Wave phenomena appear in many fields of science such as acoustics, geophysics, and quantum mechanics. They can often be described by partial differential equations (PDEs). As PDEs typically are too difficult to solve by hand, the only option is to compute approximate solutions by implementing numerical methods on computers. Ideally, the numerical methods should produce accurate solutions at low computational cost. For wave propagation problems, high-order finite difference methods are known to be computationally cheap, but historically it has been difficult to construct stable methods. Thus, they have not been guaranteed to produce reasonable results. In this thesis we consider finite difference methods on summation-by-parts (SBP) form. To impose boundary and interface conditions we use the simultaneous approximation term (SAT) method. The SBP-SAT technique is designed such that the numerical solution mimics the energy estimates satisfied by the true solution. Hence, SBP-SAT schemes are energy-stable by construction and guaranteed to converge to the true solution of well-posed linear PDE. The SBP-SAT framework provides a means to derive high-order methods without jeopardizing stability. Thus, they overcome most of the drawbacks historically associated with finite difference methods. This thesis consists of three parts. The first part is devoted to improving existing SBP-SAT methods. In Papers I and II, we derive schemes with improved accuracy compared to standard schemes. In Paper III, we present an embedded boundary method that makes it easier to cope with complex geometries. The second part of the thesis shows how to apply the SBP-SAT method to wave propagation problems in acoustics (Paper IV) and quantum mechanics (Papers V and VI). The third part of the thesis, consisting of Paper VII, presents an efficient, fully explicit time-integration scheme well suited for locally refined meshes.

Flexible beam structures are popular in civil and mechanical engineering. Many of these structures undergo large displacements and finite rotations, but with small deformations. Their dynamic behaviors are usually investigated using finite beam elements. A well known method to derive such beam elements is the corotational approach. This method has been extensively used in nonlinear static analysis. However, its application in nonlinear dynamics is rather limited. The purpose of this thesis is to investigate the nonlinear dynamic behavior of flexible beam structures using the corotational method. For the 2D case, a new dynamic corotational beam formulation is presented. The idea is to adopt the same corotational kinetic description in static and dynamic parts. The main novelty is to use cubic interpolations to derive both inertia terms and internal terms in order to capture correctly all inertia effects. This new formulation is compared with two classic formulations using constant Timoshenko and constant lumped mass matrices. This work is presented in the first appended journal paper. For the 3D case, update procedures of finite rotations, which are central issues in development of nonlinear beam elements in dynamic analysis, are discussed. Three classic and one new formulations of beam elements based on the three different parameterizations of the finite rotations are presented. In these formulations, the corotational method is used to develop expressions of the internal forces and the tangent stiffness matrices, while the dynamic terms are formulated into a total Lagrangian context. Many aspects of the four formulations are investigated. First, theoretical derivations as well as practical implementations are given in details. The similarities and differences between the formulations are pointed out. Second, numerical accuracy and computational efficiency of these four formulations are compared. Regarding efficiency, the choice of the predictor at each time step and the possibility to simplify the tangent inertia matrix are carefully investigated. This work is presented in the second appended journal paper. To make this thesis self-contained, two chapters concerning the parametrization of the finite rotations and the derivation of the 3D corotational beam element in statics are added. / QC 20120521

A two-dimensional numerical model (RiverFLO-2D) has been enhanced to simulate flooding of urban areas by developing an innovative wet and dry surface algorithm, accounting for variable rainfall, and recoding the model computer program for parallel computing. The model formulation is based on the shallow water equations solved with an explicit time-stepping element-by-element finite element method. The dry-wet surface algorithm is based on a local approximation of the continuity and momentum equations for elements that are completely dry. This algorithm achieves global volume conservation in the finite element, even for flows over complex topographic surfaces. A new module was implemented to account for variable rainfall in space and time using NEXRAD precipitation estimates. The resulting computer code was parallelized using OpenMP Application Program Interface, which allows the model to run up to 5 times faster on multiple core computers. The model was verified with analytical solutions and validated with laboratory and field data. Model application to the Malpasset dam break and Sumacarcel flooding event show that the model accurately predicts flood wave travel times and water depths for these numerically demanding real cases. To illustrate the predictive capability of the enhanced model, an application was made of the city of Sweetwater flooding in Miami-Dade County, FL caused by the Hurricane Irene. The simulation starts with dry bed and rainfall is provided by NEXRAD estimates. Integrating NEXRAD rainfall estimates, developing a novel dry-wet area algorithm and parallelizing RiverFLO-2D code, this dissertation presents a proof of concept to accurately and efficiently predict floods in urban areas, identifying future improvements along this line of research.

The purpose of this thesis is to propose several corotational beam formulations for both 2D and 3D nonlinear dynamic analyse of flexible structures. The main novelty of these formulations is that the cubic interpolation functions are used to derive not only the internal force vector and the tangent stiffness matrix but also the inertial force vector and the dynamic matrix. By neglecting the quadratic terms of the local transversal displacements, closed-form expressions for the inertial terms are obtained for 2D problems. Based on an extensive comparative study of the parameterizations of the finite rotations and the time stepping method, and by adopting an approximation of the local rotations, two consistent and effective beam formulations for 3D dynamics are developed. In contrast with the first formulation, the second one takes into account the warping deformations and the shear center eccentricity. The accuracy of these formulations is demonstrated through several numerical examples.

Submitted by Renata Lopes (renatasil82@gmail.com) on 2017-02-24T17:38:27Z No. of bitstreams: 1 jonathanestebanarroyosilva.pdf: 3851364 bytes, checksum: 7341b01ce42c37de611bb2df24f9012c (MD5) / Approved for entry into archive by Adriana Oliveira (adriana.oliveira@ufjf.edu.br) on 2017-03-06T19:29:14Z (GMT) No. of bitstreams: 1 jonathanestebanarroyosilva.pdf: 3851364 bytes, checksum: 7341b01ce42c37de611bb2df24f9012c (MD5) / Made available in DSpace on 2017-03-06T19:29:14Z (GMT). No. of bitstreams: 1 jonathanestebanarroyosilva.pdf: 3851364 bytes, checksum: 7341b01ce42c37de611bb2df24f9012c (MD5) Previous issue date: 2014-02-24 / FAPEMIG - Fundação de Amparo à Pesquisa do Estado de Minas Gerais / Este trabalho apresenta um novo esquema de marcha no tempo capaz de reduzir oscilações espúrias através de amortecimento numérico para problemas de propagação de ondas elásticas no âmbito da Aproximação Explícita de Green (\Explicit Green's Approach" (ExGA)) [1]. A expressão integral referente ao ExGA é escrita em termos das funções de Green e Degrau. Seus cálculos são realizados de forma independente por meio da formulação semi-discreta do MEF e o método Diferença Central. Devido ao princípio da causalidade, as funções de Green e Degrau possuem um suporte compacto ao redor dos pontos fonte para um intervalo de tempo suficientemente pequeno que é usualmente Empregado nos métodos explícitos clássicos de integração temporal aplicados à modelagem de propagação de ondas. Neste sentido, as funções de Green e Degrau em t = Δt podem ser eficientemente calculadas localmente através de subdomínios pequenos. Cada subdomínio local com sua respectiva submalha cobre somente pontos nodais onde os valores das funções de Green e Degrau são não nulos. A precisão e eficiência da metodologia proposta é demostrada ao analisar três exemplos numéricos. / This work presents a new time-marching scheme able to reduce spurious oscillations by means of numerical damping for elastic wave propagation problems in the framework of the Explicit Green's Approach (ExGA) [1]. The integral expression concerned with the ExGA is written in terms of the Green's and the Step response functions. Their computations are carried out independently by means of the semidiscrete FEM and the Central difference method. Due to the principle of causality, the Green's and Step response functions admit a compact support surround the source points for a small enough time step that is usually employed in common explicit time integration methods applied to wave propagation modeling. In this sense, the Green's and Step response functions at t = Δt can be e ciently computed locally through small subdomains. Each local subdomain with its respective submesh covers only nodes whose Green's and Step response function values do not vanish. The accuracy and e ciency of the proposed methodology are demonstrated by analyzing three numerical examples.

