The solution of large and sparse eigenvalue problems arising from the discretization of the diffusion equation is considered. The multigroup diffusion equation is discretized by means of the Nodal expansion Method (NEM) [9, 10]. A new formulation of the higher order NEM variants revealing the true nature of the problem, that is, a generalized eigenvalue problem, is proposed. These generalized eigenvalue problems are solved using the Jacobi-Davidson (JD) method [26]. The most expensive part of the method consists of solving a linear system referred to as correction equation. It is solved using Krylov subspace methods in combination with aggregation-based Algebraic Multigrid (AMG) techniques. In that context, a particular aggregation technique used in combination with classical smoothers, referred to as oblique geometric coarsening, has been derived. Its particularity is that it aggregates unknowns that are not coupled, which has never been done to our knowledge. A modular code, combining JD with an AMG preconditioner, has been developed. The code comes with many options, that have been tested. In particular, the instability of the Rayleigh-Ritz [33] acceleration procedure in the non-symmetric case has been underlined. Our code has also been compared to an industrial code extracted from ARTEMIS.

In the field of biology, MD simulations are continuously used to investigate biological studies. A Molecular Dynamics (MD) system is defined by the position and momentum of particles and their interactions. The dynamics of a system can be evaluated by an N-body problem and the simulation is continued until the energy reaches equilibrium. Thus, solving the dynamics numerically and evaluating the interaction is computationally expensive even for a small number of particles in the system. We are focusing on long-ranged interactions, since the calculation time is O(N^2) for an N particle system. In this dissertation, we are proposing two research directions for the MD simulation. First, we design a new variation of Multigrid (MG) algorithm called Multi-level charge assignment (MCA) that requires O(N) time for accurate and efficient calculation of the electrostatic forces. We apply MCA and back interpolation based on the structure of molecules to enhance the accuracy of the simulation. Our second research utilizes reconfigurable models to achieve fast calculation time. We have been working on exploiting two reconfigurable models. We design FPGA-based MD simulator implementing MCA method for Xilinx Virtex-IV. It performs about 10 to 100 times faster than software implementation depending on the simulation accuracy desired. We also design fast and scalable Reconfigurable mesh (R-Mesh) algorithms for MD simulations. This work demonstrates that the large scale biological studies can be simulated in close to real time. The R-Mesh algorithms we design highlight the feasibility of these models to evaluate potentials with faster calculation times. Specifically, we develop R-Mesh algorithms for both Direct method and Multigrid method. The Direct method evaluates exact potentials and forces, but requires O(N^2) calculation time for evaluating electrostatic forces on a general purpose processor. The MG method adopts an interpolation technique to reduce calculation time to O(N) for a given accuracy. However, our R-Mesh algorithms require only O(N) or O(logN) time complexity for the Direct method on N linear R-Mesh and N¡¿N R-Mesh, respectively and O(r)+O(logM) time complexity for the Multigrid method on an X¡¿Y¡¿Z R-Mesh. r is N/M and M = X¡¿Y¡¿Z is the number of finest grid points.

