Automated CAD Model Generation for Structural Optimisation

Computer-aided design (CAD) models play a crucial role in the design, manufacturing and maintenance of products. Therefore, the mesh-based finite element descriptions common in structural optimisation must be first translated into CAD models. Currently, this translation either can be performed semi-manually or fails to reserve the structural optimality found by the structural optimisation due to the intrinsic difference in geometric representation between finite element mesh and CAD model. This thesis propose a fully automated and topologically accurate approach to synthesise structurally sound parametric CAD models from topology-optimised finite element models to fill the long-existing gap between structural optimisation and CAD systems. This approach successfully preserves the optimal structural performance during the mesh-CAD conversion. The solution provided in this thesis is to first convert the topology-optimised structure into a spatial frame structure and then to regenerate it in a CAD system using standard constructive solid geometry (CSG) operations. The obtained parametric CAD models are compact, that is, have as few as possible geometric parameters, which makes them ideal for editing and further processing within a CAD system. The critical task of converting the topology-optimised structure into an optimal spatial frame structure is accomplished in several steps. The first step is to generate a one-voxel-wide voxel chain model from the topology-optimised voxel model using a topology-preserving skeletonisation algorithm from digital topology. The undirected graph defined by the voxel chain model yields a spatial frame structure after processing it with the proposed graph algorithms. Subsequently, the cross-sections and layout of the frame members are optimised to recover its optimality, which may have been compromised during the conversion process. At last, the obtained frame structure is generated in a CAD system by repeatedly combining primitive solids, like cylinders and spheres, using boolean operations. The resulting solid model is a boundary representation (B-Rep) consisting of trimmed non-uniform rational B-spline (NURBS) curves and surfaces. The numerical studies in this thesis clarify that the converted spatial frame structures are with equivalent structural performance. Moreover, CAD models generated from the spatial frame structures have significantly fewer geometric degree of freedom compared to the topology-optimised structures. Though the numerical studies use topology-optimised structures as input and compact CSG models as output, this thesis also provides the way to extend the proposed generation process to taking other optimised meshes and producing outputs of various geometric representations. This offers a wide range of possible applications and brings new thoughts to industrial design and manufacturing.Chinese Scholarship Counci

Efficient CAD based adjoint optimization of turbomachinery using adaptive shape parameterization

