The parallel version of precondition iterative techniques is developed for matrices arising from the panel boundary element method for three-dimensional simple connected domains with Dirichlet boundary conditions. Results were obtained on an nCUBE-2 parallel computer showing that iterative solution methods are very well suited also in three-dimensional case for implementation on a MIMD computer and that they are much more efficient than usual direct solution techniques.

This paper presents three new coupling methods for interior penalty discontinuous Galerkin finite element methods and boundary element methods. The new methods allow one to use discontinuous basis functions on the interface between the subdomains represented by the finite element and boundary element methods. This feature is particularly important when discontinuous Galerkin finite element methods are used. Error and stability analysis is presented for some of the methods. Numerical examples suggest that all three methods exhibit very similar convergence properties, consistent with available theoretical results.:1. Introduction 2. Model Problem and Background 3. New Coupling Methods 4. Stability and Error Analysis 5. Numerical Examples 6. Summary A. Appendix

<p>Droplet dispersions with adsorbed exotic surface active species (proteins, fatty alcohol, fatty acids, solid particulates, lipids, or polymers) find an immense number of applications in the field of engineering and bioscience. Interfacial rheology plays an essential role in the dynamics of many of these systems, yet little is understood about how these effects alter droplet dynamics. Most surfactants studied historically have been simple enough that the droplet dynamics can be described by Marangoni effects (surfactant concentration gradients), surface dilution, and adsorption/desorption kinetics without including the intrinsic surface rheology. One of the challenges in examining droplet systems with complex interfaces is that the intrinsic rheological effects are strongly coupled with surfactant transport effects (surface convection, diffusion, dilution and adsorption/desorption). The surface rheology can impact the ability of surfactant to transport along the surface, while surfactant transport can alter the surface rheology by changing the surface concentration. In this work, we develop axisymmetric boundary-integral simulations that allow us to quantitatively explore the combined effect of intrinsic surface rheology and surfactant transport on droplet dynamics in the Stokes flow limit. We assume that the droplet interface is predominantly viscous and that the Boussinesq Scriven constitutive relationship describes the properties of the viscous membrane. The key questions that we address in this work are:</p> <p><br></p> <ul> <li>How do viscous membranes impact droplet deformation, breakup and relaxation?      </li> </ul> <p>     When a droplet is placed under external flow, it can either attain a stable shape under flow or stretch indefinitely above a critical flow rate and break apart. In this topic, we first discuss the breakup conditions for a droplet suspended in an unbounded immiscible fluid under a general linear flow field using perturbation theories for surface viscosity in the limit of small droplet deformation. We neglect the inhomogeneity in surfactant concentration and surface tension for this part. We find that the surface shear/dilational viscosity increases/decreases the critical capillary number for droplet breakup compared to a clean droplet at the same capillary number and droplet viscosity ratio value. In the second part of this topic, we solve the problem using boundary integral simulations for the case of axisymmetric extensional flow. Numerically solving this problem allows us to examine the effect of Marangoni stresses, pressure thickening/thinning surface viscosities, and stronger flows. We compare the droplet breakup results from our simulations to results from second-order perturbation theories. We present the physical mechanism behind our observations using traction arguments from interfacial viscosities. We conclude this topic by examining the combined role of surface viscosity and surfactant transport on the relaxation of an initially extended droplet in a quiescent external fluid.</p> <p><br></p> <ul> <li>How do viscous membranes alter droplet sedimentation?</li> </ul> <p>      When an initially deformed droplet sediment under gravity, it can either revert to a spherical shape or undergo instability where the droplet develops a long tail or cavity at its rear end. Here, we use numerical simulations to discuss how interfacial viscosity alters the breakup criterion and the formation of threads/cavities under gravity. We examine the combined influence of intrinsic surface viscosity and surfactant transport on droplet stability by assuming a linear dependence of surface tension on surfactant concentration and an exponential dependence of interfacial viscosities on surface pressure. We find that surface shear viscosity inhibits the tail/cavity growth at the droplet’s rear end and increases the critical capillary number compared to a clean droplet. In contrast, surface dilational viscosity promotes tail/cavity growth and lowers the critical capillary number compared to a clean droplet.