Parallel unstructured solvers for linear partial differential equations

This thesis presents the development of a parallel algorithm to solve symmetric systems of linear equations and the computational implementation of a parallel partial differential equations solver for unstructured meshes. The proposed method, called distributive conjugate gradient - DCG, is based on a single-level domain decomposition method and the conjugate gradient method to obtain a highly scalable parallel algorithm. An overview on methods for the discretization of domains and partial differential equations is given. The partition and refinement of meshes is discussed and the formulation of the weighted residual method for two- and three-dimensions presented. Some of the methods to solve systems of linear equations are introduced, highlighting the conjugate gradient method and domain decomposition methods. A parallel unstructured PDE solver is proposed and its actual implementation presented. Emphasis is given to the data partition adopted and the scheme used for communication among adjacent subdomains is explained. A series of experiments in processor scalability is also reported. The derivation and parallelization of DCG are presented and the method validated throughout numerical experiments. The method capabilities and limitations were investigated by the solution of the Poisson equation with various source terms. The experimental results obtained using the parallel solver developed as part of this work show that the algorithm presented is accurate and highly scalable, achieving roughly linear parallel speed-up in many of the cases tested

A novel Shifted Boundary Method (SBM) for embedded domains based on Multi-Point Constraints.

This project presents a recent finite element approach for embedded domains that belongs to the category of approximate boundary methods: the Shifted Boundary Method (SBM). This is an unfitted approach used to represent embedded boundaries. SBM suggests shifting the location of boundary conditions from the true boundary to a surrogate boundary and adjusting the Dirichlet boundary conditions appropriately to preserve the optimal convergence rate. The theory and implementation details of the developed formulations in the open-source Kratos Multiphysics software have been comprehensively covered for Poisson and convection-diffusion problems. In addition, we introduced a novel approach based on Multi-Point constraints. The new technique allows the method to impose the new Dirichlet boundary conditions, which depend on the gradient of the solution, without introducing any nonlinear loop. The new shifted boundary method was validated by performing convergence studies on stationary and transient convection-diffusion problems, also testing embedded complex inner and outer boundaries.This project presents a recent finite element approach for embedded domains that belongs to the category of approximate boundary methods: the Shifted Boundary Method (SBM). This is an unfitted approach used to represent embedded boundaries. SBM suggests shifting the location of boundary conditions from the true boundary to a surrogate boundary and adjusting the Dirichlet boundary conditions appropriately to preserve the optimal convergence rate. The theory and implementation details of the developed formulations in the open-source Kratos Multiphysics software have been comprehensively covered for Poisson and convection-diffusion problems. In addition, we introduced a novel approach based on Multi-Point constraints. The new technique allows the method to impose the new Dirichlet boundary conditions, which depend on the gradient of the solution, without introducing any nonlinear loop. The new shifted boundary method was validated by performing convergence studies on stationary and transient convection-diffusion problems, also testing embedded complex inner and outer boundaries

Parallel unstructured solvers for linear partial differential equations

This thesis presents the development of a parallel algorithm to solve symmetric systems of linear equations and the computational implementation of a parallel partial differential equations solver for unstructured meshes. The proposed method, called distributive conjugate gradient - DCG, is based on a single-level domain decomposition method and the conjugate gradient method to obtain a highly scalable parallel algorithm. An overview on methods for the discretization of domains and partial differential equations is given. The partition and refinement of meshes is discussed and the formulation of the weighted residual method for two- and three-dimensions presented. Some of the methods to solve systems of linear equations are introduced, highlighting the conjugate gradient method and domain decomposition methods. A parallel unstructured PDE solver is proposed and its actual implementation presented. Emphasis is given to the data partition adopted and the scheme used for communication among adjacent subdomains is explained. A series of experiments in processor scalability is also reported. The derivation and parallelization of DCG are presented and the method validated throughout numerical experiments. The method capabilities and limitations were investigated by the solution of the Poisson equation with various source terms. The experimental results obtained using the parallel solver developed as part of this work show that the algorithm presented is accurate and highly scalable, achieving roughly linear parallel speed-up in many of the cases tested.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

A compressible Lagrangian framework for the simulation of underwater implosion problems

