Self-Adaptive Methods for PDE

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Anisotropic Adaptivity and Subgrid Scale Modelling for the Solution of the Neutron Transport Equation with an Emphasis on Shielding Applications

This thesis demonstrates advanced new discretisation and adaptive meshing technologies that improve the accuracy and stability of using finite element discretisations applied to the Boltzmann transport equation (BTE). This equation describes the advective transport of neutral particles such as neutrons and photons within a domain. The BTE is difficult to solve, due to its large phase space (three dimensions of space, two of angle and one each of energy and time) and the presence of non-physical oscillations in many situations. This work explores the use of a finite element method that combines the advantages of the two schemes: the discontinuous and continuous Galerkin methods. The new discretisation uses multiscale (subgrid) finite elements that work locally within each element in the finite element mesh in addition to a global, continuous, formulation. The use of higher order functions that describe the variation of the angular flux over each element is also explored using these subgrid finite element schemes. In addition to the spatial discretisation, methods have also been developed to optimise the finite element mesh in order to reduce resulting errors in the solution over the domain, or locally in situations where there is a goal of specific interest (such as a dose in a detector region). The chapters of this thesis have been structured to be submitted individually for journal publication, and are arranged as follows. Chapter 1 introduces the reader to motivation behind the research contained within this thesis. Chapter 2 introduces the forms of the BTE that are used within this thesis. Chapter 3 provides the methods that are used, together with examples, of the validation and verification of the software that was developed as a result of this work, the transport code RADIANT. Chapter 4 introduces the inner element subgrid scale finite element discretisation of the BTE that forms the basis of the discretisations within RADIANT and explores its convergence and computational times on a set of benchmark problems. Chapter 5 develops the error metrics that are used to optimise the mesh in order to reduce the discretisation error within a finite element mesh using anisotropic adaptivity that can use elongated elements that accurately resolves computational demanding regions, such as in the presence of shocks. The work of this chapter is then extended in Chapter 6 that forms error metrics for goal based adaptivity to minimise the error in a detector response. Finally, conclusions from this thesis and suggestions for future work that may be explored are discussed in Chapter 7.Open Acces

Large-scale tree-based unfitted finite elements for metal additive manufacturing

This thesis addresses large-scale numerical simulations of partial differential equations posed on evolving geometries. Our target application is the simulation of metal additive manufacturing (or 3D printing) with powder-bed fusion methods, such as Selective Laser Melting (SLM), Direct Metal Laser Sintering (DMLS) or Electron-Beam Melting (EBM). The simulation of metal additive manufacturing processes is a remarkable computational challenge, because processes are characterised by multiple scales in space and time and multiple complex physics that occur in intricate three-dimensional growing-in-time geometries. Only the synergy of advanced numerical algorithms and high-performance scientific computing tools can fully resolve, in the short run, the simulation needs in the area. The main goal of this Thesis is to design a a novel highly-scalable numerical framework with multi-resolution capability in arbitrarily complex evolving geometries. To this end, the framework is built by combining three computational tools: (1) parallel mesh generation and adaptation with forest-of-trees meshes, (2) robust unfitted finite element methods and (3) parallel finite element modelling of the geometry evolution in time. Our numerical research is driven by several limitations and open questions in the state-of-the-art of the three aforementioned areas, which are vital to achieve our main objective. All our developments are deployed with high-end distributed-memory implementations in the large-scale open-source software project FEMPAR. In considering our target application, (4) temporal and spatial model reduction strategies for thermal finite element models are investigated. They are coupled to our new large-scale computational framework to simplify optimisation of the manufacturing process. The contributions of this Thesis span the four ingredients above. Current understanding of (1) is substantially improved with rigorous proofs of the computational benefits of the 2:1 k-balance (ease of parallel implementation and high-scalability) and the minimum requirements a parallel tree-based mesh must fulfil to yield correct parallel finite element solvers atop them. Concerning (2), a robust, optimal and scalable formulation of the aggregated unfitted finite element method is proposed on parallel tree-based meshes for elliptic problems with unfitted external contour or unfitted interfaces. To the author’s best knowledge, this marks the first time techniques (1) and (2) are brought together. After enhancing (1)+(2) with a novel parallel approach for (3), the resulting framework is able to mitigate a major performance bottleneck in large-scale simulations of metal additive manufacturing processes by powder-bed fusion: scalable adaptive (re)meshing in arbitrarily complex geometries that grow in time. Along the development of this Thesis, our application problem (4) is investigated in two joint collaborations with the Monash Centre for Additive Manufacturing and Monash University in Melbourne, Australia. The first contribution is an experimentally-supported thorough numerical assessment of time-lumping methods, the second one is a novel experimentally-validated formulation of a new physics-based thermal contact model, accounting for thermal inertia and suitable for model localisation, the so-called virtual domain approximation. By efficiently exploiting high-performance computing resources, our new computational framework enables large-scale finite element analysis of metal additive manufacturing processes, with increased fidelity of predictions and dramatical reductions of computing times. It can also be combined with the proposed model reductions for fast thermal optimisation of the manufacturing process. These tools open the path to accelerate the understanding of the process-to-performance link and digital product design and certification in metal additive manufacturing, two milestones that are vital to exploit the technology for mass-production.Aquesta tesi tracta la simulació a gran escala d'equacions en derivades parcials sobre geometries variables. L'aplicació principal és la simulació de procesos de fabricació additiva (o impressió 3D) amb metalls i per mètodes de fusió de llit de pols, com ara Selective Laser Melting (SLM), Direct Metal Laser Sintering (DMLS) o Electron-Beam Melting (EBM). La simulació d'aquests processos és un repte computacional excepcional, perquè els processos estan caracteritzats per múltiples escales espaitemporals i múltiples físiques que tenen lloc sobre geometries tridimensionals complicades que creixen en el temps. La sinèrgia entre algorismes numèrics avançats i eines de computació científica d'alt rendiment és la única via per resoldre completament i a curt termini les necessitats en simulació d'aquesta àrea. El principal objectiu d'aquesta tesi és dissenyar un nou marc numèric escalable de simulació amb capacitat de multiresolució en geometries complexes i variables. El nou marc es construeix unint tres eines computacionals: (1) mallat paral·lel i adaptatiu amb malles de boscs d'arbre, (2) mètodes d'elements finits immersos robustos i (3) modelització en paral·lel amb elements finits de geometries que creixen en el temps. Algunes limitacions i problemes oberts en l'estat de l'art, que són claus per aconseguir el nostre objectiu, guien la nostra recerca. Tots els desenvolupaments s'implementen en arquitectures de memòria distribuïda amb el programari d'accés obert FEMPAR. Quant al problema d'aplicació, (4) s'investiguen models reduïts en espai i temps per models tèrmics del procés. Aquests models reduïts s'acoplen al nostre marc computacional per simplificar l'optimització del procés. Les contribucions d'aquesta tesi abasten els quatre punts de dalt. L'estat de l'art de (1) es millora substancialment amb proves riguroses dels beneficis computacionals del 2:1 balancejat (fàcil paral·lelització i alta escalabilitat), així com dels requisits mínims que aquest tipus de mallat han de complir per garantir que els espais d'elements finits que s'hi defineixin estiguin ben posats. Quant a (2), s'ha formulat un mètode robust, òptim i escalable per agregació per problemes el·líptics amb contorn o interface immerses. Després d'augmentar (1)+(2) amb un nova estratègia paral·lela per (3), el marc de simulació resultant mitiga de manera efectiva el principal coll d'ampolla en la simulació de processos de fabricació additiva en llits de pols de metall: adaptivitat i remallat escalable en geometries complexes que creixen en el temps. Durant el desenvolupament de la tesi, es col·labora amb el Monash Centre for Additive Manufacturing i la Universitat de Monash de Melbourne, Austràlia, per investigar el problema d'aplicació. En primer lloc, es fa una anàlisi experimental i numèrica exhaustiva dels mètodes d'aggregació temporal. En segon lloc, es proposa i valida experimental una nova formulació de contacte tèrmic que té en compte la inèrcia tèrmica i és adequat per a localitzar el model, l'anomenada aproximació per dominis virtuals. Mitjançant l'ús eficient de recursos computacionals d'alt rendiment, el nostre nou marc computacional fa possible l'anàlisi d'elements finits a gran escala dels processos de fabricació additiva amb metalls, amb augment de la fidelitat de les prediccions i reduccions significatives de temps de computació. Així mateix, es pot combinar amb els models reduïts que es proposen per l'optimització tèrmica del procés de fabricació. Aquestes eines contribueixen a accelerar la comprensió del lligam procés-rendiment i la digitalització del disseny i certificació de productes en fabricació additiva per metalls, dues fites crucials per explotar la tecnologia en producció en massa.Postprint (published version

Computational fluid dynamics of coupled free/porous regimes: a specialised case of pleated cartridge filter

The multidisciplinary project AEROFIL has been defined and coordinated with the idea of developing novel filter designs to be employed in aeronautic hydraulic systems. The cartridge filters would be constructed using eco-friendly filtration media supported by unconventional disposable or reusable solid components. My main contribution to this project is the development of a robust and cost-effective design and analysis tool for simulating the hydrodynamics in these pleated cartridge filters. The coupled free and porous flow regimes are generally observed in filtration processes. These processes have been the subject of intense investigation for researchers over the decades who are striving hard to resolve some of the critical issues related to the free/porous interfacial constraints and their mathematical representations concerning its industrial applications. [Continues.

Adaptive discontinuous Galerkin methods for the neutron transport equation

In this thesis we study the neutron transport (Boltzmann transport equation) which is used to model the movement of neutrons inside a nuclear reactor. More specifically we consider the mono-energetic, time independent neutron transport equation. The neutron transport equation has predominantly been solved numerically by employing low order discretisation methods, particularly in the case of the angular domain. We proceed by surveying the advantages and disadvantages of common numerical methods developed for the numerical solution of the neutron transport equation before explaining our choice of using a discontinuous Galerkin (DG) discretisation for both the spatial and angular domain. The bulk of the thesis describes an arbitrary order in both angle and space solver for the neutron transport equation. We discuss some implementation issues, including the use of an ordered solver to facilitate the solution of the linear systems resulting from the discretisation. The resulting solver is benchmarked using both source and critical eigenvalue computations. In the pseudo three--dimensional case we employ our solver for the computation of the critical eigenvalue for three industrial benchmark problems. We then employ the Dual Weighted Residual (DWR) approach to adaptivity to derive and implement error indicators for both two--dimensional and pseudo three--dimensional neutron transport source problems. Finally, we present some preliminary results on the use of a DWR indicator for the eigenvalue problem

Adaptive discontinuous Galerkin methods for the neutron transport equation

In this thesis we study the neutron transport (Boltzmann transport equation) which is used to model the movement of neutrons inside a nuclear reactor. More specifically we consider the mono-energetic, time independent neutron transport equation. The neutron transport equation has predominantly been solved numerically by employing low order discretisation methods, particularly in the case of the angular domain. We proceed by surveying the advantages and disadvantages of common numerical methods developed for the numerical solution of the neutron transport equation before explaining our choice of using a discontinuous Galerkin (DG) discretisation for both the spatial and angular domain. The bulk of the thesis describes an arbitrary order in both angle and space solver for the neutron transport equation. We discuss some implementation issues, including the use of an ordered solver to facilitate the solution of the linear systems resulting from the discretisation. The resulting solver is benchmarked using both source and critical eigenvalue computations. In the pseudo three--dimensional case we employ our solver for the computation of the critical eigenvalue for three industrial benchmark problems. We then employ the Dual Weighted Residual (DWR) approach to adaptivity to derive and implement error indicators for both two--dimensional and pseudo three--dimensional neutron transport source problems. Finally, we present some preliminary results on the use of a DWR indicator for the eigenvalue problem

Turbulence models with adaptive meshing for industrial CFD

Computational fluid dynamics (CFD) and affordable computing power have advanced considerably in recent years, bringing full 3D simulation of complex high Reynolds number flows within reach of industry. However, providing accurate and trustworthy results in diverse flows with constraints on computational resources is still a considerable challenge. Owing to the complexity of commonly-encountered turbulent flows, robust turbulence models are required which do not have to be manually tuned to specific flow conditions to ensure their accuracy. In this regard, a highly effective approach is unstructured mesh adaptivity which automatically refines or coarsens the mesh locally in order to achieve a desired accuracy with minimum computational effort. However, the use of such adaptive meshes with turbulence models raises questions about the origins and interactions of various errors. This thesis describes the development, verification and validation of robust turbulence models suited to high Reynolds number single-phase turbulent flow using unstructured adaptive meshes. The main focus of this thesis is a new tensorial dynamic large eddy simulation (LES) model. The novel combination of the dynamic LES method with a tensorial definition of filter width is ideal for capturing the anisotropy and inhomogeneity of turbulence. This model is designed for use with unstructured mesh adaptivity, which enables the simulation of turbulent flow with high efficiency in terms of mesh resolution. Furthermore, the model is robust since both the resolution and the sub-filter-scale (SFS) stresses adapt to local flow conditions so that little a priori knowledge of the flow is required. Verification tests of the filtering method and validation of the new LES model in the 3D backward-facing step are presented. To provide context for the research, the contribution made by CFD simulations to the analysis of nuclear reactor safety and performance is discussed. The practicalities of performing simulations on high performance computing (HPC) facilities are also discussed. Background theory necessary to understand the research is presented, including a mathematical description of turbulent flow and the classes of CFD methods used to approximate it. A review of turbulence models, discretisation methods, boundary conditions and adaptive meshing methods is included. The construction and testing of a Reynolds-averaged Navier-Stokes (RANS) k - ε turbulence model and a scale-adaptive very large eddy simulation (VLES) model in the open-source CFD code Fluidity are also described. The development of a law-of-the-wall boundary condition for turbulent flow in variational (weak) form is also presented. Verification tests are performed to establish that the k - ε model has been coded correctly. Validation of the RANS model and the wall function using fixed and adaptive meshes is carried out in the 2D backward-facing step. Finally, results of simulations of a vortex diode device using various turbulence models are presented and compared to results from the commercial CFD code CFX and experimental results. This study was carried out during the industrial component of the Engineering Doctorate, which was intended to further the development and understanding of CFD at Rolls-Royce Nuclear. The device presents a challenging test case for CFD but some useful conclusions are reached about how to model it. The thesis concludes with a summary of findings and proposals for further research

Efficient "black-box" multigrid solvers for convection-dominated problems

The main objective of this project is to develop a "black-box" multigrid preconditioner for the iterative solution of finite element discretisations of the convection-diffusion equation with dominant convection. This equation can be considered a stand alone scalar problem or as part of a more complex system of partial differential equations, such as the Navier-Stokes equations. The project will focus on the stand alone scalar problem. Multigrid is considered an optimal preconditioner for scalar elliptic problems. This strategy can also be used for convection-diffusion problems, however an appropriate robust smoother needs to be developed to achieve mesh-independent convergence. The focus of the thesis is on the development of such a smoother. In this context a novel smoother is developed referred to as truncated incomplete factorisation (tILU) smoother. In terms of computational complexity and memory requirements, the smoother is considerably less expensive than the standard ILU(0) smoother. At the same time, it exhibits the same robustness as ILU(0) with respect to the problem and discretisation parameters. The new smoother significantly outperforms the standard damped Jacobi smoother and is a competitor to the Gauss-Seidel smoother (and in a number of important cases tILU outperforms the Gauss-Seidel smoother). The new smoother depends on a single parameter (the truncation ratio). The project obtains a default value for this parameter and demonstrated the robust performance of the smoother on a broad range of problems. Therefore, the new smoothing method can be regarded as "black-box". Furthermore, the new smoother does not require any particular ordering of the nodes, which is a prerequisite for many robust smoothers developed for convection-dominated convection-diffusion problems. To test the effectiveness of the preconditioning methodology, we consider a number of model problems (in both 2D and 3D) including uniform and complex (recirculating) convection fields discretised by uniform, stretched and adaptively refined grids. The new multigrid preconditioner within block preconditioning of the Navier-Stokes equations was also tested. The numerical results gained during the investigation confirm that tILU is a scalable, robust smoother for both geometric and algebraic multigrid. Also, comprehensive tests show that the tILU smoother is a competitive method.EThOS - Electronic Theses Online ServiceEPSRCGBUnited Kingdo

Fault tolerant flight control system design for unmanned aerial vehicles

Safety and reliability of air vehicles is of the utmost importance. This is particularly true for large civil transport aircraft where a large number of human lives depend on safety critical design. With the increase in the use of unmanned aerial vehicles (UAVs) in our airspace it is essential that UAV safety is also given attention to prevent devastating failures which could ultimately lead to loss of human lives. While civil aircraft have human operators, the pilot, to counteract any unforeseen faults, autonomous UAVs are only as good as the on board flight computer. Large civil aircraft also have the luxury of weight hence redundant actuators (control surfaces) can be installed and in the event of a faulty set of actuators the redundant actuators can be brought into action to negate the effects of any faults. Again weight is a luxury that UAVs do not have. The main objective of this research is to study the design of a fault tolerant flight controller that can exploit the mathematical redundancies in the flight dynamic equations as opposed to adding hardware redundancies that would result in significant weight increase. This thesis presents new research into fault tolerant control for flight vehicles. Upon examining the flight dynamic equations it can be seen, for example, that an aileron, which is primarily used to perform a roll manoeuvre, can be used to execute a limited pitch moment. Hence a control method is required that moves away from the traditional fixed structure model where control surface roles are clearly defined. For this reason, in this thesis, I have chosen to study the application of model predictive control (MPC) to fault tolerant control systems. MPC is a model based method where a model of the plant forms an integral part of the controller. An optimisation is performed based on model estimations of the plant and the inputs are chosen via an optimisation process. One of the main contributions of this thesis is the development of a nonlinear model predictive controller for fault tolerant flight control. An aircraft is a highly nonlinear system hence if a nonlinear model can be integrated into the control process the cross-coupling effects of the control surface contributions can be easily exploited. An active fault tolerant control system comprises not only of the fault tolerant controller but also a fault detection and isolation subsystem. A common fault detection method is based on parameter estimation using filtering techniques. The solution proposed in this thesis uses an unscented Kalman filter (UKF) for parameter estimation and controller updates. In summary the main contribution of this thesis is the development of a new active fault tolerant flight control system. This new innovative controller exploits the idea of analytical redundancy as opposed to hardware redundancy. It comprises of a nonlinear model predictive based controller using pseudospectral discretisation to solve the nonlinear optimal control problem. Furthermore a UKF is incorporated into the design of the active fault tolerant flight control system