Strides in modern computational fluid dynamics and leaps in high-power computing have led to unprecedented capabilities for handling large aerodynamic problem. In particular, the emergence of adjoint design methods has been a break-through in the field of aerodynamic shape optimization. It enables expensive, high-dimensional optimization problems to be tackled efficiently using gradient-based methods in CFD; a task that was previously inconceivable. However, adjoint design methods are intended for gradient-based optimization; the curse of dimensionality is still very much alive when it comes to design space exploration, where gradient-free methods cannot be avoided. This research describes a novel approach for reducing dimensionality in large, computationally expensive design problems to a point where gradient-free methods become possible. This is done using an innovative application of Principal Component Analysis (PCA), where the latter is applied to the gradient distribution of the objective function; something that had not been done before. This yields a linear transformation that maps a high-dimensional problem onto an equivalent low-dimensional subspace. None of the original variables are discarded; they are simply linearly combined into a new set of variables that are fewer in number. The method is tested on a range of analytical functions, a two-dimensional staggered airfoil test problem and a three-dimensional Over-Wing Nacelle (OWN) integration problem. In all cases, the method performed as expected and was found to be cost effective, requiring only a relatively small number of samples to achieve large dimensionality reduction.

In the present thesis a novel numerical approach for the optimization of composite structures fabricated from woven composite materials is developed. The aim is to increase the ultimate strength of the structure while at the same time decreasing its weight. The numerical approach is based on a combination of the numerical algorithm of progressive damage modelling (PDM), along with shape optimization (SO) in an iterative subroutine. PDM, which is comprised of three steps, namely stress analysis, failure analysis and material property degradation, is used to predict the initiation and propagation of failure in the structure. During the phase of SO certain geometrical parameters are varied within limits in order to minimize the stresses that lead the structure to ultimate failure as indicated by PDM results. Finally the resulting geometry is solved with PDM to ensure the enhancement in the ultimate strength and the decrease in ultimate weight. Within the frame of this approach, a new methodology for the numerical modeling and the simulation of mechanical behavior of woven composite materials is proposed. The highly inhomogeneous nature of woven composite materials in the micro-scale is taken under consideration to create accurate representative volume element (RVE) FE models which represent the actual material. Then PDM is used for the simulation of their mechanical response. The calculated properties, in terms of stiffnesses and strengths, are then inserted as inputs in the global FE model of the composite structure. Additionally, the reliability and applicability of a continuum damage model (CDM), in comparison with cohesive zone model (CZM), are assessed in order to use the CDM for the modeling of the adhesive’s mechanical behavior. The mentioned numerical approach is applied