The Construction of Optimized High-Order Surface Meshes by Energy-Minimization

Despite the increasing popularity of high-order methods in computational fluid dynamics, their application to practical problems still remains challenging. In order to exploit the advantages of high-order methods with geometrically complex computational domains, coarse curved meshes are necessary, i.e. high-order representations of the geometry. This dissertation presents a strategy for the generation of curved high-order surface meshes. The mesh generation method combines least-squares fitting with energy functionals, which approximate physical bending and stretching energies, in an incremental energy-minimizing fitting strategy. Since the energy weighting is reduced in each increment, the resulting surface representation features high accuracy. Nevertheless, the beneficial influence of the energy-minimization is retained. The presented method aims at enabling the utilization of the superior convergence properties of high-order methods by facilitating the construction of coarser meshes, while ensuring accuracy by allowing an arbitrary choice of geometric approximation order. Results show surface meshes of remarkable quality, even for very coarse meshes representing complex domains, e.g. blood vessels

IST Austria Thesis

Fabrication of curved shells plays an important role in modern design, industry, and science. Among their remarkable properties are, for example, aesthetics of organic shapes, ability to evenly distribute loads, or efficient flow separation. They find applications across vast length scales ranging from sky-scraper architecture to microscopic devices. But, at the same time, the design of curved shells and their manufacturing process pose a variety of challenges. In this thesis, they are addressed from several perspectives. In particular, this thesis presents approaches based on the transformation of initially flat sheets into the target curved surfaces. This involves problems of interactive design of shells with nontrivial mechanical constraints, inverse design of complex structural materials, and data-driven modeling of delicate and time-dependent physical properties. At the same time, two newly-developed self-morphing mechanisms targeting flat-to-curved transformation are presented. In architecture, doubly curved surfaces can be realized as cold bent glass panelizations. Originally flat glass panels are bent into frames and remain stressed. This is a cost-efficient fabrication approach compared to hot bending, when glass panels are shaped plastically. However such constructions are prone to breaking during bending, and it is highly nontrivial to navigate the design space, keeping the panels fabricable and aesthetically pleasing at the same time. We introduce an interactive design system for cold bent glass façades, while previously even offline optimization for such scenarios has not been sufficiently developed. Our method is based on a deep learning approach providing quick and high precision estimation of glass panel shape and stress while handling the shape multimodality. Fabrication of smaller objects of scales below 1 m, can also greatly benefit from shaping originally flat sheets. In this respect, we designed new self-morphing shell mechanisms transforming from an initial flat state to a doubly curved state with high precision and detail. Our so-called CurveUps demonstrate the encodement of the geometric information into the shell. Furthermore, we explored the frontiers of programmable materials and showed how temporal information can additionally be encoded into a flat shell. This allows prescribing deformation sequences for doubly curved surfaces and, thus, facilitates self-collision avoidance enabling complex shapes and functionalities otherwise impossible. Both of these methods include inverse design tools keeping the user in the design loop

Numerical quadrature for Gregory quads

We investigate quadrature rules in the context of quadrilateral Gregory patches, in short Gregory quads. We provide numerical and where possible symbolic quadrature rules for the space spanned by the twenty polynomial/rational functions associated with Gregory quads, as well as the derived spaces including derivatives, products, and products of derivatives of these functions. This opens up the possibility for a wider adoption of Gregory quads in numerical simulations

Efficient and High-Quality Rendering of Higher-Order Geometric Data Representations

