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Electromagnetic scattering by 3-D arbitrarily shaped homogeneous dielectric objects is characterized. In the analysis, the method of moments is first employed to solve the combined field integral equation for scattering properties of these three-dimensional homogeneous dielectric objects of arbitrary shape. The fast multipole method, and the multi-level fast multipole algorithm are implemented into our codes for matrix-vector manipulations. Specifically, four proposals are made and discussed to increase convergence and accuracy of iterative procedures (conjugate gradient method). Numerical results are obtained using various methods and compared to each other. / Singapore-MIT Alliance (SMA)

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Este trabalho teve como motivação experimental problemas surgidos nos laboratórios de biofísica do IF-USP em medidas com vesículas carregadas, que podem ser usadas para estudar membranas biológicas. As propriedades destes sistemas, e, em particular, como função da temperatura, só podem ser investigadas indiretamente. A interpretação dos resultados depende de uma modelagem coerente. Entre as exigências de coerência, estariam a justificativa para a discrepância entre resultados para as medidas de raio dos macroíons lipídicos, no intervalo de temperaturas próximas à transição gelfluido, obtidas por técnicas experimentais diferentes (Static Light Scattering (SLS) e Dynamic Light Scattering (DLS)) e as anomalias no calor específico, na condutividade e na mobilidade eletroforética da solução coloidal iônica, no mesmo intervalo de temperatura. Estudos anteriores a este trabalho sugeriam a formação de poros em tais vesículas, como tentativa de explicar diferenças nos resultados das técnicas de espalhamento, bem como o papel da análise do equilíbrio termodinâmico da dissociação sobre as propriedades térmicas e termoelétricas. Para interpretar e dar coerência aos diversos resultados experimentais existentes, é necessário desenvolver modelos teóricos. É objetivo deste trabalho desenvolver técnicas de tratamento de modelos teóricos quanto às propriedades de transporte. Assim, neste estudo utilizamos o método computacional conhecido como ``Lattice Boltzmann\'\' (LBM) procurando focar no estudo de propriedades de meios porosos e de coloides carregados. Para melhor compreensão dos limites e justificativas do modelo, realizamos um breve estudo sobre a equação de Boltzmann e suas propriedades. Assim, depois de desenvolver um código em linguagem C para o LBM, e testá-lo com resultados conhecidos, utilizamos o ``Lattice Boltzmann\'\' para determinar o coeficiente de arrasto de esferas e cascas esféricas porosas, comparando com resultados analíticos e experimentais conhecidos. Para o estudo de sistemas coloidais carregados, acoplamos o ``Lattice Boltzmann`` a outra técnica computacional, ``Fast Multipole Method\'\' (FMM), para poder estudar efeitos elétricos e hidrodinâmicos associados aos coloides com carga. Foram feitas simulações de fluxo eletrosmótico e eletrólitos entre placas carregadas que apresentaram resultados animadores ao comparar com resultados analíticos, constatando que FMM pode ser uma alternativa à resolução da equação de Laplace para determinar o potencial eletrostático em simulações com LBM. Além disso foram feitas simulações de mobilidade eletroforética em meios sem sal, que mostram que o código pode ser utilizado como ferramenta na busca da solução para as dúvidas surgidas no estudo de vesículas carregadas. / This study was inspired by the problem of interpreting experimental results arising in the Biophysics Laboratory of the Institute of Physics - USP. Different techniques are used to investigate charged vesicles that are used as an experimental model for biological membranes. Careful measurements of vesicle radius, in the range of gel-fluid transition temperature, through different experimental techniques, namely Static and Dynamic Light Scattering (SLS and DLS) led to very different results. Previous studies of the same system suggested the formation of pores in such vesicles. In addition, specific heat and conductivity measurements on charged vesicles displayed an anomalous region, in the range of gel-fluid transition temperature, as compared to neutral vesicles. In an attempt to make progress in the understanding of the above problems, we use the computational method known as Lattice Boltzmann Method (LBM) seeking to focus on the study of transport properties of porous and charged colloids. To better understand the limits of the model and justifications, we make a brief study of the Boltzmann equation and its properties. Thus, after developing a code in $C$ language for LBM, and testing it with known results, we use the Lattice Boltzmann method to obtain the drag coefficient of spheres and porous spherical shells. We compare our results with analytical and experimental results from the literature and obtain good fitting. For the study of charged colloidal systems, we associate the Lattice Boltzmann method with a computational technique for the calculation of the eletrostatic potential: the Fast Multipole Method (FMM), which enables us to study electrical and hydrodynamic effects on charged colloids. We simulate electroosmotic flow and electrolytes between charged plates, with encouraging results in the comparison with known analytical result. This suggests that FMM may be a good alternative to resolution of the Laplace equation to determine the electrostatic potential simulations with LBM. Moreover we have obtained the electrophoretic mobility for charged colloids in saltless solutions, which makes our code a possible instrument for the interpretation of experimental results on charged vesicles.

