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Holmquist, Sonia. "AN EXAMINATION OF THE EFFECTIVENESS OF THE ADOMIAN DECOMPOSITION METHOD IN FLUID DYNAMIC APPLICATIONS." Doctoral diss., University of Central Florida, 2007. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/2524.
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Since its introduction in the 1980's, the Adomian Decomposition Method (ADM) has proven to be an efficient and reliable method for solving many types of problems. Originally developed to solve nonlinear functional equations, the ADM has since been used for a wide range of equation types (like boundary value problems, integral equations, equations arising in flow of incompressible and compressible fluids etc...). This work is devoted to an evaluation of the effectiveness of this method when used for fluid dynamic applications. In particular, the ADM has been applied to the Blasius equation, the Falkner-Skan equation, and the Orr-Sommerfeld equation. This study is divided into five Chapters and an Appendix. The first chapter is devoted to an introduction of the Adomian Decomposition method (ADM) with simple illustrations. The Second Chapter is devoted to the application of the ADM to generalized Blasius Equation and our result is compared to other published results when the parameter values are appropriately set. Chapter 3 presents the solution generated for the Falkner-Skan equation. Finally, the Orr-Sommerfeld equation is dealt with in the fourth Chapter. Chapter 5 is devoted to the findings and recommendations based on this study. The Appendix contains details of the solutions considered as well as an alternate solution for the generalized Blasius Equation using Bender's delta-perturbation method.Ph.D.
Department of Mathematics
Sciences
Mathematics PhD
2
McKee, Alex Clive Seymoore. "Analytical solutions of orientation aggregation models, multiple solutions and path following with the Adomian decomposition method." Thesis, Brunel University, 2011. http://bura.brunel.ac.uk/handle/2438/7349.
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In this work we apply the Adomian decomposition method to an orientation aggregation problem modelling the time distribution of filaments. We find analytical solutions under certain specific criteria and programmatically implement the Adomian method to two variants of the orientation aggregation model. We extend the utility of the Adomian decomposition method beyond its original capability to enable it to converge to more than one solution of a nonlinear problem and further to be used as a corrector in path following bifurcation problems.
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Šustková, Apolena. "Řešení obyčejných diferenciálních rovnic neceločíselného řádu metodou Adomianova rozkladu." Master's thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2021. http://www.nusl.cz/ntk/nusl-445455.
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This master's thesis deals with solving fractional-order ordinary differential equations by the Adomian decomposition method. A part of the work is therefore devoted to the theory of equations containing differential operators of non-integer order, especially the Caputo operator. The next part is devoted to the Adomian decomposition method itself, its properties and implementation in the case of Chen system. The work also deals with bifurcation analysis of this system, both for integer and non-integer case. One of the objectives is to clarify the discrepancy in the literature concerning the fractional-order Chen system, where experiments based on the use of the Adomian decomposition method give different results for certain input parameters compared with numerical methods. The clarification of this discrepancy is based on recent theoretical knowledge in the field of fractional-order differential equations and their systems. The conclusions are supported by numerical experiments, own code implementing the Adomian decomposition method on the Chen system was used.
4
Ladeia, Cibele Aparecida. "A equação de transferência radiativa condutiva em geometria cilíndrica para o problema do escape do lançamento de foguetes." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2016. http://hdl.handle.net/10183/152738.
