Dissertations / Theses on the topic 'And Fast Level Set Method'

The level-set method is an implicit interface capturing method that can be used in two or more dimensions. The method is popular e.g. in computer graphics, and as here, in simulations of two-phase flow. The motivation for the simulations performed here is to obtain a better understanding of the complex two-phase flow phenomena ocurring in heat exchangers used for liquefaction of natural gas, including the study of droplet-film interactions and coalescence.One of the main advantages of the level-set method is that it handles changes in the interface topology in a natural way. In the present work, the calculation of the curvature and normal vectors of an interface represented by the level-set method is considered. The curvature and normal vectors are usually calculated using central-difference stencils, but this standard method fails when the interface undergoesa topological change, e.g. when two droplets collide and merge. Several methodshave previously been developed to handle this problem. In the present work,a new method is presented, which is a development on existing methods. The newmethod handles more general cases than previous methods. In contrast to someprevious methods, the present method retains the implicit formulation and can easily be extended to three-dimensional simulations, as demonstrated in this work.Briefly, the new method consists in extracting one or more local level sets forbodies close to the grid point considered, reinitializing these local level setsto remove kinks, and using these to calculate the curvature and normal vector atthe grid point considered. For the curvature, multiple values are averaged,while for the normal vector, the one corresponding to the closest interface isselected.With this new method, several two-phase flow simulations are performed that arerelevant for understanding the liquefaction of natural gas. The new methodenables simulations that are more general than previous ones. A two-dimensionalsimulation was performed of a 0.6 mm diameter methanol droplet falling through air and merging with a deep pool of methanol. The new method gave good results in this case, but unphysical oscillations in the pressure field rendered this result unsuitablefor comparison with experimental results.Several similar cases with significantly lower density differences between thetwo fluids were also considered; in these cases, the pressure field behavedphysically, but the results are less applicable to the understanding of naturalgas liquefication, and better suited for validation of the new method. Inparticular, an axisymmetric simulation of a 0.11 mm diameter water droplet in decanemerging with a deep pool of water has been considered. The results of thissimulation show a very close agreement with experimental data. Attempts werealso made to simulate a larger droplet, but in this case finer grids were neededthan what could be achieved here due to the computational time cost of gridrefinement. Purely geometrical results are also presented in order to validate the results of the new method, and three-dimensional results are given for a static interface configuration, demonstrating that the method is easily extended to higher dimensions.

As computer systems become increasingly complex and diverse, so too do the architectures they implement. This leads to an increase in complexity in the tools used to design new hardware and software. One particularly important tool in hardware and software design is the Instruction Set Simulator, which is used to prototype new architectures and hardware features, verify hardware, and test and debug software. Many Architecture Description Languages exist which facilitate the description of new architectural or hardware features, and generate a tools such as simulators. However, these typically suffer from poor performance, are difficult to test effectively, and may be limited in functionality. This thesis considers three objectives when developing Instruction Set Simulators: performance, correctness, and completeness, and presents techniques which contribute to each of these. Performance is obtained by combining Dynamic Binary Translation techniques with a novel analysis of high level architecture descriptions. This makes use of partial evaluation techniques in order to both improve the translation system, and to improve the quality of the translated code, leading a performance improvement of over 2.5x compared to a naïve implementation. This thesis also presents techniques which contribute to the correctness objective. Each possible behaviour of each described instruction is used to guide the generation of a test case. Constraint satisfaction techniques are used to determine the necessary instruction encoding and context for each behaviour to be produced. It is shown that this is a significant improvement over benchmark-driven testing, and this technique has led to the discovery of several bugs and inconsistencies in multiple state of the art instruction set simulators. Finally, several challenges in ‘Full System’ simulation are addressed, contributing to both the performance and completeness objectives. Full System simulation generally carries significant performance costs compared with other simulation strategies. Crucially, instructions which access memory require virtual to physical address translation and can now cause exceptions. Both of these processes must be correctly and efficiently handled by the simulator. This thesis presents novel techniques to address this issue which provide up to a 1.65x speedup over a state of the art solution.

<p>The Particle Level set method is a successful extension to Level set methods to improve thevolume preservation in fluid simulations. This thesis will analyze how sparse volume data structures can be used to store both the signed distance function and the particles in order to improve access speed and memory efficiency. This Particle Level set implementation will be evaluated against Digital Domains current Particle Level set implementation. Different degrees of quantization will be used to implement particle representations with varying accuracy. These particles will be tested and both visual results and error measurments will be presented. The sparse volume data structures DB-Grid and Field3D will be evaluated in terms of speed and memory efficiency.</p>

Accurate segmemation of spinal vertebrae is important in the study of spinal related disease or disorders such as vertebral fractures. Identifying the severity of fractures and understanding its cause will help physicians determine the most effective pharmacological treatments and clinical management strategies for spinal disorders. Detection and segmentation are the crucial steps towards this quantitative framework. Although these quantitative image analysis techniques have received increasing interest recently, accurate detection and segmentation methods are still lacking. The complexity of vertebrae shapes, gaps in the cortical bone, internal boundaries, as well as the noisy. incomplete or missing information from the medical images have undoubtedly increased the challenge. Level set methods are effective for image segmentation. They are well-known for their ability to handle abrupt topological changes. However, the methods suffer from limitations such as slow convergence and leaking problems. As such, over the past two decades, the original level set method has evolved in many directions, including integration of prior shape models into the segmentation framework. For highly challenging medical images, incorporating prior shape into the segmentation framework have shown great success in recent years. In this work, a new shape model and an associated energy for level set segmentation is proposed. The proposed shape model presents a new dimension to extract local shape parameter directly from the shape model, which is different from previous work that focused on an indirect manner of feature extractions and in global sense. For the 2D and 3D segmentation of spinal vertebrae, a new segmentation framework combining the use of Willmore functional and kernel density estimation is proposed. While the kernel density estimator provide a dissimilarity measure with the shape prior during the level set evolution, the Willmore energy helps to regularize and smoothen the surface in the process. The introduction of Willmore functional into the 3D segmentation framework has solved the commonly encountered irregularity problem on extracted surface. Experimental results clearly demonstrate the feasibility of the proposed framework when tested on CT images of spinal vertebrae. The ultimate goal of this work is to provide a quantitative platform for efficient and accurate diagnosis of spinal disorder related diseases.

Geometric deformable models for active contours have brought tremendous impact to classical problems in image processing such as providing ways to devise efficient compu­tational algorithms for automatic segmentation. This is achieved by using the level set method, which allows handling automatic changes in topology while providing a frame­work for very fast numerical schemes. However, topological flexibility is not desired when an object with known topology is sought. It is natural to capture the target in a way that gives the correct topology. A geometric deformable model with topology preserving is developed that can guarantee the topology will be preserved while all the computational advantages of the level set approach are maintained. A key issue in object detection using the shape of the object’s boundary and surface reconstruction using slice contours is the ability to identify the complete boundaries of the segmented objects in the scene. The segmentation results provided by geometric de­formable models are usually dependent on the contour initialization, and in most cases, the results of the segmentation will only provide partial objects boundaries. A new method based on digital topology is proposed to detect the complete boundary informa­tion of the segmented objects. By carrying out a topological analysis of the objects, this method can provide the right initialization that can capture all the boundaries of the objects in certain cases.

Interface tracking methods for segregated flows such as breaking ocean waves are an important tool in marine engineering. With the development in marine renewable devices increasing and a multitude of other marine flow problems that benefit from the possibility of simulation on computer, the need for accurate free surface solvers capable of solving wave simulations has never been greater. An important component of successfully simulating segregated flow of any type is accurately tracking the position of the separating interface between fluids. It is desirable to represent the interface as a sharp, smooth, continuous entity in simulations. Popular Eulerian interface tracking methods appropriate for segregated flows such as the Marker and Cell Method (MAC) and the Volume of Fluid (VOF) were considered. However these methods have drawbacks with smearing of the interface and high computational costs in 3D simulations being among the most prevalent. This PhD project uses a level set method to implicitly represent an interface. The level set method is a signed distance function capable of both sharp and smooth representations of a free surface. It was found, over time, that the level set function ceases to represent a signed distance due to interaction of local velocity fields. This affects the accuracy to which the level set can represent a fluid interface, leading to mass loss. An advection solver, the Cubic Interpolated Polynomial (CIP) method, is presented and tested for its ability to transport a level set interface around a numerical domain in 2D. An advection problem of the level set function demonstrates the mass loss that can befall the method. To combat this, a process known as reinitialisation can be used to re-distance the level set function between time-steps, maintaining better accuracy. The goal of this PhD project is to present a new numerical gradient approximation that allows for the extension of the reinitialisation method to unstructured numerical grids. A particular focus is the Cartesian cut cell grid method. It allows geometric boundaries of arbitrary complexity to be cut from a regular Cartesian grid, allowing for flexible high quality grid generation with low computational cost. A reinitialisation routine using 1st order gradient approximation is implemented and demonstrated with 1D and 2D test problems. An additional area-conserving constraint is introduced to improve accuracy further. From the results, 1st order gradient approximation is shown to be inadequate for improving the accuracy of the level set method. To obtain higher accuracy and the potential for use on unstructured grids a novel gradient approximation based on a slope limited least squares method, suitable for level set reinitialisation, is developed. The new gradient scheme shows a significant improvement in accuracy when compared with level set reinitialisation methods using a lower order gradient approximation on a structured grid. A short study is conducted to find the optimal parameters for running 2D level set interface tracking and the new reinitialisation method. The details of the steps required to implement the current method on a Cartesian cut cell grid are discussed. Finally, suggestions for future work using the methods demonstrated in the thesis are presented.