The present thesis incorporates the CAD model into an adjoint-based optimization loop and uses it for the shape optimization of a 2D transonic turbine blade mid-section (profile). This is demonstrated by performing a single and multipoint optimization of the LS89 turbine, originally designed at the VKI. Substantial aerodynamic improvements are reported for both design point and off-design conditions.The case is deeply analysed from the flow analysis point of view. The present thesis is a step forward in three main aspects. First, the way the CAD model (for turbomachinery applications) is used within the shape optimization loop.To include the CAD model into the optimization loop, the CAD kernel and the grid generator (multiblock structured) are differentiated using the Algorithmic Differentiation (AD) tool ADOL-C. The advantage of including the CAD model in the design system is that assembly or manufacturing constraints can be imposed on the shape, allowing the optimized model or component to be manufactured. Second, a new definition of the parametric effectiveness indicator is proposed, based on the ability of a set of CAD-based design variables to produce a shape change using the adjoint sensitivities. An interesting thing is that parametric effectiveness considers the design variables can be non-orthogonal to each other and it can be applied to any type of constrained or unconstrained problems. If, in the beginning of the optimization, the parametric effectiveness is high, it is expected to reach a final solution with increased performance. Third, a new adaptive shape parameterization strategy is adopted, which is assisted by the above parametric effectiveness indicator in order to explore the design space more efficiently. The parametric effectiveness, which rates the quality of a CAD based parameterization for optimization, is used in a novel multilevel shape refinement procedure to: (1) introduce the minimum amount of design variables required to modify the shape in the direction the adjoint sensitivities dictate; (2) to create the best parameterization to be used during the optimization. By using the proposed methods and tools, not only the optimal geometry is defined by the CAD, which is the industry adopted standard for the design of components, but also, the designer avoids the use of either too few (slow improvements from cycle to cycle) or too many (increase the computational burden) design variables. The proposed methodology results to be an effective strategy to explore rich design spaces, to improve convergence rate, robustness and final solution of the adjoint-based optimization.Aquesta tesi incorpora el model de CAD en un procés iteratiu d'optimització basat en el mètode adjunt i l'utilitza per a l'optimització de la secció d'una turbina transónica 2D (perfil). Això es demostra realitzant una optimització de punt únic i multipunt de la turbina LS89, originalment dissenyada en el VKI. Es reporten millores aerodinàmiques substancials tant per al punt de disseny com per les condicions fora del disseny. El cas s'analitza en profunditat des del punt de vista aerodinàmic. Aquesta tesi representa un avanç en tres aspectes principals. Primer, la forma en què es fa servir el model CAD (per a aplicacions de turbomàquines) dins el procés d'optimització. Per incloure el model CAD en el bucle d'optimització, s'apliquen tècniques de diferenciació algorítmica (l'eina ADOL-C) al kernel del CAD i el generador de la malla (estructurada i multibloc). L'avantatge d'incloure el model CAD en el sistema de disseny és que es poden imposar restriccions de fabricació a la geometria, i això permet que el disseny ja optimitzat es pugui fabricar. En segon lloc, es proposa una nova definició de l'indicador d'efectivitat paramètrica, basat en la capacitat de produir el canvi en la geometria que dicta el mètode adjunt mitjançant l'ús de les variables de disseny que defineixen el model CAD. Cal destacar que l'efectivitat paramètrica considera que les variables de disseny poden ser no ortogonals entre si i es pot aplicar a qualsevol tipus de problemes restringits o no restringits. Si, al començament de l'optimització, l'efectivitat paramètrica és alta, s'espera que l'optimització arribi a una solució final amb major rendiment. En tercer lloc, s'adopta una nova estratègia de parametrització adaptativa, que és assistida per l'indicador d'efectivitat paramètrica anterior per explorar l'espai de disseny de manera més eficient. L'efectivitat paramètrica, que classifica la qualitat d'una parametrització basada en CAD per a l'optimització, s'utilitza en un nou procediment de refinament multinivell per: (1) introduir la quantitat mínima de variables de disseny requerides per modificar la geometria en la direcció que dicten les sensibilitats del mètode adjunt; (2) per crear la millor parametrització que s'utilitzarà durant l'optimització. En utilitzar els mètodes i eines proposats, no només la geometria òptima està definida en el model CAD, que és l'estàndard adoptat per la indústria per al disseny de components, sinó que també el dissenyador evita l'ús de molt poques (millores lentes de cicle a cicle) o massa variables de disseny (augmenten la càrrega computacional). La metodologia proposada resulta ser una estratègia efectiva per explorar espais de disseny enriquits, millora la taxa de convergència, la solidesa i la solució final de l'optimització basada en el mètode adjunt

Efficient CAD based adjoint optimization of turbomachinery using adaptive shape parameterization

