@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{Malik,
  author    = {Malik, Irfan},
  title     = {An adaptive contact formulation for Isogeometric Finite Element Analysis},
  doi       = {10.25643/bauhaus-universitaet.4612},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20220324-46129},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  pages     = {124},
  abstract  = {Numerical simulation of physical phenomena, like electro-magnetics, structural and fluid mechanics is essential for the cost- and time-efficient development of mechanical products at high quality. It allows to investigate the behavior of a product or a system far before the first prototype of a product is manufactured. This thesis addresses the simulation of contact mechanics. Mechanical contacts appear in nearly every product of mechanical engineering. Gearboxes, roller bearings, valves and pumps are only some examples. Simulating these systems not only for the maximal/minimal stresses and strains but for the stress-distribution in case of tribo-contacts is a challenging task from a numerical point of view. Classical procedures like the Finite Element Method suffer from the nonsmooth representation of contact surfaces with discrete Lagrange elements. On the one hand, an error due to the approximate description of the surface is introduced. On the other hand it is difficult to attain a robust contact search because surface normals can not be described in a unique form at element edges. This thesis introduces therefore a novel approach, the adaptive isogeometric contact formulation based on polynomial Splines over hierarchical T-meshes (PHT-Splines), for the approximate solution of the non-linear contact problem. It provides a more accurate, robust and efficient solution compared to conventional methods. During the development of this method the focus was laid on the solution of static contact problems without friction in 2D and 3D in which the structures undergo small deformations. The mathematical description of the problem entails a system of partial differential equations and boundary conditions which model the linear elastic behaviour of continua. Additionally, it comprises side conditions, the Karush-Kuhn-Tuckerconditions, to prevent the contacting structures from non-physical penetration. The mathematical model must be transformed into its integral form for approximation of the solution. Employing a penalty method, contact constraints are incorporated by adding the resulting equations in weak form to the overall set of equations. For an efficient space discretization of the bulk and especially the contact boundary of the structures, the principle of Isogeometric Analysis (IGA) is applied. Isogeometric Finite Element Methods provide several advantages over conventional Finite Element discretization. Surface approximation with Non-Uniform Rational B-Splines (NURBS) allow a robust numerical solution of the contact problem with high accuracy in terms of an exact geometry description including the surface smoothness. The numerical evaluation of the contact integral is challenging due to generally non-conforming meshes of the contacting structures. In this work the highly accurate Mortar Method is applied in the isogeometric setting for the evaluation of contact contributions. This leads to an algebraic system of equations that is linearized and solved in sequential steps. This procedure is known as the Newton Raphson Method. Based on numerical examples, the advantages of the isogeometric approach with classical refinement strategies, like the p- and h-refinement, are shown and the influence of relevant algorithmic parameters on the approximate solution of the contact problem is verified. One drawback of the Spline approximations of stresses though is that they lack accuracy at the contact edge where the structures change their boundary from contact to no contact and where the solution features a kink. The approximation with smooth Spline functions yields numerical artefacts in the form of non-physical oscillations. This property of the numerical solution is not only a drawback for the simulation of e.g. tribological contacts, it also influences the convergence properties of iterative solution procedures negatively. Hence, the NURBS discretized geometries are transformed to Polynomial Splines over Hierarchical T-meshes (PHT-Splines), for the local refinement along contact edges to reduce the artefact of pressure oscillations. NURBS have a tensor product structure which does not allow to refine only certain parts of the geometrical domain while leaving other parts unchanged. Due to the B{\´e}zier Extraction, lying behind the transformation from NURBS to PHT-Splines, the connected mesh structure is broken up into separate elements. This allows an efficient local refinement along the contact edge. Before single elements are refined in a hierarchical form with cross-insertion, existing basis functions must be modified or eliminated. This process of truncation assures local and global linear independence of the refined basis which is needed for a unique approximate solution. The contact boundary is a priori unknown. Local refinement along the contact edge, especially for 3D problems, is for this reason not straight forward. In this work the use of an a posteriori error estimation procedure, the Super Convergent Recovery Solution Based Error Estimation Scheme, together with the D{\"o}rfler Marking Method is suggested for the spatial search of the contact edge. Numerical examples show that the developed method improves the quality of solutions along the contact edge significantly compared to NURBS based approximate solutions. Also, the error in maximum contact pressures, which correlates with the pressure artefacts, is minimized by the adaptive local refinement. In a final step the practicability of the developed solution algorithm is verified by an industrial application: The highly loaded mechanical contact between roller and cam in the drive train of a high-pressure fuel pump is considered.},