</p> <p><br></p> <ul> <li>How do viscous membranes affect droplet coalescence?</li> </ul> <p>      When two droplets approach under external flow, a thin film is formed between the two droplets. Here, we develop numerical simulations to model the full coalescence process from the collision of two droplets under uniaxial compressional flow to the point where the film approaches rupture. We investigate the role of interfacial viscosity on the film profiles and drainage time. We observe that both surface shear and dilational viscosity significantly delay the film drainage time relative to a clean droplet. Interestingly, we find that the film drainage behaviour of a droplet with surface viscosity is not altered by the relative ratio of shear to dilational viscosity but rather depends on the sum of shear and dilational Boussinesq numbers. This is in contrast to the effect of surface viscosity observed in the previous processes (droplet breakup and sedimentation), where surface shear viscosity increases the critical capillary number compared to a clean droplet, while surface dilatational viscosity has the opposite effect.</p>

A method for solving the inverse geometrical problem is presented by reconstructing the unknown subsurface cavity geometry using boundary element methods, a genetic algorithm, and Nelder-Mead non-linear simplex optimization. The heat conduction problem is solved utilizing the boundary element method, which calculates the difference between the measured temperature at the exposed surface and the computed temperature under the current update of the unknown subsurface flaws and cavities. In a first step, clusters of singularities are utilized to solve the inverse problem and to identify the location of the centroid(s) of the subsurface cavity(ies)/flaw(s). In a second step, the reconstruction of the estimated cavity(ies)/flaw(s) geometry(ies) is accomplished by utilizing an anchored grid pattern upon which cubic spline knots are restricted to move in the search for unknown geometry. Solution of the inverse problem is achieved using a genetic algorithm accelerated with the Nelder-Mead non-linear simplex. To optimize the cubic spline interpolated geometry, the flux (Neumann) boundary conditions are minimized using a least squares functional. The automated algorithm successfully reconstructs single and multiple subsurface cavities within two dimensional mediums. The solver is also shown to accurately predict cavity geometries with random noise in the boundary condition measurements. Subsurface cavities can be difficult to detect based on their location. By applying different boundary conditions to the same geometry, more information is supplied at the boundary, and the subsurface cavity is easily detected despite its low heat signature effect at the boundaries. Extensions to three-dimensional applications are outlined

[pt] O Método Híbrido dos Elementos de Contorno foi formulado em 1987. Desde então, este método tem sido aplicado com sucesso a diversos tipos de problemas de elasticidade e potencial, inclusive problemas dependentes do tempo. Porém, alguns aspectos importantes do método permaneceram abertos a investigação. Esta dissertação apresenta três contribuições, com desenvolvimentos feitos para problemas de elasticidade, mas prontamente extensíveis a problemas de potencial. Numa primeira etapa, desenvolve-se uma expressão para os resultados de deslocamentos no domínio, levando-se em conta corretamente a parcela de deslocamentos de corpo rígido. A partir deste primeiro desenvolvimento, é proposta uma formulação simplificada do método, na qual uma matriz de flexibilidade é obtida diretamente, num procedimento que dispensa qualquer tipo de integração. Esta nova formulação, como mostrado nos exemplos numéricos, é extremamente precisa e de simples implementação computacional. No entanto, por não ter uma base variacional, esta formulação conduz a uma matriz de rigidez não-simétrica. Na terceira contribuição, o Método Híbrido dos Elementos de Contorno e o Método Híbrido Simplificado dos Elementos de Contorno são aplicados a problemas gerais de meio infinito, para qualquer tipo de condições de contorno. Para isto é mostrado que as propriedades espectrais de ambos os métodos estão interrelacionadas. Apresenta-se um grande número de resultados numéricos de problemas bidimensionais, para validação dos desenvolvimentos teóricos realizados. / [en] The hybrid boundary element method was introduced in 1987. Since then, the method has been applied successfully to different problems of elasticity and potential, including time-dependent problems. However, some important aspects of the method have remained open to investigation. This dissertation consists in a threefold contribution, with developments outlined for elasticity, but readily extensible to potential problems. The first step is aimed at improving the expression of displacement results in the domain by taking correctly into account the amount of rigid body movements. Based on the assessment