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Nesta contribuição apresentamos uma solução para a equação de transferência radiativa condutiva em geometria cilíndrica. Esta solução é aplicada para simular a radiação e campo de temperatura juntamente com o transporte de energia radiativa e condutiva proveniente do escape liberado em lançamentos de foguetes. Para este fim, discutimos uma abordagem semianalítica reduzindo a equação original, que é contínua nas variáveis angulares, numa equação semelhante ao problema SN da transferência radiativa condutiva. A solução é construída usando um método de composição por transformada de Laplace e o método da decomposição de Adomian. O esquema recursivo ´e apresentado para o sistema de equações de ordenadas duplamente discretas juntamente com as dependências dos parâmetros e suas influências sobre a convergência heurística da solução. A solução obtida, em seguida, permite construir o campo próximo relevante para caracterizar o termo fonte para problemas de dispersão ao ajustar os parâmetros do modelo, tais como, emissividade, refletividade, albedo e outros, em comparação com a observação, que são relevantes para os processos de dispersão de campo distante e podem ser manipulados de forma independente do presente problema. Além do método de solução, também relatamos sobre algumas soluções e simulações numéricas.In this contribution we present a solution for the radiative conductive transfer equation in cylinder geometry. This solution is applied to simulate the radiation and temperature field together with conductive and radiative energy transport originated from the exhaust released in rocket launches. To this end we discuss a semi-analytical approach reducing the original equation, which is continuous in the angular variables, into an equation similar to the SN radiative conductive transfer problem. The solution is constructed using a composite method by Laplace transform and Adomian decomposition method. The recursive scheme is presented for the doubly discrete ordinate equations system together with parameter dependencies and their influence on heuristic convergence of the solution. The obtained solution allows then to construct the relevant near field to characterize the source term for dispersion problems when adjusting the model parameters such as emissivity, reflectivity, albedo and others in comparison to the observation, that are relevant for far field dispersion processes and may be handled independently from the present problem. In addition to the solution method we also report some solutions and numerical simulations.
5
Archalousová, Olga. "Singulární počáteční úloha pro obyčejné diferenciální a integrodiferenciální rovnice." Doctoral thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2011. http://www.nusl.cz/ntk/nusl-233525.
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The thesis deals with qualitative properties of solutions of singular initial value problems for ordinary differential and integrodifferential equations which occur in the theory of linear and nonlinear electrical circuits and the theory of therminionic currents. The research is concentrated especially on questions of existence and uniqueness of solutions, asymptotic estimates of solutions and modications of Adomian decomposition method for singular initial problems. Solution algoritms are derived for scalar differential equations of Lane-Emden type using Taylor series and modication of the Adomian decomposition method. For certain classes of nonlinear of integrodifferential equations asymptotic expansions of solutions are constructed in a neighbourhood of a singular point. By means of the combination of Wazewski's topological method and Schauder xed-point theorem there are proved asymptotic estimates of solutions in a region which is homeomorphic to a cone having vertex coinciding with the initial point. Using Banach xed-point theorem the uniqueness of a solution of the singular initial value problem is proved for systems of integrodifferential equations of Volterra and Fredholm type including implicit systems. Moreover, conditions of continuous dependence of a solution on a parameter are determined. Obtained results are presented in illustrative examples.
6
Basto, Mário João Freitas de Sousa. "Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation." Doctoral thesis, 2006. http://hdl.handle.net/10216/12555.
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Basto, Mário João Freitas de Sousa. "Adomian decomposition method, nonlinear equations and spectral solutions of burgers equation." Tese, 2006. http://hdl.handle.net/10216/12555.
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Pa-YeeTsai and 蔡培毅. "Applications of the Hybrid Laplace Adomian Decomposition Method to Nonlinear Physical Systems." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/08208509569007412239.
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博士國立成功大學
機械工程學系碩博士班
98
The Laplace Adomian decomposition method (LADM) combines the numerical Laplace transform algorithm and the Admoian decomposition method (ADM). The truncated series solution solved by the LADM diverges rapidly as the applicable domain increases. However, the Pad? approximant extends the domain of the truncated series solution to obtain better accuracy and convergence. In this paper, a hybrid method of the LADM combined with the Pad? approximant, named the hybrid Laplace Adomian decomposition method is proposed to solve the nonlinear physical systems to demonstrate efficient and reliable results. The linearization and small parameter assumptions are unnecessary for solving the nonlinear system problems by the hybrid Laplace Adomian decomposition method. The LADM─Pad? approximant solution is easy to obtain to demonstrate a real nonlinear physical phenomenon, and the transformation of the boundary value conditions into an initial value problem is also unnecessary when solving a boundary value problem. Furthermore, the LADM─Pad? approximant solution is able to demonstrate a nonlinear physical system by an algebra form. So the calculation is no like the numerical method that every value needs to be known every time. The hybrid Laplace Adomian decomposition method has been successfully applied to solve various nonlinear problems such as, nonlinear pendulum systems, nonlinear oscillation systems, nonlinear control systems, and nonlinear fluid dynamic systems. The LADM-Pad? approximant solutions demonstrate efficient and reliable results and have been shown a good accuracy and convergence in comparison with the exact solutions and other numerical method solutions. Moreover, the LADM─Pad? approximant solutions have been demonstrated not only the superiority of the accuracy and convergence over both the ADM and LADM solutions, but also extended the applicable domain to overcome their drawbacks.