  subject      = {Isogeometrische Analyse},
  language  = {en}
}
@phdthesis{ShaabanMohamed,
  author    = {Shaaban Mohamed, Ahmed Mostafa},
  title     = {Isogeometric boundary element analysis and structural shape optimization for Helmholtz acoustic problems},
  doi       = {10.25643/bauhaus-universitaet.4703},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20220816-47030},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  abstract  = {In this thesis, a new approach is developed for applications of shape optimization on the time harmonic wave propagation (Helmholtz equation) for acoustic problems. This approach is introduced for different dimensional problems: 2D, 3D axi-symmetric and fully 3D problems. The boundary element method (BEM) is coupled with the isogeometric analysis (IGA) forming the so-called (IGABEM) which speeds up meshing and gives higher accuracy in comparison with standard BEM. BEM is superior for handling unbounded domains by modeling only the inner boundaries and avoiding the truncation error, present in the finite element method (FEM) since BEM solutions satisfy the Sommerfeld radiation condition automatically. Moreover, BEM reduces the space dimension by one from a volumetric three-dimensional problem to a surface two-dimensional problem, or from a surface two-dimensional problem to a perimeter one-dimensional problem. Non-uniform rational B-splines basis functions (NURBS) are used in an isogeometric setting to describe both the CAD geometries and the physical fields. IGABEM is coupled with one of the gradient-free optimization methods, the Particle Swarm Optimization (PSO) for structural shape optimization problems. PSO is a straightforward method since it does not require any sensitivity analysis but it has some trade-offs with regard to the computational cost. Coupling IGA with optimization problems enables the NURBS basis functions to represent the three models: shape design, analysis and optimization models, by a definition of a set of control points to be the control variables and the optimization parameters as well which enables an easy transition between the three models. Acoustic shape optimization for various frequencies in different mediums is performed with PSO and the results are compared with the benchmark solutions from the literature for different dimensional problems proving the efficiency of the proposed approach with the following remarks: - In 2D problems, two BEM methods are used: the conventional isogeometric boundary element method (IGABEM) and the eXtended IGABEM (XIBEM) enriched with the partition-of-unity expansion using a set of plane waves, where the results are generally in good agreement with the linterature with some computation advantage to XIBEM which allows coarser meshes. -In 3D axi-symmetric problems, the three-dimensional problem is simplified in BEM from a surface integral to a combination of two 1D integrals. The first is the line integral similar to a two-dimensional BEM problem. The second integral is performed over the angle of revolution. The discretization is applied only to the former integration. This leads to significant computational savings and, consequently, better treatment for higher frequencies over the full three-dimensional models. - In fully 3D problems, a detailed comparison between two BEM methods: the conventional boundary integral equation (CBIE) and Burton-Miller (BM) is provided including the computational cost. The proposed models are enhanced with a modified collocation scheme with offsets to Greville abscissae to avoid placing collocation points at the corners. Placing collocation points on smooth surface enables accurate evaluation of normals for BM formulation in addition to straightforward prediction of jump-terms and avoids singularities in \$\mathcal{O} (1/r)\$ integrals eliminating the need for polar integration. Furthermore, no additional special treatment is required for the hyper-singular integral while collocating on highly distorted elements, such as those containing sphere poles. The obtained results indicate that, CBIE with PSO is a feasible alternative (except for a small number of fictitious frequencies) which is easier to implement. Furthermore, BM presents an outstanding treatment of the complicated geometry of mufflers with internal extended inlet/outlet tube as an interior 3D Helmholtz acoustic problem instead of using mixed or dual BEM.},
  subject      = {Randelemente-Methode},
  language  = {en}
}
@phdthesis{Valizadeh,
  author    = {Valizadeh, Navid},