Computer-Aided Design (CAD) bezeichnet den Entwurf industrieller Produkte mit Hilfe von virtuellen 3D Modellen. Ein CAD-Modell besteht aus parametrischen Kurven und Flächen, in den meisten Fällen non-uniform rational B-Splines (NURBS). Diese mathematische Beschreibung wird ebenfalls zur Analyse, Optimierung und Präsentation des Modells verwendet. In jeder dieser Entwicklungsphasen wird eine unterschiedliche visuelle Darstellung benötigt, um den entsprechenden Nutzern ein geeignetes Feedback zu geben. Designer bevorzugen beispielsweise illustrative oder realistische Darstellungen, Ingenieure benötigen eine verständliche Visualisierung der Simulationsergebnisse, während eine immersive 3D Darstellung bei einer Benutzbarkeitsanalyse oder der Designauswahl hilfreich sein kann. Die interaktive Darstellung von NURBS-Modellen und -Simulationsdaten ist jedoch aufgrund des hohen Rechenaufwandes und der eingeschränkten Hardwareunterstützung eine große Herausforderung. Diese Arbeit stellt vier neuartige Verfahren vor, welche sich mit der interaktiven Darstellung von NURBS-Modellen und Simulationensdaten befassen. Die vorgestellten Algorithmen nutzen neue Fähigkeiten aktueller Grafikkarten aus, um den Stand der Technik bezüglich Qualität, Effizienz und Darstellungsgeschwindigkeit zu verbessern. Zwei dieser Verfahren befassen sich mit der direkten Darstellung der parametrischen Beschreibung ohne Approximationen oder zeitaufwändige Vorberechnungen. Die dabei vorgestellten Datenstrukturen und Algorithmen ermöglichen die effiziente Unterteilung, Klassifizierung, Tessellierung und Darstellung getrimmter NURBS-Flächen und einen interaktiven Ray-Casting-Algorithmus für die Isoflächenvisualisierung von NURBSbasierten isogeometrischen Analysen. Die weiteren zwei Verfahren beschreiben zum einen das vielseitige Konzept der programmierbaren Transparenz für illustrative und verständliche Visualisierungen tiefenkomplexer CAD-Modelle und zum anderen eine neue hybride Methode zur Reprojektion halbtransparenter und undurchsichtiger Bildinformation für die Beschleunigung der Erzeugung von stereoskopischen Bildpaaren. Die beiden letztgenannten Ansätze basieren auf rasterisierter Geometrie und sind somit ebenfalls für normale Dreiecksmodelle anwendbar, wodurch die Arbeiten auch einen wichtigen Beitrag in den Bereichen der Computergrafik und der virtuellen Realität darstellen. Die Auswertung der Arbeit wurde mit großen, realen NURBS-Datensätzen durchgeführt. Die Resultate zeigen, dass die direkte Darstellung auf Grundlage der parametrischen Beschreibung mit interaktiven Bildwiederholraten und in subpixelgenauer Qualität möglich ist. Die Einführung programmierbarer Transparenz ermöglicht zudem die Umsetzung kollaborativer 3D Interaktionstechniken für die Exploration der Modelle in virtuellenUmgebungen sowie illustrative und verständliche Visualisierungen tiefenkomplexer CAD-Modelle. Die Erzeugung stereoskopischer Bildpaare für die interaktive Visualisierung auf 3D Displays konnte beschleunigt werden. Diese messbare Verbesserung wurde zudem im Rahmen einer Nutzerstudie als wahrnehmbar und vorteilhaft befunden.In computer-aided design (CAD), industrial products are designed using a virtual 3D model. A CAD model typically consists of curves and surfaces in a parametric representation, in most cases, non-uniform rational B-splines (NURBS). The same representation is also used for the analysis, optimization and presentation of the model. In each phase of this process, different visualizations are required to provide an appropriate user feedback. Designers work with illustrative and realistic renderings, engineers need a comprehensible visualization of the simulation results, and usability studies or product presentations benefit from using a 3D display. However, the interactive visualization of NURBS models and corresponding physical simulations is a challenging task because of the computational complexity and the limited graphics hardware support. This thesis proposes four novel rendering approaches that improve the interactive visualization of CAD models and their analysis. The presented algorithms exploit latest graphics hardware capabilities to advance the state-of-the-art in terms of quality, efficiency and performance. In particular, two approaches describe the direct rendering of the parametric representation without precomputed approximations and timeconsuming pre-processing steps. New data structures and algorithms are presented for the efficient partition, classification, tessellation, and rendering of trimmed NURBS surfaces as well as the first direct isosurface ray-casting approach for NURBS-based isogeometric analysis. The other two approaches introduce the versatile concept of programmable order-independent semi-transparency for the illustrative and comprehensible visualization of depth-complex CAD models, and a novel method for the hybrid reprojection of opaque and semi-transparent image information to accelerate stereoscopic rendering. Both approaches are also applicable to standard polygonal geometry which contributes to the computer graphics and virtual reality research communities. The evaluation is based on real-world NURBS-based models and simulation data. The results show that rendering can be performed directly on the underlying parametric representation with interactive frame rates and subpixel-precise image results. The computational costs of additional visualization effects, such as semi-transparency and stereoscopic rendering, are reduced to maintain interactive frame rates. The benefit of this performance gain was confirmed by quantitative measurements and a pilot user study

Finite element analysis enhanced with subdivision surface boundary representations