in an H-shaped joining element fabricated from two different woven composite materials for the loading case of tension. In the first case NCF composite is used while in the second case the joint is made of 3D fully interlaced weave (FIW) composite. The purpose of the H-shaped element is the joining of two composite plates via the method of adhesive bonding. / Στην παρούσα διατριβή αναπτύχθηκε μια νέα μέθοδος αριθμητικής βελτιστοποίησης δομικών στοιχείων από σύνθετα υλικά με σκοπό την αύξηση της αντοχής τους. Η μέθοδος βασίζεται σε έναν αριθμητικό αλγόριθμο Προοδευτικής Εξέλιξης της Βλάβης (ΠΕΒ) και τη Βελτιστοποίηση Σχήματος (ΒΣ) τα οποία συνδυάζονται σε μια επαναληπτική υπό-ρουτίνα. Στην ΠΕΒ περιλαμβάνονται τα βήματα της ανάλυσης τάσεων, ανάλυσης αστοχίας και υποβάθμιση των ιδιοτήτων των στοιχείων. Η χρησιμότητα της έγκειται στην πρόβλεψη της έναρξης και εξέλιξης της αστοχίας στο δομικό στοιχείο κάτι απαραίτητο για την κατανόηση της μηχανικής συμπεριφοράς. Η ΒΣ έχει ως σκοπό την μεταβολή συγκεκριμένων γεωμετρικών παραμέτρων για να επιτευχθεί ελαχιστοποίηση των κρίσιμών τάσεων που προκύπτουν από τα αποτελέσματα της ΠΕΒ και οδηγούν στην αστοχία του στοιχείου. Παράλληλα, για την μοντελοποίηση και τον υπολογισμό των μηχανικών ιδιοτήτων πρωτότυπων πλεγμένων σύνθετων υλικών προτείνεται καινούργια μια μεθοδολογία η οποία λαμβάνει υπ’ όψιν την υψηλή ανομοιογένεια των υλικών στην μικρό-κλίμακα για να υπολογίσει τις ιδιότητες τους. Η μεθοδολογία εφαρμόστηκε σε ένα νέο συνδετικό στοιχείο σχήματος H κατασκευασμένο από δύο διαφορετικά πλεγμένα σύνθετα υλικά, τα μη πτυχωτά και τα τρισδιάστατα πλεγμένα σύνθετα υλικά, για την περίπτωση του εφελκυσμού. Σκοπός του συνδέσμου είναι η ένωση δύο πλακών από σύνθετα υλικά χρησιμοποιώντας κόλλα. Αρχικά το μοντέλο πεπερασμένων στοιχείων του συνδέσμου δημιουργείται και επιλύεται με την μέθοδο ΠΕΒ. Για την προσομοίωση της μη-γραμμικής συμπεριφοράς της κόλλας αναπτύσσεται ένα δι-γραμμικό μοντέλο. Για την προσομοίωση της πλήρης μηχανικής συμπεριφοράς των μη πτυχωτών και τρισδιάστατα πλεγμένων συνθέτων υλικών, αναπτύσσεται μια διαδικασία η οποία περιλαμβάνει τα βήματα της γεωμετρικής μοντελοποίησης, της κατασκευής του μοντέλου πεπερασμένων στοιχειών και την επίλυση αυτού με την μέθοδο ΠΕΒ. Τα αποτελέσματα, σε όρους διαγραμμάτων τάσεων-παραμορφώσεων, χρησιμοποιούνται ως δεδομένα στο μοντέλο πεπερασμένων στοιχείων του συνδέσμου το οποίο επιλύεται και υπολογίζεται το διάγραμμα δύναμης-μετατόπισης. Στην συνέχεια, λαμβάνει μέρος η γεωμετρική βελτιστοποίηση βασιζόμενη στα αποτελέσματα της επίλυσης της αρχικής γεωμετρίας. Σε αυτό το σημείο επιλέγεται η μεταβλητή προς ελαχιστοποίηση στην διαδικασία της βελτιστοποίησης. Το μέγεθος αυτό ονομάζεται Συνάρτηση Σκοπού (ΣΣ) και ορίζεται ως ο συντελεστή βλάβης που ευθύνεται για την τελική αστοχία του δομικού στοιχείου. Ως ένα επιπλέον κριτήριο για την επιλογή της βέλτιστης γεωμετρίας επιλέγεται η μείωση βάρους δεδομένου ότι πρόκειται για αεροπορική κατασκευή. Η γεωμετρία που ελαχιστοποιεί την συνάρτηση σκοπού και ταυτόχρονα είναι ελαφρύτερη από την αρχική, επιλέγεται ως η τελική γεωμετρία. Τέλος, γίνεται η επιτυχής επικύρωση της βελτιστοποίησης με την σύγκριση των αριθμητικών αποτελεσμάτων μεταξύ της αρχικής και τελικής γεωμετρίας. Η μεθοδολογία της ΠΕΒ εφαρμόζεται στην τελική γεωμετρία και τα διαγράμματα δύναμης μετατόπισης συγκρίνονται για να διαπιστωθεί η αύξηση στο μέγιστο φορτίο που μπορεί να φέρει το συνδετικό στοιχείο πριν την τελική αστοχία.