of displacements, a simplified formulation of the method is proposed, in which a flexibility-like matrix is directly obtained, in a procedure that requires no integration at all. This novel formulation, as shown in the numerical examples, is extremely accurate and rather inexpensive. Since it lacks a variational basis, however, the method leads to a non-symmetric stiffness matrix. In a third step, both hybrid and simplified boundary element methods are extended to general problems in an infinite domain, for any type of boundary conditions. It is shown that the matrices of both methods are spectrally interrelated. A large number of numerical results of two-dimensional problems validate the theoretical achievements. / [es] El Método Híbrido de los Elementos de Contorno fue formulado en 1987. Desde entonces, este método ha sido aplicado con éxito a diversos tipos de problemas de elasticidad y potencial, incluso en problemas dependientes del tiempo. No obstante, algunos aspectos importantes del método han permanecido abiertos a la investigación. Esta disertación presenta tres contribuciones, desarrolladas para problemas de elasticidad, pero perfectamente extendibles a problemas de potencial. En una primeira etapa, se desarrolla una expresión para los resultados de deslocamientos en el dominio, teniendo en cuenta la parcela correcta de deslocamientos del cuerpo rígido. A partir de este primer desarrollo, se propone una formulación simplificada del método, en el cual, se obtiene una matriz de flexibilidad diretamente, a través de un procedimiento que dispensa cualquier tipo de integración. Esta nueva formulación, como muestran los ejemplos numéricos, es extremamente precisa y de simple implementación computacional. Sin embargo, por no tener una base variacional, esta formulación conduce a una matriz de rígidez no simétrica. En la tercera contribución, se aplican el Método Híbrido de los Elementos de Contorno y el Método Híbrido Simplificado de los Elementos de Contorno a problemas gerales de medio infinito, para cualquier tipo de condiciones de contorno. Para ello, se demuestra que las propriedads espectrales de ambos métodos están interrelacionadas. Con el objetivo de evaluar los desarrollos teóricos aquí abordados se presentan un gran número de resultados numéricos de problemas bidimensionales.

[pt] Este trabalho apresenta um estudo do cálculo de sensibilidades necessário para a análise de problemas inversos e de otimização, usando o método híbrido dos elementos de contorno. Com esta finalidade, é desenvolvida uma formulação que permite obter as sensibilidades à mudança de forma, por diferenciação implícita das integrais de contorno, de uma estrutura já discretizada. Demonstra-se que as sensibilidades das matrizes obtidas desta formulação apresentam propriedades espectrais definidas, que são derivadas da formulação básica do método híbrido dos elementos de contorno. Todo o desenvolvimento é feito para um problema da elastostática tridimensional, embora sejam apresentadas apenas aplicações de problemas bidimensionais e de potencial, como casos particulares. As singularidades que surgem na integração no cálculo das sensibilidades são facilmente solucionáveis a partir das integrais da formulação básica do método híbrido dos elementos de contorno. As implementações numéricas são feitas utilizando a linguagem de programação Maple V release 3. Para ambos os casos, de potencial e elasticidade bidimensional, são usados elementos lineares para a representação do contorno. São apresentadas comparações entre os resultados analíticos obtidos através desta formulação com os resultados obtidos usando a técnica de diferenças finitas (centradas), com o objetivo de demonstrar a eficiência e precisão da metodologia aqui desenvolvida. / [en] The present work describes a formulation for computing design sensitivities required in inverse problems and shape optimization of solid objects, in the frame of the hybrid boundary element method. The so-called direct differentiation method is applied in order to calculate the gradients, i.e. the implicit diferentiation of the discretized boundary is performed, resulting in a general and efficient analysis technique for shape design sensitivity analysis of all structural quantities. It is demonstrated that the resulting sensitivities matrices present some useful spectral properties, which are related to the matrix spectral properties of the basic hybrid formulation. This formulation is valid for tridimensional solids, although only potential and bidimensional applications are considered as particular cases. The singularities that appear in the resulting boundary integrals are exactly the same which have already been dealt with in the basic formulation. The analytical and numerical procedures were performed by using the mathematical package Maple V release 3. Linear boundary elements were used for both potential and elasticity problems. Numerical results obtained by the present procedure are compared to finite differences results to demonstrate the effectiveness of the present formulation. / [es] Este trabajo presenta un estudio del cálculo de sensibilidades, que tiene gran