Hyperbolic balance laws: residual distribution, local and global fluxes

This review paper describes a class of scheme named "residual distribution schemes" or "fluctuation splitting schemes". They are a generalization of Roe's numerical flux [61] in fluctuation form. The so-called multidimensional fluctuation schemes have historically first been developed for steady homogeneous hyperbolic systems. Their application to unsteady problems and conservation laws has been really understood only relatively recently. This understanding has allowed to make of the residual distribution framework a powerful playground to develop numerical discretizations embedding some prescribed constraints. This paper describes in some detail these techniques, with several examples, ranging from the compressible Euler equations to the Shallow Water equations

BEST : Bézier-Enhanced Shell Triangle : a new rotation-free thin shell finite element

A new thin shell finite element is presented. This new element doesn’ t have rotational degrees of freedom. Instead, in order to overcome the C1 continuity requirement across elements, the author resorts to enhance the geometric description of the flat triangles of a mesh made out of linear triangles, by means of Bernstein polynomials and triangular Bernstein-Bézier patches. The author estimates the surface normals at the nodes of a mesh of triangles, in order to use them to define the Bernstein-Bézier patches. Ubach, Estruch and García-Espinosa performed a comprehensive statistical comparison of different weighting factors. The conclusion of that work is that the inverse of the area of the circumscribed circle to the triangle and the internal angle of the triangle at the node considered, should be used as weighting factor. Using this new weighting factor, we reduce by about 10% the root mean square error in the estimation of normals of randomly generated surfaces with respect to the previous best weighting factor found in the literature. The author uses the information of the normal vectors at the nodes and the triangular Bernstein-Bézier patches to build cubic Bézier triangles. These cubic Bézier triangles are surface interpolants; C1 continuous at the nodes and C0 continuous across the edges. Owing to this approach, the new element is called Bézier-enhanced shell triangle (BEST). The BEST element takes advantage of all the nodes’ connectivities in each triangle of the mesh. The computation of the normal vectors at the nodes doesn’ t depend on the number of triangles surrounding each node of the mesh. The BEST element is independent from the mesh topology. A new paradigm is presented consisting on the reconstruction of the geometry of a cubic triangular element. This geometric reconstruction exploits the properties of cubic B-spline functions (cubic Bézier triangle). This way, the author builds a conforming continuum-based shell finite element. A cubic Bézier triangle has 30 parameters (3 coordinates for each of the 10 control points). Therefore it needs to apply 30 independent conditions. 15 of these conditions are given directly by the positions of the 3 vertices of the triangle and the orientations of the normal vectors at the 3 vertices. 8 of the remaining conditions are imposed introducing energy minimization considerations. These energy minimization considerations serve also to define a well-posed element. The author defines 3 different reduced problems for the 3 different shell deformation modes: bending deformation, membrane (in-plane extension) deformation and in-plane shear (drilling rotation) deformation. The only degrees of freedom of the BEST element are the vertices’ coordinates (9 variables). The remaining 21 parameters are solved internally. In order to fix the values of these 21 internal parameters, each BEST element solves 9 systems of linear equations of rank 3. The BEST element is successfully applied to the analysis of thin shells in linear and geometrically non-linear regimes using an implicit method. The non-linearity is solved using a Total Lagrangian formulation. The author succeeds at pre-integrating through-the-thickness efficiently and accurately. The through-the-thickness integrals are evaluated just once: at the reference configuration. There are just 