Baigiamajame darbe atliktas tiltin÷s santvaros, iš metalinių profiliuočių, konstrukcijos formos ir strypų skerspjūvių optimizavimas, esant vienam ir dviems apkrovų variantams. Konstrukcija projektuota atsižvelgiant į STR reikalavimus. Tam buvo spręstas netiesinio programavimo uždavinys. Sudarytos nagrinėjamo modelio pagrindinės priklausomybės taikant pusiausviruosius baigtinius elementus. Jų pagrindu sudaryta konstrukcijos optimizavimo programa MATLAB aplinkoje. Naudojantis ją, nustatyta optimali konstrukcijos struktūra ir optimalūs strypų skerspjūviai, esant minimaliam konstrukcijos tūriui, taikant ir netaikant santvaros aukščio ribojimą. Suformuluotos išvados. Darbą sudaro 6 dalys: įvadas, santvaros skaičiuojamoji schema ir apkrovų deriniai, santvaros įtempių ir deformacijų būvio analizė taikant pusiausviruosius baigtinius elementus, konstrukcijos masės minimizavimo uždavinys,optimizavimo rezultatai, išvados, literatūros sąrašas. Darbo apimtis – 67 p. teksto be priedų, 19 iliustr., 11 lent., 26 bibliografiniai šaltiniai. Atskirai pridedami darbo priedai. / In the work bridge truss made of steel cross sections was optimized. There was both form of truss and cross sections optimized according to one, and two loads combinations. Construction was design by STR (Lithuanian national design codes). For the mathematical nonlinear programing problem was solved. The main equations inequality of structure analyze discrete model was made using finite elements method. Basing on them program for optimization in Matlab language was made. By that program, optimal truss form and cross sections was found by minimizing structure volume. Six projects were found including and not including limitation of truss high. Structure: introduction, truss analytical scheme and load combinations, truss analysis using finite element method, construction mass minimization, optimal structure results, conclusions and suggestions, references. Thesis consist of: 67 p. text without appendixes, 19 pictures, 11 tables, 26 bibliographical entries. Appendixes included.

Gradient-based aerodynamic shape optimization, based on Computational Fluid Dynamics analysis of the flow, is a method that can automatically improve designs of aircraft components. The prospect is to reduce a cost function that reflects aerodynamic performances. When the shape is described by a large number of parameters, the calculation of one gradient of the cost function is only feasible by recourse to techniques that are derived from the theory of optimal control. In order to obtain the best computational efficiency, the so called adjoint method is applied here on the complete mapping, from the parameters of design to the values of the cost function. The mapping considered here includes the Euler equations for compressible flow discretized on unstructured meshes by a median-dual finite-volume scheme, the primal-to-dual mesh transformation, the mesh deformation, and the parameterization. The results of the present research concern the detailed derivations of expressions, equations, and algorithms that are necessary to calculate the gradient of the cost function. The discrete adjoint of the Euler equations and the exact dual-to-primal transformation of the gradient have been implemented for 2D and 3D applications in the code Edge, a program of Computational Fluid Dynamics used by Swedish industries. Moreover, techniques are proposed here in the aim to further reduce the computational cost of aerodynamic shape optimization. For instance, an interpolation scheme is derived based on Radial Basis Functions that can execute the deformation of unstructured meshes faster than methods based on an elliptic equation. In order to improve the accuracy of the shape, obtained by numerical optimization, a moving mesh adaptation scheme is realized based on a variable diffusivity equation of Winslow type. This adaptation has been successfully applied on a simple case of shape optimization involving a supersonic flow. An interpolation technique has been derived based on a mollifier in order to improve the convergence of the coupled mesh-flow equations entering the adaptive scheme. The method of adjoint derived here has also been applied successfully when coupling the Euler equations with the boundary-layer and parabolized stability equations, with the aim to delay the laminar-to-turbulent transition of the flow. The delay of transition is an efficient way to reduce the drag due to viscosity at high Reynolds numbers.

Advances in numerical optimization have raised the possibility that efficient and novel aircraft configurations may be ``discovered'' by an algorithm. To begin exploring this possibility, a fast and robust set of tools for aerodynamic shape optimization is developed. Parameterization and mesh-movement are integrated to accommodate large changes in the geometry. This integrated approach uses a coarse B-spline control grid to represent the geometry and move the computational mesh; consequently, the mesh-movement algorithm is two to three orders faster than a node-based linear elasticity approach, without compromising mesh quality. Aerodynamic analysis is performed using a flow solver for the Euler equations. The governing equations are discretized using summation-by-parts finite-difference operators and simultaneous approximation terms, which permit nonsmooth mesh continuity at block interfaces. The discretization results in a set of nonlinear algebraic equations, which are solved using an efficient parallel Newton-Krylov-Schur strategy. A gradient-based optimization algorithm is adopted. The gradient is evaluated using adjoint variables for the flow and mesh equations in a sequential approach. The flow adjoint equations are solved using a novel variant of the Krylov solver GCROT. This variant of GCROT is flexible to take advantage of non-stationary preconditioners and is shown to outperform restarted flexible GMRES. The aerodynamic optimizer is applied to several studies of induced-drag minimization. An elliptical lift distribution is recovered by varying spanwise twist, thereby validating the algorithm. Planform optimization based on the Euler equations produces a nonelliptical lift distribution, in contrast with the predictions of lifting-line theory. A study of spanwise vertical shape optimization confirms that a winglet-up configuration is more efficient than a winglet-down configuration. A split-tip geometry is used to explore nonlinear wake-wing interactions: the optimized split-tip demonstrates a significant reduction in induced drag relative to a single-tip wing. Finally, the optimal spanwise loading for a box-wing configuration is investigated.

Boundary shape optimization is a technique to search for an optimal shape by modifying the boundary of a device with a pre-specified topology. We consider boundary shape optimization of acoustic horns in loudspeakers and brass wind instruments. A horn is an interfacial device, situated between a source, such as a waveguide or a transducer, and surrounding space. Horns are used to control both the transmission properties from the source and the spatial power distribution in the far-field (directivity patterns). Transmission and directivity properties of a horn are sensitive to the shape of the horn flare. By changing the horn flare we design transmission efficient horns. However, it is difficult to achieve both controllability of directivity patterns and high transmission efficiency by using only changes in the horn flare. Therefore we use simultaneous shape and so-called topology optimization to design a horn/acoustic-lens combination to achieve high transmission efficiency and even directivity. We also design transmission efficient interfacial devices without imposing an upper constraint on the mouth diameter. The results demonstrate that there appears to be a natural limit on the optimal mouth diameter. We optimize brasswind instruments with respect to its intonation properties. The instrument is modeled using a hybrid method between a one-dimensional transmission line analogy for the slowly flaring part of the instrument, and a finite element model for the rapidly flaring part. An experimental study is carried out to verify the transmission properties of optimized horn. We produce a prototype of an optimized horn and then measure the input impedance of the horn. The measured values agree reasonably well with the predicted optimal values. The finite element method and the boundary element method are used as discretization methods in the thesis. Gradient-based optimization methods are used for optimization, in which the gradients are supplied by the adjoint methods.

Le problème de l'optimisation des valeurs propres du Laplacien est ancien puisqu'à la fin du XIXème siècle Lord Rayleigh conjecturait que la première valeur propre avec condition de Dirichlet était minimisée par le disque. Depuis le problème a été beaucoup étudié. Et les possibilités de recherches sont multiples : diverses conditions, ajout de contraintes, existence, description des optima ... Dans ce document on se limite aux conditions de Dirichlet et de Neumann, dans R^2 et dans R^3. On procède dans un premier temps à un état de l'art. On se focalise ensuite sur les disques et les boules. En effet, ils font partie des rares formes pour lesquelles il est possible de calculer explicitement et relativement facilement les valeurs propres. On verra malheureusement que ces formes ne sont la plupart du temps pas des minimiseurs. Enfin on s'intéresse aux simulations numériques possibles. En effet, puisque peu de calculs théoriques peuvent être faits il est intéressant d'obtenir numériquement des candidats. Cela permet ensuite d'avoir des hypothèses de travail théorique. `{A} cet effet nous donnerons des éléments de compréhension sur une méthode de simulation numérique ainsi que des résultats obtenus. / The optimization of Laplacian eigenvalues is a classical problem. In fact, at the end of the nineteenth century, Lord Rayleigh conjectured that the first eigenvalue with Dirichlet boundary condition is minimized by a disk. This problem received a lot of attention since this first study and research possibilities are numerous: various conditions, geometrical constraints added, existence, description of optimal shapes... In this document we restrict us to Dirichlet and Neumann boundary conditions in R^2 and R^3. We begin with a state of the art. Then we focus our study on disks and balls. Indeed, these are some of the only shapes for which it is possible to explicitly and relatively easily compute the eigenvalues. But we show in one of the main result of this document that they are not minimizers for most eigenvalues. Finally we take an interest in the possible numerical experiments. Since we can do very few theoretical computations, it is interesting to get numerical candidates. Then we can deduce some theoretical working assumptions. With this in mind we give some keys to understand our numerical method and we also give some results obtained.

