Incompressible Lagrangian fluid flow with thermal coupling

In this monograph is presented a method for the solution of an incompressible viscous fluid flow with heat transfer and solidification usin a fully Lagrangian description on the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation.Postprint (published version

Incompressible lagrangian fluid flow with thermal coupling

A method is presented for the solution of an incompressible viscous fluid flow with heat transfer and solidification using a fully Lagrangian description of the motion. The originality of this method consists in assembling various concepts and techniques which appear naturally due to the Lagrangian formulation. First of all, the Navier-Stokes equations of motion coupled with the Boussinesq approximation must be reformulated in the Lagrangian framework, whereas they have been mostly derived in an Eulerian context. Secondly, the Lagrangian formulation implies to follow the material particles during their motion, which means to convect the mesh in the case of the Finite Element Method (FEM), the spatial discretisation method chosen in this work. This provokes various difficulties for the mesh generation, mainly in three dimensions, whereas it eliminates the classical numerical difficulty to deal with the convective term, as much in the Navier-Stokes equations as in the energy equation. Even without the discretization of the convective term, an efficient iterative solver, which constitutes the only viable alternative for three dimensional problems, must be designed for the class of Generalized Stokes Problems (GSP), which could be able to behave well independently of the mesh Reynolds number, as it can vary greatly for coupled fluid-thermal analysis. Moreover, it offers a natural framework to treat free-surface problems like wave breaking and rough fluid-structure contact. On one hand, the convection of the mesh during one time step after the resolution of the non-linear system provides explicitly the locus of the domain to be considered. On the other hand, fluid-to-fluid and fluid-to-wall contact, as well as the update of the domain due to the remeshing, must be accurately and efficiently performed. Finally, the solidification of the fluid coupled with its motion through a variable viscosity is considered An efficient overall algorithm must be designed to bring the method effective, particularly in a three dimensional context, which is the ambition of this monograph. Various numerical examples are included to validate and highlight the potential of the method

Locally optimal Delaunay-refinement and optimisation-based mesh generation

The field of mesh generation concerns the development of efficient algorithmic techniques to construct high-quality tessellations of complex geometrical objects. In this thesis, I investigate the problem of unstructured simplicial mesh generation for problems in two- and three-dimensional spaces, in which meshes consist of collections of triangular and tetrahedral elements. I focus on the development of efficient algorithms and computer programs to produce high-quality meshes for planar, surface and volumetric objects of arbitrary complexity. I develop and implement a number of new algorithms for mesh construction based on the Frontal-Delaunay paradigm - a hybridisation of conventional Delaunay-refinement and advancing-front techniques. I show that the proposed algorithms are a significant improvement on existing approaches, typically outperforming the Delaunay-refinement technique in terms of both element shape- and size-quality, while offering significantly improved theoretical robustness compared to advancing-front techniques. I verify experimentally that the proposed methods achieve the same element shape- and size-guarantees that are typically associated with conventional Delaunay-refinement techniques. In addition to mesh construction, methods for mesh improvement are also investigated. I develop and implement a family of techniques designed to improve the element shape quality of existing simplicial meshes, using a combination of optimisation-based vertex smoothing, local topological transformation and vertex insertion techniques. These operations are interleaved according to a new priority-based schedule, and I show that the resulting algorithms are competitive with existing state-of-the-art approaches in terms of mesh quality, while offering significant improvements in computational efficiency. Optimised C++ implementations for the proposed mesh generation and mesh optimisation algorithms are provided in the JIGSAW and JITTERBUG software libraries