The development of efficient algorithms to understand implosion dynamics presents a number of challenges. The foremost challenge is to efficiently represent the coupled compressible fluid dynamics of internal air and surrounding water. Secondly, the method must allow one to accurately detect or follow the interface between the phases. Finally, it must be capable of resolving any shock waves which may be created in air or water during the final stage of the collapse. We present a fully Lagrangian compressible numerical framework for the simulation of underwater implosion. Both air and water are considered compressible and the equations for the Lagrangian shock hydrodynamics are stabilized via a variationally consistent multiscale method [109]. A nodally perfect matched definition of the interface is used [57, 25] and then the kinetic variables, pressure and density, are duplicated at the interface level. An adaptive mesh generation procedure, which respects the interface connectivities, is applied to provide enough refinement at the interface level. This framework is then used to simulate the underwater implosion of a large cylindrical bubble, with a size in the order of cm. Rapid collapse and growth of the bubble occurred on very small spatial (0.3mm), and time (0.1ms) scales followed by Rayleigh-Taylor instabilities at the interface, in addition to the shock waves traveling in the fluid domains are among the phenomena that are observed in the simulation. We then extend our framework to model the underwater implosion of a cylindrical aluminum container considering a monolithic fluid-structure interaction (FSI). The aluminum cylinder, which separates the internal atmospheric-pressure air from the external high-pressure water, is modeled by a three node rotation-free shell element. The cylinder undergoes fast transient deformations, large enough to produce self-contact along it. A novel elastic frictionless contact model is used to detect contact and compute the non-penetrating forces in the discretized domain between the mid-planes of the shell. Two schemes are tested, implicit using the predictor/multi-corrector Bossak scheme, and explicit, using the forward Euler scheme. The results of the two simulations are compared with experimental data.El desarrollo de métodos eficientes para modelar la dinámica de implosión presenta varios desafíos. El primero es una representación eficaz de la dinámica del sistema acoplado de aire-agua. El segundo es que el método tiene que permitir una detección exacta o un seguimiento adecuado de la interfase entre ambas fases. Por último el método tiene que ser capaz de resolver cualquier choque que podría generar en el aire o en el agua, sobre todo en la última fase del colapso. Nosotros presentamos un método numérico compresible y totalmente Lagrangiano para simular la implosión bajo el agua. Tanto el aire como el agua se consideran compresibles y las ecuaciones Lagrangianos para la hidrodinámica del choque se estabilizan mediante un método multiescala que es variacionalmente consistente [109]. Se utiliza una definición de interfase que coincide perfectamente con los nodos [57, 25]. Ésta, nos facilita duplicar eficazmente las variables cinéticas como la presión y la densidad en los nodos de la interfase. Con el fin de obtener suficiente resolución alrededor de la interfase, la malla se genera de forma adaptativa y respetando la posición de la interfase. A continuación el método desarrollado se utiliza para simular la implosión bajo el agua de una burbuja cilíndrica del tamaño de un centímetro. Varios fenómenos se han capturado durante el colapso: un ciclo inmediato de colapso-crecimiento de la burbuja que ocurre en un espacio (0.3mm) y tiempo (0.1ms) bastante limitado, aparición de inestabilidades de tipo Rayleigh-Taylor en la interfase y formaron de varias ondas de choque que viajan tanto en el agua como en el aire. Después, seguimos el desarrollo del método para modelar la implosión bajo el agua de un contenedor metálico considerando una interacción monolítica de fluido y estructura. El cilindro de aluminio, que a su vez contiene aire a presión atmosférica y está rodeada de agua en alta presión, se modelando con elementos de lámina de tres nodos y sin grados de libertad de rotación. El cilindro se somete a deformaciones transitorias suficientemente rápidos y enormes hasta llegar a colapsar. Un nuevo modelo elástico de contacto sin considerar la fricción se ha desarrollado para detectar el contacto y calcular las fuerzas en el dominio discretizado entre las superficies medianas de las laminas. Dos esquemas temporales están considerados, uno es implícito utilizando el método de Bossak y otro es explícito utilizando Forward Euler. Al final los resultados de ambos casos se comparan con los resultados experimentales

A compressible Lagrangian framework for the simulation of underwater implosion problems

The development of efficient algorithms to understand implosion dynamics presents a number of challenges. The foremost challenge is to efficiently represent the coupled compressible fluid dynamics of internal air and surrounding water. Secondly, the method must allow one to accurately detect or follow the interface between the phases. Finally, it must be capable of resolving any shock waves which may be created in air or water during the final stage of the collapse. We present a fully Lagrangian compressible numerical framework for the simulation of underwater implosion. Both air and water are considered compressible and the equations for the Lagrangian shock hydrodynamics are stabilized via a variationally consistent multiscale method. A nodally perfect matched definition of the interface is used and then the kinetic variables, pressure and density, are duplicated at the interface level. An adaptive mesh generation procedure, which respects the interface connectivities, is applied to provide enough refinement at the interface level. This framework is then used to simulate the underwater implosion of a large cylindrical bubble, with a size in the order of cm. Rapid collapse and growth of the bubble occurred on very small spatial (0.3mm), and time (0.1ms) scales followed by Rayleigh-Taylor instabilities at the interface, in addition to the shock waves traveling in the fluid domains are among the phenomena that are observed in the simulation. We then extend our framework to model the underwater implosion of a cylindrical aluminum container considering a monolithic fluid-structure interaction (FSI). The aluminum cylinder, which separates the internal atmospheric-pressure air from the external high-pressure water, is modeled by a three node rotation-free shell element. The cylinder undergoes fast transient deformations, large enough to produce self-contact along it. A novel elastic frictionless contact model is used to detect contact and compute the non-penetrating forces in the discretized domain between the mid-planes of the shell. Two schemes are tested, implicit using the predictor/multi-corrector Bossak scheme, and explicit, using the forward Euler scheme. The results of the two simulations are compared with experimental data

