The importance of the Stokes system stems from the fact that the Stokes system is the stationary linearised form of the Navier Stokes system [Te01, Chapter1]. This linearisation is allowed when neglecting the inertial terms at a low Reinolds numbers Re << 1. The Stokes system essentially models the behaviour of a non - turbulent viscous fluid. The mixed interior boundary value problem related to the compressible Stokes system is reduced to two different BDIES which are equivalent to the original boundary value problem. These boundary-domain integral equation systems (BDIES) can be expressed in terms of surface and volume parametrix-based potential type operators whose properties are also analysed in appropriate Sobolev spaces. The invertibility and Fredholm properties related to the matrix operators that de ne the BDIES are also presented. Furthermore, we also consider the mixed compressible Stokes system with variable viscosity in unbounded domains. An analysis of the similarities and differences with regards to the bounded domain case is presented. Furthermore, we outline the mapping properties of the surface and volume parametrix-based potentials in weighted Sobolev spaces. Equivalence and invertibility results still hold under certain decay conditions on the variable coeffi cient The last part of the thesis refers to the mixed boundary value problem for the stationary heat transfer partial di erential equation with variable coe cient. This BVP is reduced to a system of direct segregated parametrix-based Boundary-Domain Integral Equations (BDIEs). We use a parametrix different from the one employed by Chkadua, Mikhailov and Natroshvili in the paper [CMN09]. Mapping properties of the potential type integral operators appearing in these equations are presented in appropriate Sobolev spaces. We prove the equivalence between the original BVP and the corresponding BDIE system. The invertibility and Fredholm properties of the boundary-domain integral operators are also analysed in both bounded and unbounded domains.

This study is to develop a 2D fully nonlinear numerical wave tank used to investigate the wave-induced dynamic properties of a dual pontoon floating structure (DPFS) with a liquid container on the top. The nonlinear numerical wave tank, developed based on the velocity potential function and the boundary element method (BEM), is to simulate dynamic properties including sway, heave, roll, and tension response. In addition, a physical model of the dual floating pontoon is tested in a hydrodynamic wave tank to validate the numerical model for simulation of wave and structure interaction. In the numerical model, a boundary integral equation method (BIEM) with linear element scheme is applied to establish a 2D fully nonlinear numerical wave tank (NWT). The nonlinear free surface condition is treated by combining the Mixed Eulerian and Lagrangian method (MEL), the fourth-order Runge-Kutta method (RK4) and a cubic spline scheme. The second-order Stokes wave theory is used to generate the velocity flux on the input boundary. Numerical damping zones are deployed at both ends of the NWT to dissipate or absorb the transmitted and reflected waves. Acceleration potential method and modal decomposition method are adopted to solve the unsteady potential functions £X1,t and £X2,t, while the system of motion equation is established according to Newton's 2nd law. Finally, the RK4 is applied to predict the motion of the platform, and the variation of free surface. As for the hydrodynamic laboratory model test, an image process scheme is applied to trace the floating structure motion and the variation of water surface inside the sloshing tank, while the mooring tension is measured by a load cell and stored in a data logger. The comparisons of numerical simulations and experimental data indicate that the numerical predictions are larger than measurements especially near the resonance frequency. This discrepancy is probably due to the fluid viscous effect. To overcome this problem and maintain the calculation efficiency, an uncoupled damping coefficient obtained through a damping ratio (£a=C/Ccr=0.02) is incorporated into the vibration system. Results reveal that responses of body motion near the resonant frequencies of each mode have significantly reduced and close to the measurements. Therefore, the suitable value of the damping ratio for the floating platform is £a=0.02. Then the numerical model with a damping ratio is applied to investigate the dynamic properties of the floating platform for different arrangements, including different mooring angle, spring constant, spacing, and the liquid container. Results demonstrate that the resonant frequency of each mode, responses of body motion and mooring tensions change along with the settings. As a whole, the platform with smaller mooring angle, longer spacing between the pontoons, higher water depth and wider width of the liquid container has relatively stable body motions and less mooring tension. Finally, the comparisons of the effects of random and regular waves on the floating structure illustrate that the variation of water surface in the liquid container is much severe in random waves than in regular waves such that the interaction between liquid and floating structure is more chaotic and thus reduces the amplitude of each response mode. As a result, the mooring tensions for random waves become much gentler than the regular waves. Key words: Boundary integral equation method; fully nonlinear numerical wave tank; dual pontoon floating structure

