RAMOS, Igor Rochaid Oliveira. Estudo de um sistema clássico de dipolos magnéticos carregados em estruturas de bicamadas. 2010. 75 f. Dissertação (Mestrado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2010. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2014-10-31T20:38:15Z No. of bitstreams: 1 2010_dis_iroramos.pdf: 3512055 bytes, checksum: ba5eb29d22412048c4f8a2972b874c7e (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2014-10-31T21:08:54Z (GMT) No. of bitstreams: 1 2010_dis_iroramos.pdf: 3512055 bytes, checksum: ba5eb29d22412048c4f8a2972b874c7e (MD5) / Made available in DSpace on 2014-10-31T21:08:54Z (GMT). No. of bitstreams: 1 2010_dis_iroramos.pdf: 3512055 bytes, checksum: ba5eb29d22412048c4f8a2972b874c7e (MD5) Previous issue date: 2010 / We study the structural and dynamical properties of a two-dimensional (2D) classical bi-layer crystal of charged magnetic dipolar particles in a setup where the dipoles are oriented perpendicular to the layers and equal density of charged dipolar particles in each layer. The energy of the system is due to the charge - charge interaction (Coulomb interaction) and the dipole - dipole interaction. Due to the long-range nature of the interactions, we use the Ewald summation method to obtain an expression for the energy involving rapidly convergent sums. By comparing the energies of a number of possible crystal geometries, we determine the phase diagram of the system as a function of the parameter η (which is related to the separation between the layers of charged magnetic dipoles and the particle density) and the relative intensity of the magnetic and electrical interactions. By changing the relative intensity of the dipole - dipole interaction with respect to electrical one, we are able to find six diferent stable crystalline structures as a function of η. An interesting feature of the present model system is the possibility to tune between the matched and staggered arrangements by varying the magnetic interaction between the dipoles, e.g. through an external magnetic field. The phase boundaries of the crystalline structures consist of both continuous and discontinuous transitions. In order to investigate the stability of the minimum energy arrangements we also calculate the phonon spectra of the system within the harmonic approximation. In this case, we resort again on the Ewald technique to obtain the rapidly convergent sums. The analysis of the phonon spectra reveals interesting features which are useful in the study of melting. / Estudamos as estruturas e as propriedades dinâmicas de um cristal clássico bidimensional (2D) em bicamadas de partículas dipolares magnéticas carregadas em um arranjo no qual os dipolos são perpendiculares às camadas e com mesma densidade de partículas em cada camada. A energia do sistema é devido à interação carga - carga (interação coulom- biana) e a interação dipolo - dipolo. Devido ao fato dessas interações serem de longo alcance, usamos o método da soma de Ewald para obter uma expressão para a energia envolvendo somas que convergem rapidamente. Comparando as energias de possíveis geometrias do cristal, determinamos o diagrama de fase do sistema em função do parâmetro η (que está relacionado com a distância entre as camadas de dipolos magnéticos carregados e a densidade de partículas) e da intensidade relativa das interações elétrica e magnética. Mudando a intensidade relativa da interacão dipolo - dipolo com respeito à interação elétrica, podemos encontrar seis diferentes estruturas cristalinas estáveis em função de η. Uma característica interessante desse sistema é a possibilidade de permanecer em arranjos nos quais as camadas são ou não deslocadas uma em relação a outra, apenas variando a interação magnética entre os dipolos, por exemplo, através de um campo magnético externo. As transições entre as estruturas cristalinas podem ser contínuas e descontínuas. No intuito de investigar a estabilidade das configurações de mínima energia, calculamos o espectro dos fônons do sistema usando a aproximação harmônica. Para isto, recorremos novamente a técnica de Ewald para obter somas que convergem rapidamente. A análise da relação de dispersão (fônons) revela características do sistema que são de grande utilidade no estudo da transição sólido-líquido (fusão).

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The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P2NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. While typically B-splines are applied in the scope of particle mesh methods, we consider for the first time also an approximation by Bessel functions. We show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that if the parameters are tuned appropriately the Bessel window function can keep up with the B-spline window and is in many cases even the better choice with respect to computational costs.

