This dissertation presents three studies addressing various modeling and computational aspects of lattice structures. The first study is concerned with characterization of the threshold behavior for very slow (subcritical) crack growth. First, it is shown that this behavior requires the presence of a healing mechanism. Then thermodynamic analysis of brittle fracture specimens near the threshold developed by Rice (1978) is extended to specimens undergoing microstructural changes. This extension gives rise to a generalization of the threshold concept that mirrors the way the resistance R-curve generalizes the fracture toughness. In the absence of experimental data, the resistance curve near the threshold is constructed using a lattice model that includes healing and rupture mechanisms. The second study is concerned with transmission of various boundary conditions through irregular lattices. The boundary conditions are parameterized using trigonometric Fourier series, and it is shown that, under certain conditions, transmission through irregular lattices can be well approximated by that through classical continuum. It is determined that such transmission must involve the wavelength of at least 12 lattice spacings; for smaller wavelength classical continuum approximations become increasingly inaccurate. Also it is shown that this restriction is much more severe than that associated with identifying the minimum size for representative volume elements. The third study is concerned with extending the use of boundary algebraic equations to problems involving irregular rather than regular lattices. Such an extension would be indispensable for solving multiscale problems defined on irregular lattices, as boundary algebraic equations provide seamless bridging between discrete and continuum models. It is shown that, in contrast to regular lattices, boundary algebraic equations for irregular lattices require a statistical rather than deterministic treatment. Furthermore, boundary algebraic equations for irregular lattices contain certain terms that require the same amount of computational effort as the original problem. / text

We consider the study of symmetry analysis of difference equations. The original work done by Lie about a century ago is known to be one of the best methods of solving differential equations. Lie's theory of difference equations on the contrary, was only first explored about twenty years ago. In 1984, Maeda [42] constructed the similarity methods for difference equations. Some work has been done in the field of symmetries of difference equations for the past years. Given an ordinary or partial differential equation (PDE), one can apply Lie algebra techniques to analyze the problem. It is commonly known that the number of independent variables can be reduced after the symmetries of the equation are obtained. One can determine the optimal system of the equation in order to get a reduction of the independent variables. In addition, using the method, one can obtain new solutions from known ones. This feature is interesting because some differential equations have apparently useless trivial solutions, but applying Lie symmetries to them, more interesting solutions are obtained. The question arises when it happens that our equation contains a discrete quantity. In other words, we aim at investigating steps to be performed when we have a difference equation. Doing so, we find symmetries of difference equations and use them to linearize and reduce the order of difference equations. In this work, we analyze the work done by some researchers in the field and apply their results to some examples. This work will focus on the topical review of symmetries of difference equations and going through that will enable us to make some contribution to the field in the near future. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2011.

Sudarytas Puasono lygties sprendimo per „rutuliukų“ potencialus algoritmas. Šiuo metodu Puasono lygties sprendimo uždavinys suvedamas į tiesinių algebrinių lygčių sistemos sprendimą. Sudaryta ir išbandyta matematiniu paketu MATHCAD to sprendimo programa. Palyginti gauti sprendiniai su tais, kurie gaunami analiziškai, įvertintas gautų sprendinių tikslumas. Šį sprendimo būdą galima panaudoti realiems fizikiniams potencialams paskaičiuoti, turint galvoje realų potencialą su kuriuo realūs krūviai. / It consists of Poisson equation solution in the "ball" potential algorithm. In this method the Poisson equation, the decision problem are reduced to linear algebraic equations system solution. Created and tested a mathematical package MATHCAD program for that decision. Compared to solutions with those obtained analytically, estimated to obtain accurate solutions. This solution can be used to calculate the real physical potentials, given the real potential of the real workloads.