Nous étudions dans cette thèse des problèmes d'optimisation de formes associés à des fonctionnelles spectrales et géométriques. L’étude porte à la fois sur des points de vue théoriques et numériques. L’idée générale est ici de proposer des résultats de Gamma-convergence qui permettent de construire des approximations numériques pour des quantités que l'on cherche à optimiser. En particulier, ces méthodes numériques sont appliquées à l’étude des minimiseurs des valeurs propres de l’opérateur Laplacien-Diriclet sous contrainte de périmètre en dimension deux et trois. Une autre classe de problèmes traités concerne les problèmes multiphasiques et les partitions optimales dans le plan et sur des surfaces tri-dimensionnelles.On présente aussi une analyse du spectre de l’opérateur Steklov en rapport avec différentes classes géométriques de domaines. Une partie de cette analyse concerne le problème de l'existence de domaines extrémaux et la stabilité spectrale sous perturbations géométriques. Une deuxième partie de l’étude est liée au développement des méthodes basées sur des solutions fondamentales qui permettent d’évaluer numériquement le spectre d'un opérateur. Une analyse détaillée de la méthode numérique montre qu'on obtient une précision de calcul importante et une économie en temps d’exécution significative par rapport aux méthodes utilisant des maillages. Cette approche est étendue au calcul du spectre des opérateurs de Wentzell et de Laplace-Beltrami. / We study some shape optimization problems associated to spectral and geometric functionals from both theoretical and numerical points of view. One of the main ideas is to provide Gamma-convergence frameworks allowing the construction of numerical approximation methods for the quantities we wish to optimize. In particular, these numerical methods are applied to the study of the Dirichlet-Laplace eigenvalues under perimeter constraint in two and three dimensions and to optimization problems concerning multiphase configurations and partitions in the plane and on three dimensional surfaces.As well, we focus on the analysis of the Steklov spectrum in different geometric classes of domains. Together with the study of existence of extremal domains and the spectral stability under geometric perturbations, we develop methods based on fundamental solutions in order to compute numerically the spectrum. A detailed analysis of the numerical method shows that we get an important precision, while the computation time is significantly decreased compared to mesh-based methods. This approach is extended to the computation of Wentzell and Laplace-Beltrami eigenvalues.

Este trabalho tem como objetivo explorar a forma de planos de fachada vinculados à incidência solar, potencializando a aplicação de sistemas fotovoltaicos. A identificação e parametrização de formas segundo os princípios geométricos de captação fotovoltaica, sugerem a aplicação de uma metodologia de projeto para superfícies de fachadas fotovoltaicas, de modo a otimizar a incidência direta da radiação solar, incorporada a volumetria da edificação. O modelo de otimização de fachadas fotovoltaicas consiste em quatro etapas. Inicialmente define-se a tecnologia fotovoltaica e a localização geográfica (1). Posteriormente, é realizada a modelagem elementar tridimensional (2) através do editor de algoritmos gráficos Grasshopper – integrado à ferramenta de modelagem Rhinoceros3D – estabelecendo assim, as restrições e variáveis da forma. Na terceira etapa, correlacionamse transformações geométricas tridimensionais (twist, taper e shear) e incidência solar (3) por meio dos softwares Ecotect Analysis e Grasshopper. Com isso, os parâmetros dimensionais atribuídos às variáveis – transformações geométricas – são vinculados aos parâmetros de radiação solar, visando à geração de formas. Após a seleção das formas com maior potencial fotovoltaico, identificam-se as zonas com maior incidência de radiação solar e realiza-se a manipulação dos pontos de controle das superfícies NURBS (4). Através das transformações geométricas taper, shear e twist foi possível gerar um conjunto de soluções otimizadas, correlacionando dados energéticos e geométricos, integrando métodos de geração de formas e avaliação performática da radiação solar. O estudo identificou que as possibilidades de articulação entre os planos fotovoltaicos e a eficiência energética têm implicações positivas, correlacionando variabilidade formal e geração de energia elétrica. / This work aims explore the shape of façade planes linked to the solar incidence, in order to optimize the use of photovoltaic systems. The identification and parameterization of forms according to geometric principles of photovoltaic capture suggest the application of a design methodology for optimizing the photovoltaic surface façade in order to optimize direct solar radiation, incorporating the volume of the building. The optimization model of photovoltaic façade consists of four steps. Initially decide on the photovoltaic technology and geographic location (1). Subsequently, three-dimensional elementary modeling is performed (2) through the graphic-algorithm editor, Grasshopper, – integrated with the modeling tool, Rhinoceros 3D, – thus establishing, restrictions and variables in shape. In the third stage, threedimensional geometric transformations are correlated (twist, taper and shear) and solar incidence (3) through the computer interfaces of Ecotect Analysis and Grasshopper software. With this, the dimensional parameters assigned to the variables – geometric transformations – are linked to parameters of solar radiation, in order to generate shapes. After the selection of potential photovoltaic shapes, zones with the greatest incident solar radiation are identified and the control points of NURBS surface are manipulated (4). Using the geometric transformations taper, shear and twist, it was possible to generate a set of optimal solutions, correlating geometric and energetic data, integrating shape generating methods and performatic evaluation of solar exposure. The work identified that possibilities of articulation between photovoltaic planes and energetic efficiency have positive results, correlating shape variability and electricity generation.