Simulacó numérica directa de la dispersió turbulenta de plomalls amb flotació en un flux en un canal Alexandre Fabregat Tomás, Tarragona, octubre del 2006 1 IntroduccióL'objectiu d'aquest treball és estudiar la dispersió turbulenta de calor en diferents configuracions basades en el canal desenvolupat mitjançant DNS (Direct Numerical Simulations). Aquesta eina ha demostrat ser de gran utilitat a l'hora d'estudiar fluxos turbulents ja que permet, donada una malla computacional capaç de capturar totes les estructures del flux i un esquema que minimitzi els errors i la dissipació numérica, descriure acuradament l'evolució temporal del flux. Permet a més, donada la descripció tridimensional i temporal del flux, determinar amb precisió qualsevol quantitat que seria impossible d'obtenir experimentalment.En el flux en un canal, el fluid esmou entre dues parets planes, llises i paral·leles separades una distància 2d impulsat per un gradient constant mitjà de pressió. El flux s'anomena desenvolupat quan ja no hi ha efectes de regió d'entrada i la única inhomogeneïtat es troba en la direcció normal a la paret. Sota aquestes condicions, les quantitats promitjades esdevenen estacionàries en el temps.En aquest treball s'ha validat el codi computacional mitjançant la reproducció d'algunes configuracions de flux prèviament estudiades per altres autors. Els nous coneixements en l'estudi de la dispersió turbulenta de calor s'han obtingut a l'incloure, en un flux totalment desenvolupat en un canal, una font lineal centrada verticalment que provoca l'aparició d'un plomall amb una temperatura més alta que la del flux del fons i que per tant, al tenir una menor densitat, experimenta flotació i es deflecteix. L'amitjanament temporal del flux permet estudiar les diferents contribucions dels diferents termes rellevants en les equacions de transport.És d'especial interés la comparativa d'aquests resultats amb els corresponents a la formació d'un plomall a partir d'una font lineal d'un escalar passiu.Per altra banda també s'ha estudiat l'eficiència en paral·lel dels mètodes multigrid en la resolució d'equacions de Poisson. Aquestes equacions són d'especial interés ja que apareixen en el càlcul de la pressió i representen un coll d'ampolla en termes de costos computacionals. Aquest mètode numèric ha estat comparat amb els mètodes de gradient conjugat (anteriorment emprats en el codi 3DINAMICS) en la resolució de diferents problemes comparant els costos en termes de temps de CPU i la seua escalabilitat en la màquina multiprocessador de memòria distribuïda del grup de recerca de Mecànica de Fluids de Tarragona.2 Descripció matemàticaUn cop adimensionalitzades mitjançant les escales adequades, les equacions de transport de quantitat de moviment i energia han estat discretitzades sobre una malla desplaçada mitjançant el mètode de volums finits emprant un esquema centrat de segon ordre. La discretització dels termes advectius en els casos amb fonts lineals ha requerit, però, d'un cura especial ja que la no-linealitat d'aquests termes pot provocar oscil·lacions artificials en el camp dels escalars. La difusió numèrica dels mètodes upwind, com el QUICK, ha estat quantificada i comparada amb resultats obtinguts per a esquemes centrats de segon ordre. Les equacions han estat integrades en el temps mitjançant un esquema implícit de segon ordre tipus Crank-Nicholson. També ha estat necessari implementar condicions de sortida per a la temperatura en els casos A i C del tipus no reflectant per tal de garantir la conservació i evitar l'aparició d'estructures artificials en el flux.3 Descripció físicaLa figura 1 presenta un esquema del domini computacional corresponent al canal desenvolupat. De l'esquema es desprén que x, y i z corresponen a les direccions principal del flux, la perpendicular i la normal a les parets respectivament. Les configuracions del flux estudiades es troben resumides a la taula 1 on s'indica la resolució de la xarxa computacional, el nombre de Reynolds (basat en la velocitat de fricció ut) i en el casos amb flotació, el nombre de Grashof, la temperatura de referència i la direcció de flotació (la direcció del vector gravetat).Les dimensions del canal s´on 8pd×2pd×2d en les direccions x, y i z respectivament.En el cas A la temperatura representa un escalar de manera que el plomall format és passiu, és a dir, no hi ha acoblament entre les equacions de quantitat de moviment i energia. A diferència d'aquest, en els casos B i C totes dues equacions queden acoblades pel terme de flotació. Aquest terme apareix quan les diferències de temperatura en el si del fluid generen diferències de densitat. En el cas B, el canal vertical amb convecció mixta, cada paret del canal es troba a una temperatura constant però diferent. El vector gravetat i la direcció del corrent estan alineades de manera que aquesta direcció continua sent homogènia. En la zona propera a la paret calenta la flotació actua en la direcció del corrent imposada pel gradient mitjà de pressió. En canvi, en la zona propera a la paret freda, la flotació s'oposa al moviment del flux.El cas C és similar al cas A però en aquesta ocasió la temperatura no es considera un escalar passiu i per tant la flotació acobla el camp dinàmic amb el de temperatures. El vector gravetat actua en aquest cas en la direcció normal. La inhomogeneïtat en la direcció del flux no permet continuar emprant condicions de contorn periòdiques i per tant, al domini presentat en la figura 1, se li ha acoblat una regió auxiliar a l'entrada on es resolen únicament les equacions de quantitat de moviment. Els camps de velocitat i pressió per a un canal totalment desenvolupat obtinguts en aquest domini auxiliar s'empraran com a condició de contorn a l'entrada del domini de computació. No és necessari cap tipus d'interpolació ja que la resolució del a xarxa d'aquest domini auxiliar és la mateixa que l'emprada en el domini de computació.4 ResultatsEls resultats per a les simulacions presentades en la taula 1 contenen, principalment, els perfils de velocitat i temperatura mitjans així com la intensitat de les fluctuacions. A més, es presenten els perfils de les diferents contribucions dels termes relevants de les equacions de transport amitjanades. Per al cas C, els camps dinàmics i de temperatura no estan desenvolupats. Els perfils mitjans a diferents posicions aigües avall permeten estudiar l'evolució del plomall ascendent a més d'analitzar com la flotació afecta al balanç de les diferents contribucions. La figure 2 presenta el camp mitjà de temperatures per al cas C amb les tres posicions en la direcció principal del flux per a les quals s'han inclòs els perfils.Finalment, es presenten els resultats corresponents a la comparativa entre els diferents solvers per a una equació de Poisson. Tots els mètodes numèrics han es-3Figura 2: Camp mitjà de temperatures per al cas C tat paral·lelitzats mitjançant les llibreries Message Passing Interface. En la figura 3 es presenten com a exemple els resultats (en termes de temps de CPU i speedup) per a la resolució de l'equació de Poisson per al desacoblament de pressió i velocitat en el cas del flux desenvolupat en un canal.Els resultats de speed-up per als diferents mètodes mostren la baixa escalabilitat del solver multigrid comparat amb els altres mètodes del tipus gradient conjugat. La raó radica en les grans necessitats de comunicació d'un algoritme construït sobre un esquema de relaxació tipus SOR. Tanmateix, multigrid és el mètode numèric que requereix menys temps de CPU per concloure la tasca. El factor respecte als mètodes de gradient conjugat pot arribar a ser de 30 i per tant és el millor candidat per a la resolució d'aquests tipus de problemes. / The main goal of this work is to study the turbulent heat transfer in a developed channel flow using Direct Numerical Simulations (DNS). These simulations solve explicitly all the scales present in the turbulent flow so, even for moderate Reynolds numbers, the discretization grids need to be fine enough to capture the smallest structures of the flow and, consequently, DNS demands large computational resources. The flow, driven by a mean constant pressure gradient in the streamwise direction, is confined between two smooth, parallel and infinite walls separated a distance 2d.The turbulent heat transport is studied for three different flow configurations.Some of them are used as benchmark results for this work. The three cases reported can be summarized as:· case A: Scalar plume from a line source in a horizontal channel.· case B:Mixed convection with the gravity vector aligned with the streamwise direction (vertical channel).· case C: Buoyant plume from a line source in a horizontal channel.In addition, preliminary results for a turbulent reacting flow in a fully developed channel are also presented.In the case B heat flux results from a temperature difference between the channel walls. The gravity vector is aligned with the streamwise direction and the Grashof, Reynolds and Prandtl numbers are Gr = 9.6 · 106, Ret = 150 and Pr = 0.71 respectively. Close to the hot wall, buoyancy acts aligned to the flow direction imposed by the mean pressure gradient so velocities are generally increased in comparison with a purely forced convection flow. Oppositely, near the cold wall, buoyancy is opposed to the flow and consequently velocities are decreased.Cases A and C are similar because in both cases a hot fluid is released within a cold background flow through a line source vertically centered in the wall-normal direction located at the inlet. The height of the source is 0.054d. The injected hot fluid disperses forming a hot plume that is convected downstream between the two adiabatic walls of the channel.The difference between cases A and C lies in the fact that for case A heat and momentum are decoupled and temperature acts as an scalar. Advection and diffusion are the only phenomena responsible for the evolution of the plume. On the other hand, in case C, buoyancy couples heat and momentum and, consequently, the plume floats drifting upward as it advances in the channel due to its lower density. In case C, the streamwise direction is not homogenous because of the coupling between heat and momentum. To guarantee developed conditions at the inlet of the channel it has been necessary to attach a buffer domain just before the computational domain. In this buffer domain, the momentum transport equations for a fully developed channel are solved with the same resolution used in the main domain.The results of cases A and B have been used to validate the 3DINAMICS CFD code by comparison with data reported in the literature. This code is written in FORTRAN 90 and parallelized using the Message Passing Interface (MPI-CHlibrary). It uses the second order in time Crank-Nicholson scheme to integrate numerically the transport equations which are discretized spatially using the centered second-order finite volume approach.The analysis of averaged turbulent quantities and the contributions of the different terms of the time-averaged transport equations is used to show how buoyancy affects the turbulent transport of momentum and heat along the channel.Finally, following a similar configuration than that of case A, a chemical reactantA released through line source reacts with a background reactant B following a second order chemical reaction with Damkh¨oler number of 1. Preliminary results for turbulent species transport are also included in this work.Special attention have been devoted to the discretization of the advective terms to avoid non-realistic values of the variables because of the non-linearities of the transport equations. The conservative non-reflecting boundary conditions have been implemented at the outlet to simulate the convected outflow when the streamwise direction can not be considered homogeneous, as in case C. For homogeneous directions, periodic boundary conditions have been used.Large grid resolutions (up to 8 million grid nodes for case C including the buffer region) demand important computational resources. A parallel Multigrid solver has substituted the previous conjugate gradient method to solve the Poisson equation in the pressure calculation. This step was the most expensive in terms of CPU costs. The Multigrid method efficiency has been compared with two different versions of the conjugate gradient approach and it has been demonstrated that this method is the most efficient in terms of CPU time although the current algorithmcan be improved to enhance the scalability inmultiprocessor computers.

Most fluid flows of practical interest involve and are affected by turbulence. One of the most promising computational methods for the prediction of turbulent flows is the so-called large eddy simulation (LES) methodology. Experience over the past decades have shown the capability of LES to provide accurate predictions for several types of flow at a reasonable computational cost. It has also become clear, however, that the LES methodology fails when applied to boundary layer flows at high Reynolds numbers. Since many engineering applications fall in exactly that category, this failure is often considered the most severe bottleneck of LES. <br /><br /> The present thesis is an attempt to move towards a solution of this problem. Inspired by the idealized picture of a turbulent boundary layer, a statistical model is used for the approximately universal turbulence in the inner boundary layer, whereas the more flow dependent outer boundary layer is solved by LES. Ideally, this results in a computational method that provides accurate predictions of rather general turbulent flows, while maintaining a tractable computational cost. In practice, the results are a vast improvement compared to LES without any inner layer modeling, but a transition layer appears where the state of the turbulence changes from being modeled statistically to resolved by LES. This so-called 'artificial buffer layer' results in the skin friction being consistently underpredicted by 10-15%. <br /><br /> The physics and dynamics of this artificial buffer layer are investigated and characterized, and it is argued that there exist several similarities with true buffer layer turbulence. Additional forcing of the momentum equations is used as a means to trigger resolved turbulence motions more quickly, and it is demonstrated that the results are better: the artificial buffer layer is smaller, the skin friction is accurately predicted, and the dynamics in the inner layer have more correct length scales. The results with the additional forcing are very sensitive to the forcing amplitude, and a simple control algorithm for this parameter is proposed and tested with favourable results.

Most fluid flows of practical interest involve and are affected by turbulence. One of the most promising computational methods for the prediction of turbulent flows is the so-called large eddy simulation (LES) methodology. Experience over the past decades have shown the capability of LES to provide accurate predictions for several types of flow at a reasonable computational cost. It has also become clear, however, that the LES methodology fails when applied to boundary layer flows at high Reynolds numbers. Since many engineering applications fall in exactly that category, this failure is often considered the most severe bottleneck of LES. <br /><br /> The present thesis is an attempt to move towards a solution of this problem. Inspired by the idealized picture of a turbulent boundary layer, a statistical model is used for the approximately universal turbulence in the inner boundary layer, whereas the more flow dependent outer boundary layer is solved by LES. Ideally, this results in a computational method that provides accurate predictions of rather general turbulent flows, while maintaining a tractable computational cost. In practice, the results are a vast improvement compared to LES without any inner layer modeling, but a transition layer appears where the state of the turbulence changes from being modeled statistically to resolved by LES. This so-called 'artificial buffer layer' results in the skin friction being consistently underpredicted by 10-15%. <br /><br /> The physics and dynamics of this artificial buffer layer are investigated and characterized, and it is argued that there exist several similarities with true buffer layer turbulence. Additional forcing of the momentum equations is used as a means to trigger resolved turbulence motions more quickly, and it is demonstrated that the results are better: the artificial buffer layer is smaller, the skin friction is accurately predicted, and the dynamics in the inner layer have more correct length scales. The results with the additional forcing are very sensitive to the forcing amplitude, and a simple control algorithm for this parameter is proposed and tested with favourable results.

The multigrid technique is one of the most effective techniques to achieve the reduction of the CPU cost for flow solvers. The multigrid strategy uses the multilevel grids which are the coarsening subsets of fine grid. An explicit solver rapidly reduces the high frequency errors on the computational grids. Since high frequency errors on coarse grids correspond to low frequency errors on fine grids, cycling through the coarse grid levels rapidly reduces the errors ranging from high-to-low frequency. The aim of this study is, therefore, to accelerate SENSE3D solver developed by TUBITAK-SAGE by implementating multigrid concept. In this work, a novel grid coarsening method suitable for cell-centered hybrid/unstructured grids is developed to provide the cells with high aspect ratio. This new grid coarsening technique relies on the agglomeration of cells based on their distribution on octree data structure. Then, the multigrid strategy is implemented to the baseline flow solver. During this implementation, the flux calculation along the face loops is modified without changing cell-centered scheme. The performance of the coarsening algorithm is investigated for all grid types in two and three dimension. The grid coarsening algorithm produces well defined, nested, body fitted coarser grids with aspect ratios of one and the coarse grids have similar characteristics of Cartesian grids. Then, the multigrid flow solutions are obtained at inviscid, laminar and turbulent flows. It is shown that, the convergence accelerations are up to 14 times for inviscid flows and in a range of 4 to 110 fold for turbulent flow solutions.

In this dissertation, we study the mixed &#64257;nite element method for the linear plane elasticity problem and iterative solvers for the resulting discrete system. We use the Arnold-Winther Element in the mixed &#64257;nite element discretization. An overlapping Schwarz preconditioner and a multigrid preconditioner for the discrete system are developed and analyzed. We start by introducing the mixed formulation (stress-displacement formulation) for the linear plane elasticity problem and its discretization. A detailed analysis of the Arnold-Winther Element is given. The &#64257;nite element discretization of the mixed formulation leads to a symmetric inde&#64257;nite linear system. Next, we study e&#64259;cient iterative solvers for the symmetric inde&#64257;nite linear system which arises from the mixed &#64257;nite element discretization of the linear plane elasticity problem. The preconditioned Minimum Residual Method is considered. It is shown that the problem of constructing a preconditioner for the inde&#64257;nite linear system can be reduced to the problem of constructing a preconditioner for the H(div) problem in the Arnold-Winther &#64257;nite element space. Our main work involves developing an overlapping Schwarz preconditioner and a multigrid preconditioner for the H(div) problem. We give condition number estimates for the preconditioned systems together with supporting numerical results.

In this paper, finite element discretizations of the degenerated operator -&omega;²(y) u_xx-&omega;²(x)u_yy=g in the unit square are investigated, where the weight function satisfies &omega;(&xi;)=&xi;^&alpha; with &alpha; &ge; 0. We propose two multi-level methods in order to solve the resulting system of linear algebraic equations. The first method is a multi-grid algorithm with line-smoother. A proof of the smoothing property is given. The second method is a BPX-like preconditioner which we call MTS-BPX preconditioner. We show that the upper eigenvalue bound of the MTS-BPX preconditioned system matrix grows proportionally to the level number.

Scientific computing and computer simulation play an increasingly important role in scientific investigation and engineering designs, supplementing traditional experiments, such as in automotive crash studies, global climate change, ocean modeling, medical imaging, and nuclear weapons. The numerical simulation is much cheaper than experimentation for these application areas and it can be used as the third way of science discovery beyond the experimental and theoretical analysis. However, the increasing demand of high resolution solutions of the Partial Differential Equations (PDEs) with less computational time has increased the importance for researchers and engineers to come up with efficient and scalable computational techniques that can solve very large-scale problems. In this dissertation, we build an efficient and highly accurate computational framework to solve PDEs using high order discretization schemes and multiscale multigrid method. Since there is no existing explicit sixth order compact finite difference schemes on a single scale grids, we used Gupta and Zhang’s fourth order compact (FOC) schemes on different scale grids combined with Richardson extrapolation schemes to compute the sixth order solutions on coarse grid. Then we developed an operator based interpolation scheme to approximate the sixth order solutions for every find grid point. We tested our method for 1D/2D/3D Poisson and convection-diffusion equations. We developed a multiscale multigrid method to efficiently solve the linear systems arising from FOC discretizations. It is similar to the full multigrid method, but it does not start from the coarsest level. The major advantage of the multiscale multigrid method is that it has an optimal computational cost similar to that of a full multigrid method and can bring us the converged fourth order solutions on two grids with different scales. In order to keep grid independent convergence for the multiscale multigrid method, line relaxation and plane relaxation are used for 2D and 3D convection diffusion equations with high Reynolds number, respectively. In addition, the residual scaling technique is also applied for high Reynolds number problems. To further optimize the multiscale computation procedure, we developed two new methods. The first method is developed to solve the FOC solutions on two grids using standardW-cycle structure. The novelty of this strategy is that we use the coarse level grid that will be generated in the standard geometric multigrid to solve the discretized equations and achieve higher order accuracy solution. It is more efficient and costs less CPU and memory compared with the V-cycle based multiscale multigrid method. The second method is called the multiple coarse grid computation. It is first proposed in superconvergent multigrid method to speed up the convergence. The basic idea of multigrid superconvergent method is to use multiple coarse grids to generate better correction for the fine grid solution than that from the single coarse grid. However, as far as we know, it has never been used to increase the order of solution accuracy for the fine grid. In this dissertation, we use the idea of multiple coarse grid computation to approximate the fourth order solutions on every coarse grid and fine grid. Then we apply the Richardson extrapolation for every fine grid point to get the sixth order solutions. For parallel implementation, we studied the parallelization and vectorization potential of the Gauss-Seidel relaxation by partitioning the grid space with four colors for solving 3D convection-diffusion equations. We used OpenMP to parallelize the loops in relaxation and residual computation. The numerical results show that the parallelized and the sequential implementation have the same convergence rate and the accuracy of the computed solutions.

This dissertation presents a multilevel algorithm to solve constant and variable coeffcient elliptic boundary value problems on adaptively refined structured meshes in 2D and 3D. Cacheaware algorithms for optimizing the operations to exploit the cache memory subsystem areshown. Keywords: Multigrid, Cache Aware, Adaptive Mesh Refinement, Partial Differential Equations, Numerical Solution.