This thesis addressed non-smooth dynamics of lumped parameter systems, and was restricted to Filippov-type systems. The main objective of this thesis was twofold. Firstly, modeling aspects of Filippov-type non-smooth dynamical systems were addressed with an emphasis on the constitutive assumptions and mathematical structure behind these models. Secondly, robust algorithms were presented to obtain numerical solutions for various Filippov-type lumped parameter systems. Governing equations were written using two different mathematical approaches. The first approach was based on differential inclusions and the second approach was based on differential-algebraic equations. The differential inclusions approach is more amenable to mathematical analysis using existing mathematical tools. On the other hand, the approach based on differential-algebraic equations gives more insight into the constitutive assumptions of a chosen model and easier to obtain numerical solutions. Bingham-type models in which the force cannot be expressed in terms of kinematic variables but the kinematic variables can be expressed in terms of force were considered. Further, Coulomb friction was considered in which neither the force can be expressed in terms of kinematic variables nor the kinematic variables in terms of force. However, one can write implicit constitutive equations in terms of kinematic quantities and force. A numerical framework was set up to study such systems and the algorithm was devised. Towards the end, representative dynamical systems of practical significance were considered. The devised algorithm was implemented on these systems and the results were obtained. The results show that the setting offered by differential-algebraic equations is appropriate for studying dynamics of lumped parameter systems under implicit constitutive models.

Thesis (Ph. D.)--Aerospace Engineering, Georgia Institute of Technology, 2009. / Committee Chair: Olivier A. Bauchau; Committee Member: Andrew Makeev; Committee Member: Carlo L. Bottasso; Committee Member: Dewey H. Hodges; Committee Member: Massimo Ruzzene. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Este estudo situa-se no campo da aprendizagem da Matemática. O objetivo da pesquisa aqui proposta foi identificar a existência de relação entre a competência cognitiva, o uso de estratégias metacognitivas e a compreensão do erro, na resolução de problemas matemáticos com equações algébricas do 1º grau. Para isso, investigou-se 38 alunos do 8º ano do Ensino Fundamental de uma escola municipal da região metropolitana de Porto Alegre, selecionados aleatoriamente em duas turmas. Adotando o método misto de pesquisa, utilizou-se como instrumentos o Whimbey Analytical Skills Inventory (WASI), como avaliador da competência cognitiva e divisor dos grupos com alto e baixo nível de competência cognitiva, tendo como referência a média geral de acertos do grupo; e a Escala de Estratégias Metacognitivas na Resolução de Problemas (E-EMRP). Realizaram-se também Observações e Entrevistas Clínicas com base na solução da Tarefa de Resolução de Problemas com Equações Algébricas do 1º Grau (TRPEA). O tratamento estatístico, realizado por meio dos testes de Correlação de Pearson e t-Student, demonstrou a existência de correlação estatisticamente significativa entre o WASI e a TRPEA, evidenciando a existência de relação entre a competência cognitiva e o desempenho na resolução de problemas e compreensão do erro pelo estudante. Verificou-se também uma diferença significativa entre os estudantes com alto e baixo nível de competência cognitiva, em relação ao desempenho na resolução dos problemas e compreensão dos erros, comprovando que estudantes com maiores níveis de competência cognitiva apresentaram melhor desempenho, cometendo menos erros e compreendendo com maior frequência os erros cometidos. Verificou-se ainda a existência de relação entre o uso de estratégias metacognitivas e a compreensão do erro, bem como em relação à competência cognitiva - por meio da observação e entrevista clínica - evidenciando que, apesar do uso das estratégias não ocorrer em todos os momentos da resolução do problema (antes, durante e após a leitura do enunciado e durante e após a resolução do problema) com a mesma frequência, os estudantes com maiores níveis de competência cognitiva demonstraram utilizar um maior conjunto de estratégias, compreendendo melhor a necessidade de sua utilização correta em todas as etapas da resolução, do que os estudantes com baixos níveis de competência cognitiva. Esses resultados alertam para a necessidade de desenvolver-se em sala de aula, atividades que tenham como propósito o treinamento do uso correto de estratégias metacognitivas, visando o aprimoramento