In this work we develop a design-through-analysis methodology by extending the concept of the NURBS-enhanced finite element method (NEFEM) to volumes bounded by Catmull-Clark subdivision surfaces. The representation of the boundary as a single watertight manifold facilitates the generation of an external curved triangular mesh, which is subsequently used to generate the interior volumetric mesh. Following the NEFEM framework, the basis functions are defined in the physical space and the numerical integration is realized with a special mapping which takes into account the exact definition of the boundary. Furthermore, an appropriate quadrature strategy is proposed to deal with the integration of elements adjacent to extraordinary vertices (EVs). Both theoretical and practical aspects of the implementation are discussed and are supported with numerical examples.</p

Towards a High Quality Real-Time Graphics Pipeline

Modern graphics hardware pipelines create photorealistic images with high geometric complexity in real time. The quality is constantly improving and advanced techniques from feature film visual effects, such as high dynamic range images and support for higher-order surface primitives, have recently been adopted. Visual effect techniques have large computational costs and significant memory bandwidth usage. In this thesis, we identify three problem areas and propose new algorithms that increase the performance of a set of computer graphics techniques. Our main focus is on efficient algorithms for the real-time graphics pipeline, but parts of our research are equally applicable to offline rendering. Our first focus is texture compression, which is a technique to reduce the memory bandwidth usage. The core idea is to store images in small compressed blocks which are sent over the memory bus and are decompressed on-the-fly when accessed. We present compression algorithms for two types of texture formats. High dynamic range images capture environment lighting with luminance differences over a wide intensity range. Normal maps store perturbation vectors for local surface normals, and give the illusion of high geometric surface detail. Our compression formats are tailored to these texture types and have compression ratios of 6:1, high visual fidelity, and low-cost decompression logic. Our second focus is tessellation culling. Culling is a commonly used technique in computer graphics for removing work that does not contribute to the final image, such as completely hidden geometry. By discarding rendering primitives from further processing, substantial arithmetic computations and memory bandwidth can be saved. Modern graphics processing units include flexible tessellation stages, where rendering primitives are subdivided for increased geometric detail. Images with highly detailed models can be synthesized, but the incurred cost is significant. We have devised a simple remapping technique that allowsfor better tessellation distribution in screen space. Furthermore, we present programmable tessellation culling, where bounding volumes for displaced geometry are computed and used to conservatively test if a primitive can be discarded before tessellation. We introduce a general tessellation culling framework, and an optimized algorithm for rendering of displaced Bézier patches, which is expected to be a common use case for graphics hardware tessellation. Our third and final focus is forward-looking, and relates to efficient algorithms for stochastic rasterization, a rendering technique where camera effects such as depth of field and motion blur can be faithfully simulated. We extend a graphics pipeline with stochastic rasterization in spatio-temporal space and show that stochastic motion blur can be rendered with rather modest pipeline modifications. Furthermore, backface culling algorithms for motion blur and depth of field rendering are presented, which are directly applicable to stochastic rasterization. Hopefully, our work in this field brings us closer to high quality real-time stochastic rendering

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

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

A Partially Randomized Approach to Trajectory Planning and Optimization for Mobile Robots with Flat Dynamics