Pendant les 20 dernières années, la méthode des fonctions de niveau (Level set method - LSM) a été développée pour résoudre à un problème complexe qui est celui des mouvements de l'interface. Cette technique de simulation représente des surfaces lisses avec précision et permet de suivre leur mouvement. Les avantages de LSM comprennent: a) les quantités géométriques sont calculées facilement, b) la solution est précise et robuste, et c) la topologie est efficacement suivie. Le problème le plus difficile est que cette méthode demande un traitement spécial des conditions aux limites sur la surface lisse. Cette thèse se concentre sur le développement numérique de la méthode des fonctions de niveau et de ses applications à différents problèmes en hydro-géologie, y compris les écoulements compressibles et incompressibles dans les milieux poreux et fracturés. Le mémoire est organisé en quatre parties où la méthodologie et la simulation sont décrites. La Partie 1concoit et développe une nouvelle génération de codes qui est capable d'analyser l'état et l'évolution des milieux poreux. Une comparaison avec le modèle de réseau de pores est présentée dans la Partie 2. La Partie 3 se propose d'analyser l'hydrogéologie sur l'île de Santa Cruz en utilisant les signaux de marée observés. La propagation de gaz compressible à Roselend est simulée dans la Partie 4 qui se concentre sur le réseau de fractures dans la roche cristalline. Le mémoire se termine avec des perspectives pour chaque application en termes de LSM<br>Over the past 20 years, a class of numerical techniques known as Level set method (LSM) has been developed to tackle one of the most complex problems, namely the interface motions. This simulation technique represents precisely smooth surfaces and it can track their motion. The advantages of the Level set method include a) geometric quantities are easily computed, b) it provides accurate and robust solutions and c) it handles topology propagation effectively. The most challenging difficulty is that this method requires a special treatment of the boundary conditions at the level set surface. This thesis is focused on the numerical development of the Level set method and on its applications to various problems in hydro-geology, including compressible and incompressible flows in porous and fractured media. The memoir is arranged in four parts where the methodology and implementation are completely addressed. Part 1 devises and develops a new generation of basic numerical tools able to analyse the state and evolution of porous media. A comparison with the Porous Network Model (PNM) is presented in Part 2. Part 3 intends to analyse the hydrogeology of the Santa Cruz Island using the observed tidal signals. The compressible gas flow in the Roselend site is simulated in Part 4 which focus on the fracture network in a crystalline rock. The work ends with possible future works for each application in terms of LSM

Level set method is a boundary tracking method that uses an implicit function to define the boundary location. By using the implicit function to define the structural boundary the level set method can be used for topology optimisation. The level set method has previously been used to solve a range of structural optimisation problems. The aim of this thesis is to extend the application of the level set method to additional applications of structural optimisation. A robust method of 3D level set topology optimisation is developed and tested. The use of a hole insertion method was found to be advantageous, but not vital, for 3D level set topology optimisation. The level set method is used to optimise the internal structure of a proximal femur. Similarities between the optimal structure and real internal trabecular bone architecture suggest that the internal bone structure may be mechanically optimal. Stress constrained level set topology optimisation is performed in 2D. Stress shape sensitivities are derived and interpolated to obtain smooth boundary sensitivity, resulting in feasible stress constrained solution in numerical examples. A new generic objective hole insertion method is used to reduce dependence on the initial solution. A level set method for optimising the design of fibre angles in composite structures is also introduced. Fibre paths are implicitly defined using the level set function. Sensitivity analysis is used to update the level set function values and optimise the fibre path. The method implicitly ensures continuous fibre paths in the optimum solution, that could be manufactured using advanced fibre placement.

Changes in retinal structure are related to different eye diseases. Various retinal imaging techniques, such as fundus imaging and optical coherence tomography (OCT) imaging modalities, have been developed for non-intrusive ophthalmology diagnoses according to the vasculature changes. However, it is time consuming or even impossible for ophthalmologists to manually label all the retinal structures from fundus images and OCT images. Therefore, computer aided diagnosis system for retinal imaging plays an important role in the assessment of ophthalmologic diseases and cardiovascular disorders. The aim of this PhD thesis is to develop segmentation methods to extract clinically useful information from these retinal images, which are acquired from different imaging modalities. In other words, we built the segmentation methods to extract important structures from both 2D fundus images and 3D OCT images. In the first part of my PhD project, two novel level set based methods were proposed for detecting the blood vessels and optic discs from fundus images. The first one integrates Chan-Vese's energy minimizing active contour method with the edge constraint term and Gaussian Mixture Model based term for blood vessels segmentation, while the second method combines the edge constraint term, the distance regularisation term and the shape-prior term for locating the optic disc. Both methods include the pre-processing stage, used for removing noise and enhancing the contrast between the object and the background. Three automated layer segmentation methods were built for segmenting intra-retinal layers from 3D OCT macular and optic nerve head images in the second part of my PhD project. The first two methods combine different methods according to the data characteristics. First, eight boundaries of the intra-retinal layers were detected from the 3D OCT macular images and the thickness maps of the seven layers were produced. Second, four boundaries of the intra-retinal layers were located from 3D optic nerve head images and the thickness maps of the Retinal Nerve Fiber Layer (RNFL) were plotted. Finally, the choroidal layer segmentation method based on the Level Set framework was designed, which embedded with the distance regularisation term, edge constraint term and Markov Random Field modelled region term. The thickness map of the choroidal layer was calculated and shown.

We give a concise explication of the theory of level set methods for modeling motion of an interface as well as the numerical implementation of these methods. We then introduce the geometry of a camera and the mathematical models for 3D reconstruction with a few examples both simulated and from a real camera. We finally describe the model for 3D surface reconstruction from n-camera views using level set methods.