Numerical studies of flow in porous media using an unstructured approach

Flow and transport in porous media is relevant to many areas of engineering and science including groundwater hydrology and the recovery of oil and gas. Porous materials are characterized by the unique shape and connectivity of the internal void structures which give rise to a large range in macroscopic transport properties. Historically an inability to accurately describe the internal pore-structure has prevented detailed study of the role of pore structure on transport. In recent decades however, the combination of high resolution imaging technologies with computational modeling has seen the development of fundamental pore-scale techniques for studying flow in porous media. Image-based pore-scale modeling of transport phenomena has become an important tool for understanding the complicated relationships between pore structure and measurable macroscopic properties, including permeability and formation factor. This has commonly been achieved by a network-based approach where the pore space is idealized as a series of pores connected by throats, or by a grid-based approach where the voxels of a 3D image represent structured quadrilateral elements or nodal locations. In this work however, image-based unstructured meshing techniques are used to represent voxelised pore spaces by grids comprising entirely of tetrahedral elements. These unstructured tetrahedral grids are used in finite element models to calculate permeability and formation factor. Solutions to the Stokes equations governing creeping, or Darcy flow, are used to validate the finite element approach employed in this work, and to assess the impact of different image-based unstructured meshing strategies on predicted permeability. Testing shows that solutions to the Stokes equations by a P2P1 tetrahedral element are significantly more accurate than solutions based on a P1P1 element, while permeability is shown to be sensitive to structural changes to the pore space induced by different meshing approaches. The modeling approach is also used to investigate the relationship of an electric and hydraulic definition of tortuosity to the Carman-Kozeny equation. The results of simulations using a number of computer generated porous structures indicate that an electrical tortuosity based on computed formation factor is well correlated with the tortuosity suggested by the Carman-Kozeny equation

Balanced-force two-phase flow modelling on unstructured and adaptive meshes

Two-phase flows occur regularly in nature and industrial processes and their understanding is of significant interest in engineering research and development. Various numerical methods to predict two-phase phase flows have been developed as a result of extensive research efforts in past decades, however, most methods are limited to Cartesian meshes. A fully-coupled implicit numerical framework for two-phase flows on unstructured meshes is presented, solving the momentum equations and a specifically constructed continuity constraint in a single equation system. The continuity constraint, derived using a momentum interpolation method, satisfies continuity, provides a strong pressure-velocity coupling and ensures a discrete balance between pressure gradient and body forces. The numerical framework is not limited to specific density ratios or a particular interface topology and includes several novelties. A further step towards a more accurate prediction of two-phase flows on unstructured meshes is taken by proposing a new method to evaluate the interface curvature. The curvature estimates obtained with this new method are shown to be as good as or better than methods reported in literature, which are mostly limited to Cartesian meshes, and the accuracy on structured and unstructured meshes is shown to be comparable. Furthermore, lasting contributions are made towards the understanding of convolution methods for two-phase flow modelling and the underlying mechanisms of parasitic currents are studied in detailed. The mesh resolution is of particular importance for two-phase flows due to the inherent first-order accuracy of the interface position using interface capturing methods. A mesh adaption algorithm for tetrahedral meshes with application to two-phase flows and its implementation are presented. The algorithm is applied to study mesh resolution requirements at interfaces and force-balancing for surface-tension-dominated two-phase flows on adaptive meshes.Open Acces

Virtual Element based formulations for computational materials micro-mechanics and homogenization

In this thesis, a computational framework for microstructural modelling of transverse behaviour of heterogeneous materials is presented. The context of this research is part of the broad and active field of Computational Micromechanics, which has emerged as an effective tool both to understand the influence of complex microstructure on the macro-mechanical response of engineering materials and to tailor-design innovative materials for specific applications through a proper modification of their microstructure. While the classical continuum approximation does not account for microstructural details within the material, computational micromechanics allows detailed modelling of a heterogeneous material's internal structural arrangement by treating each constituent as a continuum. Such an approach requires modelling a certain material microstructure by considering most of the microstructure's morphological features. The most common numerical technique used in computational micromechanics analysis is the Finite Element Method (FEM). Its use has been driven by the development of mesh generation programs, which lead to the quasi-automatic discretisation of the artificial microstructure domain and the possibility of implementing appropriate constitutive equations for the different phases and their interfaces. In FEM's applications to computational micromechanics, the phase arrangements are discretised using continuum elements. The mesh is created so that element boundaries and, wherever required, special interface elements are located at all interfaces between material's constituents. This approach can be effective in modelling many microstructures, and it is readily available in commercial codes. However, the need to accurately resolve the kinematic and stress fields related to complex material behaviours may lead to very large models that may need prohibitive processing time despite the increasing modern computers' performance. When rather complex microstructure's morphologies are considered, the quasi-automatic discretisation process stated before might fail to generate high-quality meshes. Time-consuming mesh regularisation techniques, both automatic and operator-driven, may be needed to obtain accurate numeric results. Indeed, the preparation of high-quality meshes is today one of the steps requiring more attention, and time, from the analyst. In this respect, the development of computational techniques to deal with complex and evolving geometries and meshes with accuracy, effectiveness, and robustness attracts relevant interest. The computational framework presented in this thesis is based on the Virtual Element Method (VEM), a recently developed numerical technique that has proven to provide robust numerical results even with highly-distorted mesh. These peculiar features have been exploited to analyse two-dimensional representations of heterogeneous materials' microstructures. Ad-hoc polygonal multi-domain meshing strategies have been developed and tested to exploit the discretisation freedom that VEM allows. To further simplify the preprocessing stage of the analysis and reduce the total computational cost, a novel hybrid formulation for analysing multi-domain problems has been developed by combining the Virtual Element Method with the well-known Boundary Element Method (BEM). The hybrid approach has been used to study both composite material's transverse behaviour in the presence of inclusions with complex geometries and damage and crack propagation in the matrix phase. Numerical results are presented that demonstrate the potential of the developed framework

