@phdthesis{LopezZermeno,
  author    = {L{\´o}pez Zerme{\~n}o, Jorge Alberto},
  title     = {Isogeometric and CAD-based methods for shape and topology optimization: Sensitivity analysis, B{\´e}zier elements and phase-field approaches},
  doi       = {10.25643/bauhaus-universitaet.4710},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20220831-47102},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  abstract  = {The Finite Element Method (FEM) is widely used in engineering for solving Partial Differential Equations (PDEs) over complex geometries. To this end, it is required to provide the FEM software with a geometric model that is typically constructed in a Computer-Aided Design (CAD) software. However, FEM and CAD use different approaches for the mathematical description of the geometry. Thus, it is required to generate a mesh, which is suitable for FEM, based on the CAD model. Nonetheless, this procedure is not a trivial task and it can be time consuming. This issue becomes more significant for solving shape and topology optimization problems, which consist in evolving the geometry iteratively. Therefore, the computational cost associated to the mesh generation process is increased exponentially for this type of applications. The main goal of this work is to investigate the integration of CAD and CAE in shape and topology optimization. To this end, numerical tools that close the gap between design and analysis are presented. The specific objectives of this work are listed below: • Automatize the sensitivity analysis in an isogeometric framework for applications in shape optimization. Applications for linear elasticity are considered. • A methodology is developed for providing a direct link between the CAD model and the analysis mesh. In consequence, the sensitivity analysis can be performed in terms of the design variables located in the design model. • The last objective is to develop an isogeometric method for shape and topological optimization. This method should take advantage of using Non-Uniform Rational B-Splines (NURBS) with higher continuity as basis functions. Isogeometric Analysis (IGA) is a framework designed to integrate the design and analysis in engineering problems. The fundamental idea of IGA is to use the same basis functions for modeling the geometry, usually NURBS, for the approximation of the solution fields. The advantage of integrating design and analysis is two-fold. First, the analysis stage is more accurate since the system of PDEs is not solved using an approximated geometry, but the exact CAD model. Moreover, providing a direct link between the design and analysis discretizations makes possible the implementation of efficient sensitivity analysis methods. Second, the computational time is significantly reduced because the mesh generation process can be avoided. Sensitivity analysis is essential for solving optimization problems when gradient-based optimization algorithms are employed. Automatic differentiation can compute exact gradients, automatically by tracking the algebraic operations performed on the design variables. For the automation of the sensitivity analysis, an isogeometric framework is used. Here, the analysis mesh is obtained after carrying out successive refinements, while retaining the coarse geometry for the domain design. An automatic differentiation (AD) toolbox is used to perform the sensitivity analysis. The AD toolbox takes the code for computing the objective and constraint functions as input. Then, using a source code transformation approach, it outputs a code for computing the objective and constraint functions, and their sensitivities as well. The sensitivities obtained from the sensitivity propagation method are compared with analytical sensitivities, which are computed using a full isogeometric approach. The computational efficiency of AD is comparable to that of analytical sensitivities. However, the memory requirements are larger for AD. Therefore, AD is preferable if the memory requirements are satisfied. Automatic sensitivity analysis demonstrates its practicality since it simplifies the work of engineers and designers. Complex geometries with sharp edges and/or holes cannot easily be described with NURBS. One solution is the use of unstructured meshes. Simplex-elements (triangles and tetrahedra for two and three dimensions respectively) are particularly useful since they can automatically parameterize a wide variety of domains. In this regard, unstructured B{\´e}zier elements, commonly used in CAD, can be employed for the exact modelling of CAD boundary representations. In two dimensions, the domain enclosed by NURBS curves is parameterized with B{\´e}zier triangles. To describe exactly the boundary of a two-dimensional CAD model, the continuity of a NURBS boundary representation is reduced to C^0. Then, the control points are used to generate a triangulation such that the boundary of the domain is identical to the initial CAD boundary representation. Thus, a direct link between the design and analysis discretizations is provided and the sensitivities can be propagated to the design domain. In three dimensions, the initial CAD boundary representation is given as a collection of NURBS surfaces that enclose a volume. Using a mesh generator (Gmsh), a tetrahedral mesh is obtained. The original surface is reconstructed by modifying the location of the control points of the tetrahedral mesh using B{\´e}zier tetrahedral