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A finite element solver has been developed for performing analysis and sensitivity analysis with Poisson's equation. An application of Poisson's equation in fluid dynamics is that of poential flow, in which case Posson's equaiton reduces to Laplace's equation. The stiffness matrix and sensitivity of the stiffness matrix are evaluated by direct integrations, as opposed to numerical integration. This allows less computational effort and minimizes the sources of computational errors. The capability of evaluating sensitivity derivatives has been added in orde to perform design sensitivity analysis of non-lifting airfoils. The discrete-direct approach to sensitivity analysis is utilized in the current work. The potential flow equations and the sensitivity equations are computed by using a preconditionaed conjugate gradient method. This method greatly reduces the time required to perfomr analysis, and the subsequent design optimization. Airfoil shape is updated at each design iteratioan by using a Bezier-Berstein surface parameterization. The unstrucured grid is adapted considering the mesh as a system of inteconnected springs. Numerical solutions from the flow solver are compared with analytical results obtained for a Joukowsky airfoil. Sensitivity derivaatives are validated using carefully determined central finite difference values. The developed software is then used to perform inverse design of a NACA 0012 and a multi-element airfoil.

<p> The thesis proposes a novel method for the computation of the design sensitivity of microwave network parameters. The approach is based on the finite-element method. When combined with the iterative update method (the Broyden method) during the gradient-based optimization process, the approach requires practically no overhead for the computation of the response Jacobian, thus accelerating the optimization. </p> <p> The efficiency and accuracy of the gradient-based optimization and the tolerance analysis greatly depend on the computation of the design sensitivity. However, common commercial full-wave electromagnetic solvers do not provide sensitivity information. With them, the design sensitivities are computed from the response themselves using finite-difference or higher-order approximations at the response level. Consequently, for each design parameter of interest, at least one additional full-wave analysis is performed. </p> <p> The proposed self-adjoint sensitivity analysis (SASA) is so far the most efficient way to extract the sensitivity information for the network parameters with the finite-element method. As an improvement of the adjoint-variable method (AVM), it eliminates the additional system analyses. With one single full-wave analysis, the sensitivities with respect to all design parameters are computed. This significantly improves the efficiency of the sensitivity computations. </p> <p> When employed in gradient-based optimization, the computational overhead of the SASA can be further reduced. Instead of the finite-difference approximation, the system matrix derivatives are updated iteratively using the Broyden update. This reduces the computational overhead of the sensitivity analysis to practically zero. Further, several switching criteria between the Broyden update and the finite-difference approximation of the system matrix derivatives is proposed to guarantee the robust convergence of the optimization algorithm. This leads to our Broyden/finite-difference SASA (B/FD-SASA). </p> <p> The efficiency in terms of CPU time as well as the accuracy of the SASA is verified by several numerical examples, where the reference results are provided through the traditional finite-difference approximations. Also, the efficiency of the B/FD-SASA is validated by a filter design example and a microwave imaging example, with implementations exploiting different gradientbased optimization algorithms. </p> / Thesis / Master of Applied Science (MASc)

In practical multibody systems, various factors such as friction, joint clearances, and external events play a significant role and can greatly influence the optimal design of the system and its controller. This research focuses on the use of gradient-based optimization methods for multibody dynamic systems with the incorporation of joint friction. The dynamic formulation has been derived in using two distinct techniques: Index-1 DAE and the tangent-space formulation in minimal coordinates. It employs a two different approaches for gradient computation: direct sensitivity approach and the adjoint sensitivity approach. After a comprehensive review of different friction models developed over time, the Brown McPhee model is selected as the most suitable due to its accuracy in dynamic simulations and its compatibility with sensitivity analysis. The proposed methodology supports the simultaneous optimization of both the system and its controller. Moreover, the sensitivities obtained using these formulations have been thoroughly validated for numerical accuracy and benchmarked against other friction models that are based on dynamic events for stiction to friction transition. The approach presented is particularly