Le but de cette thèse est de définir une méthode d'optimisation de formes qui conjugue l'utilisation de la dérivée seconde de forme et la méthode des lignes de niveaux pour la représentation d'une forme.On considèrera d'abord deux cas plus simples : un cas d'optimisation paramétrique et un cas d'optimisation discrète.Ce travail est divisé en quatre parties.La première contient le matériel nécessaire à la compréhension de l'ensemble de la thèse.Le premier chapitre rappelle des résultats généraux d'optimisation, et notamment le fait que les méthodes d'ordre deux ont une convergence quadratique sous certaines hypothèses.Le deuxième chapitre répertorie différentes modélisations pour l'optimisation de formes, et le troisième se concentre sur l'optimisation paramétrique puis l'optimisation géométrique.Les quatrième et cinquième chapitres introduisent respectivement la méthode des lignes de niveaux (level-set) et la méthode des éléments-finis.La deuxième partie commence par les chapitres 6 et 7 qui détaillent des calculs de dérivée seconde dans le cas de l'optimisation paramétrique puis géométrique.Ces chapitres précisent aussi la structure et certaines propriétés de la dérivée seconde de forme.Le huitième chapitre traite du cas de l'optimisation discrète.Dans le neuvième chapitre on introduit différentes méthodes pour un calcul approché de la dérivée seconde, puis on définit un algorithme de second ordre dans un cadre général.Cela donne la possibilité de faire quelques premières simulations numériques dans le cas de l'optimisation paramétrique (Chapitre 6) et dans le cas de l'optimisation discrète (Chapitre 7).La troisième partie est consacrée à l'optimisation géométrique.Le dixième chapitre définit une nouvelle notion de dérivée de forme qui prend en compte le fait que l'évolution des formes par la méthode des lignes de niveaux, grâce à la résolution d'une équation eikonale, se fait toujours selon la normale.Cela permet de définir aussi une méthode d'ordre deux pour l'optimisation.Le onzième chapitre détaille l'approximation d'intégrales de surface et le douzième chapitre est consacré à des exemples numériques.La dernière partie concerne l'analyse numérique d'algorithmes d'optimisation de formes par la méthode des lignes de niveaux.Le Chapitre 13 détaille la version discrète d'un algorithme d'optimisation de formes.Le Chapitre 14 analyse les schémas numériques relatifs à la méthodes des lignes de niveaux.Enfin le dernier chapitre fait l'analyse numérique complète d'un exemple d'optimisation de formes en dimension un, avec une étude des vitesses de convergence<br>The main purpose of this thesis is the definition of a shape optimization method which combines second-order differentiationwith the representation of a shape by a level-set function. A second-order method is first designed for simple shape optimization problems : a thickness parametrization and a discrete optimization problem. This work is divided in four parts.The first one is bibliographical and contains different necessary backgrounds for the rest of the work. Chapter 1 presents the classical results for general optimization and notably the quadratic rate of convergence of second-order methods in well-suited cases. Chapter 2 is a review of the different modelings for shape optimization while Chapter 3 details two particular modelings : the thickness parametrization and the geometric modeling. The level-set method is presented in Chapter 4 and Chapter 5 recalls the basics of the finite element method.The second part opens with Chapter 6 and Chapter 7 which detail the calculation of second-order derivatives for the thickness parametrization and the geometric shape modeling. These chapters also focus on the particular structures of the second-order derivative. Then Chapter 8 is concerned with the computation of discrete derivatives for shape optimization. Finally Chapter 9 deals with different methods for approximating a second-order derivative and the definition of a second-order algorithm in a general modeling. It is also the occasion to make a few numerical experiments for the thickness (defined in Chapter 6) and the discrete (defined in Chapter 8) modelings.Then, the third part is devoted to the geometric modeling for shape optimization. It starts with the definition of a new framework for shape differentiation in Chapter 10 and a resulting second-order method. This new framework for shape derivatives deals with normal evolutions of a shape given by an eikonal equation like in the level-set method. Chapter 11 is dedicated to the numerical computation of shape derivatives and Chapter 12 contains different numerical experiments.Finally the last part of this work is about the numerical analysis of shape optimization algorithms based on the level-set method. Chapter 13 is concerned with a complete discretization of a shape optimization algorithm. Chapter 14 then analyses the numerical schemes for the level-set method, and the numerical error they may introduce. Finally Chapter 15 details completely a one-dimensional shape optimization example, with an error analysis on the rates of convergence

Tracking multiple articulated objects (such as a human body) and handling occlusion between them is a challenging problem in automated video analysis. This work proposes a new approach for accurately and steadily visual tracking people, which should function even if the system encounters occlusion in video sequences. In this approach, targets are represented with a Gaussian mixture, which are adapted to regions of the target automatically using an EM-model algorithm. Field speeds are defined for changed pixels in each frame based on the probability of their belonging to a particular person's blobs. Pixels are matched to the models using a fast numerical level set method. Since each target is tracked with its blob's information, the system is capable of handling partial or full occlusion during tracking. Experimental results on a number of challenging sequences that were collected in non-experimental environments demonstrate the effectiveness of the approach.

The level set method offers a simple and robust approach to modeling liquid-vapor interfaces that arise in boiling and condensing flows. The current liquid-vapor phase-transition techniques used with the level set method are not able to account for different thermal conductivities and specific heats in each respective phase, nor are they able to accurately account for latent heat absorption and release. This paper presents a new level set based technique for liquid-vapor phase-transition that accounts for different material properties in each respective phase, such as thermal conductivity and specific heat, while maintaining the interface at the saturation temperature. The phase-transition technique is built on the ghost fluid framework coupled with the standard level set method. A new technique is presented for constructing ghost nodes that implicitly captures the immersed boundary conditions and is second order accurate. The method is tested against analytical solutions, and it is used to model film boiling. The new phase-transition technique will greatly assist efforts to accurately capture the physics of boiling and condensing flows. In addition to presenting a new phase transition technique, a coupled level set volume of fluid advection scheme is developed for phase transition flows. The new scheme resolves the mass loss problem associated with the level set method, and the method provides an easy way to accurately calculate the curvature of an interface, which can be difficult with the volume of fluid method. A film boiling simulation is performed to illustrate the superior performance of the coupled level set volume of fluid approach over the level set method and the volume of fluid method.

L'apparition des techniques d'imagerie par résonance magnétique (IRM) à la fin du XXe siècle a révolutionné le monde de la médecine moderne, en permettant de visualiser avec précision l'intérieur de structures anatomiques de manière non invasive. Cette technique d'imagerie a fortement contribué à l'étude du cerveau humain, en permettant de discerner avec précision les différentes structures anatomique de la tête, notament le cortex cérébral. Le discernement de structures anatomiques de la tête porte le nom segmentation, et consiste à ``extraire'' des régions dans les IRMs. Plusieurs méthodes de segmentation existent, et cette thèse portent sur les méthodes à base d'évolution d'hyper-surfaces: une hyper-surface (surface en 3D) est progressivement déformée pour finalement épouser les frontières de la region à segmenter. Un modèle de tête correspond au partitionnement de la tête en plusieurs structures anatomiques préalablement segmentées. Un modèle de tête classique comprends en général 5 structures anatomiques (peau, crane , liquide céphalo rachidien, matière grise, matière blanche), imbriquées les unes dans les autres à la manière de ``poupée russes''. Néanmoins de par la complexité de leurs formes, segmenter ces structures manuellement s'avère pénible et extrêmement difficile. Cette thèse se consacre à la mise en place de nouveaux modèles de segmentation robustes aux altérations d'IRMs, et à l'application de ces modèles pour la création automatique de modèles anatomiques de la tête. Apres avoir brièvement survolé un état de l'art des differentes méthodes de segmentation d'image, deux contributions à la segmentation par évolution d'hypersurface sont proposées. La première constitue une nouvelle représentation et un nouveau schémas numérique pour la méthode des ensembles de niveaux, en utilisant des éléments finis quadrilateraux. Cette representation vise à améliorer la qualité et la robustesse du modèle. La seconde contribution constitue un nouveau modèle de segmentation basé sur des statistiques locales, robuste aux altérations présentes dans les IRMs. Ce nouveau modèle vise à unifier plusieurs modèles ``état de l'art'' en segmentation d'image. Enfin, un cadre pour la création automatique de modèle de tête est proposé, utilisant princpalement le precédent modèle de segmentation par statistiques locales<br>Magnetic Resonance Images (MRI) have been introduced at the end of the XXth century and have revolutionized the world of modern medicine, allowing to view with precision the inside of anatomical structures in a non-invasive way. This imaging technique has greatly contributed to the study and comprehension of the human brain, allowing to discern with precision the different anatomical structures composing the head, especially the cerebral cortex. Discernment between these anatomical structures is called segmentation, and consist in “extracting” structures of interest from MRIs. Several models exists to perform image segmentation, and this thesis focus on those based on hypersurface evolutions: an hypersurface (surface in 3D) is incrementally adjusted to finally fit the border of the region of interest. A head model corresponds to the partitioning of the head into several segmented anatomical structures. A classic head model generally includes 5 anatomical structures (skin, skull, cerebrospinal fluid, grey matter, white matter), nested inside each other in the manner of “Russian nested dolls”. Nevertheless because of the complexity of their shapes, manual segmentation of these structures is tedious and extremely difficult. This thesis is dedicated to the creation of new segmentation models robust to MRI alterations, and to the application of these models in the purpose of automatic creation of anatomical head models. After briefly reviewing some classical models in image segmentation, two contributions to segmentation based on hypersurface evolution are proposed. The first one corresponds to a new representation and a new numerical scheme for the level-sets method, based on quadrilateral finite elements. This representation aims at improving the accuracy and robustness of the model. The second contribution corresponds to a new segmentation model based on local statistics, and robust to standard MRI alterations. This model aims at unifying several 'state-of-the-art' models in image segmentation. Finally, a framework for automatic creation of anatomical head models is proposed, mainly using the previous local-statistic based segmentation model

The level set method is a powerful way of tracking surfaces by defining the surface as a zero level set of a continuous function that is usually a signed distance function. The level set method is one of the best methods for simulating multi-phase flow because it can easily handle fast topological changes, as well as splitting and merging of fluids. In this thesis, a standard level set method was implemented in C++, using the finite element method library deal.II, to simulate incompressible two-phase flow on some benchmark problems. The results show a significant change of mass in the simulations, something that should not be allowed to happen when simulating incompressible fluids. The mass changes mainly occur in the reinitialization phase, where the level set function is rebuilt to look more like a signed distance function.