14 through-the-thickness scalar integrals to perform for each Gauss point. The numerical examples results show that the BEST element has the potential to achieve cubic convergence. Although they also cast doubts on the possibility of reproducing this result for a wide range of problems. For in-plane shear dominated problems, the formulation used in this thesis only achieves linear convergence. For membrane oriented tests with curvature, the convergence is quadratic. The BEST element exhibits membrane locking behavior. The author suggests exploiting further the drilling rotations kinematics in order to solve membrane locking.Se presenta un nuevo elemento finito de lámina delgada. Este nuevo elemento no usa rotaciones como grados de libertad. En su lugar, para sortear el requisito de mantener continuidad C1 entre elementos, el autor mejora la descripción geométrica de los triángulos planos de una malla de triángulos lineales, por medio de polinomios de Bernstein y particiones triangulares de Bernstein-Bézier. Para definir las particiones de Bernstein-Bézier, el autor estima las normales a la superficie en los nodos de una malla de triángulos. Ubach, Estruch y García-Espinosa hicieron una comparación estadística exhaustiva entre distintos factores de ponderación. La conclusión de dicho trabajo conduce a usar como factor de ponderación: el inverso del área de la circunferencia circunscrita al triángulo y el ángulo interno del triángulo en el nodo considerado. Con este nuevo factor de ponderación, se reduce en aproximadamente un 10% el error medio cuadrático cometido en la estimación de las normales de superficies generadas aleatoriamente, respecto del mejor factor usado previamente en la literatura. Con la información de los vectores normales en los nodos, el autor construye triángulos cúbicos de Bézier. Estos triángulos cúbicos de Bézier interpolan la superficie; con continuidad C1 en los nodos y C0 en las aristas. En virtud a este planteamiento, el nuevo elemento recibe el nombre de BEST. El elemento BEST aprovecha todas las conectividades nodales de cada triángulo de la malla. El número de triángulos que rodean cada nodo de la malla no afecta al cálculo de los vectores normales. El elemento BEST es independiente de la topología de la malla. Se propone un nuevo paradigma que consiste en reconstruir la geometría de un elemento triangular cúbico. Esta reconstrucción geométrica aprovecha las propiedades de las funciones cúbicas B-spline (triángulo cúbico de Bézier). Así, el autor crea un elemento de lámina conforme basado en el continuo. Un triángulo cúbico de Bézier tiene 30 parámetros (3 coordenadas para cada uno de los 10 puntos de control). Es necesario aplicar 30 condiciones independientes. 15 de estas condiciones se deducen de la posición de los 3 vértices del triángulo y de los vectores normales en los 3 vértices. De las otras 15 condiciones, 8 se obtienen a partir de criterios de minimización de la energía. Estos criterios de minimización de la energía sirven para definir un elemento bien planteado. El autor desarrolla 3 problemas reducidos para los 3 modos de deformación de la lámina: deformación de flexión, de membrana (extensión en el plano) y de cortante en el plano (rotación de taladro). Los únicos grados de libertad del elemento BEST son las posiciones de los vértices (9 variables). Los otros 21 parámetros se resuelven internamente. Para obtener estos 21 parámetros internos, hay que resolver 9 sistemas de ecuaciones lineales de rango 3 para cada elemento BEST. Se ha aplicado el elemento BEST con éxito al cálculo de láminas delgadas en régimen lineal y geométricamente no-lineal con un método implícito. La no-linealidad se plantea con una formulación Lagrangiana total. Se demuestra cómo pre-integrar en el espesor de manera eficiente y precisa. Solo es preciso evaluar las integrales en el espesor una vez: en la configuración de referencia. Solo hay 14 integrales escalares en el espesor para cada punto de Gauss. Los ejemplos numéricos muestran que el elemento BEST tiene potencial para converger cúbicamente. Pero también existen dudas sobre la capacidad de reproducir de manera consistente este resultado en un amplio rango de problemas. En problemas dominados por la deformación de cortante