valuable in applications like robotics and servo-mechanical systems where the design and actuation are closely interconnected. To obtain numerical results, a new implementation of the MBSVT (Multi-Body Systems at Virginia Tech) software package, known as MBSVT 2.0, is reprogrammed in Julia and MATLAB to ensure ease of implementation while maintaining high computational efficiency. The research includes multiple case studies that illustrate the advantages of the concurrent optimization of design and control for specific applications. Efficient techniques for control signal parameterization are presented using linear basis functions. A special focus has been made on the computational efficiency of the formulation and various techniques like sparse-matrix algebra and Jacobian-free products have been employed in the implementation. The dissertation concludes with a summary of key results and contributions and the future scope for this research. / Doctor of Philosophy / In simpler terms, this research focuses on improving the design and control of complex mechanical systems, like robots and automotive systems, by considering factors such as friction in the joints. Friction in a system can greatly affect how it performs for the desired task. The research uses a method called gradient-based optimization, which essentially means finding the most optimal parameters of the system and its controller such that they achieve a desired goal in the most optimal way. Before a model for such a system can be developed, various techniques need to be researched for incorporation of friction mathematically. A model known as Brown McPhee friction is one such model suitable for such an analysis. When optimizing any system on a computer, an iterative process needs to be performed which may prove to be very expensive in terms of computational resources required and the time taken to achieve a solution. Hence, proper mathematical and computational techniques need to be employed to ensure that the resources of a computer are utilized in the most efficient way to get the solution is the quickest way possible. Among the various novelties of this research, it is worth noting that this method that allows for simultaneous design and control optimization, which is particularly useful for applications such as robotics and servo-mechanical systems. Considering the design and control together, leads to more efficient and effective systems. The approach is tested using a software package called MBSVT 2.0, which was specifically developed as part of this research. The software is available in 3 languages: Julia, MATLAB and Fortran for universal access to people from various communities. The results from various case studies are presented that demonstrate this simultaneous design and control approach and highlights its effectiveness making the systems more robust and better performing.

The main focus of this research is to use graph-theoretic formulations to develop an automated algorithm for the generation of sensitivity equations. The idea is to combine the benefits of direct differentiation with that of graph-theoretic formulation. The primary deliverable of this work is the developed software module which can derive the system equations and the sensitivity equations directly from the linear graph of the system. Sensitivity analysis refers to the study of changes in system behaviour brought about by the changes in model parameters. Due to the rapid increase in the sizes and complexities of the models being analyzed, it is important to extend the capabilities of the current tools of sensitivity analysis, and an automated, efficient, and accurate method for the generation of sensitivity equations is highly desirable. In this work, a graph-theoretic algorithm is developed to generate the sensitivity equations. In the current implementation, the proposed algorithm uses direct differentiation to generate sensitivity equations at the component level and graph-theoretic methods to assemble the equation fragments to form the sensitivity equations. This way certain amount of control can be established over the size and complexity of the generated sensitivity equations. The implementation of the algorithm is based on a commercial software package \verb MapleSim[Multibody] and can generate governing and sensitivity equations for multibody models created in MapleSim. In this thesis, the algorithm is tested on various mechanical, hydraulic, electro-chemical, multibody, and multi-domain systems. The generated sensitivity information are used to perform design optimization and parametric importance studies. The sensitivity results are validated using finite difference formulations. The results demonstrate that graph-theoretic sensitivity analysis is an automated, accurate, algorithmic method of generation for sensitivity equations, which enables the user to have some control over the form and complexity of the generated equations. The results show that the graph-theoretic method is more efficient than the finite difference approach. It is also demonstrated that the efficiency of the generated equations are at par or better than the equation obtained by direct differentiation.

