@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{Jaouadi,
  author    = {Jaouadi, Zouhour},
  title     = {Pareto and Reliability-Oriented Aeroelastic Shape Optimization of Bridge Decks},
  doi       = {10.25643/bauhaus-universitaet.4935},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20230303-49352},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {167},
  abstract  = {Due to the development of new technologies and materials, optimized bridge design has recently gained more attention. The aim is to reduce the bridge components materials and the CO2 emission from the cement manufacturing process. Thus, most long-span bridges are designed to be with high flexibility, low structural damping, and longer and slender spans. Such designs lead, however, to aeroelastic challenges. Moreover, the consideration of both the structural and aeroelastic behavior in bridges leads to contradictory solutions as the structural constraints lead to deck prototypes with high depth which provide high inertia to material volume ratios. On the other hand, considering solely the aerodynamic requirements, slender airfoil-shaped bridge box girders are recommended since they prevent vortex shedding and exhibit minimum drag. Within this framework comes this study which provides approaches to find optimal bridge deck cross-sections while considering the aerodynamic effects. Shape optimization of deck cross-section is usually formulated to minimize the amount of material by finding adequate parameters such as the depth, the height, and the thickness and while ensuring the overall stability of the structure by the application of some constraints. Codes and studies have been implemented to analyze the wind phenomena and the structural responses towards bridge deck cross-sections where simplifications have been adopted due to the complexity and the uniqueness of such components besides the difficulty of obtaining a final model of the aerodynamic behavior. In this thesis, two main perspectives have been studied; the first is fully deterministic and presents a novel framework on generating optimal aerodynamic shapes for streamlined and trapezoidal cross-sections based on the meta-modeling approach. Single and multi-objective optimizations were both carried out and a Pareto Front is generated. The performance of the optimal designs is checked afterwards. In the second part, a new strategy based on Reliability-Based Design Optimization (RBDO) to mitigate the vortex-induced vibration (VIV) on the Trans-Tokyo Bay bridge is proposed. Small changes in the leading and trailing edges are presented and uncertainties are considered in the structural system. Probabilistic constraints based on polynomial regression are evaluated and the problem is solved while applying the Reliability Index Approach (RIA) and the Performance Measure Approach (PMA). The results obtained in the first part showed that the aspect ratio has a significant effect on the aerodynamic behavior where deeper cross-sections have lower resistance against flutter and should be avoided. In the second part, the adopted RBDO approach succeeded to mitigate the VIV, and it is proven that designs with narrow or prolonged bottom-base length and featuring an abrupt surface change in the leading and trailing edges can lead to high vertical vibration amplitude. It is expected that this research will help engineers with the selections of the adequate deck cross-section layout, and encourage researchers to apply concepts of optimization regarding this field and develop the presented approaches for further studies.},
  subject      = {Gestaltoptimierung},
  language  = {en}
}