Austrian High-Performance-Computing meeting (AHPC2020)

This booklet is a collection of abstracts presented at the AHPC conference

LES and Hybrid RANS/LES turbulence modelling in unstructured finite volume code and applications to nuclear reactor fuel bundle

Rod bundle is a typical constitutive element of a very wide range of nuclear reactor designs. This thesis describes the investigation of such geometry with wall-resolved Large Eddy Simulation (LES). In order to alleviate the mesh constraint, imposed by the near wall resolution, the usage of embedded refinements and polyhedral meshes is analysed firstly with a inviscid laminar case (Taylor Green vortices) and secondly with a fully turbulent case (channel flow only with embedded refinement). The inviscid test case shows that the addition of embedded refinements decreases the conservation properties of the code. Indeed the accuracy decreases from second order in a structured conformal mesh, to something in between first and second order depending on the quality of the unstructured mesh. Better results are obtained when the interface between refined and coarse areas presents a more regular and structured pattern, reducing the generation of skewed and stretched cells. The channel flow simulation shows that the Reynolds stresses, of some embedded refined meshes, are affected by spurious oscillations. Surprisingly this effect is present in the unstructured meshes with the best orthogonal properties. Indeed analysis of Reynolds stress budgets shows that terms, where the gradient in the wall normal direction is dominant, have a largely oscillatory behaviour. The cause of the problem is attributed to the convective term and in particular in the method used for the gradient reconstruction. As a consequence of these contradictory signs between the inviscid and the fully turbulent cases, the rod bundle test case is analysed using a conventional body fitted multiblock mesh. Two different Reynolds numbers are investigated reporting Reynolds stresses and budgets. The flow is characterised by an energetic and almost periodic azimuthal flow pulsation in the gap region between adjacent sub-channels, which makes turbulent quantities largely different from those in plane channel and pipes and enhances mixing. Experiments found that a constant Strouhal number, with the variation of the Reynolds number, characterises the phenomenon. The frequency analysis finds that present simulations are distinguished by three dominant frequencies, the first in agreement with the experimental value and two higher ones, which might be due to the correlation of the azimuthal velocity in the streamwise direction. Several passive temperature fields are added at the simulations in order to study the effects of the variation of the Prandtl number and the change in boundary conditions (Neumann and Dirichlet). A simplified case where an imbalance of the scalar between adjacent sub-channels is also investigated in order to evaluate the variation of the heat fluxes with respect to the homogeneous case. An alternative solution, to reduce the mesh constraint imposed by the wall, is to hybridize LES with RANS. The main achievement of this work is to integrate the heat transfer modelling to the already existing model for the dynamic part. Further investigations of the blending function, used to merge the two velocity fields, are carried out in conjunction with a study of the model dependency on the mesh resolution. The validation is performed on a fully developed channel flow at different Reynolds numbers and with constant wall heat flux. On coarse meshes the model shows an improvement of the results for both thermal and hydraulic parts with respect to a standard LES. On refined meshes, suitable for wall-resolved LES, the model suffers from a problem of double counting of modelled Reynolds stresses and heat fluxes because the RANS contribution does not naturally disappear as the mesh resolution increases.EThOS - Electronic Theses Online ServiceEPSRCGBUnited Kingdo

Modelling and numerical simulation of combustion and multi-phase flows using finite volume methods on unstructured meshes