<div>Positive displacement machines are used in a wide variety of applications, ranging from fluid power where they act as a transmission of power, to lubrication and fluid transport. As the core of the fluid system responsible for mechanical--hydraulic energy conversion, the efficiencies of these units are a major driver of the total efficiency of the system. Furthermore, the durability of these units is a strong decider in the useful life of the system in which they operate.</div><div><br></div><div>The key challenge in designing these units comes from understanding their working principles and designing their lubricating interfaces, which must simultaneously perform a load carrying and sealing function as the unit operates. While most of the physical phenomena relevant to these machines have been studied previously in some capacity, the significance of their mutual interactions has not. For this reason, the importance of these mutual interactions is a fundamental question in these machines that this thesis answers for the first time. In analysis of two different machine types, it is confirmed that mutual interactions of both physical phenomena and neighboring fluid domains of the unit contribute significantly to the overall performance of the machine. Namely, these analyses demonstrate load sharing owing to mutual interactions on average of 20% and as high as 50%, and mutual flow interactions of at least 10%.</div><div><br></div><div>In this thesis, the behavior of the thin films of fluid in the lubricating interfaces of the units, the bodies that make up these films, and the volumes which interface with them will be considered. The resulting coupled problem requires a model that can consider the effects of motion of all floating bodies on all films and volumes, and collect the resulting loads applied by the fluid as it responds. This will require a novel 6 degree of freedom dynamics model including the inertia of the bodies and the transient pressure and shear loads of all interfaces of the body and the fluid domain.</div><div><br></div><div>During operation, fluid cavitation and aeration can occur in both the displacement chambers of the machine and its lubricating interfaces. To capture this, a novel cavitation algorithm is developed in this thesis, which considers the release of bubbles due to both gas trapped within the fluid and vaporization of the operating fluid in localized low pressure regions of the films. In the absence of cavitation, this model will also be used to find the pressures and flows over the film, communicating this information with the remainder of the fluid domain.</div><div><br></div><div>Due to the high pressures that form in these units, the bodies deform. The resulting deformation changes the shape of the films and therefore its pressure distribution. This coupled effect will be captured in one of two ways, the first relying on existing geometric information of the unit, and the other using a novel analytical approach that is developed to avoid this necessity. In either case, the added damping due to the shear of the materials will be considered for the first time. Additionally in regions of low gap height, mixed lubrication occurs and the effects of the surface asperities of the floating bodies cannot be neglected. Accurate modeling of this condition is necessary to predict wear that leads to failure in these units. This work will then develop a novel implementation for mixed lubrication modeling that is directly integrated into the cavitation modeling approach.</div><div><br></div><div>Finally, effort is made to maintain a generic tools, such that the model can be applied to any positive displacement machine. This thesis will present the first toolbox of its kind, which accounts for all the mentioned aspects in such a way that they can be captured for any machine. Using both multithreaded and sequential implementations, the tool will be capable of fully utilizing a machine on which it is run for both low latency (design) and high throughput (optimization) applications respectively. In order to make these applications feasible, the various modules of the tool will be strongly coupled using asynchronous time stepping. This approach is made possible with the development of a novel impedance tensor of the mixed universal Reynolds equation, and shows marked improvements in simulation time by requiring at most 50% of the simulation time of existing approaches.</div><div><br></div><div>In the present thesis, the developed tool will be validated using experimental data collected from 3 fundamentally different machines. Individual advancements of the tool will also be verified in isolation with comparison to the state of the art and commercial software in the relevant fields. As a demonstration of the use of the tool for design, detailed analysis of the displacing actions and lubricating interfaces of these same units will be performed. These validations demonstrate the ability of the tool to predict machine efficiencies with error averaging around 1% over all operating conditions for multiple machine types, and capture transient behavior of the units. To demonstrate the utility as a virtual optimization tool, design of a complete external gear machine design will be performed. This demonstration will start from only analytical parameters, and will track a route to a complete prototype.</div>

Computational fluid dynamics (CFD) has become a significant tool routinely used in design and optimization in aerospace industry. Typical flows may be characterized by high-speed and compressible flow features and, in many cases, by massive flow separation and unsteadiness. Accurate and efficient numerical solution of time-dependent problems is hence required, and the efficiency of standard dual-time stepping methods used for unsteady flows in many CFD codes has been found inadequate for large-scale industrial problems. This has motivated the present work, in which major effort is made to replace the explicit relaxation methods with implicit time integration schemes. The CFD flow solver considered in this work is Edge, a node-based solver for unstructured grids based on a dual, edge-based formulation. Edge is the Swedish national CFD tool for computing compressible flow, used at the Swedish aircraft manufacturer SAAB, and developed at FOI, lately in collaboration with external national and international partners. The work is initially devoted to the implementation of an implicit Lower-Upper Symmetric Gauss-Seidel (LU-SGS) type of relaxation in Edge with the purpose to speed up the convergence to steady state. The convergence of LU-SGS has been firstly accelerated by basing the implicit operator on a flux splitting method of matrix dissipation type. An increase of the diagonal dominance of the system matrix was the principal motivation. Then the code has been optimized by means of performance tools Intel Vtune and CrayPAT, improving the run time. It was found that the ordering of the unknowns significantly influences the convergence. Thus, different ordering techniques have been investigated. Finding the optimal ordering method is a very hard problem and the results obtained are mostly illustrative. Finally, to improve convergence speed on the stretched computational grids used for boundary layers LU-SGS has been combined with the line-implicit method. / QC 20120626