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Hsu, Jung-Chang, and 徐榮昌. "Application of the Adomian Modified Decomposition Method to the Free Vibrations of Beams." Thesis, 2009. http://ndltd.ncl.edu.tw/handle/45660886473944714312.
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博士國立成功大學
機械工程學系碩博士班
97
The paper solves the eigenvalue problems and deals with the free vibration problems by using the Adomian decomposition method (ADM) and Adomian modified decomposition method (AMDM). First, using the ADM, the eigenvalues and normalized eigenfunctions for the Strum-Liouville eigenvalue problem are solved, and the governing differential equation becomes a recursive algebraic equation and boundary conditions become simple algebraic frequency equations which are suitable for symbolic computation. Moreover, after some simple algebraic operations on these frequency equations any th natural frequency, the closed form series solution of any th mode shape can be obtained. Second, the free vibration problems of Euler-Bernoulli beam under various supporting conditions are discussed. Third, using the AMDM, the free vibration problems of an elastically restrained non-uniform Euler-Bernoulli beam with tip mass of rotatory inertia and eccentricity resting on an elastic foundation and subjected to an axial load are proposed. Some numerical results are given to illustrate the influence of the physical parameters on the natural frequencies of the dynamic system. Finally, this paper deals with free vibration problems of non-uniform Timoshenko beams. In this paper, the computed results agree well with those analytical and numerical results given in the literature. These results indicate that the present analysis is accurate, and provides a unified and systematic procedure which is simple and more straightforward than the other analyses.
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Yu-ShengChang and 張又升. "Applications of the Hybrid Laplace Adomian Decomposition Method to Nonlinear Heat Transfer Problems." Thesis, 2014. http://ndltd.ncl.edu.tw/handle/02149396582383943770.
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碩士國立成功大學
機械工程學系
102
When we try to solve physical problems ,we usually build math models to approach our problems. Nonlinear terms are very common both in physical problems and math models, but they will complicate the solving process. In this paper , we use LADM method to solve nonlinear heat transfer problems. We have two cases , nonlinear fin system and nonlinear continuously moving plates system. Some parameters like Convection, radiation, slope of the thermal conductivity-temperature curve , slope of the surface emissivity-temperature curve are discussed. We found when dimensionless numbers as following increase：Nc、Nr 、B(which presents the conventional intensity to conductional intensity、radiative intensity to conductional intensity、surface emission , respectively)will speed up heat transfer in fin or plate. Dimensionless number A(which presents slope of the thermal conductivity-temperature curve)increases will make heat transfer more faster in fin or plate. Dimensionless number Pe(which presents peclet number) increases (if we have a constant fin length or plate length and constant thermal diffusivity )will make final temperature higher. Assuming a power law variation (decided by parameter m ) of the convection coefficient . Nonlinear terms can also be increased by parameter m . In conclusion, LADM is an effective way to solve nonlinear system. Following this paper , we can know more in material or fluid selecting.
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Po-WeiChen and 陳柏維. "Applications of the Hybrid Laplace Adomian Decomposition Method to Falkner-Skan wedge flow." Thesis, 2011. http://ndltd.ncl.edu.tw/handle/62792525754510336574.
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Chih-YiDeng and 鄧智宜. "Apply Laplace Adomian Decomposition Method to the Vibration Analysis of linear and nonlinear Beams." Thesis, 2017. http://ndltd.ncl.edu.tw/handle/pztjje.