The present thesis is devoted to the development and implementation of mathematical models and numerical methods in order to carry out computational simulations of complex heat and mass transfer phenomena. Several areas and topics in the field of Computational Fluid Dynamics (CFD) have been treated and covered during the development of the current thesis, specially combustion and dispersed multi-phase flows. This type of simulations requires the implementation and coupling of different physics. The numerical simulation of multiphysics phenomena is challenging due to the wide range of spatial and temporal scales which can characterize each one of the physics involved in the problem. Moreover, when solving turbulent flows, turbulence itself is a very complex physical phenomenon that can demand a huge computational effort. Hence, in order to make turbulent flow simulations computationally affordable, the turbulence should be modelled. Therefore, throughout this thesis different numerical methods and algorithms have been developed and implemented aiming to perform multiphysics simulations in turbulent flows. The first topic addressed is turbulent combustion. Chapter 2 presents a combustion model able to notably reduce the computational cost of the simulation. The model, namely the Progress-Variable (PV) model, relies on a separation of the spatio-temporal scales between the flow and the chemistry. Moreover, in order to account for the influence of the sub-grid species concentrations and energy fluctuations, the PV model is coupled to the Presumed Conditional Moment (PCM) model. Chapter 2 also shows the development of a smart load-balancing method for the evaluation of chemical reaction rates in parallel combustion simulations. Chapter 3 is devoted to dispersed multiphase flows. This type of flows are composed of a continuous phase and a dispersed phase in the form of unconnected particles or droplets. In this thesis, the Eulergian-Lagrangian approach has been selected. This type of model is the best-suited for dispersed multiphase flows with thousands or millions of particles, and with a flow regime ranging from the very dilute up to relatively dense. In Chapter 4, a new method capable of performing parallel numerical simulations using non-overlapping disconnected mesh domains with adjacent boundaries is presented. The presented algorithm stitches at each iteration independent meshes and solves them as a unique domain. Finally, Chapter 5 addresses a transversal aspect to the previously covered topics throughout the thesis. In this chapter, a self-adaptive strategy for the maximisation of the time-step for the numerical solution of convection-diffusion equations is discussed. The method is capable of determining dynamically at each iteration which is the maximum allowable time-step which assures a stable time integration. Moreover, the method also smartly modifies the temporal integration scheme in order to maximize its stability region depending on the properties of the system matrix.La present tesis està dedicada al desenvolupament e implementació de models matemàtics i mètodes numèrics amb l’objectiu de realitzar simulacions computacionals de fenòmens complexos de transferència de calor i massa. Diverses àrees i temes en el camp de la Dinàmica de Fluids Computacional (CFD) han sigut tractats i coberts durant el desenvolupament de la present tesi, en especial, la combustió i els fluxos multi-fase dispersos. Aquest tipus de simulacions de fenòmens multi-físics es desafiant degut al gran rang d’escales espaio-temporals que poden caracteritzar cada una de les físiques involucrades en el problema. D’altra banda, quan es resolen fluxos turbulents, la pròpia turbulència ja és un fenomen físic molt complex que pot requerir un gran esforç computacional. Per tant, amb l’objectiu de fer les simulacions computacionals de fluxos turbulents computacionalment assequibles, la turbulència ha de ser modelada. Per tant, durant aquesta tesis diferents mètodes i algoritmes han sigut desenvolupats e implementats amb l’objectiu de realitzar simulacions multi-físiques en fluxos turbulents. El primer tema abordat és la combustió turbulenta. El Capítol 2 presenta un model de combustió capaç de reduir notablement el cost computacional de la simulació. El model, anomenat el model Progress-Variable (PV), està basat en la separació d’escales espaio-temporals entre el fluid i la química. A més, amb l’objectiu de tenir en compte l’influencia de les fluctuacions a nivell sub-grid d’energia i concentracions d'espècies, el model PV s’acobla amb el model Presumed Conditional Moment (PCM). El Capítol 2 també mostra el desenvolupament d’un mètode intel·ligent de balanceig de càrrega per l'avaluació de el rati de reacció químic en simulacions de combustió paral·leles. El Capítol 3 està dedicat als fluxos multi-fase dispersos. Aquest tipus de fluids estan formats per una fase continua i una fase dispersa en forma de partícules o gotes inconnexes. En aquesta tesis, l’aproximació Euleriana-Lagrangiana ha sigut la seleccionada. Aquest tipus de model és el més adequat per fluxos multi-fase dispersos amb milers o milions de partícules, i amb règims que van des del molt diluït fins al relativament dens. Al Capítol 4, es presenta un nou mètode capaç de realitzar simulacions numèriques paral·leles utilitzant malles inconnexes no solapades que tenen fronteres adjacents. L’algoritme presentat cus a cada iteració les malles independents i les resol com un únic domini. Finalment, el Capítol 5 tracta un aspecte transversal a tots els temes coberts al llarg de la tesi. En aquest capítol es discuteix una estratègia auto-adaptativa destinada a la maximització del pas de temps per a la solució numèrica d’equacions de convecció-difusió. El mètode es capaç de determinar dinàmicament a cada iteració quin és el màxim pas de temps possible que assegura una integració temporal estable. A més, el mètode també modifica de forma intel·ligent la regió d’estabilitat en funció de les propietats de la matriu del sistema.Postprint (published version