da capacidade de resolução de problemas matemáticos, assim como a prevenção e compreensão dos erros cometidos. / This study belongs to the field of mathematics learning. The research aimed to confirm the existence of a relationship among cognitive ability, usage of metacognitive strategies and comprehension of error, in mathematical problem solving 1st degree algebraic equations. For this purpose, 38 students attending the 8th year of primary education in a public school in the metropolitan area of Porto Alegre (Brazil), were randomly divided into two groups. A mixed method research was adopted, in which the tools chosen were: Whimbey Analytical Skills Inventory (WASI), used as a measure of cognitive competence and also as criteria for splitting the group according to cognitive competence level (high and low), considering the overall average of the group's score as guideline, and Metacognitive Strategies Scale in Problem Solving (MSSPS). The method included observations and clinic interviews based on solution of Problem Solving 1st Degree Algebraic Equations (PSAET) as well. Statistical procedure, through Pearson correlation and Student’s t-tests, showed a statistically significant correlation between WASI and PSAET, which demonstrated the existence of relationship between cognitive ability and performance in problem solving and comprehension of error by the student. In addition, there was detected a significant difference between students with high and low levels of cognitive ability, referent to the performance in problem solving and comprehension of errors, proving that students with higher levels of cognitive ability showed best performance, by making fewer errors and comprehending them more frequently. Furthermore, there was confirmed the existence of a relationship between the use of metacognitive strategies and comprehension of the error, and relatively to cognitive competence - through observation and clinical interview. This is an evidence that, despite the strategies were not employed at all stages of problem solving (before, during and after reading the statement and during and after the resolution of the problem) with the same frequency, students with higher levels of cognitive competence demonstrated using a larger set of strategies, aware of the need to the right utilization at all stages of resolution, compared to students with low levels of cognitive competence. These results emphasized the importance of developing classroom activities with purpose of training the correct usage of metacognitive strategies, in order to improve the ability to solve mathematical problems, besides prevention and understanding of errors.

Este estudo situa-se no campo da aprendizagem da Matemática. O objetivo da pesquisa aqui proposta foi identificar a existência de relação entre a competência cognitiva, o uso de estratégias metacognitivas e a compreensão do erro, na resolução de problemas matemáticos com equações algébricas do 1º grau. Para isso, investigou-se 38 alunos do 8º ano do Ensino Fundamental de uma escola municipal da região metropolitana de Porto Alegre, selecionados aleatoriamente em duas turmas. Adotando o método misto de pesquisa, utilizou-se como instrumentos o Whimbey Analytical Skills Inventory (WASI), como avaliador da competência cognitiva e divisor dos grupos com alto e baixo nível de competência cognitiva, tendo como referência a média geral de acertos do grupo; e a Escala de Estratégias Metacognitivas na Resolução de Problemas (E-EMRP). Realizaram-se também Observações e Entrevistas Clínicas com base na solução da Tarefa de Resolução de Problemas com Equações Algébricas do 1º Grau (TRPEA). O tratamento estatístico, realizado por meio dos testes de Correlação de Pearson e t-Student, demonstrou a existência de correlação estatisticamente significativa entre o WASI e a TRPEA, evidenciando a existência de relação entre a competência cognitiva e o desempenho na resolução de problemas e compreensão do erro pelo estudante. Verificou-se também uma diferença significativa entre os estudantes com alto e baixo nível de competência cognitiva, em relação ao desempenho na resolução dos problemas e compreensão dos erros, comprovando que estudantes com maiores níveis de competência cognitiva apresentaram melhor desempenho, cometendo menos erros e compreendendo com maior frequência os erros cometidos. Verificou-se ainda a existência de relação entre o uso de estratégias metacognitivas e a compreensão do erro, bem como em relação à competência cognitiva - por meio da observação e entrevista clínica - evidenciando que, apesar do uso das estratégias não ocorrer em todos os momentos da resolução do problema (antes, durante e após a leitura do enunciado e durante e