There is a need for reliable methods to determine approximate solutions of nonlinear continuous systems. Recently, it has been proved that finite-degree-of-freedom Galerkin-type discretization procedures applied to some distributed-parameter systems may fail to predict the correct dynamics. By contrast, direct procedures yield reliable approximate solutions. Starting from these results and extending some of these concepts and procedures, we compare the outcomes of these two approaches (the Galerkin discretization and the direct application of a reduction method to the original governing equations) with experimental results. The nonlinear planar vibrations of a buckled beam around its first buckling mode shape are investigated. Frequency-response curves characterizing single-mode responses of the beam under a primary resonance are generated using both approaches and contrasted with experimentally obtained frequency-response curves. It is shown that discretization leads to erroneous quantitative as well as qualitative results in certain ranges of the buckling level, whereas the direct approach predicts the correct dynamics of the system. / Master of Science

Design Space Exploration (DSE) is the exploration of a space of possible designs with the goal of finding some optimal design according to some constraints and criteria. Within embedded systems design, automated DSE in particular can allow the system designer to efficiently find good solutions in highly complex design spaces. One particular tool for performing automated DSE is IDeSyDe which uses Constraint Programming (CP) and constraint optimization for modelling and optimization. The constraint models of DSE often include some real-valued parameters, but optimized CP-solvers typically require integer arguments. This makes it necessary to discretize the problem in order to make the approach useful in practice, effectively limiting the size of the search space significantly. The effects of this discretization procedure on the quality of the solutions have not previously been well studied. An investigation into how this kind of discretization affects the approximate solutions could make the approach more rigorous, and possibly also uncover exploitable details that could facilitate the development of even more efficient algorithms. This project presents a convergence proof based in CP and Multiresolutional analysis (MRA), including a practically useful error bound for solutions obtained with different discretizations. In particular, the mapping and scheduling of Syncronous Data Flow (SDF) models for streaming applications onto tile-based multiple processor system-on-chip platforms with a common time-division multiplexing bus interconnect is studied. The theoretical results are also verified using IDeSyDe for a few different configurations of applications and platforms. It can be seen that the experiments behave as predicted, with first order convergence in total error and adherence to the bound. / Designrymdsutforskning är benämningen för en systematisk utforskning av en rymd av möjliga designer i syfte att hitta bra eller optimala lösningar som optimerar något mål och som uppfyller krav och begränsningar. Automatiserad designrymdsutforskning har i synnerhet sett utveckling för tillämpningar inom design av inbyggda system, där den ständigt ökande komplexiteten hos moderna plattformar motiverat utvecklingen av nya metoder. Två stora delar är nödvändiga för att kunna tillämpa designrymdsutforskning för design av inbyggda system: en modell av systemet och en optimiseringsprocess. Beroende på situation kan systemmodeller variera från detaljerade simuleringar på transistornivå till övergripande analytiska modeller på applikationsnivå eller högre. Detaljerade simuleringar gör det möjligt att utvärdera en viss lösning mycket noggrant, men till en hög beräkningskostnad. Med analytiska modeller är det istället billigt att utvärdera enskilda lösningar, men på bekostnad av noggrannhet. På samma sätt kan olika optimeringsprocesser också användas: snabbare approximativa algoritmer kan användas för att hitta lösningar relativt snabbt men utan garantier för optimalitet, medans mer uttömmande algoritmer typiskt kräver mycket beräkningskraft. Ett verktyg för automatiserad designrymdsutforskning är IDeSyDe. IDeSyDe använder villkorsbaserade modeller och uttömmande sökning genom Branch and Bound. Optimerade algoritmiska lösare för villkorsprogrammeringsproblem kräver ofta heltalsparametrar. Modeller för designrymdsutforskning innehåller å andra sidan ofta kontinuerliga parametrar. På grund av detta är det ofta nödvändigt att disktretisera problemet för att effektivt kunna hitta lösningar. Eftersom en diskretisering begränsar mängden lösningar i sökrymden riskerar en sådan omformulering att ta bort även optimala lösningar. En designrymdsutforskningsalgoritm som utnyttjar diskretisering av designrymden måste på grund av detta generellt ses som en approximativ algoritm. Hur en sådan diskretisering påverkar lösningarna -- dvs. hur nära de approximativa lösningarna kan förväntas komma den optimala lösningen utan diskretisering -- har dock inte studerats i närmare detalj. En bättre förståelse för hur diskreta, approximativa problem och lösningar relaterar till sina exakta motsvarigheter kan ge metoden mer rigör. En undersökning av den underliggande matematiken har också potential att belysa andra samband och strukturer som potentiellt skulle kunna användas för att utveckla bättre eller mer effektiva algoritmer. I den här rapporten presenteras ett konvergensbevis baserat på villkorsprogrammering och multiupplösningsanalys med ett begränsat felintervall i termer av probleminstansspecifika parametrar och en diskretiseringsparameter. Beviset är framtaget för tillämpning med IDeSyDe och är därför begränsat till en kombination av modeller som verktyget för närvarande stödjer, nämligen strömmande-dataflödesapplikationer beskrivna som synkrona dataflödesmodeller (Synchronous Data Flow, SDF) samt en ''tile''-baserad modell för system med flera processorer på ett chip (MPSoC) med en gemensam tidspartitionerad multiplexor-bus för kommunikation mellan processor-''tiles''. De teoretiska resultaten är verifierade och tillämpade på ett flertal exempelfall beräknade med IDeSyDe, där konvergensen studerats experimentellt.

