@phdthesis{Chan,
  author    = {Chan, Chiu Ling},
  title     = {Smooth representation of thin shells and volume structures for isogeometric analysis},
  doi       = {10.25643/bauhaus-universitaet.4208},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20200812-42083},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {162},
  abstract  = {The purpose of this study is to develop self-contained methods for obtaining smooth meshes which are compatible with isogeometric analysis (IGA). The study contains three main parts. We start by developing a better understanding of shapes and splines through the study of an image-related problem. Then we proceed towards obtaining smooth volumetric meshes of the given voxel-based images. Finally, we treat the smoothness issue on the multi-patch domains with C1 coupling. Following are the highlights of each part. First, we present a B-spline convolution method for boundary representation of voxel-based images. We adopt the filtering technique to compute the B-spline coefficients and gradients of the images effectively. We then implement the B-spline convolution for developing a non-rigid images registration method. The proposed method is in some sense of "isoparametric", for which all the computation is done within the B-splines framework. Particularly, updating the images by using B-spline composition promote smooth transformation map between the images. We show the possible medical applications of our method by applying it for registration of brain images. Secondly, we develop a self-contained volumetric parametrization method based on the B-splines boundary representation. We aim to convert a given voxel-based data to a matching C1 representation with hierarchical cubic splines. The concept of the osculating circle is employed to enhance the geometric approximation, where it is done by a single template and linear transformations (scaling, translations, and rotations) without the need for solving an optimization problem. Moreover, we use the Laplacian smoothing and refinement techniques to avoid irregular meshes and to improve mesh quality. We show with several examples that the method is capable of handling complex 2D and 3D configurations. In particular, we parametrize the 3D Stanford bunny which contains irregular shapes and voids. Finally, we propose the B´ezier ordinates approach and splines approach for C1 coupling. In the first approach, the new basis functions are defined in terms of the B´ezier Bernstein polynomials. For the second approach, the new basis is defined as a linear combination of C0 basis functions. The methods are not limited to planar or bilinear mappings. They allow the modeling of solutions to fourth order partial differential equations (PDEs) on complex geometric domains, provided that the given patches are G1 continuous. Both methods have their advantages. In particular, the B´ezier approach offer more degree of freedoms, while the spline approach is more computationally efficient. In addition, we proposed partial degree elevation to overcome the C1-locking issue caused by the over constraining of the solution space. We demonstrate the potential of the resulting C1 basis functions for application in IGA which involve fourth order PDEs such as those appearing in Kirchhoff-Love shell models, Cahn-Hilliard phase field application, and biharmonic problems.},
  subject      = {Modellierung},
  language  = {en}
}
@phdthesis{Stein,
  author    = {Stein, Peter},
  title     = {Procedurally generated models for Isogeometric Analysis},
  publisher = {Universit{\"a}tsverlag},
  address   = {Weimar},
  isbn      = {978-3-86068-488-7},
  doi       = {10.25643/bauhaus-universitaet.1848},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20130212-18483},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {82},
  abstract  = {Increasingly powerful hard- and software allows for the numerical simulation of complex physical phenomena with high levels of detail. In light of this development the definition of numerical models for the Finite Element Method (FEM) has become the bottleneck in the simulation process. Characteristic features of the model generation are large manual efforts and a de-coupling of geometric and numerical model. In the highly probable case of design revisions all steps of model preprocessing and mesh generation have to be repeated. This includes the idealization and approximation of a geometric model as well as the definition of boundary conditions and model parameters. Design variants leading to more resource-efficient structures might hence be disregarded due to limited budgets and constrained time frames. A potential solution to above problem is given with the concept of Isogeometric Analysis (IGA). Core idea of this method is to directly employ a geometric model for numerical simulations, which allows to circumvent model transformations and the accompanying data losses. Basis for this method are geometric models described in terms of Non-uniform rational B-Splines (NURBS). This class of piecewise continuous rational polynomial functions is ubiquitous in computer graphics and Computer-Aided Design (CAD). It allows the description of a wide range of geometries using a compact mathematical representation. The shape of an object thereby results from the interpolation of a set of control points by means of the NURBS functions, allowing efficient representations for curves, surfaces and solid bodies alike. Existing software applications, however, only support the modeling and manipulation of the former two. The description of three-dimensional solid bodies consequently requires significant manual effort, thus essentially forbidding the setup of complex models. This thesis proposes a procedural approach for the generation of volumetric NURBS models. That is, a model is not described in terms of its data structures but as a sequence of modeling operations applied to a simple initial shape. In a sense this