após a resolução do problema) com a mesma frequência, os estudantes com maiores níveis de competência cognitiva demonstraram utilizar um maior conjunto de estratégias, compreendendo melhor a necessidade de sua utilização correta em todas as etapas da resolução, do que os estudantes com baixos níveis de competência cognitiva. Esses resultados alertam para a necessidade de desenvolver-se em sala de aula, atividades que tenham como propósito o treinamento do uso correto de estratégias metacognitivas, visando o aprimoramento da capacidade de resolução de problemas matemáticos, assim como a prevenção e compreensão dos erros cometidos. / This study belongs to the field of mathematics learning. The research aimed to confirm the existence of a relationship among cognitive ability, usage of metacognitive strategies and comprehension of error, in mathematical problem solving 1st degree algebraic equations. For this purpose, 38 students attending the 8th year of primary education in a public school in the metropolitan area of Porto Alegre (Brazil), were randomly divided into two groups. A mixed method research was adopted, in which the tools chosen were: Whimbey Analytical Skills Inventory (WASI), used as a measure of cognitive competence and also as criteria for splitting the group according to cognitive competence level (high and low), considering the overall average of the group's score as guideline, and Metacognitive Strategies Scale in Problem Solving (MSSPS). The method included observations and clinic interviews based on solution of Problem Solving 1st Degree Algebraic Equations (PSAET) as well. Statistical procedure, through Pearson correlation and Student’s t-tests, showed a statistically significant correlation between WASI and PSAET, which demonstrated the existence of relationship between cognitive ability and performance in problem solving and comprehension of error by the student. In addition, there was detected a significant difference between students with high and low levels of cognitive ability, referent to the performance in problem solving and comprehension of errors, proving that students with higher levels of cognitive ability showed best performance, by making fewer errors and comprehending them more frequently. Furthermore, there was confirmed the existence of a relationship between the use of metacognitive strategies and comprehension of the error, and relatively to cognitive competence - through observation and clinical interview. This is an evidence that, despite the strategies were not employed at all stages of problem solving (before, during and after reading the statement and during and after the resolution of the problem) with the same frequency, students with higher levels of cognitive competence demonstrated using a larger set of strategies, aware of the need to the right utilization at all stages of resolution, compared to students with low levels of cognitive competence. These results emphasized the importance of developing classroom activities with purpose of training the correct usage of metacognitive strategies, in order to improve the ability to solve mathematical problems, besides prevention and understanding of errors.

Este estudo propÃe a discussÃo sobre EquaÃÃes AlgÃbricas, objetivando realizar um estudo sobre as demonstraÃÃes das fÃrmulas, abordando desde aspectos histÃricos atà os diversos mÃtodos de resoluÃÃo de problemas, neste caso, os mÃtodos trabalhados foram o AlgÃbrico, o GeomÃtrico e o Computacional. Esta pesquisa se baseou num estudo bibliogrÃfico sobre as dificuldades de realizar as demonstraÃÃes das fÃrmulas trabalhadas nos conteÃdos de matemÃtica, bem como nas demonstraÃÃes propriamente ditas, aliadas a diversos exemplos resolvidos. A anÃlise do material bibliogrÃfico permitiu distribuir este estudo atravÃs do MÃtodo AlgÃbrico de resoluÃÃo de problemas, em que se discutiu a demonstraÃÃo e aplicaÃÃo das fÃrmulas resolutivas das equaÃÃes polinomiais de 1Â, 2Â, 3 e 4 graus, e ainda citando a impossibilidade da existÃncia de fÃrmulas para equaÃÃes de grau n > 4. No estudo sobre o MÃtodo GeomÃtrico, percebeu-se como a geometria està eficientemente presente na resoluÃÃo de problemas e que as soluÃÃes sÃo possÃveis apenas atravÃs de rÃgua e compasso, neste tÃpico foram abordados mÃtodos para resoluÃÃo de equaÃÃes polinomiais de 1 e 2 graus. Sobre o MÃtodo Computacional, foi enfatizado o estudo sobre os mÃtodos iterativos de resoluÃÃo, que sÃo processos de aproximaÃÃes sucessivas, para o cÃlculo de zeros da funÃÃo, neste item foram discutidos os mÃtodos de Newton, bisseÃÃo, secante, cordas e ponto fixo, de modo que ao final do tÃpico foram comparados os mÃtodos sob os aspectos de garantia e agilidade de convergÃncia e esforÃo