The present thesis incorporates the CAD model into an adjoint-based optimization loop and uses it for the shape optimization of a 2D transonic turbine blade mid-section (profile). This is demonstrated by performing a single and multipoint optimization of the LS89 turbine, originally designed at the VKI. Substantial aerodynamic improvements are reported for both design point and off-design conditions.The case is deeply analysed from the flow analysis point of view. The present thesis is a step forward in three main aspects. First, the way the CAD model (for turbomachinery applications) is used within the shape optimization loop.To include the CAD model into the optimization loop, the CAD kernel and the grid generator (multiblock structured) are differentiated using the Algorithmic Differentiation (AD) tool ADOL-C. The advantage of including the CAD model in the design system is that assembly or manufacturing constraints can be imposed on the shape, allowing the optimized model or component to be manufactured. Second, a new definition of the parametric effectiveness indicator is proposed, based on the ability of a set of CAD-based design variables to produce a shape change using the adjoint sensitivities. An interesting thing is that parametric effectiveness considers the design variables can be non-orthogonal to each other and it can be applied to any type of constrained or unconstrained problems. If, in the beginning of the optimization, the parametric effectiveness is high, it is expected to reach a final solution with increased performance. Third, a new adaptive shape parameterization strategy is adopted, which is assisted by the above parametric effectiveness indicator in order to explore the design space more efficiently. The parametric effectiveness, which rates the quality of a CAD based parameterization for optimization, is used in a novel multilevel shape refinement procedure to: (1) introduce the minimum amount of design variables required to modify the shape in the direction the adjoint sensitivities dictate; (2) to create the best parameterization to be used during the optimization. By using the proposed methods and tools, not only the optimal geometry is defined by the CAD, which is the industry adopted standard for the design of components, but also, the designer avoids the use of either too few (slow improvements from cycle to cycle) or too many (increase the computational burden) design variables. The proposed methodology results to be an effective strategy to explore rich design spaces, to improve convergence rate, robustness and final solution of the adjoint-based optimization.Aquesta tesi incorpora el model de CAD en un procés iteratiu d'optimització basat en el mètode adjunt i l'utilitza per a l'optimització de la secció d'una turbina transónica 2D (perfil). Això es demostra realitzant una optimització de punt únic i multipunt de la turbina LS89, originalment dissenyada en el VKI. Es reporten millores aerodinàmiques substancials tant per al punt de disseny com per les condicions fora del disseny. El cas s'analitza en profunditat des del punt de vista aerodinàmic. Aquesta tesi representa un avanç en tres aspectes principals. Primer, la forma en què es fa servir el model CAD (per a aplicacions de turbomàquines) dins el procés d'optimització. Per incloure el model CAD en el bucle d'optimització, s'apliquen tècniques de diferenciació algorítmica (l'eina ADOL-C) al kernel del CAD i el generador de la malla (estructurada i multibloc). L'avantatge d'incloure el model CAD en el sistema de disseny és que es poden imposar restriccions de fabricació a la geometria, i això permet que el disseny ja optimitzat es pugui fabricar. En segon lloc, es proposa una nova definició de l'indicador d'efectivitat paramètrica, basat en la capacitat de produir el canvi en la geometria que dicta el mètode adjunt mitjançant l'ús de les variables de disseny que defineixen el model CAD. Cal destacar que l'efectivitat paramètrica considera que les variables de disseny poden ser no ortogonals entre si i es pot aplicar a qualsevol tipus de problemes restringits o no restringits. Si, al començament de l'optimització, l'efectivitat paramètrica és alta, s'espera que l'optimització arribi a una solució final amb major rendiment. En tercer lloc, s'adopta una nova estratègia de parametrització adaptativa, que és assistida per l'indicador d'efectivitat paramètrica anterior per explorar l'espai de disseny de manera més eficient. L'efectivitat paramètrica, que classifica la qualitat d'una parametrització basada en CAD per a l'optimització, s'utilitza en un nou procediment de refinament multinivell per: (1) introduir la quantitat mínima de variables de disseny requerides per modificar la geometria en la direcció que dicten les sensibilitats del mètode adjunt; (2) per crear la millor parametrització que s'utilitzarà durant l'optimització. En utilitzar els mètodes i eines proposats, no només la geometria òptima està definida en el model CAD, que és l'estàndard adoptat per la indústria per al disseny de components, sinó que també el dissenyador evita l'ús de molt poques (millores lentes de cicle a cicle) o massa variables de disseny (augmenten la càrrega computacional). La metodologia proposada resulta ser una estratègia efectiva per explorar espais de disseny enriquits, millora la taxa de convergència, la solidesa i la solució final de l'optimització basada en el mètode adjunt.Postprint (published version

Transverse Isotropic and Orthotropic Composites: Experiments, Identification and Finite Element Analysis