A aplicação de modelos hidrológicos vem sendo bastante utilizada como apoio à tomada de decisão no planejamento dos recursos hídricos. Tendo em vista que os dados que servem de entrada para esses modelos estão sujeitos a erros diversos, o presente estudo teve o intuito de contribuir com o conhecimento do impacto desses erros no desempenho do modelo e na estimativa de seus parâmetros. O modelo analisado foi o IPH II fazendo uso do programa computacional WIN_IPH2. Entendendo que a avaliação da sensibilidade ainda é uma área que requer mais estudos, o presente trabalho é focado na utilização das análises de sensibilidade estática e dinâmica. Para isso foram geradas diversas séries temporais de dados de entradas do modelo hidrológico obtidas pela perturbação da série de dados observados. A perturbação foi representada por erros aleatórios (seguindo uma distribuição normal ou uniforme) ou sistemáticos incorporados ás séries temporais das variáveis: precipitação e evapotranspiração. Posteriormente, as análises de sensibilidade estática e dinâmica foram executadas. Para efetuar o acompanhamento da interferência dos erros, na eficiência do modelo, foi feita a avaliação dos resultados obtidos com a aplicação do modelo WIN_IPH2 para diferentes medidas de desempenho, e verificado o impacto dos erros nos dados de entrada no desempenho do modelo (sensibilidade estática) e no desempenho do modelo e na estimativa dos parâmetros (sensibilidade dinâmica). Na análise de sensibilidade estática verificou-se o decaimento mais acentuado da eficiência do modelo, em comparação com a análise de sensibilidade dinâmica, onde o modelo consegue contornar os erros nos dados de entrada com a alteração dos valores dos parâmetros. Por fim, o presente estudo confirmou as conclusões obtidas em estudos anteriores: Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). Além disso, o presente estudo apontou para outros fatores, na medida em que, observa-se junto à tendência do desempenho do modelo cair quando a intensidade do erro gerado é elevada, a importância de avaliar o possível comprometimento de dados em eventos extremos, uma vez que, nessa situação o desempenho do modelo passa a ser afetado de forma mais acentuada. / The hydrologic models had been used to support the decision making in water resources management. Since the input data of those models are subject to several kinds of errors, this study aimed to assess how this errors affect the model performance and the estimate of its parameters. The hydrologic model IPH II was used. Perceiving that the sensitivity analysis is still a field that requires further knowledge, this study was focused in the use of the dynamic and the static sensitivity procedures. In this sense, several time series of input data were obtained through the perturbations of an observed time serie. The perturbation was represented by the addition of random errors (with a normal or uniform distribution) or systematic errors to the observed time series of evapotranspiration and precipitation. Then, the static and dynamic sensibility analysis were performed. The effect of input data errors was assessed for several calibration processes of the IPH II using several performance measures. Thus, modification of the model performance (static sensitivity analysis) and model performance and parameter estimation (dynamic sensitivity analysis) were estimated. In the static sensitivity analysis it was found a most pronounced decay of the model efficiency in comparison with the dynamic sensitivity analysis, where the model can circumvent the errors in the input data with modification of the optimum parameter values. Finally, this study confirmed the conclusions of other previous studies as Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). In addition this study found other factors, as was observed that if the intensity of the error is high in an extreme event of precipitation, it reduced the model performance more than when it is low, in spite of the time series of errors have the same statistics.