computacional. Os resultados conseguidos indicaram a importÃncia do tema de resoluÃÃo de problemas com Ãnfase nas demonstraÃÃes das fÃrmulas, e que a contextualizaÃÃo histÃrica pode contribuir para desmitificar o processo de criaÃÃo e humanizaÃÃo da matemÃtica. / This study proposes a discussion of Algebraic Equations, aiming to conduct a study on the statements of the formulas, addressing the historic aspects to the various methods of problem solving, in this case, the methods were worked Algebraic, Geometric and Computational. This research was based on a literature study of the difficulties of performing demonstrations of formulas worked in the contents of mathematics as well as in the statements themselves, together with many worked examples. The analysis of the bibliographic material allowed to distribute this study by the method Algebraic problem-solving, in which they discussed the demonstration and application of resolving formulas of polynomial equations of 1st, 2nd, 3rd and 4th grades, and even citing the impossibility of the existence of formulas equations above 4 degree. In the study of the geometric method, we noticed how this geometry efficiently present in solving problems and those solutions are possible only by ruler and compass, this topic was discussed methods for solving equations of 1st and 2nd grade. About Computational Method, the study on the iterative resolution methods that are processes of successive approximations for the calculation of zeros of the function, this item was discussed methods of Newton, bisection, secant, and ropes fixed point was emphasized in so that at the end of the topic the methods under warranty and agility aspects of convergence and computational effort were compared. The achieved results show the importance of the topic of problem solving with emphasis on the statements of the formulas, and the historical context can help to demystify the process of creating and humanization of mathematics.

Este estudo situa-se no campo da aprendizagem da Matemática. O objetivo da pesquisa aqui proposta foi identificar a existência de relação entre a competência cognitiva, o uso de estratégias metacognitivas e a compreensão do erro, na resolução de problemas matemáticos com equações algébricas do 1º grau. Para isso, investigou-se 38 alunos do 8º ano do Ensino Fundamental de uma escola municipal da região metropolitana de Porto Alegre, selecionados aleatoriamente em duas turmas. Adotando o método misto de pesquisa, utilizou-se como instrumentos o Whimbey Analytical Skills Inventory (WASI), como avaliador da competência cognitiva e divisor dos grupos com alto e baixo nível de competência cognitiva, tendo como referência a média geral de acertos do grupo; e a Escala de Estratégias Metacognitivas na Resolução de Problemas (E-EMRP). Realizaram-se também Observações e Entrevistas Clínicas com base na solução da Tarefa de Resolução de Problemas com Equações Algébricas do 1º Grau (TRPEA). O tratamento estatístico, realizado por meio dos testes de Correlação de Pearson e t-Student, demonstrou a existência de correlação estatisticamente significativa entre o WASI e a TRPEA, evidenciando a existência de relação entre a competência cognitiva e o desempenho na resolução de problemas e compreensão do erro pelo estudante. Verificou-se também uma diferença significativa entre os estudantes com alto e baixo nível de competência cognitiva, em relação ao desempenho na resolução dos problemas e compreensão dos erros, comprovando que estudantes com maiores níveis de competência cognitiva apresentaram melhor desempenho, cometendo menos erros e compreendendo com maior frequência os erros cometidos. Verificou-se ainda a existência de relação entre o uso de estratégias metacognitivas e a compreensão do erro, bem como em relação à competência cognitiva - por meio da observação e entrevista clínica - evidenciando que, apesar do uso das estratégias não ocorrer em todos os momentos da resolução do problema (antes, durante e após a leitura do enunciado e durante e após a resolução do problema) com a mesma frequência, os estudantes com maiores níveis de competência cognitiva demonstraram utilizar um maior conjunto de estratégias, compreendendo melhor a necessidade de sua utilização correta em todas as etapas da resolução, do que os estudantes com baixos níveis de competência cognitiva. Esses resultados alertam para a necessidade de desenvolver-se em sala de aula, atividades que tenham como propósito o treinamento do uso correto de estratégias metacognitivas, visando o aprimoramento da capacidade de resolução de problemas matemáticos, assim como a prevenção e compreensão dos erros cometidos. / This study belongs to the field of mathematics learning. The research aimed