importancia en el análisis de problemas inversos y de optimización, usando el método híbrido de los elementos de contorno. Con esta finalidad, se desarrolla una formulación que permite obtener las sensibilidades al cambio de forma de una extructura ya discretizada, por diferenciación implícita de las integrales de contorno. Se demuestra que las sensibilidades de las matrices obtenidas por esta formulación presentan propriedades espectrales definidas, que son derivadas de la formulación básica del método híbrido de los elementos de contorno. El desarrollo de la formulación se realiza para un problema de elastostática tridimensional, aunque se presentan apenas las aplicaciones de problemas bidimensionales y de potencial, como casos particulares. Las singularidades que surgen en la integración en el cálculo de las sensibilidades pueden ser fácilmente resueltas a partir de las integrales de la formulación básica del método híbrido de los elementos de contorno. La implementación numérica utiliza el lenguaje de programación Maple V release 3. Para los casos de potencial y elasticidad bidimensional, se utilizan elementos lineales para la representación del contorno. Se comparan los resultados analíticos obtenidos a través de esta formulación con los resultados obtenidos usando la técnica de diferencias finitas (centradas), con el objetivo de demostrar la eficiencia y precisión de la metodología aqui desarrollada.

No description available.

Ultrasound directed self-assembly (DSA) utilizes the acoustic radiation force associated with a standing ultrasound wave field to organize particles dispersed in a fluid medium into specific patterns. The ability to tailor the organization and packing density of spherical particles using ultrasound DSA in a viscous fluid medium is crucial in the context of (additive) manufacturing of engineered materials with tailored properties. However, the fundamental physics of the ultrasound DSA process in a viscous fluid medium, and the relationship between the ultrasound DSA process parameters and the specific patterns of particles that result from it, are not well-understood. Researchers have theoretically described the acoustic radiation force and the acoustic interaction force that act on spherical particles in a standing ultrasound wave field in both inviscid and viscous media. In addition, they have solved the forward and inverse ultrasound DSA problem in an inviscid medium, in which they relate the patterns of particles and the ultrasound DSA operating parameters. However, no theoretical model exists that allows simulating the steady-state and transient local particle packing density in a viscous medium during ultrasound DSA. Thus, in this dissertation, we (i) theoretically derive and experimentally validate a model to determine the steady-state locations where spherical particles assemble during ultrasound DSA as a function of medium viscosity and particle volume fraction. (ii) We also theoretically derive and experimentally validate a model to quantify the steady-state and transient local packing density of spherical particles within the pattern features that result from ultrasound DSA. Using these models, we quantify and predict the locations where spherical particles assemble during ultrasound DSA in a viscous medium, considering the effects of medium viscosity and particle volume fraction. We demonstrate that the deviation between locations where particles assemble in viscous and inviscid media first increases and then decreases with increasing particle volume fraction and medium viscosity, which we explain by means of the sound propagation velocity of the mixture. In addition, we quantify and predict the steady-state and transient local packing density of spherical particles within the pattern features, using ultrasound DSA in combination with vat photopolymerization (VP). We show that the steady-state local particle packing density increases with increasing particle volume fraction and increases with decreasing particle size. We also show that the transient local particle packing density increases with increasing particle volume fraction, decreasing particle size, and decreasing fluid medium viscosity. Increasing particle size and decreasing fluid medium viscosity decreases the time to reach steady-state. Finally, we implement single and multiple scattering in the calculation of the acoustic radiation force for spherical particles in a viscous medium and quantify their relative contributions to the calculation of the acoustic radiation force as a function of ultrasound DSA operating parameters and material properties. We demonstrate that the deviation between considering single and multiple scattering may reach up to 100%, depending on the ultrasound DSA process parameters and material properties. Also, increasing the particle volume fraction increases the need to account for multiple scattering. Quantifying and predicting the local packing density of spherical particles during ultrasound DSA in a viscous medium, as a function of ultrasound DSA process parameters is crucial towards using ultrasound DSA in engineering applications, in particular (additive) manufacturing