Mathematical Models for the Design of Electrical Machines

This book is a comprehensive set of articles reflecting the latest advances and developments in mathematical modeling and the design of electrical machines for different applications. The main models discussed are based on the: i) Maxwell–Fourier method (i.e., the formal resolution of Maxwell’s equations by using the separation of variables method and the Fourier’s series in 2-D or 3-D with a quasi-Cartesian or polar coordinate system); ii) electrical, thermal and magnetic equivalent circuit; iii) hybrid model. In these different papers, the numerical method and the experimental tests have been used as comparisons or validations

Composite Finite Elements for Trabecular Bone Microstructures

In many medical and technical applications, numerical simulations need to be performed for objects with interfaces of geometrically complex shape. We focus on the biomechanical problem of elasticity simulations for trabecular bone microstructures. The goal of this dissertation is to develop and implement an efficient simulation tool for finite element simulations on such structures, so-called composite finite elements. We will deal with both the case of material/void interfaces (complicated domains) and the case of interfaces between different materials (discontinuous coefficients). In classical finite element simulations, geometric complexity is encoded in tetrahedral and typically unstructured meshes. Composite finite elements, in contrast, encode geometric complexity in specialized basis functions on a uniform mesh of hexahedral structure. Other than alternative approaches (such as e.g. fictitious domain methods, generalized finite element methods, immersed interface methods, partition of unity methods, unfitted meshes, and extended finite element methods), the composite finite elements are tailored to geometry descriptions by 3D voxel image data and use the corresponding voxel grid as computational mesh, without introducing additional degrees of freedom, and thus making use of efficient data structures for uniformly structured meshes. The composite finite element method for complicated domains goes back to Wolfgang Hackbusch and Stefan Sauter and restricts standard affine finite element basis functions on the uniformly structured tetrahedral grid (obtained by subdivision of each cube in six tetrahedra) to an approximation of the interior. This can be implemented as a composition of standard finite element basis functions on a local auxiliary and purely virtual grid by which we approximate the interface. In case of discontinuous coefficients, the same local auxiliary composition approach is used. Composition weights are obtained by solving local interpolation problems for which coupling conditions across the interface need to be determined. These depend both on the local interface geometry and on the (scalar or tensor-valued) material coefficients on both sides of the interface. We consider heat diffusion as a scalar model problem and linear elasticity as a vector-valued model problem to develop and implement the composite finite elements. Uniform cubic meshes contain a natural hierarchy of coarsened grids, which allows us to implement a multigrid solver for the case of complicated domains. Besides simulations of single loading cases, we also apply the composite finite element method to the problem of determining effective material properties, e.g. for multiscale simulations. For periodic microstructures, this is achieved by solving corrector problems on the fundamental cells using affine-periodic boundary conditions corresponding to uniaxial compression and shearing. For statistically periodic trabecular structures, representative fundamental cells can be identified but do not permit the periodic approach. Instead, macroscopic displacements are imposed using the same set as before of affine-periodic Dirichlet boundary conditions on all faces. The stress response of