A numerical method for computing three-dimensional Stokes flow driven by a doubly-periodic array of regularized forces is presented. In the non-periodic direction either a free boundary or a homogeneous Dirichlet condition is enforced. The method consists of finding a regularized Green's function in Fourier space analytically. Then only an inverse fast Fourier transform (inverse FFT) has to be computed. Accuracy is verified by comparing numerical results to a solution that is independent of the method. In an Ewald splitting, the FFT method can be used to compute the smooth component of the flow, which allows for a splitting parameter as small as a few grid cells. This selection makes the sum in physical space converge extremely fast. Numerical examples demonstrate that fact. Since the forces are regularized, in some cases splitting is not even needed, depending on the relative sizes of the numerical parameters. The method is applied to model the flow created by carpets of nodal cilia based on cilium shape. / acase@tulane.edu

Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Estudamos as estruturas e as propriedades dinÃmicas de um cristal clÃssico bidimensional (2D) em bicamadas de partÃculas dipolares magnÃticas carregadas em um arranjo no qual os dipolos sÃo perpendiculares Ãs camadas e com mesma densidade de partÃculas em cada camada. A energia do sistema à devido à interaÃÃo carga - carga (interaÃÃo coulom- biana) e a interaÃÃo dipolo - dipolo. Devido ao fato dessas interaÃÃes serem de longo alcance, usamos o mÃtodo da soma de Ewald para obter uma expressÃo para a energia envolvendo somas que convergem rapidamente. Comparando as energias de possÃveis geometrias do cristal, determinamos o diagrama de fase do sistema em funÃÃo do parÃmetro η (que està relacionado com a distÃncia entre as camadas de dipolos magnÃticos carregados e a densidade de partÃculas) e da intensidade relativa das interaÃÃes elÃtrica e magnÃtica. Mudando a intensidade relativa da interacÃo dipolo - dipolo com respeito à interaÃÃo elÃtrica, podemos encontrar seis diferentes estruturas cristalinas estÃveis em funÃÃo de η. Uma caracterÃstica interessante desse sistema à a possibilidade de permanecer em arranjos nos quais as camadas sÃo ou nÃo deslocadas uma em relaÃÃo a outra, apenas variando a interaÃÃo magnÃtica entre os dipolos, por exemplo, atravÃs de um campo magnÃtico externo. As transiÃÃes entre as estruturas cristalinas podem ser contÃnuas e descontÃnuas. No intuito de investigar a estabilidade das configuraÃÃes de mÃnima energia, calculamos o espectro dos fÃnons do sistema usando a aproximaÃÃo harmÃnica. Para isto, recorremos novamente a tÃcnica de Ewald para obter somas que convergem rapidamente. A anÃlise da relaÃÃo de dispersÃo (fÃnons) revela caracterÃsticas do sistema que sÃo de grande utilidade no estudo da transiÃÃo sÃlido-lÃquido (fusÃo). / We study the structural and dynamical properties of a two-dimensional (2D) classical bi-layer crystal of charged magnetic dipolar particles in a setup where the dipoles are oriented perpendicular to the layers and equal density of charged dipolar particles in each layer. The energy of the system is due to the charge - charge interaction (Coulomb interaction) and the dipole - dipole interaction. Due to the long-range nature of the interactions, we use the Ewald summation method to obtain an expression for the energy involving rapidly convergent sums. By comparing the energies of a number of possible crystal geometries, we determine the phase diagram of the system as a function of the parameter η (which is related to the separation between the layers of charged magnetic dipoles and the particle density) and the relative intensity of the magnetic and electrical interactions. By changing the relative intensity of the dipole - dipole interaction with respect to electrical one, we are able to find six diferent stable crystalline structures as a function of η. An interesting feature of the present model system is the possibility to tune between the matched and staggered arrangements by varying the magnetic interaction between the dipoles, e.g. through an external magnetic field. The phase boundaries of the crystalline structures consist of both continuous and discontinuous transitions. In order to investigate the stability of the minimum energy arrangements we also calculate the phonon spectra of the system within the harmonic approximation. In this case, we resort again on the Ewald technique to obtain the rapidly convergent sums. The analysis of the phonon spectra reveals interesting features which are useful in the study of melting.

The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P2NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. While typically B-splines are applied in the scope of particle mesh methods, we consider for the first time also an approximation by Bessel functions. We show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that if the parameters are tuned appropriately the Bessel window function can keep up with the B-spline window and is in many cases even the better choice with respect to computational costs.