Die konstitutive Modellierung und numerische Analyse des Verhaltens von Verbundwerkstoffen, insbesondere von transversal isotropen und orthotropen Werkstoffen, hat in der Industrie große Aufmerksamkeit bekommen. Dies ist vor allem durch die Verwendung von Verbundwerkstoffen für ein breites Spektrum von Anwendungen in verschiedenen Branchen erkennbar. Vorteile von Verbundwerkstoffen wie hohe Festigkeit und Flexibilität bei der Konstruktion machen diese attraktiv. Aufgrund vieler Designfaktoren bei Verbundwerkstoffen, wie zum Beispiel das Verbinden mit anderen Bauteilen, sind Löcher in Laminaten unvermeidlich. Die Fasern werden in der Regel durch Bohren eines Lochs im Laminat bzw. unterbrochen. Alternativ können die Fasern um die Löcher herum gelegt werden. Eines der Ziele dieser Arbeit ist es, herauszufinden, ob die Tendenz zum Bruch, d.h. die zugehörige Spannungsverteilung zu untersuchen. Um die beiden Fälle (Faserumlenkung versus gerader Faser) zu vergleichen und einen tieferen Einblick in den Prozess durch Simulationen zu erhalten, wird ein konstitutives Modell der transversalen Isotropie für den Fall kleiner Verzerrungen hergleitet. Das Modell ist in das in-house Finite-Elemente Programm TASAFEM implementiert worden. Eine große Herausforderung stellt die Beschreibung der räumlich verteilten Faserorientierungen für den Fall, dass die Fasern um das Loch herumgelegt werden. Zunächst wird die Verteilung der Fasern mit Hilfe der Stromlinienfunktion modelliert, um die inhomogenen Faserorientierungen für die FE-Simulationen zu erhalten. Um die Genauigkeit der Simulationen zu erhöhen, werden B-Splines verwendet, um die Faserrichtungen entsprechend den experimentellen Beobachtungen zu modellieren. Im sehr breiten Bereich der geometrischen Modellierung insbesondere bei CAD-Anwendungen (Computer-Aided Design) werden B-Splines häufig zur Beschreibung von Kurven und Flächen verwendet, vor allem aufgrund ihrer mathematischen Eigenschaften und ihrer hohen Flexibilität. Hierbei werden die Eigenschaften von Tangentenvektoren an Koordinatenflächen ausgenutzt, um die Richtungen zu bestimmen. Eine weitere Herausforderung bei den durchgeführten Simulationen ist die Identifikation der erforderlichen Materialparameter für das verwendete Materialmodell. Zu diesem Zweck werden verschiedene Experimente durchgeführt, um die Parameter eindeutig zu bestimmen. Zum Schluss wird der gesamte Prozess der Modellierung, Simulation und Identifizierung der Materialparameter durch spezielle Tests validiert. Orthotrope Laminate gehören zu den am häufigsten verwendeten Laminaten in industriellen Anwendungen. Die Untersuchungen werden daher auf orthotrope Laminate ausgeweitet. Das Ziel ist es, das Verhalten auch auf orthotrope Laminate auf der Grundlage identifizierter Parameter zu übertragen. Es wird ein konstitutives Modell der Orthotropie für den Fall kleiner Dehnungen angewandt und in den in-house-Code TASAFEM implementiert. Auch hier besteht die Herausforderung, der Materialparameter von orthotropen Laminaten bereitzustellen, die für die erforderlichen FE-Simulationen notwendig sind. Die Materialparameter werden im Rahmen eines Least-Square-Ansatzes mit Hilfe von Messdaten eines digitalen Bildkorrelationssystems identifiziert. Zu diesem Zweck sind verschiedene Versuche wie Zug-, Scher-, Druck- und Zugschertests durchgeführt worden. Diese sind zur Identifikation der neun Materialparameter der linearen, orthotropen Elastizität herangezogen worden. Im nächsten Schritt ist es notwendig, den numerischen Ansatz mit experimentellen Messungen zu validieren. Zur Validierung werden Proben verwendet, bei denen die Proben mit zwei senkrechten Faserrichtungen ausgestattet sind. Hierbei wird das Loch nach dem Herstellungsprozess der Proben gebohrt. Zum Schluss wird ein Vergleich zwischen den Ergebnissen der Finite-Elemente-Simulationen und den experimentellen Ergebnissen vorgestellt.In today’s engineering industry, constitutive modeling and numerical analysis of the behavior of composite materials, particularly transversely isotropic and orthotropic materials, have gained a lot of attention. This is mainly due to the usage of composites for a wide range of applications in different industries. Moreover, the advantages of composites such as high strength and flexibility in design make these materials attractive. Due to many factors in the design of composites, holes in laminates are unavoidable. Fibers are usually cut by drilling a hole into laminates. Alternatively, fiber can be bypassed around holes in order to reduce the fracture tendency around a hole, or to achieve different stress distributions. One of the goals of this work is to compare these cases: In one case, fibers were bypassed around the hole while fibers were cut in the other case by drilling a hole. In order to compare these cases and to get a deeper insight into the process using simulations, a constitutive model of transverse isotropic for the small strain case is applied based on large strain theory. The model is implemented in the in-house finite element program TASAFEM. One major challenge of this simulation is to determine the fiber orientations. To begin with, the circumplacement of fibers is modeled using the streamline function to obtain the inhomogeneous fiber direction for finite element simulations. In order to increase the precision of simulations, the B-spline method is used to model the fiber directions according to the experimental observations. In the broad field of geometric modeling and computer-aided design (CAD), it is common to use B-splines to describe curves and surfaces which is mainly due to their mathematical properties and their flexibility. Another challenge regarding the simulations is to identify the required parameters for the presented material model. Several different experiments are carried out in this regard. Finally, the whole process of modeling, simulation, and material parameter identification is validated by means of validation tests. Orthotropic laminates belong to the most commonly used laminates in industrial applications. The investigation is extended to orthotropy laminates, where we have fibers in two directions, and our aim is to predict the behavior of orthotropy laminates based on the calculated parameters. A constitutive model of orthotropy for the small strain case is applied and implemented in the inhouse code TASAFEM. Another challenge in this work is to calculate the material parameters of orthotropy laminates as a basis for finite element simulations. The material parameters are identified within a least-square approach with the help of optical results of a digital image correlation system. For this purpose, different experiments such as tensile, three rail shear, lap shear and compression tests are carried out. Nine material parameters of linear elastic for orthotropy case are identified. In the next step, it is necessary to validate the numerical approach with experimental observations. The validation examples are performed as theses samples have fibers in two perpendicular directions, where the hole is drilled after the production process. Finally, a comparison between the finite element simulations and the experimental results is provided