en el plano, la formulación utilizada en esta tesis solo alcanza convergencia lineal. En ejemplos orientados a la deformación de membrana que incluyen curvatura, la convergencia es cuadrática. El elemento BEST sufre de bloqueo por membrana. El autor sugiere desarrollar más profundamente la cinemática de las rotaciones de taladro para resolver el bloqueo por membrana.Es presenta un nou element finit de làmina prima. Aquest nou element no fa servir rotacions com a graus de llibertat. Enlloc d'això, per esquivar el requisit de mantenir continuïtat C1 entre els elements, l'autor millora la descripció geomètrica dels triangles plans d'una malla de triangles lineals, mitjançant polinomis de Bernstein i particions triangulars de Bernstein-Bézier.Per definir les particions de Bernstein-Bézier, l'autor estima les normals a la superfície en els nodes d'una malla de triangles. Ubach, Estruch i García-Espinosa varen fer una comparació estadística exhaustiva entre diferents factors de ponderació. La conclusió d'aquest treball condueix a fer servir com a factor de ponderació: l'invers de l'àrea de la circumferència circumscrita al triangle i l'angle intern del triangle en el node considerat. Amb aquest nou factor de ponderació, es redueix aproximadament en un 10% l'error quadràtic mig comès en l'estimació de les normals de superfícies generades aleatòriament, respecte del millor factor usat prèviament a la literatura.Amb la informació dels vectors normals en els nodes, l'autor construeix triangles cúbics de Bézier. Aquests triangles cúbics de Bézier interpolen la superfície; amb continuïtat C1 als nodes i C0 a les arestes. En virtut d'aquest plantejament, el nou element rep el nom de BEST (Bézier-enhanced shell triangle).L'element BEST aprofita totes les connectivitats nodals de cada triangle de la malla. El nombre de triangles que envolten cada node de la malla no afecta al càlcul dels vectors normals. L'element BEST és independent de la topologia de la malla.Es proposa un nou paradigma que consisteix en reconstruir la geometria d'un element triangular cúbic. Aquesta reconstrucció geomètrica aprofita les propietats de les funcions cúbiques B-spline (triangle cúbic de Bézier). D'aquesta manera l'autor crea un element de làmina que és conforme i basat en el continu.Un triangle cúbic de Bézier té 30 paràmetres (3 coordenades per cadascun dels 10 punts de control). Cal aplicar 30 condicions independents. 15 d'aquestes condicions es dedueixen de la posició dels 3 vèrtexs del triangle i dels vectors normals en els 3 vèrtexs.De les 15 condicions restants, 8 s'obtenen a partir de criteris de minimització de l'energia. Aquests criteris de minimització de l'energia serveixen per definir un element ben plantejat. L'autor desenvolupa 3 problemes reduïts per als 3 modes de deformació de la làmina: deformació de flexió, de membrana (extensió en el pla) i de tallant en el pla (rotació de barrina).Els únics graus de llibertat de l'element BEST són les posicions dels vèrtexs (9 variables). Els altres 21 paràmetres es resolen internament. Per obtenir aquests 21 paràmetres interns, cal resoldre 9 sistemes d'equacions lineals de rang 3 per cada element BEST.S'ha aplicat l'element BEST amb èxit al càlcul de làmines primes en règim lineal i geomètricament no-lineal fent servir un mètode implícit. La no-linealitat es planteja amb una formulació Lagrangiana total. Es demostra com es pot pre-integrar a través del gruix de manera eficient i precisa. Només cal avaluar les integrals a través del gruix un cop: a la configuració de referència. Només hi ha 14 integrals escalars a través del gruix per a cada punt de Gauss. Els exemples numèrics mostren que l'element BEST té potencial per convergir cúbicament. Però també hi ha dubtes de que aquest resultat es pugui reproduir de manera consistent per un ventall ampli de problemes. En problemes dominats per la deformació de tallant en el pla, la formulació emprada en aquesta tesi només assoleix convergència lineal. En exemples orientats a la deformació de membrana que incloguin curvatura, la convergència és quadràtica. L'element BEST pateix de bloqueig per membrana. L'autor suggereix desenvolupar en més profunditat la cinemàtica de les rotacions de barrina per resoldre el bloqueig per membrana

Splines for damage and fracture in solids

This thesis addresses different aspects of numerical fracture mechanics and spline technology for analysis. An energy-based arc-length control for physically non-linear problems is proposed. It switches between an internal energy-based and a dissipation-based arc-length method. The arc-length control allows to trace an equilibrium path with multiple snap-through and/or snap-back phenomena and only requires two parameters. Phase field models for brittle and cohesive fracture are numerically assessed. The impact of different parameters and boundary conditions on the phase field model for brittle fracture is investigated. It is demonstrated that Γ-convergence is not attained numerically for the phase field model for brittle fracture and that the phase field model for cohesive fracture does not pass a two-dimensional patch test when using an unstructured mesh. The properties of the Bézier extraction operator for T-splines are exploited for the determination of linear dependencies, partition of unity properties, nesting behaviour and local refinement. Unstructured T-spline meshes with extraordinary points are modified such that the blending functions fulfil the partition of unity property and possess a higher continuity. Bézier extraction for Powell-Sabin B-splines is introduced. Different spline technologies are compared when solving Kirchhoff-Love plate theory on a disc with simply supported and clamped boundary conditions. Powell-Sabin B-splines are utilised for smeared and discrete approaches to fracture. Due to the higher continuity of Powell-Sabin B-splines, the implicit fourth order gradient damage model for quasi-brittle materials can be solved and stresses can be computed directly at the crack tip when considering the cohesive zone method

System- and Data-Driven Methods and Algorithms

An increasing complexity of models used to predict real-world systems leads to the need for algorithms to replace complex models with far simpler ones, while preserving the accuracy of the predictions. This two-volume handbook covers methods as well as applications. This first volume focuses on real-time control theory, data assimilation, real-time visualization, high-dimensional state spaces and interaction of different reduction techniques

B-Spline based uncertainty quantification for stochastic analysis

The consideration of uncertainties has become inevitable in state-of-the-art science and technology. Research in the field of uncertainty quantification has gained much importance in the last decades. The main focus of scientists is the identification of uncertain sources, the determination and hierarchization of uncertainties, and the investigation of their influences on system responses. Polynomial chaos expansion, among others, is suitable for this purpose, and has asserted itself as a versatile and powerful tool in various applications. In the last years, its combination with any kind of dimension reduction methods has been intensively pursued, providing support for the processing of high-dimensional input variables up to now. Indeed, this is also referred to as the curse of dimensionality and its abolishment would be considered as a milestone in uncertainty quantification. At this point, the present thesis starts and investigates spline spaces, as a natural extension of polynomials, in the field of uncertainty quantification. The newly developed method 'spline chaos', aims to employ the more complex, but thereby more flexible, structure of splines to counter harder real-world applications where polynomial chaos fails. Ordinarily, the bases of polynomial chaos expansions are orthogonal polynomials, which are replaced by B-spline basis functions in this work. Convergence of the new method is proved and emphasized by numerical examples, which are extended to an accuracy analysis with multi-dimensional input. Moreover, by solving several stochastic differential equations, it is shown that the spline chaos is a generalization of multi-element Legendre chaos and superior to it. Finally, the spline chaos accounts for solving partial differential equations and results in a stochastic Galerkin isogeometric analysis that contributes to the efficient uncertainty quantification of elliptic partial differential equations. A general framework in combination with an a priori error estimation of the expected solution is provided

Reduced Models for Optimal Control, Shape Optimization and Inverse Problems in Haemodynamics

The objective of this thesis is to develop reduced models for the numerical solution of optimal control, shape optimization and inverse problems. In all these cases suitable functionals of state variables have to be minimized. State variables are solutions of a partial differential equation (PDE), representing a constraint for the minimization problem. The solution of these problems induce large computational costs due to the numerical discretization of PDEs and to iterative procedures usually required by numerical optimization (many-query context). In order to reduce the computational complexity, we take advantage of the reduced basis (RB) approximation for parametrized PDEs, once the state problem has been reformulated in parametrized form. This method enables a rapid and reliable approximation of parametrized PDEs by constructing low-dimensional, problem-specific approximation spaces. In case of PDEs defined over domains of variable shapes (e.g. in shape optimization problems) we need to introduce suitable, low-dimensional shape parametrization techniques in order to tackle the geometrical complexity. Free-Form Deformations and Radial-Basis Functions techniques have been analyzed and successfully applied with this aim. We analyze the reduced framework built by coupling these tools and apply it to the solution of optimal control and shape optimization problems. Robust optimization problems under uncertain conditions are also taken into consideration. Moreover, both deterministic and Bayesian frameworks are set in order to tackle inverse identification problems. As state equations, we consider steady viscous flow problems described by Stokes or Navier-Stokes equations, for which we provide a detailed analysis and construction of RB approximation and a posteriori error estimation. Several numerical test cases are also illustrated to show efficacy and reliability of RB approximations. We exploit this general reduced framework to solve some optimization and inverse problems arising in haemodynamics. More specifically, we focus on the optimal design of cardiovascular prostheses, such as bypass grafts, and on inverse identification of pathological conditions or flow/shape features in realistic parametrized geometries, such as carotid artery bifurcations

