Thesis (Ph. D.)--Ohio State University, 2006. / Title from first page of PDF file. Includes bibliographical references (p. 137-143).

This thesis concerns the approximation of optimally controlled partial differential equations for inverse problems in optimal design. Important examples of such problems are optimal material design and parameter reconstruction. In optimal material design the goal is to construct a material that meets some optimality criterion, e.g. to design a beam, with fixed weight, that is as stiff as possible. Parameter reconstrucion concerns, for example, the problem to find the interior structure of a material from surface displacement measurements resulting from applied external forces. Optimal control problems, particularly for partial differential equations, are often ill-posed and need to be regularized to obtain good approximations. We here use the theory of the corresponding Hamilton-Jacobi-Bellman equations to construct regularizations and derive error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method where the first, analytical, step is to regularize the Hamiltonian. Next its Hamiltonian system is computed efficiently with the Newton method using a sparse Jacobian. An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L² projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems. Another treated issue is the relevance of input data for parameter reconstruction problems, where the goal is to determine a spacially distributed coefficient of a partial differential equation from partial observations of the solution. It is here shown that the choice of input data, that generates the partial observations, affects the reconstruction, and that it is possible to formulate meaningful optimality criteria for the input data that enhances the quality of the reconstructed coefficient. In the thesis we present solutions to various applications in optimal material design and reconstruction. / Denna avhandling handlar om approximation av optimalt styrda partiella differentialekvationer för inversa problem inom optimal design. Viktiga exempel på sådana problem är optimal materialdesign och parameterskattning. Inom materialdesign är målet att konstruera ett material som uppfyller vissa optimalitetsvillkor, t.ex. att konstruera en så styv balk som möjligt under en given vikt, medan ett exempel på parameterskattning är att hitta den inre strukturen hos ett material genom att applicera ytkrafter och mäta de resulterande förskjutningarna. Problem inom optimal styrning, speciellt för styrning av partiella differentialekvationer,är ofta illa ställa och måste regulariseras för att kunna lösas numeriskt. Teorin för Hamilton-Jacobi-Bellmans ekvationer används här för att konstruera regulariseringar och ge feluppskattningar till problem inom optimaldesign. Den konstruerade Pontryaginmetoden är en enkel och generell metod där det första analytiska steget är att regularisera Hamiltonianen. I nästa steg löses det Hamiltonska systemet effektivt med Newtons metod och en gles Jacobian. Vi härleder även en feluppskattning för skillnaden mellan den exakta och den approximerade målfunktionen. Denna uppskattning beror endast på skillnaden mellan den sanna och den regulariserade, ändligt dimensionella, Hamiltonianen, båda utvärderade längst lösningsbanan och dessL²-projektion. Felet beror alltså ej på skillnaden mellan den exakta och denapproximativa lösningen till det Hamiltonska systemet. Ett annat fall som behandlas är frågan hur indata ska väljas för parameterskattningsproblem. För sådana problem är målet vanligen att bestämma en rumsligt beroende koefficient till en partiell differentialekvation, givet ofullständiga mätningar av lösningen. Här visas att valet av indata, som genererarde ofullständiga mätningarna, påverkar parameterskattningen, och att det är möjligt att formulera meningsfulla optimalitetsvillkor för indata som ökar kvaliteten på parameterskattningen. I avhandlingen presenteras lösningar för diverse tillämpningar inom optimal materialdesign och parameterskattning. / QC 20100712

There are three distinct processes that are predominant in models of flowing media with interacting components: advection, reaction, and diffusion. Collectively, these processes are typically modelled with partial differential equations (PDEs) known as advection-reaction-diffusion (ARD) equations.<p> To solve most PDEs in practice, approximation methods known as numerical methods are used. The method of lines is used to approximate PDEs with systems of ordinary differential equations (ODEs) by a process known as semi-discretization. ODEs are more readily analysed and benefit from well-developed numerical methods and software. Each term of an ODE that corresponds to one of the processes of an ARD equation benefits from particular mathematical properties in a numerical method. These properties are often mutually exclusive for many basic numerical methods.<p> A limitation to the widespread use of more complex numerical methods is that the development of the appropriate software to provide comparisons to existing numerical methods is not straightforward. Scientific and numerical software is often inflexible, motivating the development of a class of software known as problem-solving environments (PSEs). Many existing PSEs such as Matlab have solvers for ODEs and PDEs but lack specific features, beyond a scripting language, to readily experiment with novel or existing solution methods. The PSE developed during the course of this thesis solves ODEs known as initial-value problems, where only the initial state is fully known. The PSE is used to assess the performance of new numerical methods for ODEs that integrate each term of a semi-discretized ARD equation. This PSE is part of the PSE pythODE that uses object-oriented and software-engineering techniques to allow implementations of many existing and novel solution methods for ODEs with minimal effort spent on code modification and integration.<p> The new numerical methods use a commutator-free exponential Runge-Kutta (CFERK) method to solve the advection term of an ARD equation. A matrix exponential is used as the exponential function, but CFERK methods can use other numerical methods that model the flowing medium. The reaction term is solved separately using an explicit Runge-Kutta method because solving it along with the diffusion term can result in stepsize restrictions and hence inefficiency. The diffusion term is solved using a Runge-Kutta-Chebyshev method that takes advantage of the spatially symmetric nature of the diffusion process to avoid stepsize restrictions from a property known as stiffness. The resulting methods, known as Integrating-factor-based 2-additive Runge-Kutta methods, are shown to be able to find higher-accuracy solutions in less computational time than competing methods for certain challenging semi-discretized ARD equations. This demonstrates the practical viability both of using CFERK methods for advection and a 3-splitting in general.

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There are three distinct processes that are predominant in models of flowing media with interacting components: advection, reaction, and diffusion. Collectively, these processes are typically modelled with partial differential equations (PDEs) known as advection-reaction-diffusion (ARD) equations.<p> To solve most PDEs in practice, approximation methods known as numerical methods are used. The method of lines is used to approximate PDEs with systems of ordinary differential equations (ODEs) by a process known as semi-discretization. ODEs are more readily analysed and benefit from well-developed numerical methods and software. Each term of an ODE that corresponds to one of the processes of an ARD equation benefits from particular mathematical properties in a numerical method. These properties are often mutually exclusive for many basic numerical methods.<p> A limitation to the widespread use of more complex numerical methods is that the development of the appropriate software to provide comparisons to existing numerical methods is not straightforward. Scientific and numerical software is often inflexible, motivating the development of a class of software known as problem-solving environments (PSEs). Many existing PSEs such as Matlab have solvers for ODEs and PDEs but lack specific features, beyond a scripting language, to readily experiment with novel or existing solution methods. The PSE developed during the course of this thesis solves ODEs known as initial-value problems, where only the initial state is fully known. The PSE is used to assess the performance of new numerical methods for ODEs that integrate each term of a semi-discretized ARD equation. This PSE is part of the PSE pythODE that uses object-oriented and software-engineering techniques to allow implementations of many existing and novel solution methods for ODEs with minimal effort spent on code modification and integration.<p> The new numerical methods use a commutator-free exponential Runge-Kutta (CFERK) method to solve the advection term of an ARD equation. A matrix exponential is used as the exponential function, but CFERK methods can use other numerical methods that model the flowing medium. The reaction term is solved separately using an explicit Runge-Kutta method because solving it along with the diffusion term can result in stepsize restrictions and hence inefficiency. The diffusion term is solved using a Runge-Kutta-Chebyshev method that takes advantage of the spatially symmetric nature of the diffusion process to avoid stepsize restrictions from a property known as stiffness. The resulting methods, known as Integrating-factor-based 2-additive Runge-Kutta methods, are shown to be able to find higher-accuracy solutions in less computational time than competing methods for certain challenging semi-discretized ARD equations. This demonstrates the practical viability both of using CFERK methods for advection and a 3-splitting in general.

With development and advancement of notebook, at the same time it brings its cooling problem, it is very important that use outside surface cooling except inside. The main in study is simulate of electronic cooling in Notebook outside surface, design force convection models and placed them under the Notebook, force convection has immediate effect on the surface and produce heat dissipation. The simulation uses software FLUENT 6.0 to analysis the result of heat dissipation, the models are constructed and described by use turbulent field of three dimensions. The study has two main parameters¡GThe form of force convection models and controlled airflow. The result of numerical analysis use Nusselt number to determine the effect of heat dissipation. According to the result of numerical analysis to increase effect of heat dissipation for the following methods¡G1. Increase airflow across the designed models, 2. Decrease the angle of elevation when using notebook, 3. Airflow enter the designed models by one entrance and leave by the side exports, 4. When airflow pass through the designed models smoothly, 5. Airflow can influence the notebook surface immediately.

The positions of high stress concentration in a bicycle frame structure made in composite materials and a way to strengthen them were investigated via SolidWorks and ANSYS, 3D picture plotting software, and numerical simulation software, respectively. The capability of productivity will be improved indirectly due to the shortening of the time in the process of customers¡¦ order, research, development and mass production. In experiment, prepreg tape (TOHO UT500 carbon fiber/AD. Group matrix) which were produced by AD. Group were made into laminates by hot-pressing machine. The material constants of the laminate and the stress-strain diagram were obtained according to the stander of ASTM D3039. The received material constants are E11 =151.55 GPa and E22 = 7.654 GPa, respectively. In simulation, the experimental data E11 and E22 were used in the numerical analysis, and obtained the stress and deformation fields of the bicycle frame structure and the front fork were plotted based on the standards provided by AD. Group. Reinforcements were made according to the positions of stress concentration in the diagrams. The results of improvements of the front fork after reinforcement include¡G the deformation of normal rigidity was improved to 9.45mm from 12.89mm, and the lateral deformation was significantly improved to 0.97mm from 13mm. Other improvements of the frame structure after reinforcement contain the deformation: dual-side rigidity was improved to 6.6mm from 11.7mm, and the deformation of single-side rigidity was improved to 12.5mm. The rigidity of the head lug was 0.46mm and there is no need to reinforce it because it was meets requirements. Keywords: Numerical analysis, bicycle frame, prepreg, stress concentration.

The purpose of thesis is aimed to predict the residual stiffness and residual strength of a composite laminate by adopting the method of cumulative damage theories numerically. In association with the experimental work the numerical result can be verified in comparison. The fatigue data in composites are well known more scattered than those in conventional metals, because the material properties are complicated due to nonhomogeneity and anisotropy. Until now there exists very few unified theories to model composite fatigue properties. Most of them are semi-empirical expressions fitted by selecting material characteristic values. This work tries to make a precise prediction with hopefully saving time, money and manpower in future experiments. On the aspect of numerical analysis, we employ finite element method incorporated with the software of ANSYS to generate 3-D finite element model and obtain the ultimate stress of cross-ply [0/90]4s and quasi-isotropic [0/+45/90/-45] laminates by Tsai-Wu failure criterion. It is assumed that the damage due to fatigue cycles is equal to the damage of stiffness and strength, in association with Miner¡¦s Rule and cumulative damage theories we obtain the residual stiffness and strength. The numerical result in comparison with the available empirical data is found acceptably well. Finally, this study can be concluded as follows. The error of ultimate stress is 3.84 % in cross-ply[0/90]4s , and 8.38 % in quasi-isotropic[0/45/90/-45]2s laminates. The error of ultimate stress in centrally notched cross-ply[0/90]4s is 0.4 %, and 22.4 % in centrally notched quasi-isotropic laminates. As the fatigue cycles increasing, the residual stiffness and residual strength of the laminates are all decreasing. The decreasing rate is very slight at first and intermediate stages, whilst it is much faster near the last stage. It is found that the prediction of residual strength is more accurate in the case of maximum stress of 60% ultimate stress than that of 80% ultimate stress.

Thesis (M. Sc.(Mathematics and Applied Mathematics))--University of Pretoria, 2009. / Summary in English. Includes bibliographical references.