elements and a point inversion algorithm. This method offers the possibility of computing the sensitivity analysis using the analysis mesh. Then, the sensitivities can be propagated into the design discretization. To reuse the mesh originally generated, a moving B{\´e}zier tetrahedral mesh approach was implemented. A gradient-based optimization algorithm is employed together with a sensitivity propagation procedure for the shape optimization cases. The proposed shape optimization approaches are used to solve some standard benchmark problems in structural mechanics. The results obtained show that the proposed approach can compute accurate gradients and evolve the geometry towards optimal solutions. In three dimensions, the moving mesh approach results in faster convergence in terms of computational time and avoids remeshing at each optimization step. For considering topological changes in a CAD-based framework, an isogeometric phase-field based shape and topology optimization is developed. In this case, the diffuse interface of a phase-field variable over a design domain implicitly describes the boundaries of the geometry. The design variables are the local values of the phase-field variable. The descent direction to minimize the objective function is found by using the sensitivities of the objective function with respect to the design variables. The evolution of the phase-field is determined by solving the time dependent Allen-Cahn equation. Especially for topology optimization problems that require C^1 continuity, such as for flexoelectric structures, the isogeometric phase field method is of great advantage. NURBS can achieve the desired continuity more efficiently than the traditional employed functions. The robustness of the method is demonstrated when applied to different geometries, boundary conditions, and material configurations. The applications illustrate that compared to piezoelectricity, the electrical performance of flexoelectric microbeams is larger under bending. In contrast, the electrical power for a structure under compression becomes larger with piezoelectricity.},
  subject      = {CAD},
  language  = {en}
}
@phdthesis{Schollmeyer,
  author    = {Schollmeyer, Andre},
  title     = {Efficient and High-Quality Rendering of Higher-Order Geometric Data Representations},
  doi       = {10.25643/bauhaus-universitaet.3823},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20181120-38234},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {143},
  abstract  = {Computer-Aided Design (CAD) bezeichnet den Entwurf industrieller Produkte mit Hilfe von virtuellen 3D Modellen. Ein CAD-Modell besteht aus parametrischen Kurven und Fl{\"a}chen, in den meisten F{\"a}llen non-uniform rational B-Splines (NURBS). Diese mathematische Beschreibung wird ebenfalls zur Analyse, Optimierung und Pr{\"a}sentation des Modells verwendet. In jeder dieser Entwicklungsphasen wird eine unterschiedliche visuelle Darstellung ben{\"o}tigt, um den entsprechenden Nutzern ein geeignetes Feedback zu geben. Designer bevorzugen beispielsweise illustrative oder realistische Darstellungen, Ingenieure ben{\"o}tigen eine verst{\"a}ndliche Visualisierung der Simulationsergebnisse, w{\"a}hrend 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 eingeschr{\"a}nkten Hardwareunterst{\"u}tzung 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 F{\"a}higkeiten aktueller Grafikkarten aus, um den Stand der Technik bez{\"u}glich Qualit{\"a}t, Effizienz und Darstellungsgeschwindigkeit zu verbessern. Zwei dieser Verfahren befassen sich mit der direkten Darstellung der parametrischen Beschreibung ohne Approximationen oder zeitaufw{\"a}ndige Vorberechnungen. Die dabei vorgestellten Datenstrukturen und Algorithmen erm{\"o}glichen die effiziente Unterteilung, Klassifizierung, Tessellierung und Darstellung getrimmter NURBS-Fl{\"a}chen und einen interaktiven Ray-Casting-Algorithmus f{\"u}r die Isofl{\"a}chenvisualisierung von NURBSbasierten isogeometrischen Analysen. Die weiteren zwei Verfahren beschreiben zum einen das vielseitige Konzept der programmierbaren Transparenz f{\"u}r illustrative und verst{\"a}ndliche Visualisierungen tiefenkomplexer CAD-Modelle und zum anderen eine neue hybride Methode zur Reprojektion halbtransparenter und undurchsichtiger Bildinformation f{\"u}r die Beschleunigung der Erzeugung von stereoskopischen Bildpaaren. Die beiden letztgenannten Ans{\"a}tze basieren auf rasterisierter Geometrie und sind somit ebenfalls f{\"u}r normale Dreiecksmodelle anwendbar, wodurch die Arbeiten auch einen wichtigen Beitrag in den Bereichen der Computergrafik und der virtuellen Realit{\"a}t darstellen. Die Auswertung der Arbeit wurde mit großen, realen NURBS-Datens{\"a}tzen durchgef{\"u}hrt. Die Resultate zeigen, dass die direkte Darstellung auf Grundlage der parametrischen Beschreibung mit interaktiven Bildwiederholraten und in subpixelgenauer Qualit{\"a}t m{\"o}glich ist. Die Einf{\"u}hrung programmierbarer Transparenz erm{\"o}glicht zudem die Umsetzung kollaborativer 3D Interaktionstechniken f{\"u}r die Exploration der Modelle in virtuellenUmgebungen sowie illustrative und verst{\"a}ndliche Visualisierungen tiefenkomplexer CAD-Modelle. Die Erzeugung stereoskopischer Bildpaare f{\"u}r die interaktive Visualisierung auf 3D Displays konnte beschleunigt werden. Diese messbare Verbesserung wurde zudem im Rahmen einer Nutzerstudie als wahrnehmbar und vorteilhaft befunden.},
  subject      = {Rendering},
  language  = {en}
}