of engineered polymer matrix composite materials with tailored properties whose properties depend on the spatial organization and packing density of particles in the matrix material. / Doctor of Philosophy / Ultrasound directed self-assembly (DSA) is a technique that uses ultrasound waves to arrange small particles submerged in a fluid into specific patterns. When combined with other manufacturing techniques, ultrasound DSA can be used to fabricate composite materials that derive their properties from the spatial organization of particles in a matrix material. However, ultrasound DSA in viscous fluids is not well-understood. Researchers have studied the forces associated with ultrasound waves that move small spherical particles in an inviscid fluid medium (fluids that experience little to no internal resistance to flow), and they have demonstrated intricate control of the patterns of particles that form using ultrasound DSA. However, that knowledge is not currently available for ultrasound DSA in viscous media. In this dissertation, we develop and evaluate theoretical models to understand ultrasound DSA of small spherical particles in a viscous fluid medium. We simulate where particles organize and how densely they pack together. We also determine the difference of the time-dependent motion of particles in a viscous fluid compared to that in an inviscid fluid medium and relate the difference to the number of particles submerged in the fluid and the viscosity of the fluid. Additionally, we examine the effect of particle size and fluid viscosity on the speed by which the particles reach their final location. We also study how ultrasound waves interact with multiple small particles in a viscous fluid, focusing on the forces that move these particles. We explore two models that account for single and multiple ultrasound wave scattering. Scattering is the process by which ultrasound waves deflect in different directions when they encounter a particle. The results show that the difference between single and multiple scattering models can be significant, depending on the ultrasound DSA process parameters and the properties of the fluid and particles. In general, the importance of accounting for multiple scattering increases with the number of particles submerged in the fluid. Understanding particle packing density when using ultrasound DSA in a viscous fluid is essential in many engineering applications, in particular manufacturing of composite materials that derive their properties from the spatial arrangement of particles in a matrix material.

A hybrid method is presented for the analysis of layers, plates, and multi-layered systems consisting of isotropic and linear elastic materials. The problem is formulated for the general case of a multi-layered system using a total potential energy formulation and employing the layerwise laminate theory of Reddy. A one-dimensional finite element model is used for the analysis of the multi-layered system through its thickness, and integral Fourier transforms are used to obtain the exact solution for the in-plane problem. Explicit expressions are obtained for the fundamental solution of the typical infinite layer, which are applied in the two-dimensional boundary integral equation model to produce the integral representation of the solution. The boundary integral equation model is two-dimensional, displacement-based and assumes piecewise continuous distribution of the displacement components through the system's thickness. The developed model describes the three-dimensional displacement field, the stress field, the strains and the interlaminar stresses over the entire domain of the problem as continuous functions of the position. This detailed three-dimensional analysis is achieved by incorporating only contour integrals. The boundary integral equations are discretized using the boundary element method and a numerical model is developed for the single numerical layer (element). This model is extended to the case of a multilayered system by introducing appropriate continuity conditions at the interfaces between the layers (firmly bonded layers, or separation, slip and friction between the layers). Assembly of the element matrices yields the global system of equations, which can be solved via iterative techniques. In addition, numerical techniques are developed for the evaluation of the boundary and domain integrals involved in the construction of the element matrices. The singular boundary integrals are computed using a special coordinate transformation, along with a subdivision of the boundary element and a transformation of the Gauss points. The domain integrals (regular, singular or near-singular) are transformed to regular definite integrals along the boundary through a semi-analytical approach. The proposed method provides a simple, efficient, and versatile model for a three-dimensional analysis of thick plates or multilayered systems. It can also be used to study plates resting on elastic foundations or plates with internal supports. The proposed method can be applied in an obvious manner to anisotropic materials and vibration problems. / Ph. D.