We present a novel approach to simulate the motion of vortex filaments in three dimensional, incompressible flows. We represent vortex filaments as thin tubes on a 3D-grid and use a Lagrangian vortex filament method to describe their motion within a level set framework. We thus obtain a hybrid method that incorporates the accuracy of vortex methods in modelling filament motion and the benefits of working on a fixed grid within the level set framework. These benefits include a general grid characterization of the topological changes that, may occur and natural filtering of high-frequency perturbations. We have validated the consistency of the hybrid method reproducing the motion of well studied configurations of vortex filaments. More importantly, we have demonstrated the capability of this hybrid method to simulate the reconnection of vortex filaments for a number of examples, including the reconnection subsequent to the head-on collision of two sinusoidally perturbed circular filaments. To our knowledge, these are the first results illustrating the use of a hybrid Level-Set Lagrangian framework to model and simulate the reconnection of vortex filaments in an inviscid fluid.

Distributed modal filters based on piezoelectric polymer have especially become popular in the field of active vibration control to reduce the problem of spillover. While distributed modal filters for one-dimensional structures can be found analytically based on the orthogonality between the mode shapes, the design for two-dimensional structures is not straightforward. It requires a continuous gain variation in two dimensions, which is not realizable from the current manufacturing point of view. In this thesis, a structural optimization problem is considered to approximate distributed modal sensors for two-dimensional plate structures, where the thickness is constant but the polarization can switch between positive and negative. The problem is solved through an explicit parametric level set method. In this framework, the boundary of a domain is represented implicitly by the zero isoline of a level set function. This allows simultaneous shape and topology changes. The level set function is approximated by a linear combination of Gaussian radial basis functions. As a result, the structural optimization problem can be directly posed in terms of the parameters of the approximation. This allows to apply standard optimization methods and bypasses the numerical drawbacks, such as reinitialization, velocity extension and regularization, which are associated with the numerical solution of the Hamilton-Jacobi equation in conventional methods.Since the level set method based on the shape derivative formally only allows shape but not topology transformation, the optimization problem is firstly tackled with a derivative-free optimization algorithm. It is shown that the approach is able to find approximate modal sensor designs with only few design variables. However, this approach becomes unsuitable as soon as the number of optimization variables is growing. Therefore, a sensitivity-based optimization approach is being applied, based on the parametric shape derivative which is with respect to the parameters of the radial basis functions. Although the shape derivatives does not exist at points where the topology changes, it is demonstrated that an optimization routine based on a SQP solver is able to perform topological changes during the optimization and finds optimal designs even from poor initial designs. In order to include the sensors' distribution as design variable, the parametric level set approach is extended to multiple level sets. It turns out that, despite the increased design space, optimal solutions always converge to full-material polarization designs. Numerical examples are provided for a simply supported as well as a cantilever square plate.<br>Doctorat en Sciences de l'ingénieur et technologie<br>info:eu-repo/semantics/nonPublished

[EN] The main topic of the present thesis is the improvement of fabrication processes simulation by means of the Level Set (LS) method. The LS is a mathematical approach used for evolving fronts according to a motion defined by certain laws. The main advantage of this method is that the front is embedded inside a higher dimensional function such that updating this function instead of directly the front itself enables a trivial handling of complex situations like the splitting or coalescing of multiple fronts. In particular, this document is focused on wet and dry etching processes, which are widely used in the micromachining process of Micro-Electro-Mechanical Systems (MEMS). A MEMS is a system formed by mechanical elements, sensors, actuators, and electronics. These devices have gained a lot of popularity in last decades and are employed in several industry fields such as automotive security, motion sensors, and smartphones. Wet etching process consists in removing selectively substrate material (e.g. silicon or quartz) with a liquid solution in order to form a certain structure. This is a complex process since the result of a particular experiment depends on many factors, such as crystallographic structure of the material, etchant solution or its temperature. Similarly, dry etching processes are used for removing substrate material, however, gaseous substances are employed in the etching stage. In both cases, the usage of a simulator capable of predicting accurately the result of a certain experiment would imply a significant reduction of design time and costs. There exist a few LS-based wet etching simulators but they have many limitations and they have never been validated with real experiments. On the other hand, atomistic models are currently considered the most advanced simulators. Nevertheless, atomistic simulators present some drawbacks like the requirement of a prior calibration process in order to use the experimental data. Additionally, a lot of effort must be invested to create an atomistic model for simulating the etching process of substrate materials with different atomistic structures. Furthermore, the final result is always formed by unconnected atoms, which makes difficult a proper visualization and understanding of complex structures, thus, usually an additional visualization technique must be employed. For its part, dry etching simulators usually employ an explicit representation technique to evolve the surface being etched according to etching models. This strategy can produce unrealistic results, specially in complex situations like the interaction of multiple surfaces. Despite some models that use implicit representation have been published, they have never been directly compared with real experiments and computational performance of the implementations have not been properly analysed. The commented limitations are addressed in the various chapters of the present thesis, producing the following contributions: - An efficient LS implementation in order to improve the visual representation of atomistic wet etching simulators. This implementation produces continuous surfaces from atomistic results. - Definition of a new LS-based model which can directly use experimental data of many etchant solutions (such as KOH, TMAH, NH4HF2, and IPA and Triton additives) to simulate wet etching processes of various substrate materials (e.g. silicon and quartz). - Validation of the developed wet etching simulator by comparing it to experimental and atomistic simulator results. - Implementation of a LS-based tool which evolves the surface being etched according to dry etching models in order to enable the simulation of complex processes. This implementation is also validated experimentally. - Acceleration of the developed wet and dry etching simulators by using Graphics Processing Units (GPUs).<br>[ES] El tema principal de la presente tesis consiste en mejorar la simulación de los procesos de fabricación utilizando el método Level Set (LS). El LS es una técnica matemática utilizada para la evolución de frentes según un movimiento definido por unas leyes. La principal ventaja de este método es que el frente está embebido dentro de una función definida en una dimensión superior. Actualizar dicha función en lugar del propio frente permite tratar de forma trivial situaciones complejas como la separación o la colisión de diversos frentes. En concreto, este documento se centra en los procesos de atacado húmedo y seco, los cuales son ampliamente utilizados en el proceso de fabricación de Sistemas Micro-Electro-Mecánicos (MEMS, de sus siglas en inglés). Un MEMS es un sistema formado por elementos mecánicos, sensores, actuadores y electrónica. Estos dispositivos hoy en día son utilizados en muchos campos de la industria como la seguridad automovilística, sensores de movimiento y teléfonos inteligentes. El proceso de atacado húmedo consiste en eliminar de forma selectiva el material del sustrato (por ejemplo, silicio o cuarzo) con una solución líquida con el fin de formar una estructura específica. Éste es un proceso complejo pues el resultado depende de muchos factores, tales como la estructura cristalográfica del material, la solución atacante o su temperatura. De forma similar, los procesos de atacado seco son utilizados para eliminar el material del sustrato, sin embargo, se utilizan sustancias gaseosas en la fase de atacado. En ambos casos, la utilización de un simulador capaz de predecir de forma precisa el resultado de un experimento concreto implicaría una reducción significativa del tiempo de diseño y de los costes. Existen unos pocos simuladores del proceso de atacado húmedo basados en el método LS, no obstante tienen muchas limitaciones y nunca han sido validados con experimentos reales. Por otro lado, los simuladores atomísticos son hoy en día considerados los simuladores más avanzados pero tienen algunos inconvenientes como la necesidad de un proceso de calibración previo para poder utilizar los datos experimentales. Además, debe invertirse mucho esfuerzo para crear un modelo atomístico para la simulación de materiales de sustrato con distintas estructuras atomísticas. Asimismo, el resultado final siempre está formado por átomos inconexos que dificultan una correcta visualización y un correcto entendimiento de aquellas estructuras complejas, por tanto, normalmente debe emplearse una técnica adicional para la visualización de dichos resultados. Por su parte, los simuladores del proceso de atacado seco normalmente utilizan técnicas de representación explícita para evolucionar, según los modelos de atacado, la superficie que está siendo atacada. Esta técnica puede producir resultados poco realistas, sobre todo en situaciones complejas como la interacción de múltiples superficies. A pesar de que unos pocos modelos son capaces de solventar estos problemas, nunca han sido comparados con experimentos reales ni el rendimiento computacional de las correspondientes implementaciones ha sido adecuadamente analizado. Las expuestas limitaciones son abordadas en la presente tesis y se han producido las siguientes contribuciones: - Implementación eficiente del método LS para mejorar la representación visual de los simuladores atomísticos del proceso de atacado húmedo. - Definición de un nuevo modelo basado en el LS que pueda usar directamente los datos experimentales de muchos atacantes para simular el proceso de atacado húmedo de diversos materiales de sustrato. - Validación del simulador comparándolo con resultados experimentales y con los de simuladores atomísticos. - Implementación de una herramienta basada en el método LS que evolucione la superficie que está siendo atacada según los modelos de atacado seco para habilitar la simulación de procesos comple<br>[CAT] El tema principal de la present tesi consisteix en millorar la simulació de processos de fabricació mitjançant el mètode Level Set (LS). El LS és una tècnica matemàtica utilitzada per a l'evolució de fronts segons un moviment definit per unes lleis en concret. El principal avantatge d'aquest mètode és que el front està embegut dins d'una funció definida en una dimensió superior. D'aquesta forma, actualitzar la dita funció en lloc del propi front, permet tractar de forma trivial situacions complexes com la separació o la col·lisió de diversos fronts. En concret, aquest document es centra en els processos d'atacat humit i sec, els quals són àmpliament utilitzats en el procés de fabricació de Sistemes Micro-Electro-Mecànics (MEMS, de les sigles en anglès). Un MEMS és un sistema format per elements mecànics, sensors, actuadors i electrònica. Aquests dispositius han guanyat molta popularitat en les últimes dècades i són utilitzats en molts camps de la indústria, com la seguretat automobilística, sensors de moviment i telèfons intel·ligents. El procés d'atacat humit consisteix en eliminar de forma selectiva el material del substrat (per exemple, silici o quars) amb una solució líquida, amb la finalitat de formar una estructura específica. Aquest és un procés complex ja que el resultat de un determinat experiment depèn de molts factors, com l'estructura cristal·logràfica del material, la solució atacant o la seva temperatura. De manera similar, els processos d'atacat sec son utilitzats per a eliminar el material del substrat, no obstant, s'utilitzen substàncies gasoses en la fase d'atacat. En ambdós casos, la utilització d'un simulador capaç de predir de forma precisa el resultat d'un experiment en concret implicaria una reducció significativa del temps de disseny i dels costos. Existeixen uns pocs simuladors del procés d'atacat humit basats en el mètode LS, no obstant tenen moltes limitacions i mai han sigut validats amb experiments reals. Per la seva part, els simuladors atomístics tenen alguns inconvenients com la necessitat d'un procés de calibratge previ per a poder utilitzar les dades experimentals. A més, deu invertir-se molt d'esforç per crear un model atomístic per a la simulació de materials de substrat amb diferents estructures atomístiques. Així mateix, el resultat final sempre està format per àtoms inconnexos que dificulten una correcta visualització i un correcte enteniment d'aquelles estructures complexes, per tant, normalment deu emprar-se una tècnica addicional per a la visualització d'aquests resultats. D'altra banda, els simuladors del procés d'atacat sec normalment utilitzen tècniques de representació explícita per evolucionar, segons els models d'atacat, la superfície que està sent atacada. Aquesta tècnica pot introduir resultats poc realistes, sobretot en situacions complexes com per exemple la interacció de múltiples superfícies. A pesar que uns pocs models son capaços de resoldre aquests problemes, mai han sigut comparats amb experiments reals ni tampoc el rendiment computacional de les corresponents implementacions ha sigut adequadament analitzat. Les exposades limitacions son abordades en els diferents capítols de la present tesi i s'han produït les següents contribucions: - Implementació eficient del mètode LS per millorar la representació visual dels simuladors atomístics del procés d'atacat humit. - Definició d'un nou model basat en el mètode LS que puga utilitzar directament les dades experimentals de molts atacants per a simular el procés d'atacat humit de diversos materials de substrat. - Validació del simulador d'atacat humit desenvolupat comparant-lo amb resultats experimentals i amb els de simuladors atomístics. - Implementació d'una ferramenta basada en el mètode LS que evolucione la superfície que està sent atacada segons els models d'atacat sec per, d'aquesta forma, habilitar la simulació de processo<br>Montoliu Álvaro, C. (2015). Study, Modelling and Implementation of the Level Set Method Used in Micromachining Processes [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/58609<br>TESIS

