subject to an inner sink, the analytical solutions of the effective stress, the pore pressure and the surface settlement in Laplace domain were derived by using Laplace transform. Then, the semi-analytical solutions of the pore pressure and the surface settlement in physical domain were obtained by implementing Laplace numerical inversion via Crump method. Findings As for the case of linear elasticity, it is shown that the simplified form of the presented solution in this study is the same as the available analytical solution in the literature. This to some degree depicts that the proposed solution in this paper is reliable. Finally, parameter studies were conducted to investigate the effects of the relevant parameters on the consolidation settlement of saturated soils. The presented solution and method are of great benefit to provide deep insights into the 1D consolidation behavior of viscoelastic saturated soils. Originality/value The presented solution and method are of great benefit to provide deep insights into the 1D consolidation behavior of viscoelastic saturated soils. Consolidation behavior of viscoelastic saturated soils could be reasonably predicted by using the proposed solution with considering variations of both flux and depth because of inner point sink.

scholarly journals The Analytical Solution of Parabolic Volterra Integro- Differential Equations in the Infinite Domain

2016 ◽  
Author(s):  
Yun Zhao ◽  
Feng-Qun Zhao
Keyword(s):  
Fourier Transform ◽  
Analytical Solution ◽  
Differential Equations ◽  
Analytical Solutions ◽  
Kernel Functions ◽  
Infinite Domain ◽  
Memory Kernel ◽  
One Dimensional ◽  
Different Types ◽  
Frictional Memory Kernel

This article focuses on obtaining the analytical solutions for parabolic Volterra integro- differential equations in d-dimensional with different types frictional memory kernel. Based on theories of Laplace transform, Fourier transform, the properties of Fox-H function and convolution theorem, analytical solutions of the equations in the infinite domain are derived under three frictional memory kernel functions respectively. The analytical solutions are expressed by infinite series, the generalized multi-parameter Mittag-Leffler function, Fox-H function and convolution form of Fourier transform. In addition, the graphical representations of the analytical solution under different parameters are given for one-dimensional parabolic Volterra integro-differential equation with power-law memory kernel. It can be seen that the solution curves subject to Gaussian decay at any given moment.

scholarly journals Adiabatic Compression of Ideal Collisionless Gas in One-Dimensional Space

2020 ◽  
pp. 6-12
Author(s):  
Д. А. Быковских ◽  
В. А. Галкин
Keyword(s):  
Analytical Solution ◽  
Exact Solutions ◽  
Particle Velocity ◽  
Software Verification ◽  
Distribution Density ◽  
Dimensional Space ◽  
Adiabatic Compression ◽  
One Dimensional ◽  
Over Time

В статье рассматривается задача об адиабатическом сжатии бесстолкновительного газа с подвижной и неподвижной границами в одномерном пространстве. Для этой задачи получен класс точных решений. Идея нахождения класса точных решений заключается в определении плотности распределения молекул в пространстве координат и скоростей с течением времени. Поскольку пространство скоростей дискретное, то для вычисления макроскопических величин необходимо суммировать плотность распределения частиц по скоростям. Представлены результаты сравнения численного исследования методом Монте-Карло и аналитического решения задачи при различном числе частиц и скоростях движения стенки. Выполнена оценка погрешностей результатов. Полученный класс аналитического решения можно использовать для верификации комплексов программ. The study focuses on adiabatic compression of collisionless gas with moving and fixedboundaries in onedimensionalspace. A class of exact solutions is found. The key concept for findingthese exact solutions is the determination of the molecule distribution density in the coordinate and velocityspaces over time. Since the velocity space is discrete, the particle velocity distribution density is integratedover the velocities to obtain the macroscopic gas properties. The analytical solution and numerical MonteCarlo solution results are compared for different particle numbers and boundary velocities, and the errorsare performed. The class of exact solutions can be used for software verification.

The Approximate Analytical Method Based on Differential Equations for Solving Problems of Statically Determinate Beam and Rigid Frame

2014 ◽  
Vol 619 ◽  
pp. 27-34
Author(s):  
Xiang Dong Zhang ◽  
Xin Gao ◽  
Lei Wang
Keyword(s):  
Differential Equation ◽  
Analytical Solution ◽  
Computer Technology ◽  
Approximate Analytical Solution ◽  
Internal Force ◽  
New Methods ◽  
Rigid Frame ◽  
Engineering Problems ◽  
Equation Group