We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.

We consider fast computation methods for simulation of dynamics of a collection of particles dispersed in an unbounded Stokesian suspension. Stokesian suspensions are of great practical interest in the manufacturing and processing of various commercial products. The most popular dynamic simulation method for these kind of suspensions was developed by Brady and Bossis (Brady and Bossis [1988]). This method uses a truncated multipole expansion to represent the fluid traction on particle surfaces. The hydrodynamic interactions in Stoke-sian suspension are long ranged in nature, resulting in strong coupled motion of all particles. For an N particle system, this method imposes an O(N3) computational cost, thus posing limitations to the number of particles that may be simulated. More recent methods (Sierou and Brady [2001], Scintilla, Darve and Shaqfeh [2005]) have attempted to solve this problem using Particle Mesh Ewald summation techniques by distributing the moments on a grid and using Fast Fourier Transform algorithms, resulting in an O(N log N) computational cost. We review these methods and propose a version that we believe is some-what superior. In the course of this study, we have identified and corrected errors in previous studies that maybe of some importance in determining the bulk properties of suspensions. Finally, we show the utility of our method in determining certain properties of suspensions and compare them to existing analytical results for the same.

The present thesis provides a modularized view on the structure of fast numerical methods for computing Coulomb interactions between charged particles in three-dimensional space. Thereby, the common structure is given in terms of three self-contained algorithmic frameworks that are built on top of each other, namely fast Fourier transform (FFT), nonequispaced fast Fourier transform (NFFT) and NFFT based particle-mesh methods (P²NFFT). For each of these frameworks algorithmic enhancement and parallel implementations are presented with special emphasis on scalability up to hundreds of thousands of parallel processes. In the context of FFT massively parallel algorithms are composed from hardware adaptive low level modules provided by the FFTW software library. The new algorithmic NFFT concepts include pruned NFFT, interlacing, analytic differentiation, and optimized deconvolution in Fourier space with respect to a mean square aliasing error. Enabled by these generalized concepts it is shown that NFFT provides a unified access to particle-mesh methods. Especially, mixed-periodic boundary conditions are handled in a consistent way and interlacing can be incorporated more efficiently. Heuristic approaches for parameter tuning are presented on the basis of thorough error estimates. / Die vorliegende Dissertation beschreibt einen modularisierten Blick auf die Struktur schneller numerischer Methoden für die Berechnung der Coulomb-Wechselwirkungen zwischen Ladungen im dreidimensionalen Raum. Die gemeinsame Struktur ist geprägt durch drei selbstständige und auf einander aufbauenden Algorithmen, nämlich der schnellen Fourier-Transformation (FFT), der nicht äquidistanten schnellen Fourier-Transformation (NFFT) und der NFFT-basierten Teilchen-Gitter-Methode (P²NFFT). Für jeden dieser Algorithmen werden Verbesserungen und parallele Implementierungen vorgestellt mit besonderem Augenmerk auf massiv paralleler Skalierbarkeit. Im Kontext der FFT werden parallele Algorithmen aus den Hardware adaptiven Modulen der FFTW Softwarebibliothek zusammengesetzt. Die neuen NFFT-Konzepte beinhalten abgeschnittene NFFT, Versatz, analytische Differentiation und optimierte Entfaltung im Fourier-Raum bezüglich des mittleren quadratischen Aliasfehlers. Mit Hilfe dieser Verallgemeinerungen bietet die NFFT einen vereinheitlichten Zugang zu Teilchen-Gitter-Methoden. Insbesondere gemischt periodische Randbedingungen werden einheitlich behandelt und Versatz wird effizienter umgesetzt. Heuristiken für die Parameterwahl werden auf Basis sorgfältiger Fehlerabschätzungen angegeben.

The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P2NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. Typically B-splines are applied in the scope of particle mesh methods, as for instance within the well-known particle-particle particle-mesh (P3M) algorithm. The publicly available P2NFFT algorithm allows the application of an oversampled FFT as well as the usage of different window functions. We consider for the first time also an approximation by Bessel functions and show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that, if the parameters are tuned appropriately, the Bessel window function is in many cases even the better choice in terms of computational costs. Moreover, the results indicate that it is often advantageous in terms of efficiency to spend some oversampling within the NFFT while using a window function with a smaller support.