Cette thèse porte sur l'optimisation de forme via la méthode des "level sets" pour deux comportements mécaniques induisant des déplacements non différentiables par rapport à la forme: le contact et la plasticité. Pour y remédier, nous utilisons des problèmes approchés issus de méthode de pénalisation et de régularisation.Dans la première partie, nous présentons quelques notions fondamentales d'optimisation de forme (chapitre 1). Puis nous exposons les résultats qui seront utiles à l'analyse des deux problèmes mécaniques considérés et nous illustrons ces résultats.La deuxième partie introduit les modèles statiques de contact (chapitre 3) et le modèle statique de plasticité (chapitre 4) que nous utilisons dans le manuscrit. Pour chacun, nous donnons les bases de la modélisation mécanique, une analyse mathématique des inéquations variationnelles associées et nous expliquons quels solveurs nous avons implémentés.La dernière partie se focalise sur l'optimisation de forme. Dans chacun des chapitres nous donnons les versions pénalisées et régularisées des modèles, prouvons, pour certains, leur convergence vers les modèles exactes, calculons leurs gradients de forme et proposons des exemples 2D et, en contact, 3D. Ainsi, dans le chapitre 5, traitons-nous du contact et considérons deux sortes de problèmes: le premier dans lequel la zone de contact est fixe, le second dans lequel la zone de contact est optimisable. Pour ce dernier, nous introduisons deux méthodes pour résoudre du contact sans discrétiser la zone de contact. Dans le chapitre 6, nous abordons le modèle de Hencky que nous approximons grâce à une pénalisation de Perzyna ainsi que grâce à un modèle de notre crue<br>The main purpose of this thesis is to perform shape optimisation, in the framework of the level set method, for two mechanical behaviours inducing displacement which are not shape differentiable: contact and plasticity. To overcome this obstacle, we use approximate problems found by penalisation and regularisation.In the first part, we present some classical notions in optimal design (chapter 1). Then we give the mathematical results needed for the analysis of the two mechanical problems in consideration and illustrate these results.The second part is meant to introduce the five static contact models (chapter 3) and the static plasticity model (chapter 4) we use in the manuscript. For each chapter we provide the basis of the mechanical modeling, a mathematical analysis of the related variational inequations and, finally, explain how we implement the associated solvers.Eventually the last part, consisting of two chapters is devoted to shape optimisation. In each of them, we state the regularised versions of the models, prove, for some of them, the convergence to the exact ones, compute shape gradients and perform some numerical experiments in 2D and, for contact, in 3D. Thus, in chapter 5, we focus on contact and consider two types of optimal design problems: one with a fixed contact zone and another one with a mobile contact zone. For this last type, we introduce two ways to solve frictionless contact without meshing the contact zone. One of them is new and the other one has never been employed in this framework. In chapter 6, we deal with the Hencky model which we approximate thanks to a Perzyna penalised problem as well as a home-made one