This paper is to investigate the wave-induced dynamic properties of the floating platform for cage aquaculture. Considering the calculation efficiency and its applicability, this problem is simplified by: (1) assuming the flow field is inviscid, incompressible and irrotational; (2) the form drag and inertia drag on the fish net is calculated by the modified Morison equation (or Morison type equation of relative motion), including the material and geometric properties; (3) the moorings is treated as a symmetric linear spring system and the influence of hydrodynamic forces on the mooring lines is neglected; and (4) the net-volume is assumed as un-deformable to avoid the inversely prolonging computing time because the mass of fish net with is too light comparing with the mass of floating platform and cause the marching time step tremendously small to reach the steady-state condition which may lead to larger numerical errors (e.g. truncation errors) in computation. The BIEM with linear element scheme is applied to establish a 2D fully nonlinear numerical wave tank (NWT). The nonlinear free surface condition is treated by combining the Mixed Eulerian and Lagrangian method (MEL), the fourth-order Runge-Kutta method (RK4) and the cubic spline scheme. The second-order Stokes wave theory is adopted to give the velocity on the input boundary. Numerical damping zones are deployed at both ends of the NWT to dissipate or absorb the transmitted and reflected wave energy. The velocity and acceleration fields should be solved simultaneously in order to obtain the wave-induced dynamic property of the floating platform. Thus, both the acceleration potential method and modal decomposition method are adopted to solve the wave forces on the floating body, while the wave forces on the fish net are calculated by the modified Morison equation. According to Newton¡¦s second law, the total forces on the gravity center of the floating platform form the equation of motion. Finally, the RK4 is applied to predict the displacement and velocity of the platform. Firstly, the NWT is validated by comparing the wave elevation, internal velocity and acceleration with those from the second-order Stokes wave theory. Moreover, the numerical damping zone is suitable for long time simulation with a wide range of wave depth. The simulated results on wave-body interactions of fixed or freely floating body also indicate good agreement with those of other published results. Secondly, in the case of the interaction of waves and the floating platform, the simulated results show well agreement with experimental data, except at the vicinity of resonant frequency of roll and heave motions. This discrepancy is due to the fluid viscous effect. To overcome this problem and maintain the calculation efficiency, an uncoupled damping coefficient obtained by a damping ratio (£i=0.1 ) is incorporated into the vibration system. Results reveal that responses of body motion near the resonant frequencies of each mode have significant reduction and close to the experimental data. Moreover, the results are also consistent well with experiments in different wave height, mooring angle, water depth either with or without fish net. Therefore, the suitable value of the damping ratio for the floating platform is £i=0.1. Finally, the present model is applied to investigate the dynamic properties of the floating platform under different draft, width, spacing, spring constant, mooring angle and depth of fish net. Results reveal that the resonant frequency and response of body motion, mooring force, reflection and transmission coefficients and wave energy will be changed. According to the resonant response, the platform with shallower draft, larger width, longer spacing between two pontoons, smaller spring constants, or deeper depth of fish net has more stable body motions and smaller mooring forces. Irregular wave cases are presented to illustrate the relationship with the regular wave cases. Results indicate that the dynamic responses of body motion and the reflection coefficient in irregular waves have similar trend with regular waves. However, in the irregular wave cases, the resonant frequency is moved to the higher frequency. Similarly, resonant response function is smaller but wider, which is due to the energy distribution in the wave spectrum.

Abstract The dissertation considers the time fractional diffusion equation (TFDE) with the Dirichlet boundary condition in the sub-diffusion case, i.e. the order of the time derivative is α ∈ (0,1). In the thesis we have studied the solvability of TFDE by the method of layer potentials. We have shown that both the single layer potential and the double layer potential approaches lead to integral equations which are uniquely solvable. The dissertation consists of four articles and a summary section. The first article presents the solution for the time fractional diffusion equation in terms of the single layer potential. In the second and third article we have studied the boundary behaviour of the layer potentials for TFDE. The fourth paper considers the spline collocation method to solve the boundary integral equation related to TFDE. In the summary part we have proved that TFDE has a unique solution and the solution is given by the double layer potential when the lateral boundary of a bounded domain admits C1 regularity. Also, we have proved that the solution depends continuously on the datum in the sense that a nontangential maximal function of the solution is norm bounded from above by the datum in L2(ΣT). If the datum belongs to the space H1,α/2(ΣT), we have proved that the nontangential function of the gradient of the solution is norm bounded from above by the datum in H1,α/2(ΣT).

Cette thèse porte sur la diffraction d’une onde plane électromagnétique par une surface lisse parfaitement conductrice (PEC). Elle présente l’analyse des propriétés d’une nouvelle formulation des trois principales équations intégrales de frontières de la théorie de la diffraction électromagnétique (EFIE, MFIE et CFIE). L’idée est d’adapter les équations intégrales conventionnelles à la diffraction d’une onde plane en supposant que la fonction de phase de l’onde plane incidente détermine la fonction de phase de la distribution de courant induit sur la surface.L’idée d’utiliser la phase dans la diffraction d’ondes planes a déjà été étudiée pour les hautes fréquences, notamment dans les thèses de Zhou (1995) et Darrigrand (2002) qui adaptèrent les espaces d’approximation des éléments finis. Dans cette thèse, cependant, nous suivons une formulation plus récente, donnée par Herberthson (2008), où la fonction de phase est incorporée dans la distribution du noyau des opérateurs intégraux.En présentant les versions modifiées de l’EFIE et de la MFIE (dénommées HEFIE et HMFIE)dans des espaces fonctionnels appropriés, nous prouvons ici l’existence d’une solution unique à cette formulation spécifique et présentons une mise en oeuvre pratique originale qui tire parti de l’expérience acquise sur l’EFIE/MFIE. Par la suite, nous explorons une propriété importante offerte par ces nouvelles formulations: la possibilité de réduire le nombre de degrés de liberté requis pour obtenir une solution précise du problème. / This thesis is about the scattering of an electromagnetic plane wave incidenton a perfectly conducting smooth surface. It presents the analysis of the properties of a newformulation of the three principal boundary integral equations of electromagnetic scattering theory(EFIE, MFIE and CFIE). The basic idea is to adapt the conventional integral equations toplane-wave scattering by supposing that the phase function of an incident plane wave determinesthe phase function of the induced boundary current distribution.This idea of using the phase in plane wave scattering has previously been studied in highfrequencyscattering, in particular in the theses by Zhou (1995) and Darrigrand (2002) whoadapt the finite element approximation spaces. In this thesis, though, we follow a more recentformulation, given by Herberthson (2008), where the phase function is incorporated in the kerneldistribution of the integral operators.Presenting the modified version of the EFIE and the MFIE (denoted HEFIE and HMFIE) inappropriate function spaces, we prove the existence of a unique solution to this specific formulationand developp an original practical implementation which takes advantage of the gainedexperience on the EFIE/MFIE. Then, we explore another important property provided by thenew formulations: the possibility to reduce the number of degrees of freedom required to get anaccurate solution of the problem.

Doctor of Philosophy / Department of Mechanical and Nuclear Engineering / Prakash Krishnaswami / Xiao J. Xin / Over the past two decades meshless methods have attracted much attention owing to their advantages in adaptivity, higher degree of solution field continuity, and capability to handle moving boundary and changing geometry. In this work, a meshless integral method based on the regularized boundary integral equation has been developed and applied to two-dimensional linear elasticity and elastoplasticity with small or large deformation. The development of the meshless integral method and its application to two-dimensional linear elasticity is described first. The governing integral equation is obtained from the weak form of elasticity over a local sub-domain, and the moving least-squares approximation is employed for meshless function approximation. This formulation incorporates: a subtraction method for singularity removal in the boundary integral equation, a special numerical integration for the calculation of integrals with weak singularity which further improves accuracy, a collocation method for the imposition of essential boundary conditions, and a method for incorporation of natural boundary conditions in the system governing equation. Next, elastoplastic material behavior with small deformation is introduced into the meshless integral method. The constitutive law is rate-independent flow theory based on von Mises yielding criterion with isotropic hardening. The method is then extended to large deformation plasticity based on Green-Naghdi’s theory using updated Lagrangian description. The Green-Lagrange strain is decomposed into the elastic and plastic part, and the elastoplastic constitutive law is employed that relates the Green-Lagrange strain to the second Piola-Kirchhoff stress. Finally, a pre- and post-processor for the meshless method using node- and pixel-based approach is presented. Numerical results from the meshless integral method agree well with available analytical solutions or finite element results, and the comparisons demonstrate that the meshless integral method is accurate and robust. This research lays the foundation for modeling and simulation of metal cutting processes.

In this paper, the complete double layer boundary integral equation formulation for Stokes flows is extended to viscoelastic fluids to solve the mobility problem for a system of particles, where the non-linearity is handled by particular solutions of the Stokes inhomogeneous equation. Some techniques of the meshless method are employed and a point-wise solver is used to solve the viscoelastic constitutive equation. Hence volume meshing is avoided. The method is tested against the numerical solution for a sphere settling in the Odroyd-B fluid and some results on a prolate motion in shear flow of the Oldroyd-B fluid are reported and compared with some theoretical and experimental results. / Singapore-MIT Alliance (SMA)

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

Acoustic and electromagnetic scattering from arbitrarily shaped structures can be numerically characterized by solving various surface integral equations (SIEs). One of the most effective techniques to solve SIEs is the Nyström method. Compared to other existing methods,the Nyström method is easier to implement especially when the geometrical discretization is non-conforming and higher-order representations of the geometry and unknowns are desired. However,singularities of the Green’s function are more difficult to”manage”since they are not ”smoothened” through the use of a testing function. This dissertation describes purely numerical schemes to account for different orders of singularities that appear in acoustic and electromagnetic SIEs when they are solved by a high-order Nyström method utilizing a mesh of curved discretization elements. These schemes make use of two sets of basis functions to smoothen singular integrals: the grid robust high-order Lagrange and the high-order Silvester-Lagrange interpolation basis functions. Numerical results comparing the convergence of two schemes are presented. Moreover, an extremely scalable implementation of fast multipole method (FMM) is developed to efficiently (and iteratively) solve the linear system resulting from the discretization of the acoustic SIEs by the Nyström method. The implementation results in O(N log N) complexity for high-frequency scattering problems. This FMM-accelerated solver can handle N =2 billion on a 200,000-core Cray XC40 with 85% strong scaling efficiency. Iterative solvers are often ineffective for ill-conditioned problems. Thus, a fast direct (LU)solver,which makes use of low-rank matrix approximations,is also developed. This solver relies on tile low rank (TLR) data compression format, as implemented in the hierarchical computations on many corearchitectures (HiCMA) library. This requires to taskify the underlying SIE kernels to expose fine-grained computations. The resulting asynchronous execution permit to weaken the artifactual synchronization points,while mitigating the overhead of data motion. We compare the obtained performance results of our TLRLU factorization against the state-of-the-art dense factorizations on shared memory systems. We achieve up to a fourfold performance speedup on a 3D acoustic problem with up to 150 K unknowns in double complex precision arithmetics.

Le Contrôle Non Destructif (CND) permet de sonder l’intérieur d’un milieu dans le but desurveiller son intégrité et son vieillissement. Assisté d’outils de simulation il permet de détecter, caractériseret localiser des défauts de structure du milieu inspecté mais sa fiabilité dépend de la précision des méthodesde simulation. Dans le cadre du CND ultrasonore, les méthodes usuelles (numériques et asymptotiques) sontbien souvent inadéquates pour simuler la diffraction par les défauts. On leur préfère des techniques hybrides.On propose dans cette thèse une nouvelle approche hybride pour la simulation numérique de la diffractionhaute fréquence en milieu étendu (configuration critique pour le CND). Combinant la méthode des équationsintégrales et la méthode des rayons, cette approche exploite le caractère multi-échelle du problème hautefréquence en proposant un modèle d’obstacle à deux échelles. Elle permet le calcul précis de la diffraction etla propagation rayon des champs. D’abord mise au point dans le cadre de la diffraction d’ondes acoustiquespar un obstacle de taille inférieure à la longueur d’onde (méthode barycentrique), l’approche est ensuiteétendue à des configurations de diffraction par des obstacles de l’ordre de la longueur d’onde grâce àl’introduction d’un partitionnement de l’unité de sa surface (méthode multi-centres). Pour accélérerl’approche hybride, on propose une procédure de résolution Online-Offline, basée sur un pré-calcul de lamatrice de diffraction associée à un ensemble réduit de directions d’incidence et d’observation et sur uneinterpolation polynomiale de ses vecteurs singuliers pour son évaluation dans des directions quelconquesd’émetteurs et de récepteurs. On étudie ensuite la stratégie dans le cadre de l’acoustique 3D puis on en faitune extension de principe à l’élastodynamique. On donne enfin un ensemble de pistes pour étendre l’approchehybride dans des cas de diffraction par un ou plusieurs obstacles pouvant être proches des bords du milieu. / Non Destructive Testing (NDT) aims at probing a medium to check its integrity and aging. Withthe help of simulation tools, it allows to detect, caracterize and locate flaws inside a material with a precisiondepending on that of the simulation methods. Usual numerical and asymptotic methods nevertheless often failat precisely computing diffraction for ultrasonic NDT. Hybrid approaches are thus prefered in this framework.In this thesis, we propose a new hybrid strategy combining the boundary integral equation method and raytracing to compute high frequency diffraction of an obstacle in a large medium (critical NDT configuration).This strategy allows to compute precisely the diffraction effects and to convert and propagate the diffractedfield as rays. The proposed strategy relies on a two-scale model of the diffracting obstacle. First developpedto simulate acoustic waves diffraction on an obstacle of size less than the wave length (barycentric method),the hybrid strategy is then extended to compute the diffraction by an obstacle of size some wave lengths(polycentric method) thanks to the introduction of a partition of unity of the obstacle surface. Besides, inorder to accelerate the hybrid approach, we propose an Online-Offline resolution procedure based on theOffline computation of the scattering matrix for a reduced set of incidence and observation directions and onthe use of a polynomial interpolation of its singular vectors for the Online evaluation of the scattering matrixfor any incidence and observation directions. We then study the possibility of extension of the hybrid strategyto 3D acoutics and elastodynamics. We finally give several perspectives for the adaptation of the approach todeal with diffraction by one or several obstacles potentially close to the propagating medium boundaries.