Motion planning problems are characterized by huge search spaces and complex obstacle structures with no concise mathematical expression. The fixed-wing airplane application considered in this thesis adds differential constraints and point-wise bounds, i. e. an infinite number of equality and inequality constraints. An optimal trajectory planning approach is presented, based on the randomized Rapidly-exploring Random Trees framework (RRT*). The local planner relies on differential flatness of the equations of motion to obtain tree branch candidates that automatically satisfy the differential constraints. Flat output trajectories, in this case equivalent to the airplane's flight path, are designed using Bézier curves. Segment feasibility in terms of point-wise inequality constraints is tested by an indicator integral, which is evaluated alongside the segment cost functional. Although the RRT* guarantees optimality in the limit of infinite planning time, it is argued by intuition and experimentation that convergence is not approached at a practically useful rate. Therefore, the randomized planner is augmented by a deterministic variational optimization technique. To this end, the optimal planning task is formulated as a semi-infinite optimization problem, using the intermediate result of the RRT(*) as an initial guess. The proposed optimization algorithm follows the feasible flavor of the primal-dual interior point paradigm. Discretization of functional (infinite) constraints is deferred to the linear subproblems, where it is realized implicitly by numeric quadrature. An inherent numerical ill-conditioning of the method is circumvented by a reduction-like approach, which tracks active constraint locations by introducing new problem variables. Obstacle avoidance is achieved by extending the line search procedure and dynamically adding obstacle-awareness constraints to the problem formulation. Experimental evaluation confirms that the hybrid approach is practically feasible and does indeed outperform RRT*'s built-in optimization mechanism, but the computational burden is still significant.Bewegungsplanungsaufgaben sind typischerweise gekennzeichnet durch umfangreiche Suchräume, deren vollständige Exploration nicht praktikabel ist, sowie durch unstrukturierte Hindernisse, für die nur selten eine geschlossene mathematische Beschreibung existiert. Bei der in dieser Arbeit betrachteten Anwendung auf Flächenflugzeuge kommen differentielle Randbedingungen und beschränkte Systemgrößen erschwerend hinzu. Der vorgestellte Ansatz zur optimalen Trajektorienplanung basiert auf dem Rapidly-exploring Random Trees-Algorithmus (RRT*), welcher die Suchraumkomplexität durch Randomisierung beherrschbar macht. Der spezifische Beitrag ist eine Realisierung des lokalen Planers zur Generierung der Äste des Suchbaums. Dieser erfordert ein flaches Bewegungsmodell, sodass differentielle Randbedingungen automatisch erfüllt sind. Die Trajektorien des flachen Ausgangs, welche im betrachteten Beispiel der Flugbahn entsprechen, werden mittels Bézier-Kurven entworfen. Die Einhaltung der Ungleichungsnebenbedingungen wird durch ein Indikator-Integral überprüft, welches sich mit wenig Zusatzaufwand parallel zum Kostenfunktional berechnen lässt. Zwar konvergiert der RRT*-Algorithmus (im probabilistischen Sinne) zu einer optimalen Lösung, jedoch ist die Konvergenzrate aus praktischer Sicht unbrauchbar langsam. Es ist daher naheliegend, den Planer durch ein gradientenbasiertes lokales Optimierungsverfahren mit besseren Konvergenzeigenschaften zu unterstützen. Hierzu wird die aktuelle Zwischenlösung des Planers als Initialschätzung für ein kompatibles semi-infinites Optimierungsproblem verwendet. Der vorgeschlagene Optimierungsalgorithmus erweitert das verbreitete innere-Punkte-Konzept (primal dual interior point method) auf semi-infinite Probleme. Eine explizite Diskretisierung der funktionalen Ungleichungsnebenbedingungen ist nicht erforderlich, denn diese erfolgt implizit durch eine numerische Integralauswertung im Rahmen der linearen Teilprobleme. Da die Methode an Stellen aktiver Nebenbedingungen nicht wohldefiniert ist, kommt zusätzlich eine Variante des Reduktions-Ansatzes zum Einsatz, bei welcher der Vektor der Optimierungsvariablen um die (endliche) Menge der aktiven Indizes erweitert wird. Weiterhin wurde eine Kollisionsvermeidung integriert, die in den Teilschritt der Liniensuche eingreift und die Problemformulierung dynamisch um Randbedingungen zur lokalen Berücksichtigung von Hindernissen erweitert. Experimentelle Untersuchungen bestätigen, dass die Ergebnisse des hybriden Ansatzes aus RRT(*) und numerischem Optimierungsverfahren der klassischen RRT*-basierten Trajektorienoptimierung überlegen sind. Der erforderliche Rechenaufwand ist zwar beträchtlich, aber unter realistischen Bedingungen praktisch beherrschbar

Advanced Techniques for Design and Manufacturing in Marine Engineering

Modern engineering design processes are driven by the extensive use of numerical simulations; naval architecture and ocean engineering are no exception. Computational power has been improved over the last few decades; therefore, the integration of different tools such as CAD, FEM, CFD, and CAM has enabled complex modeling and manufacturing problems to be solved in a more feasible way. Classical naval design methodology can take advantage of this integration, giving rise to more robust designs in terms of shape, structural and hydrodynamic performances, and the manufacturing process.This Special Issue invites researchers and engineers from both academia and the industry to publish the latest progress in design and manufacturing techniques in marine engineering and to debate the current issues and future perspectives in this research area. Suitable topics for this issue include, but are not limited to, the following:CAD-based approaches for designing the hull and appendages of sailing and engine-powered boats and comparisons with traditional techniques;Finite element method applications to predict the structural performance of the whole boat or of a portion of it, with particular attention to the modeling of the material used;Embedded measurement systems for structural health monitoring;Determination of hydrodynamic efficiency using experimental, numerical, or semi-empiric methods for displacement and planning hulls;Topology optimization techniques to overcome traditional scantling criteria based on international standards;Applications of additive manufacturing to derive innovative shapes for internal reinforcements or sandwich hull structures