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A detailed code verification study of an unstructured finite volume Computational Fluid Dynamics (CFD) code is performed. The Method of Manufactured Solutions is used to generate exact solutions for the Euler and Navier-Stokes equations to verify the correctness of the code through order of accuracy testing. The verification testing is performed on different mesh types which include triangular and quadrilateral elements in 2D and tetrahedral, prismatic, and hexahedral elements in 3D. The requirements of systematic mesh refinement are discussed, particularly in regards to unstructured meshes. Different code options verified include the baseline steady state governing equations, transport models, turbulence models, boundary conditions and unsteady flows. Coding mistakes, algorithm inconsistencies, and mesh quality sensitivities uncovered during the code verification are presented. In recent years, there has been significant work on the development of algorithms for the compressible Navier-Stokes equations on unstructured grids. One of the challenging tasks during the development of these algorithms is the formulation of consistent and accurate diffusion operators. The robustness and accuracy of diffusion operators depends on mesh quality. A survey of diffusion operators for compressible CFD solvers is conducted to understand different formulation procedures for diffusion fluxes. A patch-wise version of the Method of Manufactured Solutions is used to test the accuracy of selected diffusion operators. This testing of diffusion operators is limited to cell-centered finite volume methods which are formally second order accurate. These diffusion operators are tested and compared on different 2D mesh topologies to study the effect of mesh quality (stretching, aspect ratio, skewness, and curvature) on their numerical accuracy. Quantities examined include the numerical approximation errors and order of accuracy associated with face gradient reconstruction. From the analysis, defects in some of the numerical formulations are identified along with some robust and accurate diffusion operators. / Ph. D.

Computational Fluid Dynamics (CFD) applications usually involve intensive computations, which can be accelerated through using open accelerators, especially GPUs due to their common use in the scientific computing community. In addition to code acceleration, it is important to ensure that the code and algorithm are implemented numerically correctly, which is called code verification. This dissertation focuses on accelerating research CFD codes on multi-CPUs/GPUs using MPI and OpenACC, as well as the code verification for turbulence model implementation using the method of manufactured solutions and code-to-code comparisons. First, a variety of performance optimizations both agnostic and specific to applications and platforms are developed in order to 1) improve the heterogeneous CPU/GPU compute utilization; 2) improve the memory bandwidth to the main memory; 3) reduce communication overhead between the CPU host and the GPU accelerator; and 4) reduce the tedious manual tuning work for GPU scheduling. Both finite difference and finite volume CFD codes and multiple platforms with different architectures are utilized to evaluate the performance optimizations used. A maximum speedup of over 70 is achieved on 16 V100 GPUs over 16 Xeon E5-2680v4 CPUs for multi-block test cases. In addition, systematic studies of code verification are performed for a second-order accurate finite volume research CFD code. Cross-term sinusoidal manufactured solutions are applied to verify the Spalart-Allmaras and k-omega SST model implementation, both in 2D and 3D. This dissertation shows that the spatial and temporal schemes are implemented numerically correctly. / Doctor of Philosophy / Computational Fluid Dynamics (CFD) is a numerical method to solve fluid problems, which usually requires a large amount of computations. A large CFD problem can be decomposed into smaller sub-problems which are stored in discrete memory locations and accelerated by a large number of compute units. In addition to code acceleration, it is important to ensure that the code and algorithm are implemented correctly, which is called code verification. This dissertation focuses on the CFD code acceleration as well as the code verification for turbulence model implementation. In this dissertation, multiple Graphic Processing Units (GPUs) are utilized to accelerate two CFD codes, considering that the GPU has high computational power and high memory bandwidth. A variety of optimizations are developed and applied to improve the performance of CFD codes on different parallel computing systems. The program execution time can be reduced significantly especially when multiple GPUs are used. In addition, code-to-code comparisons with some NASA CFD codes and the method of manufactured solutions are utilized to verify the correctness of a research CFD code.