to confirm the existence of a relationship among cognitive ability, usage of metacognitive strategies and comprehension of error, in mathematical problem solving 1st degree algebraic equations. For this purpose, 38 students attending the 8th year of primary education in a public school in the metropolitan area of Porto Alegre (Brazil), were randomly divided into two groups. A mixed method research was adopted, in which the tools chosen were: Whimbey Analytical Skills Inventory (WASI), used as a measure of cognitive competence and also as criteria for splitting the group according to cognitive competence level (high and low), considering the overall average of the group's score as guideline, and Metacognitive Strategies Scale in Problem Solving (MSSPS). The method included observations and clinic interviews based on solution of Problem Solving 1st Degree Algebraic Equations (PSAET) as well. Statistical procedure, through Pearson correlation and Student’s t-tests, showed a statistically significant correlation between WASI and PSAET, which demonstrated the existence of relationship between cognitive ability and performance in problem solving and comprehension of error by the student. In addition, there was detected a significant difference between students with high and low levels of cognitive ability, referent to the performance in problem solving and comprehension of errors, proving that students with higher levels of cognitive ability showed best performance, by making fewer errors and comprehending them more frequently. Furthermore, there was confirmed the existence of a relationship between the use of metacognitive strategies and comprehension of the error, and relatively to cognitive competence - through observation and clinical interview. This is an evidence that, despite the strategies were not employed at all stages of problem solving (before, during and after reading the statement and during and after the resolution of the problem) with the same frequency, students with higher levels of cognitive competence demonstrated using a larger set of strategies, aware of the need to the right utilization at all stages of resolution, compared to students with low levels of cognitive competence. These results emphasized the importance of developing classroom activities with purpose of training the correct usage of metacognitive strategies, in order to improve the ability to solve mathematical problems, besides prevention and understanding of errors.

Systems of differential-algebraic equations (DAEs) arise in many areas including chemical engineering, electrical circuit simulation, and robotics. Such systems are routinely generated by simulation and modeling environments, like MapleSim, Matlab/Simulink, and those based on the Modelica language. Before a simulation starts and a numerical solution method is applied, some kind of structural analysis (SA) is performed to determine the structure and the index of a DAE system. Structural analysis methods serve as a necessary preprocessing stage, and among them, Pantelides's graph-theory-based algorithm is widely used in industry. Recently, Pryce's Σ-method is becoming increasingly popular, owing to its straightforward approach and capability of analyzing high-order systems. Both methods are equivalent in the sense that (a) when one succeeds, producing a nonsingular Jacobian, the other also succeeds, and that (b) the two give the same structural index in the case of either success or failure. When SA succeeds, the structural results can be used to perform an index reduction process, or to devise a stage-by-stage solution scheme for computing derivatives or Taylor coefficients up to some order. Although such a success occurs on fairly many problems of interest, SA can fail on some simple, solvable DAEs with an identically singular Jacobian, and give incorrect structural information that usually includes the index. In this thesis, we focus on the Σ-method and investigate its failures. Aiming at making this SA more reliable, we develop two conversion methods for fixing SA failures. These methods reformulate a DAE on which the Σ-method fails into an equivalent problem on which SA is more likely to succeed with a nonsingular Jacobian. The implementation of our methods requires symbolic computations. We also combine our conversion methods with block triangularization of a DAE. Using a block triangular form of a Jacobian sparsity pattern, we identify which diagonal block(s) of the Jacobian is identically singular, and then perform a conversion on each singular block. This approach can reduce the computational cost and improve the efficiency of finding a suitable conversion for fixing SA's failures. / Thesis / Doctor of Philosophy (PhD)