  title     = {Developments in Isogeometric Analysis and Application to High-Order Phase-Field Models of Biomembranes},
  doi       = {10.25643/bauhaus-universitaet.4565},
  url       = {http://nbn-resolving.de/urn:nbn:de:gbv:wim2-20220114-45658},
  school      = {Bauhaus-Universit{\"a}t Weimar},
  abstract  = {Isogeometric analysis (IGA) is a numerical method for solving partial differential equations (PDEs), which was introduced with the aim of integrating finite element analysis with computer-aided design systems. The main idea of the method is to use the same spline basis functions which describe the geometry in CAD systems for the approximation of solution fields in the finite element method (FEM). Originally, NURBS which is a standard technology employed in CAD systems was adopted as basis functions in IGA but there were several variants of IGA using other technologies such as T-splines, PHT splines, and subdivision surfaces as basis functions. In general, IGA offers two key advantages over classical FEM: (i) by describing the CAD geometry exactly using smooth, high-order spline functions, the mesh generation process is simplified and the interoperability between CAD and FEM is improved, (ii) IGA can be viewed as a high-order finite element method which offers basis functions with high inter-element continuity and therefore can provide a primal variational formulation of high-order PDEs in a straightforward fashion. The main goal of this thesis is to further advance isogeometric analysis by exploiting these major advantages, namely precise geometric modeling and the use of smooth high-order splines as basis functions, and develop robust computational methods for problems with complex geometry and/or complex multi-physics. As the first contribution of this thesis, we leverage the precise geometric modeling of isogeometric analysis and propose a new method for its coupling with meshfree discretizations. We exploit the strengths of both methods by using IGA to provide a smooth, geometrically-exact surface discretization of the problem domain boundary, while the Reproducing Kernel Particle Method (RKPM) discretization is used to provide the volumetric discretization of the domain interior. The coupling strategy is based upon the higher-order consistency or reproducing conditions that are directly imposed in the physical domain. The resulting coupled method enjoys several favorable features: (i) it preserves the geometric exactness of IGA, (ii) it circumvents the need for global volumetric parameterization of the problem domain, (iii) it achieves arbitrary-order approximation accuracy while preserving higher-order smoothness of the discretization. Several numerical examples are solved to show the optimal convergence properties of the coupled IGA-RKPM formulation, and to demonstrate its effectiveness in constructing volumetric discretizations for complex-geometry objects. As for the next contribution, we exploit the use of smooth, high-order spline basis functions in IGA to solve high-order surface PDEs governing the morphological evolution of vesicles. These governing equations are often consisted of geometric PDEs, high-order PDEs on stationary or evolving surfaces, or a combination of them. We propose an isogeometric formulation for solving these PDEs. In the context of geometric PDEs, we consider phase-field approximations of mean curvature flow and Willmore flow problems and numerically study the convergence behavior of isogeometric analysis for these problems. As a model problem for high-order PDEs on stationary surfaces, we consider the Cahn-Hilliard equation on a sphere, where the surface is modeled using a phase-field approach. As for the high-order PDEs on evolving surfaces, a phase-field model of a deforming multi-component vesicle, which consists of two fourth-order nonlinear PDEs, is solved using the isogeometric analysis in a primal variational framework. Through several numerical examples in 2D, 3D and axisymmetric 3D settings, we show the robustness of IGA for solving the considered phase-field models. Finally, we present a monolithic, implicit formulation based on isogeometric analysis and generalized-alpha time integration for simulating hydrodynamics of vesicles according to a phase-field model. Compared to earlier works, the number of equations of the phase-field model which need to be solved is reduced by leveraging high continuity of NURBS functions, and the algorithm is extended to 3D settings. We use residual-based variational multi-scale method (RBVMS) for solving Navier-Stokes equations, while the rest of PDEs in the phase-field model are treated using a standard Galerkin-based IGA. We introduce the resistive immersed surface (RIS) method into the formulation which can be employed for an implicit description of complex geometries using a diffuse-interface approach. The implementation highlights the robustness of the RBVMS method for Navier-Stokes equations of incompressible flows with non-trivial localized forcing terms including bending and tension forces of the vesicle. The potential of the phase-field model and isogeometric analysis for accurate simulation of a variety of fluid-vesicle interaction problems in 2D and 3D is demonstrated.},
  subject      = {Phasenfeldmodell},
  language  = {en}
}