the material is subsequently computed only on an interior subdomain to prevent artificial stiffening near the boundary. We finally check for orthotropy of the macroscopic elasticity tensor and identify its axes.Zusammengesetzte finite Elemente für trabekuläre Mikrostrukturen in Knochen In vielen medizinischen und technischen Anwendungen werden numerische Simulationen für Objekte mit geometrisch komplizierter Form durchgeführt. Gegenstand dieser Dissertation ist die Simulation der Elastizität trabekulärer Mikrostrukturen von Knochen, einem biomechanischen Problem. Ziel ist es, ein effizientes Simulationswerkzeug für solche Strukturen zu entwickeln, die sogenannten zusammengesetzten finiten Elemente. Wir betrachten dabei sowohl den Fall von Interfaces zwischen Material und Hohlraum (komplizierte Gebiete) als auch zwischen verschiedenen Materialien (unstetige Koeffizienten). In klassischen Finite-Element-Simulationen wird geometrische Komplexität typischerweise in unstrukturierten Tetraeder-Gittern kodiert. Zusammengesetzte finite Elemente dagegen kodieren geometrische Komplexität in speziellen Basisfunktionen auf einem gleichförmigen Würfelgitter. Anders als alternative Ansätze (wie zum Beispiel fictitious domain methods, generalized finite element methods, immersed interface methods, partition of unity methods, unfitted meshes und extended finite element methods) sind die zusammengesetzten finiten Elemente zugeschnitten auf die Geometriebeschreibung durch dreidimensionale Bilddaten und benutzen das zugehörige Voxelgitter als Rechengitter, ohne zusätzliche Freiheitsgrade einzuführen. Somit können sie effiziente Datenstrukturen für gleichförmig strukturierte Gitter ausnutzen. Die Methode der zusammengesetzten finiten Elemente geht zurück auf Wolfgang Hackbusch und Stefan Sauter. Man schränkt dabei übliche affine Finite-Element-Basisfunktionen auf gleichförmig strukturierten Tetraedergittern (die man durch Unterteilung jedes Würfels in sechs Tetraeder erhält) auf das approximierte Innere ein. Dies kann implementiert werden durch das Zusammensetzen von Standard-Basisfunktionen auf einem lokalen und rein virtuellen Hilfsgitter, durch das das Interface approximiert wird. Im Falle unstetiger Koeffizienten wird die gleiche lokale Hilfskonstruktion verwendet. Gewichte für das Zusammensetzen erhält man hier, indem lokale Interpolationsprobleme gelöst werden, wozu zunächst Kopplungsbedingungen über das Interface hinweg bestimmt werden. Diese hängen ab sowohl von der lokalen Geometrie des Interface als auch von den (skalaren oder tensorwertigen) Material-Koeffizienten auf beiden Seiten des Interface. Wir betrachten Wärmeleitung als skalares und lineare Elastizität als vektorwertiges Modellproblem, um die zusammengesetzten finiten Elemente zu entwickeln und zu implementieren. Gleichförmige Würfelgitter enthalten eine natürliche Hierarchie vergröberter Gitter, was es uns erlaubt, im Falle komplizierter Gebiete einen Mehrgitterlöser zu implementieren. Neben Simulationen einzelner Lastfälle wenden wir die zusammengesetzten finiten Elemente auch auf das Problem an, effektive Materialeigenschaften zu bestimmen, etwa für mehrskalige Simulationen. Für periodische Mikrostrukturen wird dies erreicht, indem man Korrekturprobleme auf der Fundamentalzelle löst. Dafür nutzt man affin-periodische Randwerte, die zu uniaxialem Druck oder zu Scherung korrespondieren. In statistisch periodischen trabekulären Mikrostrukturen lassen sich ebenfalls Fundamentalzellen identifizieren, sie erlauben jedoch keinen periodischen Ansatz. Stattdessen werden makroskopische Verschiebungen zu denselben affin-periodischen Randbedingungen vorgegeben, allerdings durch Dirichlet-Randwerte auf allen Seitenflächen. Die Spannungsantwort des Materials wird anschließend nur auf einem inneren Teilbereich berechnet, um künstliche Versteifung am Rand zu verhindern. Schließlich prüfen wir den makroskopischen Elastizitätstensor auf Orthotropie und identifizieren deren Achsen