A aplicação de modelos hidrológicos vem sendo bastante utilizada como apoio à tomada de decisão no planejamento dos recursos hídricos. Tendo em vista que os dados que servem de entrada para esses modelos estão sujeitos a erros diversos, o presente estudo teve o intuito de contribuir com o conhecimento do impacto desses erros no desempenho do modelo e na estimativa de seus parâmetros. O modelo analisado foi o IPH II fazendo uso do programa computacional WIN_IPH2. Entendendo que a avaliação da sensibilidade ainda é uma área que requer mais estudos, o presente trabalho é focado na utilização das análises de sensibilidade estática e dinâmica. Para isso foram geradas diversas séries temporais de dados de entradas do modelo hidrológico obtidas pela perturbação da série de dados observados. A perturbação foi representada por erros aleatórios (seguindo uma distribuição normal ou uniforme) ou sistemáticos incorporados ás séries temporais das variáveis: precipitação e evapotranspiração. Posteriormente, as análises de sensibilidade estática e dinâmica foram executadas. Para efetuar o acompanhamento da interferência dos erros, na eficiência do modelo, foi feita a avaliação dos resultados obtidos com a aplicação do modelo WIN_IPH2 para diferentes medidas de desempenho, e verificado o impacto dos erros nos dados de entrada no desempenho do modelo (sensibilidade estática) e no desempenho do modelo e na estimativa dos parâmetros (sensibilidade dinâmica). Na análise de sensibilidade estática verificou-se o decaimento mais acentuado da eficiência do modelo, em comparação com a análise de sensibilidade dinâmica, onde o modelo consegue contornar os erros nos dados de entrada com a alteração dos valores dos parâmetros. Por fim, o presente estudo confirmou as conclusões obtidas em estudos anteriores: Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). Além disso, o presente estudo apontou para outros fatores, na medida em que, observa-se junto à tendência do desempenho do modelo cair quando a intensidade do erro gerado é elevada, a importância de avaliar o possível comprometimento de dados em eventos extremos, uma vez que, nessa situação o desempenho do modelo passa a ser afetado de forma mais acentuada. / The hydrologic models had been used to support the decision making in water resources management. Since the input data of those models are subject to several kinds of errors, this study aimed to assess how this errors affect the model performance and the estimate of its parameters. The hydrologic model IPH II was used. Perceiving that the sensitivity analysis is still a field that requires further knowledge, this study was focused in the use of the dynamic and the static sensitivity procedures. In this sense, several time series of input data were obtained through the perturbations of an observed time serie. The perturbation was represented by the addition of random errors (with a normal or uniform distribution) or systematic errors to the observed time series of evapotranspiration and precipitation. Then, the static and dynamic sensibility analysis were performed. The effect of input data errors was assessed for several calibration processes of the IPH II using several performance measures. Thus, modification of the model performance (static sensitivity analysis) and model performance and parameter estimation (dynamic sensitivity analysis) were estimated. In the static sensitivity analysis it was found a most pronounced decay of the model efficiency in comparison with the dynamic sensitivity analysis, where the model can circumvent the errors in the input data with modification of the optimum parameter values. Finally, this study confirmed the conclusions of other previous studies as Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). In addition this study found other factors, as was observed that if the intensity of the error is high in an extreme event of precipitation, it reduced the model performance more than when it is low, in spite of the time series of errors have the same statistics.

A aplicação de modelos hidrológicos vem sendo bastante utilizada como apoio à tomada de decisão no planejamento dos recursos hídricos. Tendo em vista que os dados que servem de entrada para esses modelos estão sujeitos a erros diversos, o presente estudo teve o intuito de contribuir com o conhecimento do impacto desses erros no desempenho do modelo e na estimativa de seus parâmetros. O modelo analisado foi o IPH II fazendo uso do programa computacional WIN_IPH2. Entendendo que a avaliação da sensibilidade ainda é uma área que requer mais estudos, o presente trabalho é focado na utilização das análises de sensibilidade estática e dinâmica. Para isso foram geradas diversas séries temporais de dados de entradas do modelo hidrológico obtidas pela perturbação da série de dados observados. A perturbação foi representada por erros aleatórios (seguindo uma distribuição normal ou uniforme) ou sistemáticos incorporados ás séries temporais das variáveis: precipitação e evapotranspiração. Posteriormente, as análises de sensibilidade estática e dinâmica foram executadas. Para efetuar o acompanhamento da interferência dos erros, na eficiência do modelo, foi feita a avaliação dos resultados obtidos com a aplicação do modelo WIN_IPH2 para diferentes medidas