Contact problem modelling using the Cartesian grid Finite Element Method

Tesis por compendio[ES] La interacción de contacto entre sólidos deformables es uno de los fenómenos más complejos en el ámbito de la mecánica computacional. La resolución de este problema requiere de algoritmos robustos para el tratamiento de no linealidades geométricas. El Método de Elementos Finitos (MEF) es uno de los más utilizados para el diseño de componentes mecánicos, incluyendo la solución de problemas de contacto. En este método el coste asociado al proceso de discretización (generación de malla) está directamente vinculado a la definición del contorno a modelar, lo cual dificulta la introducción en la simulación de superficies complejas, como las superficies NURBS, cada vez más utilizadas en el diseño de componentes. Esta tesis está basada en el "Cartesian grid Finite Element Method" (cgFEM). En esta metodología, encuadrada en la categoría de métodos "Immersed Boundary", se extiende el problema a un dominio de aproximación (cuyo mallado es sencillo de generar) que contiene al dominio de análisis completamente en su interior. Al desvincular la discretización de la definición del contorno del problema se reduce drásticamente el coste de generación de malla. Es por ello que el método cgFEM es una herramienta adecuada para la resolución de problemas en los que es necesario modificar la geometría múltiples veces, como el problema de optimización de forma o la simulación de desgaste. El método cgFEM permite también crear de manera automática y eficiente modelos de Elementos Finitos a partir de imágenes médicas. La introducción de restricciones de contacto habilitaría la posibilidad de considerar los diferentes estados de integración implante-tejido en procesos de optimización personalizada de implantes. Así, en esta tesis se desarrolla una formulación para resolver problemas de contacto 3D con el método cgFEM, considerando tanto modelos de contacto sin fricción como problemas con rozamiento de Coulomb. La ausencia de nodos en el contorno en cgFEM impide la aplicación de métodos tradicionales para imponer las restricciones de contacto, por lo que se ha desarrollado una formulación estabilizada que hace uso de un campo de tensiones recuperado para asegurar la estabilidad del método. Para una mayor precisión de la solución, se ha introducido la definición analítica de las superficies en contacto en la formulación propuesta. Además, se propone la mejora de la robustez de la metodología cgFEM en dos aspectos: el control del mal condicionamiento del problema numérico mediante un método estabilizado, y la mejora del campo de tensiones recuperado, utilizado en el proceso de estimación de error. La metodología propuesta se ha validado a través de diversos ejemplos numéricos presentados en la tesis, mostrando el gran potencial de cgFEM en este tipo de problemas.[CA] La interacció de contacte entre sòlids deformables és un dels fenòmens més complexos en l'àmbit de la mecànica computacional. La resolució d'este problema requerix d'algoritmes robustos per al tractament de no linealitats geomètriques. El Mètode dels Elements Finits (MEF) és un dels més utilitzats per al disseny de components mecànics, incloent la solució de problemes de contacte. En este mètode el cost associat al procés de discretització (generació de malla) està directament vinculat a la definició del contorn a modelar, la qual cosa dificulta la introducció en la simulació de superfícies complexes, com les superfícies NURBS, cada vegada més utilitzades en el disseny de components. Esta tesi està basada en el "Cartesian grid Finite Element Method" (cgFEM). En esta metodologia, enquadrada en la categoria de mètodes "Immersed Boundary", s'estén el problema a un domini d'aproximació (el mallat del qual és senzill de generar) que conté al domini d'anàlisi completament en el seu interior. Al desvincular la discretització de la definició del contorn del problema es reduïx dràsticament el cost de generació de malla. És per això que el mètode cgFEM és una ferramenta adequada per a la resolució de problemes en què és necessari modificar la geometria múltiples vegades, com el problema d'optimització de forma o la simulació de desgast. El mètode cgFEM permet també crear de manera