With the application of computer technology in civil engineering more and more widely, it is important to find new methods suitable for computer programming to solve the engineering problems. In this paper, a new method based on differential equation group is introduced to analyze statically determinate beam and rigid frame. Firstly, the division method of member system is given and differential equation group is established. Secondly, the determination of boundary conditions is discussed in different situations. And the approximate analytical solution of internal force of statically determinate beam and rigid frame is obtained. At last, two calculating examples are given. The result shows that this method is easy to be programmed and suitable for application in engineering and teaching.

scholarly journals Electrocaloric devices part I: Analytical solution of one-dimensional transient heat conduction in a multilayer electrocaloric system

2020 ◽  
Vol 10 (06) ◽  
pp. 2050028
Author(s):  
Farrukh Najmi ◽  
Wenxian Shen ◽  
Lorenzo Cremaschi ◽  
Z.-Y. Cheng
Keyword(s):  
Heat Conduction ◽  
Analytical Solution ◽  
Analytical Solutions ◽  
Initial Temperature ◽  
Transient Heat Conduction ◽  
Typical Structure ◽  
One Dimensional ◽  
Conduction Heat Transfer ◽  
Materials Used

The analytical solution is reported for one-dimensional (1D) dynamic conduction heat transfer within a multilayer system that is the typical structure of electrocaloric devices. Here, the multilayer structure of typical electrocaloric devices is simplified as four layers in which two layers of electrocaloric materials (ECMs) are sandwiched between two semi-infinite bodies representing the thermal sink and source. The temperature of electrocaloric layers can be instantaneously changed by external electric field to establish the initial temperature profile. The analytical solution includes the temperatures in four bodies as a function of both time and location and heat flux through each of the three interfaces as a function of time. Each of these analytical solutions includes five infinite series. It is proved that each of these series is convergent so that the sum of each series can be calculated using the first [Formula: see text] terms of the series. The formula for calculating the value of [Formula: see text] is presented so that the simulation of an electrocaloric device, such as the temperature distribution and heat transferred from one body to another can be performed. The value of [Formula: see text] is dependent on the thickness of electrocaloric material layers, the time of heat conduction, and thermal properties of the materials used. Based on a case study, it is concluded that the [Formula: see text] is mostly less than 20 and barely reaches more than 70. The application of the analytical solutions for the simulation of real electrocaloric devices is discussed.

Transient convective diffusion to a uniformly accessible surface under aparent slip conditions

1991 ◽  
Vol 56 (2) ◽  
pp. 334-343
Author(s):  
Ondřej Wein
Keyword(s):  
Power Law ◽  
Analytical Solutions ◽  
Convective Diffusion ◽  
Active Surface ◽  
Velocity Profiles ◽  
One Dimensional ◽  
Slip Conditions ◽  
Accessible Surface ◽  
Diffusion Problems

Analytical solutions are given to a class of unsteady one-dimensional convective-diffusion problems assuming power-law velocity profiles close to the transport-active surface.

scholarly journals Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

scholarly journals Prediction of Dead Oil Viscosity: Machine Learning vs. Classical Correlations

Energies ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 930
Author(s):  
Fahimeh Hadavimoghaddam ◽  
Mehdi Ostadhassan ◽  
Ehsan Heidaryan ◽  
Mohammad Ali Sadri ◽  
Inna Chapanova ◽  
...  
Keyword(s):  
Machine Learning ◽  
Viscosity Data ◽  
Oil Viscosity ◽  
Pvt Data ◽  
Proposed Model ◽  
Wide Range ◽  
Engineering Problems ◽  
Functional Forms ◽  
Insight Into

Dead oil viscosity is a critical parameter to solve numerous reservoir engineering problems and one of the most unreliable properties to predict with classical black oil correlations. Determination of dead oil viscosity by experiments is expensive and time-consuming, which means developing an accurate and quick prediction model is required. This paper implements six machine learning models: random forest (RF), lightgbm, XGBoost, multilayer perceptron (MLP) neural network, stochastic real-valued (SRV) and SuperLearner to predict dead oil viscosity. More than 2000 pressure–volume–temperature (PVT) data were used for developing and testing these models. A huge range of viscosity data were used, from light intermediate to heavy oil. In this study, we give insight into the performance of different functional forms that have been used in the literature to formulate dead oil viscosity. The results show that the functional form f(γAPI,T), has the best performance, and additional correlating parameters might be unnecessary. Furthermore, SuperLearner outperformed other machine learning (ML) algorithms as well as common correlations that are based on the metric analysis. The SuperLearner model can potentially replace the empirical models for viscosity predictions on a wide range of viscosities (any oil type). Ultimately, the proposed model is capable of simulating the true physical trend of the dead oil viscosity with variations of oil API gravity, temperature and shear rate.

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.

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