describes the "evolution" of the geometric model under the sequence of operations. In order to adapt this concept to NURBS geometries, only a compact set of commands is necessary which, in turn, can be adapted from existing algorithms. A model then can be treated in terms of interpretable model parameters. This leads to an abstraction from its data structures and model variants can be set up by variation of the governing parameters. The proposed concept complements existing template modeling approaches: templates can not only be defined in terms of modeling commands but can also serve as input geometry for said operations. Such templates, arranged in a nested hierarchy, provide an elegant model representation. They offer adaptivity on each tier of the model hierarchy and allow to create complex models from only few model parameters. This is demonstrated for volumetric fluid domains used in the simulation of vertical-axis wind turbines. Starting from a template representation of airfoil cross-sections, the complete "negative space" around the rotor blades can be described by a small set of model parameters, and model variants can be set up in a fraction of a second. NURBS models offer a high geometric flexibility, allowing to represent a given shape in different ways. Different model instances can exhibit varying suitability for numerical analyses. For their assessment, Finite Element mesh quality metrics are regarded. The considered metrics are based on purely geometric criteria and allow to identify model degenerations commonly used to achieve certain geometric features. They can be used to decide upon model adaptions and provide a measure for their efficacy. Unfortunately, they do not reveal a relation between mesh distortion and ill-conditioning of the equation systems resulting from the numerical model.},
  subject      = {NURBS},
  language  = {en}
}
@phdthesis{NguyenThanh,
  author    = {Nguyen-Thanh, Nhon},
  title     = {Isogeometric analysis based on rational splines over hierarchical T-mesh and alpha finite element method for structural analysis},
  issn      = {1610-7381},
  doi       = {10.25643/bauhaus-universitaet.2078},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20131125-20781},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {196},
  abstract  = {This thesis presents two new methods in finite elements and isogeometric analysis for structural analysis. The first method proposes an alternative alpha finite element method using triangular elements. In this method, the piecewise constant strain field of linear triangular finite element method models is enhanced by additional strain terms with an adjustable parameter a, which results in an effectively softer stiffness formulation compared to a linear triangular element. In order to avoid the transverse shear locking of Reissner-Mindlin plates analysis the alpha finite element method is coupled with a discrete shear gap technique for triangular elements to significantly improve the accuracy of the standard triangular finite elements. The basic idea behind this element formulation is to approximate displacements and rotations as in the standard finite element method, but to construct the bending, geometrical and shear strains using node-based smoothing domains. Several numerical examples are presented and show that the alpha FEM gives a good agreement compared to several other methods in the literature. Second method, isogeometric analysis based on rational splines over hierarchical T-meshes (RHT-splines) is proposed. The RHT-splines are a generalization of Non-Uniform Rational B-splines (NURBS) over hierarchical T-meshes, which is a piecewise bicubic polynomial over a hierarchical T-mesh. The RHT-splines basis functions not only inherit all the properties of NURBS such as non-negativity, local support and partition of unity but also more importantly as the capability of joining geometric objects without gaps, preserving higher order continuity everywhere and allow local refinement and adaptivity. In order to drive the adaptive refinement, an efficient recovery-based error estimator is employed. For this problem an imaginary surface is defined. The imaginary surface is basically constructed by RHT-splines basis functions which is used for approximation and interpolation functions as well as the construction of the recovered stress components. Numerical investigations prove that the proposed method is capable to obtain results with higher accuracy and convergence rate than NURBS results.},
  subject      = {Isogeometric analysis},
  language  = {en}
}
@phdthesis{Schwedler,
  author    = {Schwedler, Michael},
  title     = {Integrated structural analysis using isogeometric finite element methods},
  doi       = {10.25643/bauhaus-universitaet.2737},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20170130-27372},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {209},
  abstract  = {The gradual digitization in the architecture, engineering, and construction industry over the past fifty years led to an extremely heterogeneous software environment, which today is embodied by the multitude of different digital tools and proprietary data formats used by the many specialists contributing to the design process in a construction project. Though these projects become increasingly complex, the demands on financial efficiency and the completion within a tight schedule grow at the same time. The digital collaboration of project partners has been identified as one key issue in successfully dealing with these challenges. Yet currently, the numerous software applications and their respective individual views on the design process severely impede that collaboration. An approach to establish a unified basis for the digital collaboration, regardless of the existing software heterogeneity, is a comprehensive digital building model contributed to by all projects partners. This type of data management known as building