A numerical treatment of non-linear higher-order geometric evolution equations with the level set and the finite element method is presented. The isotropic, weak anisotropic and strong anisotropic situation is discussed. Most of the equations considered in this work arise from the field of thin film growth. A short introduction to the subject is given. Four different models are discussed: mean curvature flow, surface diffusion, a kinetic model, which combines the effects of mean curvature flow and surface diffusion and includes a further kinetic component, and an adatom model, which incorporates in addition free adatoms. As an introduction to the numerical schemes, first the isotropic and weak anisotropic situation is considered. Then strong anisotropies (non-convex anisotropies) are used to simulate the phenomena of faceting and coarsening. The experimentally observed effect of corner and edge roundings is reached in the simulation through the regularization of the strong anisotropy with a higher-order curvature term. The curvature regularization leads to an increase by two in the order of the equations, which results in highly non-linear equations of up to 6th order. For the numerical solution, the equations are transformed into systems of second order equations, which are solved with a Schur complement approach. The adatom model constitutes a diffusion equation on a moving surface. An operator splitting approach is used for the numerical solution. In difference to other works, which restrict to the isotropic situation, also the anisotropic situation is discussed and solved numerically. Furthermore, a treatment of geometric evolution equations on implicitly given curved surfaces with the level set method is given. In particular, the numerical solution of surface diffusion on curved surfaces is presented. The equations are discretized in space by standard linear finite elements. For the time discretization a semi-implicit discretization scheme is employed. The derivation of the numerical schemes is presented in detail, and numerous computational results are given for the 2D and 3D situation. To keep computational costs low, the finite element grid is adaptively refined near the moving curves and surfaces resp. A redistancing algorithm based on a local Hopf-Lax formula is used. The algorithm has been extended by the authors to the 3D case. A detailed description of the algorithm in 3D is presented in this work<br>In der Arbeit geht es um die numerische Behandlung nicht-linearer geometrischer Evolutionsgleichungen höherer Ordnung mit Levelset- und Finite-Elemente-Verfahren. Der isotrope, schwach anisotrope und stark anisotrope Fall wird diskutiert. Die meisten in dieser Arbeit betrachteten Gleichungen entstammen dem Gebiet des Dünnschicht-Wachstums. Eine kurze Einführung in dieses Gebiet wird gegeben. Es werden vier verschiedene Modelle diskutiert: mittlerer Krümmungsfluss, Oberflächendiffusion, ein kinetisches Modell, welches die Effekte des mittleren Krümmungsflusses und der Oberflächendiffusion kombiniert und zusätzlich eine kinetische Komponente beinhaltet, und ein Adatom-Modell, welches außerdem freie Adatome berücksichtigt. Als Einführung in die numerischen Schemata, wird zuerst der isotrope und schwach anisotrope Fall betrachtet. Anschließend werden starke Anisotropien (nicht-konvexe Anisotropien) benutzt, um Facettierungs- und Vergröberungsphänomene zu simulieren. Der in Experimenten beobachtete Effekt der Ecken- und Kanten-Abrundung wird in der Simulation durch die Regularisierung der starken Anisotropie durch einen Krümmungsterm höherer Ordnung erreicht. Die Krümmungsregularisierung führt zu einer Erhöhung der Ordnung der Gleichung um zwei, was hochgradig nicht-lineare Gleichungen von bis zu sechster Ordnung ergibt. Für die numerische Lösung werden die Gleichungen auf Systeme zweiter Ordnungsgleichungen transformiert, welche mit einem Schurkomplement-Ansatz gelöst werden. Das Adatom-Modell bildet eine Diffusionsgleichung auf einer bewegten Fläche. Zur numerischen Lösung wird ein Operatorsplitting-Ansatz verwendet. Im Unterschied zu anderen Arbeiten, die sich auf den isotropen Fall beschränken, wird auch der anisotrope Fall diskutiert und numerisch gelöst. Außerdem werden geometrische Evolutionsgleichungen auf implizit gegebenen gekrümmten Flächen mit Levelset-Verfahren behandelt. Insbesondere wird die numerische Lösung von Oberflächendiffusion auf gekrümmten Flächen dargestellt. Die Gleichungen werden im Ort mit linearen Standard-Finiten-Elementen diskretisiert. Als Zeitdiskretisierung wird ein semi-implizites Diskretisierungsschema verwendet. Die Herleitung der numerischen Schemata wird detailliert dargestellt, und zahlreiche numerische Ergebnisse für den 2D und 3D Fall sind gegeben. Um den Rechenaufwand gering zu halten, wird das Finite-Elemente-Gitter adaptiv an den bewegten Kurven bzw. den bewegten Flächen verfeinert. Es wird ein Redistancing-Algorithmus basierend auf einer lokalen Hopf-Lax Formel benutzt. Der Algorithmus wurde von den Autoren auf den 3D Fall erweitert. In dieser Arbeit wird der Algorithmus für den 3D Fall detailliert beschrieben

A numerical treatment of non-linear higher-order geometric evolution equations with the level set and the finite element method is presented. The isotropic, weak anisotropic and strong anisotropic situation is discussed. Most of the equations considered in this work arise from the field of thin film growth. A short introduction to the subject is given. Four different models are discussed: mean curvature flow, surface diffusion, a kinetic model, which combines the effects of mean curvature flow and surface diffusion and includes a further kinetic component, and an adatom model, which incorporates in addition free adatoms. As an introduction to the numerical schemes, first the isotropic and weak anisotropic situation is considered. Then strong anisotropies (non-convex anisotropies) are used to simulate the phenomena of faceting and coarsening. The experimentally observed effect of corner and edge roundings is reached in the simulation through the regularization of the strong anisotropy with a higher-order curvature term. The curvature regularization leads to an increase by two in the order of the equations, which results in highly non-linear equations of up to 6th order. For the numerical solution, the equations are transformed into systems of second order equations, which are solved with a Schur complement approach. The adatom model constitutes a diffusion equation on a moving surface. An operator splitting approach is used for the numerical solution. In difference to other works, which restrict to the isotropic situation, also the anisotropic situation is discussed and solved numerically. Furthermore, a treatment of geometric evolution equations on implicitly given curved surfaces with the level set method is given. In particular, the numerical solution of surface diffusion on curved surfaces is presented. The equations are discretized in space by standard linear finite elements. For the time discretization a semi-implicit discretization scheme is employed. The derivation of the numerical schemes is presented in detail, and numerous computational results are given for the 2D and 3D situation. To keep computational costs low, the finite element grid is adaptively refined near the moving curves and surfaces resp. A redistancing algorithm based on a local Hopf-Lax formula is used. The algorithm has been extended by the authors to the 3D case. A detailed description of the algorithm in 3D is presented in this work.<br>In der Arbeit geht es um die numerische Behandlung nicht-linearer geometrischer Evolutionsgleichungen höherer Ordnung mit Levelset- und Finite-Elemente-Verfahren. Der isotrope, schwach anisotrope und stark anisotrope Fall wird diskutiert. Die meisten in dieser Arbeit betrachteten Gleichungen entstammen dem Gebiet des Dünnschicht-Wachstums. Eine kurze Einführung in dieses Gebiet wird gegeben. Es werden vier verschiedene Modelle diskutiert: mittlerer Krümmungsfluss, Oberflächendiffusion, ein kinetisches Modell, welches die Effekte des mittleren Krümmungsflusses und der Oberflächendiffusion kombiniert und zusätzlich eine kinetische Komponente beinhaltet, und ein Adatom-Modell, welches außerdem freie Adatome berücksichtigt. Als Einführung in die numerischen Schemata, wird zuerst der isotrope und schwach anisotrope Fall betrachtet. Anschließend werden starke Anisotropien (nicht-konvexe Anisotropien) benutzt, um Facettierungs- und Vergröberungsphänomene zu simulieren. Der in Experimenten beobachtete Effekt der Ecken- und Kanten-Abrundung wird in der Simulation durch die Regularisierung der starken Anisotropie durch einen Krümmungsterm höherer Ordnung erreicht. Die Krümmungsregularisierung führt zu einer Erhöhung der Ordnung der Gleichung um zwei, was hochgradig nicht-lineare Gleichungen von bis zu sechster Ordnung ergibt. Für die numerische Lösung werden die Gleichungen auf Systeme zweiter Ordnungsgleichungen transformiert, welche mit einem Schurkomplement-Ansatz gelöst werden. Das Adatom-Modell bildet eine Diffusionsgleichung auf einer bewegten Fläche. Zur numerischen Lösung wird ein Operatorsplitting-Ansatz verwendet. Im Unterschied zu anderen Arbeiten, die sich auf den isotropen Fall beschränken, wird auch der anisotrope Fall diskutiert und numerisch gelöst. Außerdem werden geometrische Evolutionsgleichungen auf implizit gegebenen gekrümmten Flächen mit Levelset-Verfahren behandelt. Insbesondere wird die numerische Lösung von Oberflächendiffusion auf gekrümmten Flächen dargestellt. Die Gleichungen werden im Ort mit linearen Standard-Finiten-Elementen diskretisiert. Als Zeitdiskretisierung wird ein semi-implizites Diskretisierungsschema verwendet. Die Herleitung der numerischen Schemata wird detailliert dargestellt, und zahlreiche numerische Ergebnisse für den 2D und 3D Fall sind gegeben. Um den Rechenaufwand gering zu halten, wird das Finite-Elemente-Gitter adaptiv an den bewegten Kurven bzw. den bewegten Flächen verfeinert. Es wird ein Redistancing-Algorithmus basierend auf einer lokalen Hopf-Lax Formel benutzt. Der Algorithmus wurde von den Autoren auf den 3D Fall erweitert. In dieser Arbeit wird der Algorithmus für den 3D Fall detailliert beschrieben.