Reduced Models for Optimal Control, Shape Optimization and Inverse Problems in Haemodynamics

The objective of this thesis is to develop reduced models for the numerical solution of optimal control, shape optimization and inverse problems. In all these cases suitable functionals of state variables have to be minimized. State variables are solutions of a partial differential equation (PDE), representing a constraint for the minimization problem. The solution of these problems induce large computational costs due to the numerical discretization of PDEs and to iterative procedures usually required by numerical optimization (many-query context). In order to reduce the computational complexity, we take advantage of the reduced basis (RB) approximation for parametrized PDEs, once the state problem has been reformulated in parametrized form. This method enables a rapid and reliable approximation of parametrized PDEs by constructing low-dimensional, problem-specific approximation spaces. In case of PDEs defined over domains of variable shapes (e.g. in shape optimization problems) we need to introduce suitable, low-dimensional shape parametrization techniques in order to tackle the geometrical complexity. Free-Form Deformations and Radial-Basis Functions techniques have been analyzed and successfully applied with this aim. We analyze the reduced framework built by coupling these tools and apply it to the solution of optimal control and shape optimization problems. Robust optimization problems under uncertain conditions are also taken into consideration. Moreover, both deterministic and Bayesian frameworks are set in order to tackle inverse identification problems. As state equations, we consider steady viscous flow problems described by Stokes or Navier-Stokes equations, for which we provide a detailed analysis and construction of RB approximation and a posteriori error estimation. Several numerical test cases are also illustrated to show efficacy and reliability of RB approximations. We exploit this general reduced framework to solve some optimization and inverse problems arising in haemodynamics. More specifically, we focus on the optimal design of cardiovascular prostheses, such as bypass grafts, and on inverse identification of pathological conditions or flow/shape features in realistic parametrized geometries, such as carotid artery bifurcations

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal