Approximate factorization for time-dependent partial differential equations

AbstractThe first application of approximate factorization in the numerical solution of time-dependent partial differential equations (PDEs) can be traced back to the celebrated papers of Peaceman and Rachford and of Douglas of 1955. For linear problems, the Peaceman–Rachford–Douglas method can be derived from the Crank–Nicolson method by the approximate factorization of the system matrix in the linear system to be solved. This factorization is based on a splitting of the system matrix. In the numerical solution of time-dependent PDEs we often encounter linear systems whose system matrix has a complicated structure, but can be split into a sum of matrices with a simple structure. In such cases, it is attractive to replace the system matrix by an approximate factorization based on this splitting. This contribution surveys various possibilities for applying approximate factorization to PDEs and presents a number of new stability results for the resulting integration methods

Coupled/combined compact IRBF schemes for fluid flow and FSI problems

The thesis is concerned with the development of compact approximation methods based on Integrated Radial Basis Functions (IRBFs) and their applications in fluid flows and FSI problems. The contributions include (i) new compact IRBF stencils where first- and second-order derivatives are included; (ii) a preconditioning technique where a preconditioner to enhance the stability of the flat IRBF solutions; and, (iii) the incorporation of the proposed stencils into the immersed boundary methods. Numerical experiments show the present schemes generally produce more accurate solutions and better convergence rates than existing methods (e.g. FDM, high-order compact FDM and compact IRBF methods)

NUMERICAL INVESTIGATION AND PARALLEL COMPUTING FOR THERMAL TRANSPORT MECHANISM DURING NANOMACHINING

Nano-scale machining, or Nanomachining is a hybrid process in which the total thermal energy necessary to remove atoms from a work-piece surface is applied from external sources. In the current study, the total thermal energy necessary to remove atoms from a work-piece surface is applied from two sources: (1) localized energy from a laser beam focused to a micron-scale spot to preheat the work-piece, and (2) a high-precision electron-beam emitted from the tips of carbon nano-tubes to remove material via evaporation/sublimation. Macro-to-nano scale heat transfer models are discussed for understanding their capability to capture and its application to predict the transient heat transfer mechanism required for nano-machining. In this case, thermal transport mechanism during nano-scale machining involves both phonons (lattice vibrations) and electrons; it is modeled using a parabolic two-step (PTS) model, which accounts for the time lag between these energy carriers. A numerical algorithm is developed for the solution of the PTS model based on explicit and implicit finite-difference methods. Since numerical solution for simulation of nanomachining involves high computational cost in terms of wall clock time consumed, performance comparison over a wide range of numerical techniques has been done to devise an efficient numerical solution procedure. Gauss-Seidel (GS), successive over relaxation (SOR), conjugate gradient (CG), d -form Douglas-Gunn time splitting, and other methods have been used to compare the computational cost involved in these methods. Use of the Douglas-Gunn time splitting in the solution of 3D time-dependent heat transport equations appears to be optimal especially as problem size (number of spatial grid points and/or required number of time steps) becomes large. Parallel computing is implemented to further reduce the wall clock time required for the complete simulation of nanomachining process. Domain decomposition with inter-processor communication using Message Passing Interface (MPI) libraries is adapted for parallel computing. Performance tuning has been implemented for efficient parallelization by overlapping communication with computation. Numerical solution for laser source and electron-beam source with different Gaussian distribution are presented. Performance of the parallel code is tested on four distinct computer cluster architecture. Results obtained for laser source agree well with available experimental data in the literature. The results for electron-beam source are self-consistent; nevertheless, they need to be validated experimentally

Numerical resolution of turbulent flows on complex geometries.

This thesis aims at developing a numerical methodology suitable for the direct numerical simulation (DNS) and large-eddy simulation (LES) of turbulent flows in order to be used in complex flows, currently encountered in industrial application. At the same time, the study of such turbulent flows can be an opportunity for gaining insight into the complex physics associated with them. To accomplish these goals, the mathematical formulation, conservative spatial discretization on unstructured grids and time- integration scheme for solving the Navier-Stokes equations are presented. The spatial discretization proposed preserves the symmetry properties of the continuous differential operator and ensure both, stability and conservation of the global kinetic energy balance on any grid. Furthermore, the time-integration technique proposed is an efficient self-adaptive strategy, based on a one-parameter second-order-explicit scheme, which has been successfully tested on both Cartesian staggered and unstructured collocated codes, leading to CPU cost reductions of up to 2.9 and 4.3, respectively. After presenting the general methodology for computing flows in complex geometries with unstructured grids, different LES models and regularization models suitable for these kind of meshes are presented and assessed by means of the analysis of different flows. First, regularization models are tested by means of the simulation of different cases with different level of complexity of the mesh. From a structured grid to a very complex mesh, with zones composed of prism and tetrahedral control volumes. It has been shown, that regularization models are very dependent on the quality of the filtering process. Although good results can be obtained with structured or smooth unstructuredmeshes, their performance is affected under fully irregular unstructured grids. A possible remedy to circumvent this issue is also presented. The main idea is to formulate the C4 model within a LES template. Although preliminary results are promising, further testing is still required. After regularization model assessment, LES models are also tested in a natural convection flow. It is shown that, although first order statistics are well solved for most of the models tested (with the exception of the Smagorinsky model), QR- and dynamic-Smagorinsky models present a better prediction of the second-order statistics. However, if CPU time is considered, then QR model is the best alternative. The second part of the thesis is devoted to the study of turbulent flows past bluff bodies. The cases studied are: the flow past a sphere, the flow past a circular cylinder and the flow past a NACA 0012 airfoil. All these cases shares some characteristics encountered in turbulent flows with massive separations, i.e., flow separation, transition to turbulence in the separated shear-layers and turbulent wakes with periodic shedding of vortices. However there are intrinsic characteristics of the turbulence in each of them, which make them interesting for the studying of the turbulence. Furthermore, the results presented for the flow past a sphere at Re = 3700 and 10000, together with the flow past a NACA 0012 at Re=50000 and AoA = 8 are the first DNS results presented in the literature for both flows. Conclusions drawn from the good results obtained point out that the use of the conservative formulation presented in this thesis, is one of the keys for the success of the SGS models used. This formulation, together with the use of unstructured grids might be a step towards the use of LES models for solving industrial flows on complex geometries at high Reynolds numbers.La present tesi proposa una metodologia apte per a realitzar simulacions directes de la turbulència (DNS) i simulacions de les grans escales (LES) de fluxos turbulents en geometries complexes. Tanmateix també s'estudia detalladament els mecanismes bàsics de funcionament dels fluxos turbulents en diferents situacions d'interès industrial i acadèmic. Per acomplir aquest objectiu s'ha desenvolupat una innovadora formulació matemàtica que permet conservar discretament les propietats continues de les equacions governants en malles no estructurades. La formulació proposada preserva la simetries originals dels operadors diferencials, assegurant així l'estabilitat i la conservació de l'energia cinètica turbulent en qualsevol mallat. Posteriorment s'ha proposat una metodologia d'integració temporal basada en una formulació explicita de segon ordre. Aquesta nova tècnica ha demostrat ser entre 2.9 i 4.3 més rapida que les tècniques anteriorment utilitzades per la comunitat. Un cop presentada la formulació per a simular fluxos turbulents en geometries complexes, s'han validat diferents models LES adaptats a malles no estructurades. Els models s'han testejat usant diferents solucions de referencia de la literatura i simulacions d'alt nivell generades en el context de la present tesi. Finalment s'ha conclòs que la conjunció de la formulació bàsica proposada amb alguns del models LES sorgits en els darrers anys es molt efectiva per a simular fluxos turbulents en situacions complexes, essent el Variational Multiscale WALE i el model QR els més adequats per a simular situacions de interes industrial. La segona part de la tesi es dedicada a l'estudi aerodinàmic del flux turbulent al voltant de diferents perfils. El perfils seleccionats son: el flux al voltant d'una esfera, flux al voltant d'un cilindre i flux al voltant d'un perfil NACA 0012. Els tres casos comparteixen fenomenologies com ara separació massiva de capes límits, esteles turbulentes i desprendiment periòdic de remolins. Tot i així cadascun d'ells es comporta diferent a nivell turbulent així que es d'interès estudiar-los i entendre quins son les causes de les diferencies físiques que es troben. Cal recordar que la física estudiada es la que es pot trobar posteriorment en ales d'avió, perfils de turbines de vent, aerodinàmica de cotxes, etc. Finalment recalcar que els resultats DNS del flux al voltant de l'esfera a Re=3700 i Re=10000 conjuntament amb els DNS del flux al voltant del perfil NACA a Re=50000 i AoA =8 son els primers presentats en la literatura internacional en el seu àmbit. Finalment es pot concloure que la formulació conservativa presentada en la tesis juntament amb els diferents models LES d'última generació testejats en la tesis, han demostrat ser una eina eficaç tan per a resoldre fluxos turbulents d'interès acadèmic com per simular situacions d'interès industrial.Postprint (published version

Advances in numerical and applied mathematics

This collection of papers covers some recent developments in numerical analysis and computational fluid dynamics. Some of these studies are of a fundamental nature. They address basic issues such as intermediate boundary conditions for approximate factorization schemes, existence and uniqueness of steady states for time dependent problems, and pitfalls of implicit time stepping. The other studies deal with modern numerical methods such as total variation diminishing schemes, higher order variants of vortex and particle methods, spectral multidomain techniques, and front tracking techniques. There is also a paper on adaptive grids. The fluid dynamics papers treat the classical problems of imcompressible flows in helically coiled pipes, vortex breakdown, and transonic flows

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described