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碩士國立成功大學
機械工程學系
105
In this study, the Laplace Adomian decomposition method (LADM) is used to analyze the free vibration and force vibration problems of uniform and non-uniform beams. First, to obtain the natural frequency and mode shape of the Euler-Bernoulli beam, we transformed the governing equation to the algebraic equation by LADM and used simple algebraic operations on the frequency equations afterwards. Moreover, investigating the effect of the natural frequency of the dynamic system by physical parameters including translational spring constant, rotational spring constant, taper ratio of beams and the magnitude of axial tensile. Furthermore, the LADM is applied to analyze the dynamic behavior including deformation, resonance and post-buckling of beams under the external forces. The results of this study show that the natural frequency upper by the upper translational spring constant, the upper rotational spring constant and the upper magnitude of axial tensile. However, the upper taper ratio would make the first frequency lower and the second frequency and the third frequency upper. When resonance occurs, the very little force can still bring about the collapse of the structure. After buckling occurs, it also indicates the deflection angular upper by the upper magnitude of concentrated force and the upper angle of concentrated force. The results of this study is consistent with analytical and numerical results given in the literature.Therefore, the LADM is simpler, faster and more straightforward than other methods.
13
Yi-ChiaChen and 陳怡嘉. "Application of the Adomian Modified Decomposition Method to the Vibrations of Vertical Column Structures." Thesis, 2017. http://ndltd.ncl.edu.tw/handle/mf7r9n.
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Chia-HsiangTseng and 曾嘉祥. "Apply Laplace Adomian Decomposition Method to Vibration and Large Deflection Analysis of Nonlinear Beam." Thesis, 2019. http://ndltd.ncl.edu.tw/handle/nxfbss.
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碩士國立成功大學
機械工程學系
107
In this study, the Laplace Adomian decomposition method (LADM) was used to analyze free vibration of a nonlinear beam and large deflection of a cantilever beam. The relationship between structure parameters and natural frequencies or deflection was also figured out. In the first section, the characteristic/eigenvalue equation and mode shape functions of a general beam were analytically derived by LADM. After that, effects of different physical parameters including geometry formula, translational or rotational spring constant, magnitude of axial tensile force, and the eccentricity of the tip mass on natural frequencies were investigated. In the second part, the governing equation of the large deflection was carried out with Euler–Bernoulli moment–curvature relationship. Next, the deflection under non-following end force and end moment was probed with LADM through this part. The results of this study indicated that natural frequencies of the beam would increase with higher translational or rotational spring constant and magnitude of axial tensile force. On the other hand, the first natural frequency would decrease, and the other frequencies would increase when the eccentricity of the tip mass was larger. As for the structure geometry, the more complex it was, the closer the natural frequencies would they be. For large deflection cases, the results revealed that the influence of end moment was more obvious than the influence of end force. Further, end moment would cause obvious deflection near the tip. The end force, by contrast, would cause the deflection through the whole beam.
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Oguntala, George A., G. Sobamowo, Y. Ahmed, and Raed A. Abd-Alhameed. "Thermal prediction of convective-radiative porous fin heatsink of functionally graded material using adomian decomposition method." 2019. http://hdl.handle.net/10454/16936.
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YesIn recent times, the subject of effective cooling have become an interesting research topic for electronic and mechanical engineers due to the increased miniaturization trend in modern electronic systems. However, fins are useful for cooling various low and high power electronic systems. For improved thermal management of electronic systems, porous fins of functionally graded materials (FGM) have been identified as a viable candidate to enhance cooling. The present study presents an analysis of a convective–radiative porous fin of FGM. For theoretical investigations, the thermal property of the functionally graded material is assumed to follow linear and power-law functions. In this study, we investigated the effects of inhomogeneity index of FGM, convective and radiative variables on the thermal performance of the porous heatsink. The results of the present study show that an increase in the inhomogeneity index of FGM, convective and radiative parameter improves fin efficiency. Moreover, the rate of heat transfer in longitudinal FGM fin increases as b increases. The temperature prediction using the Adomian decomposition method is in excellent agreement with other analytical and method.
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Ungani, Tendani Patrick. "The adomian decomposition method applied to blood flow through arteries in the presence of a magnetic field." Thesis, 2015. http://hdl.handle.net/10539/17648.
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A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. February 16, 2015.The Adomian decomposition method is an effective procedure for the analytical solution of a wide class of dynamical systems without linearization or weak nonlinearity assumptions, closure approximations, perturbation theory, or restrictive assumptions on stochasticity. Our aim here is to apply the Adomian decomposition method to steady two-dimensional blood flow through a constricted artery in the presence of a uniform transverse magnetic field. Blood flow is the study of measuring blood pressure and determining flow through arteries. Blood flow is assumed to be Newtonian and is governed by the equation of continuity and the momentum balanced equation (which are known as the Navier-Stokes equations). This model is consistent with the principles of ferro-hydrodynamics and magnetohydrodynamics and takes into account both magnetization and electrical conductivity of blood. We apply the Adomian decomposition method to the equations governing blood flow through arteries in the presence of an external transverse magnetic field. The results show that the e ect of a uniform external transverse magnetic field applied to blood flow through arteries favors the physiological condition of blood. The motion of blood in stenosed arteries can be regulated by applying a magnetic field externally and increasing/decreasing the intensity of the applied field.
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Ncube, Mahluli Naisbitt. "The natural transform decomposition method for solving fractional differential equations." Diss., 2018. http://hdl.handle.net/10500/25348.
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In this dissertation, we use the Natural transform decomposition method to obtain approximate
analytical solution of fractional differential equations. This technique is a combination
of decomposition methods and natural transform method. We use the Adomian decomposition,
the homotopy perturbation and the Daftardar-Jafari methods as our decomposition
methods. The fractional derivatives are considered in the Caputo and Caputo-
Fabrizio sense.Mathematical Sciences
M. Sc. (Applied Mathematics)
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Mdziniso, Madoda Majahonkhe. "The new spectral Adomian decomposition method and its higher order based iterative schemes for solving highly nonlinear two-point boundary value problems." Thesis, 2014. http://hdl.handle.net/10210/11351.
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M.Sc. (Applied Mathematics)A comparison between the recently developed spectral relaxation method (SRM) and the spectral local linearisation method (SLLM) is done for the first time in this work. Both spectral hybrid methods are employed in finding the solution to the non isothermal mass and heat balance model of a catalytic pellet boundary value problem (BVP) with finite mass and heat transfer resistance, which is a coupled system of singular nonlinear ordinary differential equations (ODEs). The SRM and the SLLM are applied, for the first time, to solve a problem with singularities. The solution by the SRM and the SLLM are validated against the results by bvp4c, a well known matlab built-in procedure for solving BVPs. Tables and graphs are used to show the comparison. The SRM and the SLLM are exceptionally accurate with the SLLM being the fastest to converge to the correct solution. We then construct a new spectral hybrid method which we named the spectral Adomian decomposition method (SADM). The SADM is used concurrently with the standard Adomian decomposition method (ADM) to solve well known models arising in fluid mechanics. These problems are the magneto hydrodynamic (MHD) Jeffery-Hamel flow model and the Darcy-Brinkman- Forchheimer momentum equations. The validity of the results by the SADM and ADM are verified by the exact solution and bvp4c solution where applicable. A simple alteration of the SADM is made to improve the performance.
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Yu-TingChen and 陳郁婷. "Laplace Adomian Decomposition Method for Analyses of Heat Transfer and Thermal Stress with the Periodic Base Temperature in Variable Profile Annular Fin." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/63014832418698761590.
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碩士國立成功大學
機械工程學系
104
In this article, the Laplace Adomian decomposition method (LADM) is used to solve the heat transfer and thermal stress analyses in variable profile annular fin. The heat transfer problem including the parameters of temperature-dependent conduction and convection, and the constant radiation coefficient, with the periodic temperature as boundary condition as well. Solving the temperature distribution in annular fin, and find out the saturated vapor pressure which is under the temperature meantime, to solve the thermal stress distribution, including the radial stress and the tangential stress. Investigating the effect of temperature distribution and thermal stress distribution by both heat transfer parameters and the diversification of fin profile and fin efficiency distribution with every heat transfer parameter. The results show that temperature distribution lower by the lower conductivity, the higher convection coefficient, radiation coefficient and decline rate of fin thickness. But change any heat transfer coefficient and the fin profile would not change the thermal stress distribution. In the high temperature heat transfer process, there have obviously impact on having the radiation coefficient. Improve the radiation coefficient can dissipate heat quickly, make the fin cooling down.
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Yu-ChenWu and 吳昱成. "Applications of the Hybrid Laplace Adomian Decomposition Method to Non-Newtonian Power-Law Fluid Falkner-Skan Boundary Layer Flow and Heat Transfer Under Magnetic Field Effect." Thesis, 2019. http://ndltd.ncl.edu.tw/handle/uxbd84.
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碩士國立成功大學
機械工程學系
107
The problem of magnetohydrodynamic flow, heat transfer and entropy generation rate for a non-Newtonian power-law fluid past a stationary wedge in the presence of a transverse magnetic field is analyzed. The Falkner-Skan equation is applied for the wedge flow. The magnetic field density and surface temperature of the wedge are assumed to vary with the distance from the origin. The governing equations are transformed to nonlinear ordinary differential equations by similarity transformation and several physical parameters related to flow behavior of power-law fluid, magnetic field, angle and surface temperature are introduced. The governing equations are solved numerically by Hybrid Laplace Adomian Decomposition Method. Laplace Adomian Decomposition Method (LADM) combines the Laplace transformation and Adomian Decomposition Method and is used to solve the nonlinear differential equations. In order to make the result of LADM converge, Padé approximant is employed and the method is named Hybrid Laplace Adomian Decomposition Method. The results of this study concerns with the velocity and temperature profiles. The Second-law of characteristic of the system is considered and computed from the velocity and temperature. Parameters effecting velocity, temperature and entropy generation rate will be discussed in this study. The local skin-friction coefficient and the local Nusselt number are also tabulated and analyzed.
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Dinesha, Disha L. "Application of Semi Analytical Methods for Large Power System Simulations." Thesis, 2018. https://etd.iisc.ac.in/handle/2005/4875.
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Power system dynamics can be accurately modelled in time domain by using nonlinear differential and algebraic equations (DAEs). The challenge lies in solving the large number of nonlinear DAEs faster than real-time in order to provide the operator with sufficient information on the unfolding critical contingencies to take preventive control measures. Research is being carried out in this regard on the application of new computing architectures and development of faster solvers including efficient parallelization techniques.
Semi analytical methods are frequently used for numerical simulations of real world systems in the applied sciences and engineering including nonlinear ODE, PDE and DAE problems. Applicability of two widely used semi-analytical methods called Adomian Decomposition Method (ADM) and Homotopy Analysis Method (HAM) have been explored for the time domain simulations of large power systems in this thesis. Since these methods have a very narrow region of convergence, they are applied over successive time intervals. These are called multi-stage methods. The multi-stage ADM and HAM (MADM and MHAM respectively) have been tested on 7 widely used test systems ranging from 10 generators, 39 buses to 4092 generators, 13659 buses. Studies on the number of terms, impact of time step on stability and accuracy and maximum step size have been conducted. An average speed up of 42% and 26% is observed in the solution time of ODEs alone using the MADM when compared to the Midpoint-Trapezoidal and Modified Euler methods respectively. This thesis also proposes a Z-transform based method to quantify the error generated by MADM application in time domain simulation. The error of MADM has been compared to the errors generated by various numerical integration
methods.
It is found that the number of algebraic equations to be solved are much less than the differential equations in large systems but they take much longer time for execution. A model of multi-machine system including the network and stator transients has been developed to convert the DAE model into an ODE model and validated with EMTP. Various numerical methods have been tested on this ODE model to improve the execution time.