The main contribution of this thesis is the implementation of manufacturing constraints in shape and topology optimization. Fabrication limitations related to the casting process are formulated as mathematical constraints and introduced in the optimization algorithm. In addition, based on the same theoretical and modelization tools, we propose a novel formulation for multi-phase optimization problems, which can be extended to the optimization of structures with functionally-graded properties. A key ingredient for the mathematical formulation of most problems throughout our work is the notion of the signed distance function to a domain. This work is divided into three parts. The rst part is bibliographical and contains the necessary background material for the understanding of the thesis' main core. It includes the rst two chapters. Chapter 1 provides a synopsis of shape and topology optimization methods and emphasizes the combination of shape sensitivity analysis and the level-set method for tracking a shape's boundary. In Chapter 2 we give a short description of the casting process, from which all our manufacturing constraints derive. We explain how industrial designers account for these limitations and propose a strategy to incorporate them in shape and topology optimization algorithms. The second part is about the mathematical formulation of manufacturing constraints. It starts with Chapter 3, where the control of thickness is discussed. Based on the signed distance function, we formulate three constraints to ensure a maximum and minimm feature size, as well as a minimal distance between structural members. Then, in Chapter 4, we propose ways to handle molding direction constraints and combine them with thickness constraints. Finally, a thermal constraint coming from the solidi cation of cast parts is treated in Chapter 5 using several thermal models. Multi-phase optimization is discussed in the third part. The general problem of shape and topology optimization using multiple phases is presented in detail in Chapter 6. A "smoothed-interface" approach, based again on the signed distance function, is proposed to avoid numerical di culties related to classical "sharp-interface" problems and a shape derivative is calculated. An extension of this novel formulation to general types of material properties' gradation is shown in the Appendix A.

This thesis proposes a probabilistic level set method to be used in segmentation of tumors with heterogeneous intensities. It models the intensities of the tumor and surrounding tissue using Gaussian mixture models. Through a contour based initialization procedure samples are gathered to be used in expectation maximization of the mixture model parameters. The proposed method is compared against a threshold-based segmentation method using MRI images retrieved from The Cancer Imaging Archive. The cases are manually segmented and an automated testing procedure is used to find optimal parameters for the proposed method and then it is tested against the threshold-based method. Segmentation times, dice coefficients, and volume errors are compared. The evaluation reveals that the proposed method has a comparable mean segmentation time to the threshold-based method, and performs faster in cases where the volume error does not exceed 40%. The mean dice coefficient and volume error are also improved while achieving lower deviation.

Thesis (Ph. D.)--University of California, San Diego, 2009.<br>Title from first page of PDF file (viewed July 14, 2009). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 59-60).

Le travail réalisé dans cette thèse introduit le concept de l'utilisation de la méthode Level-Set pour simuler l'interface Glace/Air au cours du temps lors du processus de givrage en vol des aéronefs. Pour cela, un code de givrage tri-dimensionnel multi-blocs et parallélisé a été implémenté au sein du solveur NSMB (Navier-Stokes-Multi-Blocks). Il comprend notamment un module de calcul des trajectoires des gouttelettes par une approche Eulérienne compatible avec l'utilisation de grilles chimères et un module thermodynamique pour le calcul des masses de glace incluant deux modèles différents : un modèle algébrique itératif et un modèle à dérivées partielles. Une attention particulière a été portée sur la vérification du code de givrage implémenté en comparant systématiquement, si possible, les résultats obtenus avec les données expérimentales et numériques existantes dans la littérature. Pour cette raison, le module de déformation de maillage existant dans NSMB a été intégré au code implémenté afin de pouvoir simuler le givrage par une méthode traditionnelle. Enfin, un nouveau principe pour le suivi de l'interface glace/air est introduit via l'utilisation d'une méthode Level-Set. Puisque dans ce travail de thèse nous nous intéressons particulièrement au concept, la méthode Level-Set développée est d'ordre un et est résolue implicitement. On montrera cependant que des résultats valides sont obtenus avec une telle approximation<br>This thesis introduces the concept of the Level-Set method for simulating the evolution through time of the ice/air interface during the process of in-flight aircraft icing. For that purpose, a three-dimensionnal multi-block and parallelized icing code have been implemented in the NSMB flow solver (Navier-Stokes-Multi-Blocks). It includes a module for calculating the droplet trajectories by an Eulerian approach compatible with the use of chimera grids and a thermodynamic module to calculate the ice masses including two different models : an iterative algebraic model and a PDE model. Particular attention was paid to the validation of the icing code irnplemented by comparing results with existing experimental and numerical data in the literature. For this reason, the existing mesh deformation algorithm in NSMB was integrated into the code to simulate icing by a traditional method. Finally a new principle to track the ice/air interface is introduced using the Level-Set method. Since we are particularly interested in the concept, the Level-Set method developped is first order and solved implicitly. However it will be shown that valid results are obtained with such an approximation

This thesis is concerned with motion planning for the classical two-phase Stefan problem in level set formulation. The interface separating the fluid phases from the solid phases is represented as the zero level set of a continuous function whose evolution is described by the level set equation. Heat conduction in the two phases is modeled by the heat equation. A quadratic tracking-type cost functional that incorporates temperature tracking terms and a control cost term that expresses the desire to have the interface follow a prescribed trajectory by adjusting the heat flux through part of the boundary of the computational domain. The formal Lagrange approach is used to establish a first-order optimality system by applying shape calculus tools. For the numerical solution, the level set equation and its adjoint are discretized in space by discontinuous Galerkin methods that are combined with suitable explicit Runge-Kutta time stepping schemes, while the temperature and its adjoint are approximated in space by the extended finite element method (which accounts for the weak discontinuity of the temperature by a dynamic local modification of the underlying finite element spaces) combined with the implicit Euler method for the temporal discretization. The curvature of the interface which arises in the adjoint system is discretized by a finite element method as well. The projected gradient method, and, in the absence of control constraints, the limited memory BFGS method are used to solve the arising optimization problems. Several numerical examples highlight the potential of the proposed optimal control approach. In particular, they show that it inherits the geometric flexibility of the level set method. Thus, in addition to unidirectional solidification, closed interfaces and changes of topology can be tracked. Finally, the Moreau-Yosida regularization is applied to transform a state constraint on the position of the interface into a penalty term that is added to the cost functional. The optimality conditions for this penalized optimal control problem and its numerical solution are discussed. An example confirms the efficacy of the state constraint<br>Die vorliegende Arbeit beschäftigt sich mit einem Optimalsteuerungsproblem für das klassische Stefan-Problem in zwei Phasen. Die Phasengrenze wird als Niveaulinie einer stetigen Funktion modelliert, was die Lösung der so genannten Level-Set-Gleichung erfordert. Durch Anpassen des Wärmeflusses am Rand des betrachteten Gebiets soll ein gewünschter Verlauf der Phasengrenze angesteuert werden. Zusammen mit dem Wunsch, ein vorgegebenes Temperaturprofil zu approximieren, wird dieses Ziel in einem quadratischen Zielfunktional formuliert. Die notwendigen Optimalitätsbedingungen erster Ordnung werden formal mit Hilfe der entsprechenden Lagrange-Funktion und unter Benutzung von Techniken aus der Formoptimierung hergeleitet. Für die numerische Lösung müssen die auftretenden partiellen Differentialgleichungen diskretisiert werden. Dies geschieht im Falle der Level-Set-Gleichung und ihrer Adjungierten auf Basis von unstetigen Galerkin-Verfahren und expliziten Runge-Kutta-Methoden. Die Wärmeleitungsgleichung und die entsprechende Gleichung im adjungierten System werden mit einer erweiterten Finite-Elemente-Methode im Ort sowie dem impliziten Euler-Verfahren in der Zeit diskretisiert. Dieser Zugang umgeht die aufwändige Adaption des Gitters, die normalerweise bei der FE-Diskretisierung von Phasenübergangsproblemen unvermeidbar ist. Auch die Krümmung der Phasengrenze wird numerisch mit Hilfe der Methode der finiten Elemente angenähert. Zur Lösung der auftretenden Optimierungsprobleme werden ein Gradienten-Projektionsverfahren und, im Fall dass keine Kontrollschranken vorliegen, die BFGS-Methode mit beschränktem Speicherbedarf eingesetzt. Numerische Beispiele beleuchten die Stärken des vorgeschlagenen Zugangs. Es stellt sich insbesondere heraus, dass sich die geometrische Flexibilität der Level-Set-Methode auf den vorgeschlagenen Zugang zur optimalen Steuerung vererbt. Zusätzlich zur gerichteten Bewegung einer flachen Phasengrenze können somit auch geschlossene Phasengrenzen sowie topologische Veränderungen angesteuert werden. Exemplarisch, und zwar an Hand einer Beschränkung an die Lage der Phasengrenze, wird auch noch die Behandlung von Zustandsbeschränkungen mittels der Moreau-Yosida-Regularisierung diskutiert. Ein numerisches Beispiel demonstriert die Wirkung der Zustandsbeschränkung