Computer simulation of wave propagation is an active research area as wave phenomena are prevalent in many applications. Examples include wireless communication, radar cross section, underwater acoustics, and seismology. For high frequency waves, this is a challenging multiscale problem, where the small scale is given by the wavelength while the large scale corresponds to the overall size of the computational domain. Research into wave equation modeling can be divided into two regimes: time domain and frequency domain. In each regime, there are two further popular research directions for the numerical simulation of the scattered wave. One relies on direct discretization of the wave equation as a hyperbolic partial differential equation in the full physical domain. The other direction aims at solving an equivalent integral equation on the surface of the scatterer. In this dissertation, we present three new techniques for the frequency domain, boundary integral equations. / text

Este trabalho teve como motivação experimental problemas surgidos nos laboratórios de biofísica do IF-USP em medidas com vesículas carregadas, que podem ser usadas para estudar membranas biológicas. As propriedades destes sistemas, e, em particular, como função da temperatura, só podem ser investigadas indiretamente. A interpretação dos resultados depende de uma modelagem coerente. Entre as exigências de coerência, estariam a justificativa para a discrepância entre resultados para as medidas de raio dos macroíons lipídicos, no intervalo de temperaturas próximas à transição gelfluido, obtidas por técnicas experimentais diferentes (Static Light Scattering (SLS) e Dynamic Light Scattering (DLS)) e as anomalias no calor específico, na condutividade e na mobilidade eletroforética da solução coloidal iônica, no mesmo intervalo de temperatura. Estudos anteriores a este trabalho sugeriam a formação de poros em tais vesículas, como tentativa de explicar diferenças nos resultados das técnicas de espalhamento, bem como o papel da análise do equilíbrio termodinâmico da dissociação sobre as propriedades térmicas e termoelétricas. Para interpretar e dar coerência aos diversos resultados experimentais existentes, é necessário desenvolver modelos teóricos. É objetivo deste trabalho desenvolver técnicas de tratamento de modelos teóricos quanto às propriedades de transporte. Assim, neste estudo utilizamos o método computacional conhecido como ``Lattice Boltzmann\'\' (LBM) procurando focar no estudo de propriedades de meios porosos e de coloides carregados. Para melhor compreensão dos limites e justificativas do modelo, realizamos um breve estudo sobre a equação de Boltzmann e suas propriedades. Assim, depois de desenvolver um código em linguagem C para o LBM, e testá-lo com resultados conhecidos, utilizamos o ``Lattice Boltzmann\'\' para determinar o coeficiente de arrasto de esferas e cascas esféricas porosas, comparando com resultados analíticos e experimentais conhecidos. Para o estudo de sistemas coloidais carregados, acoplamos o ``Lattice Boltzmann`` a outra técnica computacional, ``Fast Multipole Method\'\' (FMM), para poder estudar efeitos elétricos e hidrodinâmicos associados aos coloides com carga. Foram feitas simulações de fluxo eletrosmótico e eletrólitos entre placas carregadas que apresentaram resultados animadores ao comparar com resultados analíticos, constatando que FMM pode ser uma alternativa à resolução da equação de Laplace para determinar o potencial eletrostático em simulações com LBM. Além disso foram feitas simulações de mobilidade eletroforética em meios sem sal, que mostram que o código pode ser utilizado como ferramenta na busca da solução para as dúvidas surgidas no estudo de vesículas carregadas. / This study was inspired by the problem of interpreting experimental results arising in the Biophysics Laboratory of the Institute of Physics - USP. Different techniques are used to investigate charged vesicles that are used as an experimental model for biological membranes. Careful measurements of vesicle radius, in the range of gel-fluid transition temperature, through different experimental techniques, namely Static and Dynamic Light Scattering (SLS and DLS) led to very different results. Previous studies of the same system suggested the formation of pores in such vesicles. In addition, specific heat and conductivity measurements on charged vesicles displayed an anomalous region, in the range of gel-fluid transition temperature, as compared to neutral vesicles. In an attempt to make progress in the understanding of the above problems, we use the computational method known as Lattice Boltzmann Method (LBM) seeking to focus on the study of transport properties of porous and charged colloids. To better understand the limits of the model and justifications, we make a brief study of the Boltzmann equation and its properties. Thus, after developing a code in $C$ language for LBM, and testing it with known results, we use the Lattice Boltzmann method to obtain the drag coefficient of spheres and porous spherical shells. We compare our results with analytical and experimental results from the literature and obtain good fitting. For the study of charged colloidal systems, we associate the Lattice Boltzmann method with a computational technique for the calculation of the eletrostatic potential: the Fast Multipole Method (FMM), which enables us to study electrical and hydrodynamic effects on charged colloids. We simulate electroosmotic flow and electrolytes between charged plates, with encouraging results in the comparison with known analytical result. This suggests that FMM may be a good alternative to resolution of the Laplace equation to determine the electrostatic potential simulations with LBM. Moreover we have obtained the electrophoretic mobility for charged colloids in saltless solutions, which makes our code a possible instrument for the interpretation of experimental results on charged vesicles.

