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The work presented in this thesis aims to provide various tools to be used during design process to make maximum use of the increasing availability of accurate engine blade measurement data for high fidelity fluid mechanic simulations at a reasonable computational expense. A new method for uncertainty propagation for geometric error has been proposed for fluid mechanics codes using adjoint error correction. Inexpensive Monte Carlo (IMC) method targets small uncertainties and provides complete probability distribution for the objective function at a significantly reduced computational cost. A brief literature survey of the existing methods is followed by the formulation of IMC. An example algebraic model is used to demonstrate the IMC method. The IMC method is extended to fluid mechanic applications using Principal Component Analysis (PCA) for reduced order modelling. Implementation details for the IMC method are discussed using an example airfoil code. Finally, the IMC method has been implemented and validated for an industrial fluid mechanic code HYDRA. A consistent methodology has been developed for the automatic generation of the linear and adjoint codes by selective use of automatic differentiation (AD) technique. The method has the advantage of keeping the linear and the adjoint codes in-sync with the changes in the underlying nonlinear fluid mechanic solver. The use of various consistency checks have been demonstrated to ease the development and maintenance process of the linear and the adjoint codes. The use of AD has been extended for the calculation of the complete Hessian using forward-on-forward approach. The complete mathematical formulation for Hessian calculation using the linear and the adjoint solutions has been outlined for fluid mechanic solvers. An efficient implementation for the Hessian calculation is demonstrated using the airfoil code. A new application of the Independent Component Analysis (ICA) is proposed for manufacturing uncertainty source identification. The mathematical formulation is outlined followed by an example application of ICA for artificially generated uncertainty for the NACA0012 airfoil.

Depuis plusieurs années à nos jours, les paliers à gaz sont utilisés avec succès sur une large gamme de turbo-machines. Certains de ces systèmes sont utilisés dans des environnements de contrôle de l’environnement par gaz réfrigérant. Dans ce travail, nous présentons un modèle théorique et numérique qui tient compte de la transition du lubrifiant vapeur / liquide, la transition de l’écoulement laminaire / turbulent et les variations 3D de la viscosité et de la température dans le fluide et les solides pour les deux situations statiques et dynamiques. Ce modèle comporte : la résolution de l'équation de Reynolds généralisée pour les fluides compressibles à viscosité variable en 3D, la description des effets de la turbulence en utilisant l'approche phénoménologique de Elrod, en utilisant un champ de viscosité turbulente 3D, la résolution de l’équation d’état non linéaire du lubrifiant, capable de décrire la transition vapeur / liquide et une approche thermique local afin d'obtenir une estimation 3D de la température du fluide, grâce à l'équation d'énergie pour film mince. La prise en compte également des effets thermiques dans les solides. Dans cette étude, nous avons montré l'importance d'une description précise des paramètres du film fluide, dont les variations influencent largement le comportement du palier. Parmi les principales théories, il y a: lubrifiant compressible, avec un comportement non-linéaire près de la transition vapeur / liquide, la transition vapeur / liquide et le calcul des paramètres équivalents du mélange, un écoulement turbulent du fluide pour le palier GFB à grande vitesse en utilisant un modèle 3D de la viscosité turbulente, un comportement 3D pour la viscosité, en particulier les variations inter-films (dépendant de la température), et un comportement 3D pour la température, en particulier dans le sens transversal du film afin d'être compatible avec la viscosité, mais également dans la direction axiale afin de tenir en compte du gradient de température potentiel qui modifie considérablement le profil 3D de la température du palier. Ces deux comportements statiques et dynamiques du palier compliant GFB sont analysés. / For years now, gas bearings are successfully used over a large panel of turbo-machineries. Some of these systems are bound to be run in controlled environments such as refrigerating gas. In this work we present a theoretical and numerical model which takes into account the vapor/liquid lubricant transition, the laminar/turbulent flow transition and both temperature and viscosity 3D variations in the fluid and the solids for both static and dynamic situations. This model involves: the resolution of the generalized Reynolds equation for compressible fluids with 3D variable viscosity, the description of the turbulence effects by the phenomenological approach of Elrod, using a 3D eddy viscosity field, the resolution of a non-linear equation of state for the lubricant, able to describe the vapor/liquid transition and a local thermal approach to obtain a 3D estimation of the fluid temperature, thanks to the thin-film energy equation. The thermal effects in solids are also taken into account. In this study, we showed the importance of an accurate description of the film parameters, which variations largely influence the bearing behaviour. Among the principal theories, there are: compressible lubricant, with an appropriate non-linear behaviour when close to the vapor/liquid transition, vapor/liquid transition and calculation of the mixture equivalent parameters, turbulent flow for high-speed GFBs with a 3D eddy viscosity mode, a 3D behaviour for viscosity, particularly the cross-film variations, (temperature dependent)and a 3D behaviour for temperature, particularly in cross-film direction in order to be consistent with viscosity, but also in the axial direction in order to account for potential temperature gradient which considerably modifies the bearing 3D temperature profile. Both static and dynamic behaviours of GFBs are analysed.

The primary requirements for a modern industrial gas turbine consist of a continuous trend of an increasing efficiency combined with very low emissions in a robust, cost-effective manner. To fulfil these tasks a high turbine inlet temperature together with advanced dry low NOX combustion chambers are employed. These dry low NOX combustion chambers generate a rather flat temperature profile compared to previous generation gas turbines, which have a rather parabolic temperature profile before the nozzle guide vane. This means that the nozzle guide vane endwall heat load for modern gas turbines is much higher compared to previous generation gas turbines. Therefore the prediction of the nozzle guide vane flow field and endwall heat transfer is crucial for the engineering task of the design layout of the vane endwall cooling system. The present study is directed towards establishing new in-depth aerodynamic and endwall heat transfer knowledge for an advanced nozzle guide vane of a modern industrial gas turbine. To reach this objective the physical processes and effects which cause the different flow fields and the endwall heat transfer pattern in a baseline configuration, a combustion chamber variant, a heat shield variant without and with additional cooling air and a cavity variant without and with additional cooling air have been investigated. The variants, which differ from the simplified baseline configuration, apply design elements which are commonly used in real modern gas turbines. This research area is crucial for the nozzle guide vane endwall heat transfer, especially for the advanced design of the nozzle guide vane of a modern industrial gas turbine and has so far hardly been investigated in the open literature. For the experimental aerodynamic and endwall heat transfer research of the baseline configuration of the advanced nozzle guide vane geometry a new low pressure, low temperature test facility has been developed, designed and constructed, since no experimental heat transfer data exist in the open literature for this type of vane configuration. The new test rig consists of a linear cascade with the baseline configuration of the advanced nozzle guide vane geometry with four upscaled airfoils and three flow passages. For the aerodynamic tests the two middle airfoils and the hub and the tip endwall are instrumented with pressure taps to monitor the Mach number distribution. For the heat transfer tests the temperature distribution on the hub endwall is measured via thermography. The analysis of these measurements, including comparisons to research in the open literature shows that the new test rig generates accurate and reproducible results which give confidence that it is a reliable tool for the experimental aerodynamic and heat transfer research on the advanced nozzle guide vane of a modern industrial gas turbine. Previous own research work together with the numerical analysis performed in another part of the project as well as conclusions from a detailed literature study lead to the conclusion that advanced Navier-Stokes CFD tools with the v2-f turbulence model are most suitable for the calculation of the flow field and the endwall heat transfer of turbine vanes and blades. Therefore this numerical tool, validated against different vane and blade geometries and for different flow conditions, has been chosen for the numerical aerodynamic and endwall heat transfer research of the advanced nozzle guide vane of a modern industrial gas turbine. The evaluation of the numerical and experimental investigations of the baseline configuration of the advanced design of a nozzle guide vane shows the flow field of an advanced mid-loaded airfoil design with the features to reduce total airfoil losses. For the hub endwall of the baseline configuration of the advanced design of a nozzle guide vane the flow characteristics and heat transfer features of the classical vane endwall secondary flow model can be detected with a very weak intensity and geometric extension compared to the studies of less advanced vane geometries in the open literature. A detailed analysis of the numerical simulations and the experimental data showed very good qualitative and quantitative agreement for the three-dimensional flow field and the endwall heat transfer. These findings, together with the evaluations obtained from the open literature, lead to the conclusions that selected CFD software Fluent together with the applied v2-f turbulence model exhibits a high level of general applicability and is not tuned to a special vane or blade geometry. Therefore the CFD code Fluent with the v2-f turbulence model has been selected for the research of the influence of the several geometric variants of the baseline configuration on the flow field and the hub endwall heat transfer of the advanced nozzle guide vane of a modern industrial gas turbine. Most of the vane endwall heat transfer research in the open literature has been carried out only for baseline configurations of the flow path between combustion chamber and nozzle guide vane. Such a simplified geometry consists of a long, planar undisturbed approach length upstream of the nozzle guide vane. The design of real modern industrial gas turbines however requires often significant variations from this baseline configuration consisting of air-cooled heat shields and purged cavities between the combustion chamber and the nozzle guide vane. A detailed evaluation of the flow field and the endwall heat transfer shows major differences between the baseline and the heat shield configuration. The heat shield in front of the airfoil of the nozzle guide vane influences the secondary flow field and the endwall heat transfer pattern strongly. Additional cooling air, released under the heat shield has a distinctive influence as well. Also the cavity between the combustion chamber and the nozzle guide vane affects the secondary flow field and the endwall heat transfer pattern. Here the influence of additional cavity cooling air is more decisive. The results of the detailed studies of the geometric variants are applied to formulate guidelines for an optimized design of the flow path between the combustion chamber and the nozzle guide vane and the nozzle guide vane endwall cooling configuration of next-generation industrial gas turbines. / QC 20100917

Η θερμοδυναμική ανάλυση του κύκλου Brayton υποδεικνύει ότι η θερμική απόδοση και το ειδικό έργο εξόδου ενός αεριοστρόβιλου μπορούν να βελτιωθούν με την αύξηση της θερμοκρασίας εισόδου των αεριών της καύσης στον στρόβιλο. Επιπλέον, οι αυξημένες θερμοκρασίες εισόδου στον στρόβιλο συνοδεύονται και από μείωση της κατανάλωσης καυσίμου, ενώ σε αεροπορικές εφαρμογές οι υψηλότερες θερμοκρασίες έχουν ώς αποτέλεσμα την αύξηση της ώσης του κινητήρα. Δυστυχώς όμως, οι υψηλές αυτές θερμοκρασίες θέτουν σε κίνδυνο την ακεραιότητα των εξαρτημάτων του στροβίλου υψηλής πίεσης και ειδικότερα τα πτερύγια (blades) του στροβίλου και το δάπεδο (endwall) στο οποίο τα πτερύγια αυτά είναι προσκολλημένα. Στους μοντέρνους κινητήρες, η θερμοκρασία εισόδου στον στρόβιλο μπορεί να φτάνει και στα επίπεδα των 1900Κ, θερμοκρασία η οποία υπερβαίνει το σημείο τήξης των υλικών από τα οποία είναι κατασκευασμένα τα εξαρτήματα του στροβίλου. Αυτό έχει ως αποτέλεσμα τα εξαρτήματα του στροβίλου να λειτουργούν σε πολύ σκληρότερο περιβάλλον απ' ότι στο παρελθόν. Η διατήρηση επαρκούς διάρκειας ζωής στις υψηλές αυτές θερμοκρασίες απαιτεί την ανάπτυξη νέων υλικών κατασκευής και αποτελεσματικών μεθόδων ψύξης για τα εξαρτήματα του στροβίλου. Για