Ce travail de recherche se situe dans le cadre du développement de la méthode des éléments de frontière (BEM) pour les milieux poreux multiphasiques. À l'heure actuelle, l'application de la BEM aux pr oblèmes des milieux poreux non-saturés est encore limitée, car l'expression analytique exacte de la solution fondamentale n'a pas été obtenue, ni dans le domaine transformé ni dans le domaine réel. Ceci provient de la complexité du système d'équations régissant le comportement des milieux poreux non-saturés. Les développements de la BEM pour les sols non-saturés effectués au cours de cette thèse sont basés sur les modèles thermo-hydro-mécanique (THHM) et hydro-mécanique (HHM) présentés dans la première partie de ce mémoire. Ces modèles phénoménologiques basés sur la théorie de la poromécanique et les acquis expérimentaux sont obtenus dans le cadre du modèle mathématique présenté par Gatmiri (1997) et Gatmiri et al. (1998). Après avoir présenté les modèles THHM et HHM, on établit pour la première fois les équations intégrales de frontière et les solutions fondamentales associées pour un milieu poreux non-saturé sous chargement quasi-statique pour les deux cas isotherme (2D dans le domaine de Laplace) et non-isotherme (2D et 3D dans les domaines de Laplace et temporel). Aussi, les équations intégrales de frontière ainsi que les solutions fondamentales 2D et 3D (dans le domaine de Laplace) pour le modèle dynamique couplé des sols non-saturés sont obtenues. Ensuite, les formulations d'éléments de frontière (BEM) basées sur la méthode quadrature de convolution (MQC) concernant les milieux poreux saturé et non-saturé sous chargement quasi-statique isotherme et dynamique sont implémentées dans le code de calcul « HYBRID ». Ayant intégrées les formulations de BEM pour les problèmes de propagation d'ondes ainsi que pour les problèmes de consolidation dans les milieux poreux saturés et non-saturés, il semble que nous ayons fourni à l'heure actuelle le premier code de calcul aux éléments de frontière (BEM) qui modélise les différents problèmes dans les sols secs, saturés et non-saturés. Une fois le code vérifié et validé, des études paramétriques portant sur des effets de site sismiques sont effectuées. Le but recherché est d'aboutir à un critère simple, directement exploitable par les ingénieurs, combinant les caractéristiques géométriques et les caractéristiques du sol, permettant de prédire l'amplification du spectre de réponse en accélération dans des vallées sédimentaires aussi bien que vides / The purpose of this dissertation is to develop a boundary element method (BEM) for multiphase porous media. Nowadays, the application of the BEM for solving problems of unsaturated porous media is still limited, because no fundamental solution exists in the published literature, neither in the frequency nor time domain. This fact rises from the complexity of the coupled partial differential equations governing the behaviour of such media. The developments of the BEM for the unsaturated soils carried out during this thesis are based on the thermo-hydro-mechanical (THHM) and hydro-mechanical (HHM) models presented in the first part of this dissertation. These phenomenological models are presented based on the experimental observations and with respect to the poromechanics theory within the framework of the suction-based mathematical model presented by Gatmiri (1997) and Gatmiri et al. (1998). After having presented the THHM and HHM models, for the first time, one establishes the boundary integral equations (BIE) and the associated fundamental solutions for the unsaturated porous media subjected to quasi-static loading for both isothermal (2D in the Laplace transform domain) and non-isothermal (2D and 3D in Laplace transform and time domains) cases. Also, the boundary integral equations as well as the fundamental solutions (2D and 3D in the Laplace transform domain) are obtained for the fully coupled dynamic model of unsaturated soils.In the next step, the boundary element formulations (BEM) based on the convolution quadrature method (CQM) regarding the saturated and unsaturated porous media subjected to isothermal quasi-static and dynamic loadings are implemented via the computer code HYBRID. Having integrated the BEM formulations for the wave propagation, as well as the consolidation problems in the saturated and unsaturated porous media, it seems that now the first boundary element code is obtained that can model the various problems in dry, saturated and unsaturated soils. Once the code is verified and validated, parametric studies on seismic site effects are carried out. The aim is to achieve a simple criterion directly usable by engineers, combining the topographical and geological characteristics of the soil, to predict the amplification of acceleration response spectra in sedimentary as well as hollow valleys