In this study, we consider electrostatics contributed from the molecules in the ionic solution. It plays a significant role in determining the binding affinity of molecules and drugs. We develop the overall framework of computing electrostatic properties for three-dimensional molecular structures, including potential, energy, and forces. These properties are derived from Poisson-Boltzmann equation, a partial differential equation that describes the electrostatic behavior of molecules in ionic solutions. In order to compute these properties, we derived new boundary integral equations and designed a boundary element algorithm based on the linear time fast multipole method for solving the linearized Poisson-Boltzmann equation. Meanwhile, a higher-order parametric formulation called algebraic spline model is used for accurate approximation of the unknown solution of the linearized Poisson-Boltzmann equation. Based on algebraic spline model, we represent the normal derivative of electrostatic potential by surrounding electrostatic potential. This representation guarantees the consistent relation between electrostatic potential and its normal derivative. In addition, accurate numerical solution and fast computation for electrostatic energy and forces are also discussed. In addition, we described our hierarchical modeling and parameter optimization of molecular structures. Based on this technique, we can control the scalability of molecular models for electrostatic computation. The numerical test and experimental results show that the proposed techniques offer an efficient and accurate solution for solving the electrostatic problem of molecules. / text

Fast algorithms for potential evaluation in N-body problems often tend to be extremely abstract and complex. This thesis presents a simple, hierarchical approach to solving the potential evaluation problem in O(n) time. The approach is developed in the field of electrostatics and can be extended to N-body problems in general. Herein, the potential vector is expressed as a product of the potential matrix and the charge vector. The potential matrix itself is a product of component matrices. The potential function satisfies the Laplace equation and is hence expressed as a linear combination of spherical harmonics, which form the general solutions of the Laplace equation. The orthogonality of the spherical harmonics is exploited to reduce execution time. The duality of the various lists in the algorithm is used to reduce storage and computational complexity. A smart tree-construction strategy leads to efficient parallelism at computation intensive stages of the algorithm. The computational complexity of the algorithm is better than that of the Fast Multipole Algorithm, which is one of the fastest contemporary algorithms to solve the potential evaluation problem. Experimental results show that accuracy of the algorithm is comparable to that of the Fast Multipole Algorithm. However, this approach uses some implementation principles from the Fast Multipole Algorithm. Parallel efficiency and scalability of the algorithms are studied by the experiments on IBM p690 multiprocessors.

This thesis deals with quantum chemistry methods for large systems. In particular, the thesis focuses on the efficient construction of the Coulomb and exchange matrices which are important parts of the Fock matrix in Hartree-Fock calculations. Density matrix purification, which is a method used to construct the density matrix for a given Fock matrix, is also discussed. The methods described are not only applicable in the Hartree-Fock case, but also in Kohn-Sham Density Functional Theory calculations, where the Coulomb and exchange matrices are parts of the Kohn-Sham matrix. Screening techniques for reducing the computational complexity of both Coulomb and exchange computations are discussed, including the fast multipole method, used for efficient computation of the Coulomb matrix. The thesis also discusses how sparsity in the matrices occurring in Hartree-Fock and Kohn-Sham Density Functional Theory calculations can be used to achieve more efficient storage of matrices as well as more efficient operations on them. / QC 20100817

Fast algorithms for potential evaluation in N-body problems often tend to be extremely abstract and complex. This thesis presents a simple, hierarchical approach to solving the potential evaluation problem in O(n) time. The approach is developed in the field of electrostatics and can be extended to N-body problems in general. Herein, the potential vector is expressed as a product of the potential matrix and the charge vector. The potential matrix itself is a product of component matrices. The potential function satisfies the Laplace equation and is hence expressed as a linear combination of spherical harmonics, which form the general solutions of the Laplace equation. The orthogonality of the spherical harmonics is exploited to reduce execution time. The duality of the various lists in the algorithm is used to reduce storage and computational complexity. A smart tree-construction strategy leads to efficient parallelism at computation intensive stages of the algorithm. The computational complexity of the algorithm is better than that of the Fast Multipole Algorithm, which is one of the fastest contemporary algorithms to solve the potential evaluation problem. Experimental results show that accuracy of the algorithm is comparable to that of the Fast Multipole Algorithm. However, this approach uses some implementation principles from the Fast Multipole Algorithm. Parallel efficiency and scalability of the algorithms are studied by the experiments on IBM p690 multiprocessors.