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黃瓊瑩. "Adomian’s Decomposition Method for Bose－Einstein Condensates in An Optical Lattice." Thesis, 2005. http://ndltd.ncl.edu.tw/handle/86275313948694131255.
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Chiu, Ching-Huang, and 邱青煌. "Application of Adomian’s Decomposition method on the Analysis of Nonlinear Heat Transfer in Fins." Thesis, 2002. http://ndltd.ncl.edu.tw/handle/62441639960624613420.
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博士國立成功大學
機械工程學系
90
Abstract The Adomian’s decomposition is extended to predict the efficiency and optimal length of a longitudinal fin with variable thermal conductivity. The solutions of the nonlinear equations have been made for the special cases where the heat exchange of the fins with the surrounding may be caused by the pure radiation or the simultaneous convection and radiation, and the thermal conductivity of the fins is variable. An analytical solution is derived and formed as an infinite power series. This considerably reduces the numerical complexity. The Temperature distributions are obtained for an annular fin of temperature dependent conductivity under periodical heat transfer condition. The heat transfer process is governed by the convectional fin parameter N, the thermal conductivity parameter ε, the frequency parameter B, and the amplitude parameter s. Many of the practical fin problems have been completely performed. （1）The surface heat dissipation include mechanisms of pure convection, pure radiation, and simultaneous convection and radiation. （2）several situations give rise to heat transfer, such as a constant base temperature, convective base boundary condition and periodic oscillating base temperature.（3）the insulated and the convective-radiative fin tip are individually considered for evaluating the effect of the fin tip conditions. The accuracy of The Adomian’s decomposition method with a varying number of terms in the series investigated. The comparison with the finite-difference method, based on a Newton linearization scheme, shown that the Adomian’s decomposition method is one of the most powerful techniques to solve nonlinear problems.
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Ghosh, Susanta. "Improved Numerical And Numeric-Analytic Schemes In Nonlinear Dynamics And Systems With Finite Rotations." Thesis, 2008. https://etd.iisc.ac.in/handle/2005/682.
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This thesis deals with different computational techniques related to some classes of nonlinear response regimes of engineering interest. The work is mainly divided into two parts. In the first part different numeric-analytic integration techniques for nonlinear oscillators are developed. In the second part, procedures for handling arbitrarily large rotations are addressed and a few novel developments are reported in the process.
To begin the first part, we have proposed an explicit numeric-analytic technique, based on the Adomian decomposition method, for integrating strongly nonlinear oscillators. Numerical experiments suggest that this method, like most other numerical techniques, is versatile and can accurately solve strongly nonlinear and chaotic systems with relatively larger step-sizes. It is then demonstrated that the procedure may also be effectively employed for solving two-point boundary value problems with the help of a shooting algorithm. This has been followed up with the derivation and numerical exploration of variants of a recently developed numeric-analytic technique, the multi-step transversal linearization (MTrL), in the context of nonlinear oscillators of relevance in engineering dynamics. A considerable generalization and improvement over the original form of a MTrL strategy is achieved in this study. Finally, we have used the concept of MTrL method on the nonlinear variational (rate) equation corresponding to a nonlinear oscillator and thus derive another family of numeric-analytic techniques, presently referred to as the multi-step tangential linearization (MTnL). A comparison of relative errors through the MTrL and MTnL techniques consistently indicate a superior quality of approximation via the MTrL route.
In the second part of the thesis, a scheme for numerical integration of rigid body rotation is proposed using only rudimentary tensor analysis. The equations of motion are rewritten in terms of rotation vectors lying in same tangent spaces, thereby facilitating vector space operations consistent with the underlying geometric structure of rotation. One of the most important findings of this part of the dissertation is that the existing constant-preserving algorithms are not necessarily accurate enough and may not be ideally applicable to cases wherein numerical accuracy is of primary importance. In contrast, the proposed rotation-algorithms, the higher order ones in particular, are significantly more accurate for conservative rotational systems for reasonably long time. Similar accuracy is expected for dissipative rotational systems as well. The operators relating rotation variables corresponding to different tangent spaces are also investigated and this should provide further insight into the understanding of rotation vector parametrization.