Molecular dynamics (MD) and computational fluid dynamics (CFD) allowresearchers to study fluid dynamics from two very different standpoints. From a microscopic standpoint, molecular dynamics uses Newton's second law of motion to simulate the interatomic behavior of individual atoms, using statistical mechanics as a tool for analysis. In contrast, CFD describes the motion of a fluid from a macroscopic level using the transport of mass, momentum, and energy of a system as a model. This thesis investigates both MD and CFD as a viable means of studying viscous flow on a nanometer scale. Specifically, we investigate a pressure-driven Poiseuille flow. The results of the MD simulations are processed using software we created to measure velocity, density, and pressure. The CFD simulations are run on numerical software that implements the MacCormack method for the Navier-Stokes equations. Additionally, the CFD simulations incorporate a local definition of viscosity, which is usually uncharacteristic of this simulation method. Based on the results of the simulations, we point out similarities and differences in the obtained steady-state solutions. / Master of Science

This work examines the Method of Nearby Problems as a way to generate analytical exact solutions to problems governed by partial differential equations (PDEs). The method involves generating a numerical solution to the original problem of interest, curve fitting the solution, and generating source terms by operating the governing PDEs upon the curve fit. Adding these source terms to the right-hand-side of the governing PDEs defines the nearby problem. In addition to its use for generating exact solutions the MNP can be extended for use as an error estimator. The nearby problem can be solved numerically on the same grid as the original problem. The nearby problem discretization error is calculated as the difference between its numerical solution and exact solution (curve fit). This is an estimate of the discretization error in the original problem of interest. The accuracy of the curve fits is quite important to this work. A method of curve fitting that takes local least squares fits and combines them together with weighting functions is used. This results in a piecewise fit with continuity at interface boundaries. A one-dimensional Burgers' equation case shows this to be a better approach then global curve fits. Six two-dimensional cases are investigated including solutions to the time-varying Burgers' equation and to the 2D steady Euler equations. The results show that the Method of Nearby Problems can be used to create realistic, analytical exact solutions to problems governed by PDEs. The resulting discretization error estimates are also shown to be reasonable for several cases examined. / Master of Science