automàtica i eficient models d'Elements Finits a partir d'imatges mèdiques. La introducció de restriccions de contacte habilitaria la possibilitat de considerar els diferents estats d'integració implant-teixit en processos d'optimització personalitzada d'implants. Així, en esta tesi es desenvolupa una formulació per a resoldre problemes de contacte 3D amb el mètode cgFEM, considerant tant models de contacte sense fricció com a problemes amb fregament de Coulomb. L'absència de nodes en el contorn en cgFEM impedix l'aplicació de mètodes tradicionals per a imposar les restriccions de contacte, per la qual cosa s'ha desenvolupat una formulació estabilitzada que fa ús d'un camp de tensions recuperat per a assegurar l'estabilitat del mètode. Per a una millor precisió de la solució, s'ha introduït la definició analítica de les superfícies en contacte en la formulació proposada. A més, es proposa la millora de la robustesa de la metodologia cgFEM en dos aspectes: el control del mal condicionament del problema numèric per mitjà d'un mètode estabilitzat, i la millora del camp de tensions recuperat, utilitzat en el procés d'estimació d'error. La metodologia proposada s'ha validat a través de diversos exemples numèrics presentats en la tesi, mostrant el gran potencial de cgFEM en este tipus de problemes.[EN] The contact interaction between elastic solids is one of the most complex phenomena in the computational mechanics research field. The solution of such problem requires robust algorithms to treat the geometrical non-linearities characteristic of the contact constrains. The Finite Element Method (FE) has become one of the most popular options for the mechanical components design, including the solution of contact problems. In this method the computational cost of the generation of the discretization (mesh generation) is directly related to the complexity of the analysis domain, namely its boundary. This complicates the introduction in the numerical simulations of complex surfaces (for example NURBS), which are being increasingly used in the CAD industry. This thesis is grounded on the Cartesian grid Finite Element Method (cgFEM). In this methodology, which belongs to the family of Immersed Boundary methods, the problem at hand is extended to an approximation domain which completely embeds the analysis domain, and its meshing is straightforward. The decoupling of the boundary definition and the discretization mesh results in a great reduction of the mesh generation's computational cost. Is for this reason that the cgFEM is a suitable tool for the solution of problems that require multiple geometry modifications, such as shape optimization problems or wear simulations. The cgFEM is also capable of automatically generating FE models from medical images without the intermediate step of generating CAD entities. The introduction of the contact interaction would open the possibility to consider different states of the union between implant and living tissue for the design of optimized implants, even in a patient-specific process. Hence, in this thesis a formulation for solving 3D contact problems with the cgFEM is presented, considering both frictionless and Coulomb's friction problems. The absence of nodes along the boundary in cgFEM prevents the enforcement of the contact constrains using the standard procedures. Thus, we develop a stabilized formulation that makes use of a recovered stress field, which ensures the stability of the method. The analytical definition of the contact surfaces (by means of NURBS) has been included in the proposed formulation in order to increase the accuracy of the solution. In addition, the robustness of the cgFEM methodology is increased in this thesis in two different aspects: the control of the numerical problem's ill-conditioning by means of a stabilized method, and the enhancement of the stress recovered field, which is used in the error estimation procedure. The proposed methodology has been validated through several numerical examples, showing the great potential of the cgFEM in these type of problems.Navarro Jiménez, JM. (2019). Contact problem modelling using the Cartesian grid Finite Element Method [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/124348TESISCompendi