information modeling (BIM) has many benefits, yet its adoption is associated with many difficulties and thus, proceeds only slowly. One aspect in the field of conflicting requirements on such a digital model is the cooperation of architects and structural engineers. Traditionally, these two disciplines use different abstractions of reality for their models that in consequence lead to incompatible digital representations thereof. The onset of isogeometric analysis (IGA) promised to ease the discrepancy in design and analysis model representations. Yet, that initial focus quickly shifted towards using these methods as a more powerful basis for numerical simulations. Furthermore, the isogeometric representation alone is not capable of solving the model abstraction problem. It is thus the intention of this work to contribute to an improved digital collaboration of architects and engineers by exploring an integrated analysis approach on the basis of an unified digital model and solid geometry expressed by splines. In the course of this work, an analysis framework is developed that utilizes such models to automatically conduct numerical simulations commonly required in construction projects. In essence, this allows to retrieve structural analysis results from BIM models in a fast and simple manner, thereby facilitating rapid design iterations and profound design feedback. The BIM implementation Industry Foundation Classes (IFC) is reviewed with regard to its capabilities of representing the unified model. The current IFC schema strongly supports the use of redundant model data, a major pitfall in digital collaboration. Additionally, it does not allow to describe the geometry by volumetric splines. As the pursued approach builds upon a unique model for both, architectural and structural design, and furthermore requires solid geometry, necessary schema modifications are suggested. Structural entities are modeled by volumetric NURBS patches, each of which constitutes an individual subdomain that, with regard to the analysis, is incompatible with the remaining full model. The resulting consequences for numerical simulation are elaborated in this work. The individual subdomains have to be weakly coupled, for which the mortar method is used. Different approaches to discretize the interface traction fields are implemented and their respective impact on the analysis results is evaluated. All necessary coupling conditions are automatically derived from the related geometry model. The weak coupling procedure leads to a linear system of equations in saddle point form, which, owed to the volumetric modeling, is large in size and, the associated coefficient matrix has, due to the use of higher degree basis functions, a high bandwidth. The peculiarities of the system require adapted solution methods that generally cause higher numerical costs than the standard procedures for symmetric, positive-definite systems do. Different methods to solve the specific system are investigated and an efficient parallel algorithm is finally proposed. When the structural analysis model is derived from the unified model in the BIM data, it does in general initially not meet the requirements on the discretization that are necessary to obtain sufficiently accurate analysis results. The consequently necessary patch refinements must be controlled automatically to allowfor an entirely automatic analysis procedure. For that purpose, an empirical refinement scheme based on the geometrical and possibly mechanical properties of the specific entities is proposed. The level of refinement may be selectively manipulated by the structural engineer in charge. Furthermore, a Zienkiewicz-Zhu type error estimator is adapted for the use with isogeometric analysis results. It is shown that also this estimator can be used to steer an adaptive refinement procedure.},
  subject      = {Finite-Elemente-Methode},
  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}
}
@article{ChakrabortyAnitescuZhuangetal.,
  author    = {Chakraborty, Ayan and Anitescu, Cosmin and Zhuang, Xiaoying and Rabczuk, Timon},
  title     = {Domain adaptation based transfer learning approach for solving PDEs on complex geometries},
  series = {Engineering with Computers},
  volume    = {2022},
  journal   = {Engineering with Computers},
  doi       = {10.1007/s00366-022-01661-2},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20220811-46776},
  pages     = {1 -- 20},
  abstract  = {In machine learning, if the training data is independently and identically distributed as the test data then a trained model can make an accurate predictions for new samples of data. Conventional machine learning has a strong dependence on massive amounts of training data which are domain specific to understand their latent patterns. In contrast, Domain adaptation and Transfer learning methods are sub-fields within machine learning that are concerned with solving the inescapable problem of insufficient training data by relaxing the domain dependence hypothesis. In this contribution, this issue has been addressed and by making a novel combination of both the methods we develop a computationally efficient and practical algorithm to solve boundary value problems based on nonlinear partial differential equations. We adopt a meshfree analysis framework to integrate the prevailing geometric modelling techniques based on NURBS and present an enhanced deep collocation approach that also plays an important role in the accuracy of solutions. We start with a brief introduction on how these methods expand upon this framework. We observe an excellent agreement between these methods and have shown that how fine-tuning a pre-trained network to a specialized domain may lead to an outstanding performance compare to the existing ones. As proof of concept, we illustrate the performance of our proposed model on several benchmark problems.},
  subject      = {Maschinelles Lernen},
  language  = {en}
}