de desempenho, e verificado o impacto dos erros nos dados de entrada no desempenho do modelo (sensibilidade estática) e no desempenho do modelo e na estimativa dos parâmetros (sensibilidade dinâmica). Na análise de sensibilidade estática verificou-se o decaimento mais acentuado da eficiência do modelo, em comparação com a análise de sensibilidade dinâmica, onde o modelo consegue contornar os erros nos dados de entrada com a alteração dos valores dos parâmetros. Por fim, o presente estudo confirmou as conclusões obtidas em estudos anteriores: Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). Além disso, o presente estudo apontou para outros fatores, na medida em que, observa-se junto à tendência do desempenho do modelo cair quando a intensidade do erro gerado é elevada, a importância de avaliar o possível comprometimento de dados em eventos extremos, uma vez que, nessa situação o desempenho do modelo passa a ser afetado de forma mais acentuada. / The hydrologic models had been used to support the decision making in water resources management. Since the input data of those models are subject to several kinds of errors, this study aimed to assess how this errors affect the model performance and the estimate of its parameters. The hydrologic model IPH II was used. Perceiving that the sensitivity analysis is still a field that requires further knowledge, this study was focused in the use of the dynamic and the static sensitivity procedures. In this sense, several time series of input data were obtained through the perturbations of an observed time serie. The perturbation was represented by the addition of random errors (with a normal or uniform distribution) or systematic errors to the observed time series of evapotranspiration and precipitation. Then, the static and dynamic sensibility analysis were performed. The effect of input data errors was assessed for several calibration processes of the IPH II using several performance measures. Thus, modification of the model performance (static sensitivity analysis) and model performance and parameter estimation (dynamic sensitivity analysis) were estimated. In the static sensitivity analysis it was found a most pronounced decay of the model efficiency in comparison with the dynamic sensitivity analysis, where the model can circumvent the errors in the input data with modification of the optimum parameter values. Finally, this study confirmed the conclusions of other previous studies as Oudin et al. (2006), Andréassian et al. (2004), Kavetski et al. (2003). In addition this study found other factors, as was observed that if the intensity of the error is high in an extreme event of precipitation, it reduced the model performance more than when it is low, in spite of the time series of errors have the same statistics.

With the advent of the personal computer, mechanistic nutrient uptake models have become widely used as research and teaching tools in plant and soil science. Three models NST 3.0, SSAND, and PCATS have evolved to represent the current state of the art. There are two major categories of mechanistic models, transient state models with numerical solutions and steady state models. NST 3.0 belongs to the former model type, while SSAND and PCATS belong to the latter. NST 3.0 has been used extensively in crop research but has not been used with woody species. Only a few studies using SSAND and PCATS are available. To better understand the similarities and differences of these three models, it would be useful to compare model predictions with experimental observations using multiple datasets from the literature to represent various situations for woody species. Therefore, the objectives of this study are to: (i) compare the predictions of uptake by the NST 3.0, SSAND, and PCATS models for a suite of nutrients against experimentally measured values, (ii) compare the behavior of the three models using a one dimensional sensitivity analysis; and (iii) compare and contrast the behavior of NST 3.0 and SSAND using a multiple dimensional sensitivity analysis approach. Predictions of nutrient uptake by the three models when run with a common data set were diverse, indicating a need for a reexamination of model structure. The failure of many of the predictions to match observations indicates the need for further studies which produce representative datasets so that the predictive accuracy of each model can be evaluated. Both types of sensitivity analyses suggest that the effect of soil moisture on simulation can be influential when nutrient concentration in the soil solution (CLi) is low. One dimensional sensitivity analysis also revealed that Imax negatively influenced the uptake estimates from the SSAND and PCATS models. Further analysis indicates that this counter intuitive response of Imax is probably related to low soil nutrient supply. The predictions of SSAND under low-nutrient-supply scenarios are generally lower than those of NST 3.0. We suspect that both of these results are artifacts of the steady state models and further studies to improve them, such as incorporating important rhizospheric effects, are needed if they are to be used successfully for the longer growth periods and lower soil nutrient supply situations more typical of woody species. / Master of Science