A rotation update is next proposed in terms of rotation vectors. This update, employed along with interpolation of relative rotations, gives a strain-objective and path independent finite element implementation of a geometrically exact beam. The method has the computational advantage of requiring considerably less nodal variables due to the use of rotation vector parametrization. We have proposed a new isoparametric interpolation of nodal quaternions for computing the rotation field within an element. This should be a computationally efficient alternative to the interpolation of local rotations. It has been proved that the proposed interpolation of rotation leads to the objectivity of strain measures. Several numerical experiments are conducted to demonstrate the frame invariance, path-independence and other superior aspects of the present approach vis-`a-vis the existing methods based on the rotation vector parametrization. It is emphasized that, in order to develop an objective finite element formulation, the use of relative rotation is not mandatory and an interpolation of total rotation variables conforming with the rotation manifold should suffice.
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Ghosh, Susanta. "Improved Numerical And Numeric-Analytic Schemes In Nonlinear Dynamics And Systems With Finite Rotations." Thesis, 2008. http://hdl.handle.net/2005/682.
Full textAbstract:
This thesis deals with different computational techniques related to some classes of nonlinear response regimes of engineering interest. The work is mainly divided into two parts. In the first part different numeric-analytic integration techniques for nonlinear oscillators are developed. In the second part, procedures for handling arbitrarily large rotations are addressed and a few novel developments are reported in the process.
To begin the first part, we have proposed an explicit numeric-analytic technique, based on the Adomian decomposition method, for integrating strongly nonlinear oscillators. Numerical experiments suggest that this method, like most other numerical techniques, is versatile and can accurately solve strongly nonlinear and chaotic systems with relatively larger step-sizes. It is then demonstrated that the procedure may also be effectively employed for solving two-point boundary value problems with the help of a shooting algorithm. This has been followed up with the derivation and numerical exploration of variants of a recently developed numeric-analytic technique, the multi-step transversal linearization (MTrL), in the context of nonlinear oscillators of relevance in engineering dynamics. A considerable generalization and improvement over the original form of a MTrL strategy is achieved in this study. Finally, we have used the concept of MTrL method on the nonlinear variational (rate) equation corresponding to a nonlinear oscillator and thus derive another family of numeric-analytic techniques, presently referred to as the multi-step tangential linearization (MTnL). A comparison of relative errors through the MTrL and MTnL techniques consistently indicate a superior quality of approximation via the MTrL route.
In the second part of the thesis, a scheme for numerical integration of rigid body rotation is proposed using only rudimentary tensor analysis. The equations of motion are rewritten in terms of rotation vectors lying in same tangent spaces, thereby facilitating vector space operations consistent with the underlying geometric structure of rotation. One of the most important findings of this part of the dissertation is that the existing constant-preserving algorithms are not necessarily accurate enough and may not be ideally applicable to cases wherein numerical accuracy is of primary importance. In contrast, the proposed rotation-algorithms, the higher order ones in particular, are significantly more accurate for conservative rotational systems for reasonably long time. Similar accuracy is expected for dissipative rotational systems as well. The operators relating rotation variables corresponding to different tangent spaces are also investigated and this should provide further insight into the understanding of rotation vector parametrization.
A rotation update is next proposed in terms of rotation vectors. This update, employed along with interpolation of relative rotations, gives a strain-objective and path independent finite element implementation of a geometrically exact beam. The method has the computational advantage of requiring considerably less nodal variables due to the use of rotation vector parametrization. We have proposed a new isoparametric interpolation of nodal quaternions for computing the rotation field within an element. This should be a computationally efficient alternative to the interpolation of local rotations. It has been proved that the proposed interpolation of rotation leads to the objectivity of strain measures. Several numerical experiments are conducted to demonstrate the frame invariance, path-independence and other superior aspects of the present approach vis-`a-vis the existing methods based on the rotation vector parametrization. It is emphasized that, in order to develop an objective finite element formulation, the use of relative rotation is not mandatory and an interpolation of total rotation variables conforming with the rotation manifold should suffice.