Level-set methods have become a valuable and well-established field of visualization over the last decades. Different implementations addressing different design goals and different data types exist. In particular, level sets can be used to extract isosurfaces from scalar volume data that fulfill certain smoothness criteria. Recently, such an approach has been generalized to operate on unstructured point-based volume data, where data points are not arranged on a regular grid nor are they connected in form of a mesh. Utilizing this new development, one can avoid an interpolation to a regular grid which inevitably introduces interpolation errors. However, the global processing of the level-set function can be slow when dealing with unstructured point-based volume data sets containing several million data points. We propose an improved level-set approach that performs the process of the level-set function locally. As for isosurface extraction we are only interested in the zero level set, values are only updated in regions close to the zero level set. In each iteration of the level-set process, the zero level set is extracted using direct isosurface extraction from unstructured point-based volume data and a narrow band around the zero level set is constructed. The band consists of two parts: an inner and an outer band. The inner band contains all data points within a small area around the zero level set. These points are updated when executing the level set step. The outer band encloses the inner band providing all those neighbors of the points of the inner band that are necessary to approximate gradients and mean curvature. Neighborhood information is obtained using an efficient kd-tree scheme, gradients and mean curvature are estimated using a four-dimensional least-squares fitting approach. Comparing ourselves to the global approach, we demonstrate that this local level-set approach for unstructured point-based volume data achieves a significant speed-up of one order of magnitude for data sets in the range of several million data points with equivalent quality and robustness.

In this paper, finite element discretizations of the degenerated operator -&omega;²(y) u_xx-&omega;²(x)u_yy=g in the unit square are investigated, where the weight function satisfies &omega;(&xi;)=&xi;^&alpha; with &alpha; &ge; 0. We propose two multi-level methods in order to solve the resulting system of linear algebraic equations. The first method is a multi-grid algorithm with line-smoother. A proof of the smoothing property is given. The second method is a BPX-like preconditioner which we call MTS-BPX preconditioner. We show that the upper eigenvalue bound of the MTS-BPX preconditioned system matrix grows proportionally to the level number.

A otimização topológica de estruturas está relacionada à concepção de projetos que executem suas funções com nível de segurança adequado empregando a quantidade mínima de material. Neste trabalho, determina-se a geometria ótima de estruturas planas por meio do acoplamento do Método dos Elementos de Contorno (MEC) ao Método Level Set (MLS). O algoritmo é composto por 3 etapas: problema mecânico, otimização topológica e reconstrução da estrutura. O problema mecânico é resolvido pelas equações algébricas do MEC. A otimização topológica é determinada pelo MLS, este representa a geometria do corpo e suas evoluções por meio da função Level Set (LS) avaliada em seu nível zero. Na reconstrução realiza-se o remalhamento, pois a cada iteração a estrutura é modificada. O acoplamento proposto resulta na geometria ótima da estrutura sem a necessidade da aplicação de filtros. Os exemplos analisados mostram que algoritmo desenvolvido capta adequadamente a geometria ótima das estruturas. Com esse trabalho, avança-se no campo das aplicações do acoplamento MEC-MLS e no desenvolvimento de soluções inovadoras para problemas complexos de engenharia.<br>In general, the topological optimization of structures is related to design projects that perform their functions with appropriate security levels using the minimum amount of material. This research determines the optimal geometry of 2D structures by coupling the Boundary Blement Method (BEM) to Level Set Method (LSM). The algorithm consists of 3 steps: mechanical model, topology optimization and structure reconstruction. The mechanical model is solved by BEM algebraic equations. The topology optimization is determined using the MLS, the geometry of the body is determined by the Level Set (LS) function evaluated at the zero level. The reconstruction achieves the remeshing, because for each iteration of the structure is modified. The proposed coupling results in the optimal geometry of the structure without the filters application. The examples show that the algorithm developed captures adequately the optimal geometry of the structures. With this dissertation, it is possible advance in the field of applications of the BEM - LSM and develop innovative solutions to complex engineering problems.

The purpose of the thesis was to explore the possibilities of using a Level-Set method to design a time-optimal path planar of a subject to direction-dependent maximum velocities. A promising application for such a planning approach lies in sailboat navigation planning, because of the dynamic ocean waves, current, wind and the characteristics of a sailboat. In the thesis, we developed an IOS application to simulate such scenario as environment properties with wind, static obstacles and the sailboat mapped into direction-dependent velocities in different locations of the environment.  Considering the wind is the main power for the sailboat, a wind speed generation function was created, based on different locations. The Level-Set method is widely used in image processing because of its various advantages, for instance, the ability to deal with topology change and stability. It also can be applied in path planning, in which the process of the Level-Set method can be considered as a continuous wave front propagating with a speed from the start location. A grid-based map was used to represent the environment. While the wave front was crossing the cell on the grid, a time was recorded for every cell, following the negative gradient direction of such crossing time, and then an optimal path could be found. In addition, we used the Narrow Band method to speed up the calculation of processing the level set equation. Finally, this report gives the results of the experiments of static obstacle avoidance, wind effects and smooth path planning.

Reservoir characterization is one of the most important problems in petroleum engineering. It involves forward reservoir modeling that predicts the fluid behavior in the reservoir and inverse problem that calibrates created reservoir models with given data. In this dissertation, we focus on two problems in the field of reservoir characterization: depth of investigation in heterogeneous reservoirs, and history matching and uncertainty quantification of channelized reservoirs. The concept of depth of investigation is fundamental to well test analysis. Much of the current well test analysis relies on analytical solutions based on homogeneous or layered reservoirs. However, such analytic solutions are severely limited for heterogeneous and fractured reservoirs, particularly for unconventional reservoirs with multistage hydraulic fractures. We first generalize the concept to heterogeneous reservoirs and provide an efficient tool to calculate drainage volume using fast marching methods and estimate pressure depletion based on geometric pressure approximation. The applicability of proposed method is illustrated using two applications in unconventional reservoirs including flow regime visualization and stimulated reservoir volume estimation. Due to high permeability contrast and non-Gaussianity of channelized permeability field, it is difficult to history match and quantify uncertainty of channelized reservoirs using traditional approaches. We treat facies boundaries as level set functions and solve the moving boundary problem (history matching) with the level set equation. In addition to level set methods, we also exploit the problem using pixel based approach. The reversible jump Markov Chain Monte Carlo approach is utilized to search the parameter space with flexible dimensions. Both proposed approaches are demonstrated with two and three dimensional examples.

碩士<br>國立臺灣大學<br>化學工程學研究所<br>94<br>The level set method has been widely used in numerics of propagating interfaces. Level set function is close to a signed distance function, and it can be used to exactly locate the interface in order to apply discretizations. Topological changes in the evolving front are handled naturally. The position of the front at time t is given by the zero-level set of a smooth, continuous function. This set needs not be connected and can break and merge as t advances. Furthermore, it can be easily extended to higher dimensions. Over the last five years, the adaptive phase field model was widely adopted to study solidification problems in our group, while many fruitful results have been reported. However, because the concept of diffusive interface is adopted, the drawback for performing phase field modeling lies on its very awful computational load. Recently, the development of level set method has become mature and was used to simulate cases with complex distortion of interface, such as dendritic growth of an alloy [1]. Since the formulation of sharp-interface model is embedded locally, in principle, the level set method can simulate these problems accurately by using relatively thicker mesh structure. In this report, we have developed an adaptive level set method based on the finite volume method (FVM) to simulate solidification problems. To check its feasibility, we have derived numerical and physical algorithms carefully and discussed the convergence of our present model. Moreover, comparisons with analytical solutions were given by testing several Stefan problems. Finally, we tried to challenge the case dendritic growth under high supercooling.

In bubble dynamics, an underwater bubble may evolve from being singly-connected to being toroidal. Furthermore, two or more individual bubbles may merge to form a single large bubble. These dynamics involve significant topological changes such as merging and breaking, which may not be handled well by front-tracking boundary element methods. In the level set method, topological changes are handled naturally through a higher-dimensional level set function. This makes it an attractive method for bubble simulation. In this paper, we present a method that combines the level set method and the boundary element method for the simulation of bubble dynamics. We propose a formulation for the update of a potential function in the level set context. This potential function is non-physical off the bubble surface but consistent with the physics on the bubble surface. We consider only axisymmetric cavitation bubbles in this paper. Included in the paper are some preliminary results and findings.<br>Singapore-MIT Alliance (SMA)