Este trabalho tem como objetivo explorar a forma de planos de fachada vinculados à incidência solar, potencializando a aplicação de sistemas fotovoltaicos. A identificação e parametrização de formas segundo os princípios geométricos de captação fotovoltaica, sugerem a aplicação de uma metodologia de projeto para superfícies de fachadas fotovoltaicas, de modo a otimizar a incidência direta da radiação solar, incorporada a volumetria da edificação. O modelo de otimização de fachadas fotovoltaicas consiste em quatro etapas. Inicialmente define-se a tecnologia fotovoltaica e a localização geográfica (1). Posteriormente, é realizada a modelagem elementar tridimensional (2) através do editor de algoritmos gráficos Grasshopper – integrado à ferramenta de modelagem Rhinoceros3D – estabelecendo assim, as restrições e variáveis da forma. Na terceira etapa, correlacionamse transformações geométricas tridimensionais (twist, taper e shear) e incidência solar (3) por meio dos softwares Ecotect Analysis e Grasshopper. Com isso, os parâmetros dimensionais atribuídos às variáveis – transformações geométricas – são vinculados aos parâmetros de radiação solar, visando à geração de formas. Após a seleção das formas com maior potencial fotovoltaico, identificam-se as zonas com maior incidência de radiação solar e realiza-se a manipulação dos pontos de controle das superfícies NURBS (4). Através das transformações geométricas taper, shear e twist foi possível gerar um conjunto de soluções otimizadas, correlacionando dados energéticos e geométricos, integrando métodos de geração de formas e avaliação performática da radiação solar. O estudo identificou que as possibilidades de articulação entre os planos fotovoltaicos e a eficiência energética têm implicações positivas, correlacionando variabilidade formal e geração de energia elétrica. / This work aims explore the shape of façade planes linked to the solar incidence, in order to optimize the use of photovoltaic systems. The identification and parameterization of forms according to geometric principles of photovoltaic capture suggest the application of a design methodology for optimizing the photovoltaic surface façade in order to optimize direct solar radiation, incorporating the volume of the building. The optimization model of photovoltaic façade consists of four steps. Initially decide on the photovoltaic technology and geographic location (1). Subsequently, three-dimensional elementary modeling is performed (2) through the graphic-algorithm editor, Grasshopper, – integrated with the modeling tool, Rhinoceros 3D, – thus establishing, restrictions and variables in shape. In the third stage, threedimensional geometric transformations are correlated (twist, taper and shear) and solar incidence (3) through the computer interfaces of Ecotect Analysis and Grasshopper software. With this, the dimensional parameters assigned to the variables – geometric transformations – are linked to parameters of solar radiation, in order to generate shapes. After the selection of potential photovoltaic shapes, zones with the greatest incident solar radiation are identified and the control points of NURBS surface are manipulated (4). Using the geometric transformations taper, shear and twist, it was possible to generate a set of optimal solutions, correlating geometric and energetic data, integrating shape generating methods and performatic evaluation of solar exposure. The work identified that possibilities of articulation between photovoltaic planes and energetic efficiency have positive results, correlating shape variability and electricity generation.