A basis for the representation, manufacturing tool path generation and scanning measurement of smooth freeform surfaces

Freeform surfaces find wide application, particularly in optics, from unique single-surface science programmes to mobile phone lenses manufactured in billions. This thesis presents research into the mathematical and algorithmic basis for the generation and measurement of smooth freeform surfaces. Two globally significant cases are reported: 1) research in this thesis created prototype segments for the world’s largest telescope; 2) research in this thesis made surfaces underpinning the redefinition of one of the seven SI base units – the kelvin - and also what will be the newly (and permanently) defined value for the Boltzmann constant. Theresearchdemonstratestwounderlyingphilosophiesofprecisionengineering, the critical roles of determinism and of precision measurement in precise manufacturing. The thesis presents methods, and reports their implementation, for the manufacture of freeform surfaces through a comprehensive strategy for tool path generation using minimum axis-count ultra-precision machine tools. In the context of freeform surface machining, the advantages of deterministic motion performance of three-axis machines are brought to bear through a novel treatment of the mathematics of variable contact point geometry. This is applied to ultra-precision diamond turning and ultra-precision large optics grinding with the Cranfield Box machine. New techniques in freeform surface representation, tool path generation, freeform tool shape representation and error compensation are presented. A comprehensive technique for very high spatial resolution CMM areal scanning of freeform surfaces is presented, with a new treatment of contact error removal, achieving interferometer-equivalent surface representation, with 1,000,000+ points and sub-200 nm rms noise without the use of any low-pass filtering

Fracture scale fluid flow models for the simulation of poroelasticity

Fractures are a common occurrence in poroelastic materials: They are created to aid in underground resource recovery, or are unwanted during failure and collapse of materials. One of the main challenges for simulating these fractures is their small opening height compared to their length, making direct simulation of the interior of the fracture computationally expensive. In this thesis, models which reduce the two-dimensional fluid flow in the interior of the fracture to the in and outflow at a one-dimensional discontinuity are extended to include the complex fluid rheology of non-Newtonian power-law and Carreau fluids. One of the main advantages of the obtained sub-grid models is their ability to reconstruct the fluid behaviour through post processing the obtained results, allowing a detailed description of the fluid within the fracture to be re-obtained. In addition, these sub-grid models are also applied to multiphase flows, allowing the interactions between the fluid phases within the fracture to be included. Finally, a numerical two-scale model is presented, coupling numerically resolved velocity profiles within the fracture to the mass balance at the discontinuity. This allows for velocity profiles for which an analytic solution is not available to be included, such as fluids displaying inertial effects. These sub-grid models are implemented using finite element methods based on standard Lagrangian elements, non-uniform rational basis splines, and T-splines. While the Lagrangian elements are convenient and commonly used, it is shown that the increased inter-element continuity of Non-Uniform Rational B-Splines (NURBS) and T-splines is required to obtain continuous fracture outflows. It is furthermore shown that this increased continuity is beneficial for the convergence rate of the non-linear solver. The benefits of using lumped integration for the fracture inflow term are demonstrated, suppressing fluid velocity oscillations, and a special fracture tip integration scheme is presented which prevents non-physical fracture inflows for NURBS. Finally, a method to generate unequal order T-spline meshes is presented, allowing for interface elements to solely be inserted for fractured elements, and making mesh refinement near the discontinuity possible. The fracture scale models and discretisation methods are used to investigate the interactions between the fluid and fracture propagation. It is shown that including a non-Newtonian fluid rheology can significantly alter the propagation velocity of the fracture, and the velocity of the fluid within the fracture. For multiphase flows, the fracture scale models show the importance of including inter-phase interactions within the fracture, providing significantly different results depending on the assumed flow model, either bubbly flow or separated flow. By comparing the fracture flow model to results obtained through direct simulation of the fracture flow the validity and accuracy of the fracture scale models are confirmed. Finally, simulations including inertial effects in the porous material show the interstitial fluid is capable of causing "stick-slip" like behaviour, and simulations using a numerical two-scale approach give an indication of the possible pressure oscillations resulting from stepwise propagation

Feasible, Robust and Reliable Automation and Control for Autonomous Systems

The Special Issue book focuses on highlighting current research and developments in the automation and control field for autonomous systems as well as showcasing state-of-the-art control strategy approaches for autonomous platforms. The book is co-edited by distinguished international control system experts currently based in Sweden, the United States of America, and the United Kingdom, with contributions from reputable researchers from China, Austria, France, the United States of America, Poland, and Hungary, among many others. The editors believe the ten articles published within this Special Issue will be highly appealing to control-systems-related researchers in applications typified in the fields of ground, aerial, maritime vehicles, and robotics as well as industrial audiences