Our knowledge of elastic wave propagation in general heterogeneous media with complex geological structures comes principally from numerical simulations. In this dissertation, I demonstrate through rigorous theoretical analyses and comprehensive numerical experiments that the Fourier method is a suitable method of choice for large scale 3D seismic modeling and imaging problems, due to its high accuracy and computational efficiency. The most attractive feature of the Fourier method is its ability to produce highly accurate solutions on relatively coarser grids, compared with other numerical methods for solving wave equations. To further advance the Fourier method, I identify two aspects of the method to focus on in this work, i.e., its implementation on modern clusters of computers and efficient high-order time stepping schemes. I propose two new parallel algorithms to improve the efficiency of the Fourier method on distributed memory systems using MPI. The first algorithm employs non-blocking all-to-all communications to optimize the conventional parallel Fourier modeling workflows by overlapping communication with computation. With a carefully designed communication-computation overlapping mechanism, a large amount of communication overhead can be concealed when implementing different kinds of wave equations. The second algorithm combines the advantages of both the Fourier method and the finite difference method by using convolutional high-order finite difference operators to evaluate the spatial derivatives in the decomposed direction. The high-order convolutional finite difference method guarantees a satisfactory accuracy and provides the flexibility of using non-blocking point-to-point communications for efficient interprocessor data exchange and the possibility of overlapping communication and computation. As a result, this hybrid method achieves an optimized balance between numerical accuracy and computational efficiency. To improve the overall accuracy of time domain Fourier simulations, I propose a family of new high-order time stepping schemes, based on a novel algorithm for designing time integration operators, to reduce temporal derivative discretization errors in a cost-effective fashion. I explore the pseudo-analytical method and propose high-order formulations to further improve its accuracy and ability to deal with spatial heterogeneities. I also extend the pseudo-analytical method to solve the variable-density acoustic and elastic wave equations. I thoroughly examine the finite difference method by conducting complete numerical dispersion and stability analyses. I comprehensively compare the finite difference method with the Fourier method and provide a series of detailed benchmarking tests of these two methods under a number of different simulation configurations. The Fourier method outperforms the finite difference method, in terms of both accuracy and efficiency, for both the theoretical studies and the numerical experiments, which provides solid evidence that the Fourier method is a superior scheme for large scale seismic modeling and imaging problems. / text

This thesis is concerned with the efficient numerical simulation of finite element based flexible multibody systems. Scaling operations are systematically applied to the governing index-3 differential algebraic equations in order to solve the problem of ill conditioning for small time step sizes. The importance of augmented Lagrangian terms is demonstrated. The use of fast sparse solvers is justified for the solution of the linearized equations of motion resulting in significant savings of computational costs. Three time stepping schemes for the integration of the governing equations of flexible multibody systems are discussed in detail. These schemes are the two-stage Radau IIA scheme, the energy decaying scheme, and the generalized-α method. Their formulations are adapted to the specific structure of the governing equations of flexible multibody systems. The efficiency of the time integration schemes is comprehensively evaluated on a series of test problems. Formulations for structural and constraint elements are reviewed and the problem of interpolation of finite rotations in geometrically exact structural elements is revisited. This results in the development of a new improved interpolation algorithm, which preserves the objectivity of the strain field and guarantees stable simulations in the presence of arbitrarily large rotations. Finally, strategies for the spatial discretization of beams in the presence of steep variations in cross-sectional properties are developed. These strategies reduce the number of degrees of freedom needed to accurately analyze beams with discontinuous properties, resulting in improved computational efficiency.

The development of numerical methods for the incompressible Navier-Stokes equations received much attention in the past 50 years. Finite element methods emerged given their robustness and reliability. In our work, we choose the P2-P1 finite element for space approximation which gives 2nd-order accuracy for velocity and 1st-order accuracy for pressure. Our research focuses on the development of several high-order semi-implicit time-stepping methods to compute unsteady flows. The methods investigated include backward difference formulae (SBDF) and defect correction strategy (DC). Using the defect correction strategy, we investigate two variants, the first one being based on high-order artificial compressibility and bootstrapping strategy proposed by Guermond and Minev (GM) and the other being a combination of GM methods with sequential regularization method (GM-SRM). Both GM and GM-SRM methods avoid solving saddle point problems as for SBDF and DC methods. This approach reduces the complexity of the linear systems at the expense that many smaller linear systems need to be solved. Next, we proposed several numerical improvements in terms of better approximations of the nonlinear advection term and high-order initialization for all methods. To further minimize the complexity of the resulting linear systems, we developed several new variants of grad-div splitting algorithms besides the one studied by Guermond and Minev. Splitting algorithm allows us to handle larger flow problems. We showed that our new methods are capable of reproducing flow characteristics (e.g., lift and drag parameters and Strouhal numbers) published in the literature for 2D lid-driven cavity and 2D flow around the cylinder. SBDF methods with grad-div stabilization terms are found to be very stable, accurate and efficient when computing flows with high Reynolds numbers. Lastly, we showcased the robustness of our methods to carry 3D computations.