This thesis gives an overview of the process that led to the development of a novel semi-implicit fluid-structure interaction model. The thesis is dedicated to the creation of a new numerical model that allows to study ship generated waves and ship manoeuvers in waterways for various vessel characteristics and speeds in different external current situations. A model like this requires a coupling between the fluid and the solid to generate the waves and the hydrodynamic forces on the hull. Since the horizontal dimensions are significantly larger than the vertical dimension, we started by employing the shallow water equations, which are based on the assumption of hydrostatic pressure. The discretization was carried out taking only the nonlinear advective terms explicitly while the pressure terms are discretized implicitly, which makes the CFL condition milder. The price to pay for this semi-implicit discretization is an increase in the algorithm complexity compared to a fully-explicit method, but it is still much simpler than a fully-implicit discretization of the governing equations. Indeed, the mass and momentum equations couple, and finding the unknowns involves solving a system of equations with dimensions equal to the number of cells. The grid supporting the discretization is staggered, overlapping and Cartesian. Since the aimed application domain is inland waterways, it is paramount to allow wetting and drying of the cells. This was achieved by acting on the depth function, the relationship between the free-surface elevation and the water depth in the cell. The main novelty of this research project is the two-way coupling of the PDE system for the water flow with the ODE system for the rigid body motion of the ship. The hull defines the ship region, and its shape can range from a simple box to an STL file of a real 3D ship geometry. Where the hull is in contact with the water, the cells are pressurized. This pressurized group of cells generates waves as it moves, and its motion is influenced by incoming external waves. This result is obtained by imposing an upper bound to the depth function, so that the water depth does not increase when it reaches the hull elevation, while the pressure is allowed to increase. This upper bound increases the nonlinearity of the system, which may have dry cells, wet free-surface cells and pressurized cells. The solution of this system is found by a single nested-Newton iterative solver of Casulli and Zanolli [36], in which with two separate linearizations the system is written in a sparse, symmetric, positive semi-definite form. This particular form allows us to employ a matrix-free conjugate gradient method, and efficiently get the unknown pressure. The integral of the pressure over the hull is applied for the hydrodynamic force and torque acting on the ship. After adding the skin friction and other external forces from the propeller or the rudder, the total force is inserted in the equation of motion of the rigid body. The ODE system is discretized with a second-order Taylor method, and it is solved for the six degrees of freedom (3 coordinates for the position vector of the barycenter and 3 rotation angles), providing the next position and orientation of the ship. The vertical translation of the rigid body is governed by the gravitational force and the restoring force from Archimedes' principle. As the ship oscillates up and down, the gravitational potential energy is partially transferred to the radiated free-surface water waves, damping and eventually stopping the motion. Also, the ship pushes and pulls the water around it, inducing the added mass force. All these elements constitute the ODE that was used for the verification of the vertical degree of freedom. The numerical simulation gave the expected results for the vertical motion. The horizontal translation, important for the manoeuvers, presented a numerical instability unseen in our previous test cases, which is connected to the relative motion between the ship and the grid. In each time step in which the ship enters a new cell, the pressure sharply increases and decreases at the ship bow. An oscillation can build up in time and create an unphysical void below the vessel. We implemented a few ideas to attenuate the oscillations. At the heart of all the following techniques is the reduction of the time derivative of the water depth, especially for those cells transitioning to a pressurized state. All these modifications were effective at controlling the oscillations, each with a different intensity, and simulations with a horizontal motion are much more stable than without these techniques. With the collaboration of the BAW research institute, we worked on the model validation. We used data from two separate experiments to compare the measurements with the numerical results. Specifically, we focused on the ship-generated wave height and the hydrodynamic forces on the hull. The comparison is satisfactory for the wave height. The force and torque prediction is plausible but underestimated compared to the measurements. The model seems to displace the water volume correctly during the ship passage, while the force and torque response might need additional work to be trusted in applications. Even though the hydrostatic assumption is mostly correct in our range of applications, the presence and the motion of a ship could generate strong vertical accelerations of the flow, which may not be negligible. For this reason, we implemented an algorithm that corrects the velocity field, introducing also dispersive effects due to a non-hydrostatic pressure. The correction consists of a higher-order Bousinnesq-type term in the momentum equation and the solution of the resulting system. The non-hydrostatic update has a small influence on the wave generation, while it alters significantly the reaction forces. The subgrid method implementation allowed to benefit from high-resolution bottom descriptions while keeping the grid size coarse. The same subgrid can also be used for a refined definition of the hull, which makes the volume computations more accurate. Furthermore, the subgrid introduces new possible states for the cells, as they can be partially dry or partially pressurized. These intermediate states translate into smoother transitions from one state to the other when the free-surface is close to the bathymetry or to the hull. Concerning the software implementation of the developed scheme, in order to improve the execution performance of the prototype script formulated initially in Matlab, the numerical method was rewritten as a Fortran program. Also, thanks to the domain decomposition technique and the MPI standard, each simulation can run in parallel on multiple CPUs, leveraging the computational power of supercomputers. The coupling of the PDE and ODE system, together with an appropriate redefinition of the depth function, proved to be a valuable method for studying fluid-structure interaction problems. The combination of efficient numerical techniques led to the development of a tool with a potential to be applied in the practice for the simulation of floating objects in wide domains.

The largest and most difficult numerical approximation error to estimate is discretization error. Residual-based discretization error estimation methods are a category of error estimators that use an estimate of the source of discretization error and information about the specific application to estimate the discretization error using only one grid level. The higher-order terms are truncated from the discretized equations and are the local source of discretization error. The accuracy of the resulting discretization error estimate depends solely on the accuracy of the estimated truncation error. Residual-based methods require only one grid level compared to the more commonly used Richardson extrapolation which requires at least two. Reducing the required number of grid levels reduces computational expense and, since only one grid level is required, can be applied to unstructured grids where multiple quality grid levels are difficult to produce. The two residual-based discretization error estimators of interest are defect correction and error transport equations. The focus of this work is the development, improvement, and evaluation of various truncation error estimation methods considering the accuracy of the truncation error estimate and the resulting discretization error estimates. The minimum requirements for accurate truncation error estimation is specified along with proper treatment for several boundary conditions. The methods are evaluated using various Euler and Navier-Stokes applications. The discretization error estimates are compared to Richardson extrapolation. The most accurate truncation error estimation method was found to be the k-exact method where the fine grid with a correction factor was considerably reliable. The single grid methods including the k-exact require that the continuous operator be modified at the boundary to be consistent with the implemented boundary conditions. Defect correction showed to be more accurate for areas of larger discretization error; however, the cost was substantial (although cheaper than the primal problem) compared to the cost of solving the ETEs which was essential free due to the linearization. Both methods showed significantly more accurate estimates compared to Richardson extrapolation especially for smooth problems. Reduced accuracy was apparent with the presence of stronger shocks and some possible modifications to adapt to singularies are proposed for future work. / Ph. D.