την αντιμετώπιση και την αποφυγή της αστοχίας των πτερυγίων (blades) και των δάπεδων (endwall) των πτερυγικών διακένων στους στροβίλους, η μέθοδος του "film cooling" έχει ενσωματωθεί στον σχεδιασμό τους. Κατά την διεργασία της ψύξης των εξαρτημάτων με την μέθοδο αυτή, ψυχρός αέρας αφαιμάσσεται από το στάδιο του συμπιεστή, διοχετεύεται μέσω εσωτερικών θαλάμων του κινητήρα στα εξαρτήματα του στροβίλου και εγχέεται μέσω διακριτών οπών στα τοιχώματα των πτερυγίων και των δαπεδικών τοιχωμάτων. Μετά την έξοδο του από τις οπές, ο ψυκτικός αέρας σχηματίζει ένα λεπτό, προστατευτικό στρώμα-φιλμ μεταξύ των θερμών αερίων της καύσης και της μεταλλικής επιφάνειας των εξαρτημάτων. Μια εκ των κρίσιμων περιοχών οι οποίες υποβάλλονται σε αυξημένους ρυθμούς μετάδοσης θερμότητας είναι και η περιοχή γύρω από την περιφέρεια σύνδεσης των πτερυγίων (blades) με τα δάπεδα (endwalls) του στροβίλου. Η περιοχή αυτή κυριαρχείται από την παρουσία ισχυρών τρισδιάστατων δευτερογενών ροών (γνωστές και ώς junction flows) οι οποίες προκαλούν αύξηση των τοπικών ρυθμών μετάδοσης θερμότητας στην περιοχή της τάξης του 350%. Επιπλέον, οι ροές αυτές, εμποδίζουν την διείσδυση ψυκτικού ρευστού στην προβληματική περιοχή εκτοπίζοντας το μακριά από την επιφάνεια του δαπέδου πριν αυτό προλάβει να παράσχει ικανοποιητική ψύξη. Αντικείμενο της παρούσας διδακτορικής διατριβής, είναι η μελέτη, ανάπτυξη και δοκιμή (τόσο πειραματικά όσο και υπολογιστικά) μιας πρωτότυπης γεωμετρίας ψύξης (με την μέθοδο του film cooling), για την αποτελεσματική αντιμετώπιση του προβλήματος της υπερθέρμανσης της περιοχής σύνδεσης του πτερυγίου – δαπέδου κυρίως γύρω από το επίπεδο του χείλους προσβολής. Το κύριο χαρακτηριστικό της πρωτότυπης μεθόδου έγχυσης είναι ότι το ψυκτικό εκχέεται κατά τέτοιο τρόπο έτσι ώστε οι ροϊκές γραμμές του ψυκτικού να υποβοηθούνται από την περιστροφική κίνηση των τοπικών τρισδιάστατων ροών. Η πολυπλοκότητα του προβλήματος ψύξης της συγκεκριμένης περιοχής προκύπτει από δύο στοιχεία. Πρώτον, όπως αναφέρθηκε και προηγουμένως, η ροή στην περιοχή σύνδεσης κοντά στο τοίχωμα χαρακτηρίζεται από πολύπλοκη τρισδιάστατη δομή. Δεύτερον, το πρόβλημα χαρακτηρίζεται από τρείς θερμοκρασίες: την θερμοκρασία της κύριας ροής, την θερμοκρασία του τοιχώματος και την θερμοκρασία του ψυκτικού αέρα. Για την πλήρη διερεύνηση των χαρακτηριστικών της προτεινόμενης μεθόδου ψύξης η εργασία περιλαμβάνει τόσο πειραματικό όσο και υπολογιστικό σκέλος: Υπολογιστικό Σκέλος (Computational part): Ο επιτυχής σχεδιασμός μιας πιθανής γεωμετρίας ψύξης για την συγκεκριμένη περιοχή του δαπέδου (endwall) απαιτεί την γνώση και κατανόηση της τοπικής ροής μέσα στην οποία το τζετ του ψυκτικού πρόκειται να εισέλθει. Επιπλέον, είναι σημαντική η κατανόηση της αλληλεπίδρασης που αναμένεται μεταξύ του ψυκτικού αέρα με την τοπική τρισδιάστατη ροή. Για τον σκοπό αυτό, χρησιμοποιήθηκε η μέθοδος της υπολογιστικής ρευστοδυναμικής (Computational Fluid Dynamics) για την πρόβλεψη του σχετικού τρισδιάστατου βασικού πεδίου ροής στην περιοχή σύνδεσης του πτερυγίου (blade) - δαπέδου (endwall). Έγιναν προσομοιώσεις τόσο για την βασική γεωμετρία απουσία έγχυσης (οι οποίες χρησιμοποιήθηκαν ως πεδίο αναφοράς) όσο και προσομοιώσεις παρουσία της πρωτότυπης έγχυσης οι οποίες αφορούσαν την επίδραση διαφόρων παραμέτρων στην αποτελεσματικότητα της ψύξης της προβληματικής περιοχής. Στις προσομοιώσεις υιοθετήθηκε η εξής θερμοκρασιακή κατανομή: Θερμό δάπεδο (endwall) - Θερμότερη κύρια ροή (mainstream) - Ψυχρός αέρας έγχυσης, η οποία είναι και αντίστοιχη με αυτήν που εμφανίζεται σε πραγματικές εφαρμογές. Τα αποτελέσματα των προσομοιώσεων βοήθησαν στην κατανόηση του ροϊκού πεδίου στην περιοχή σύνδεσης τόσο ποιοτικά όσο και ποσοτικά σε ότι αφορά τα σχετικά μεγέθη των ροϊκών δομών και των αεροδυναμικών χαρακτηριστικών τις περιοχής. Αυτό είχε ώς αποτέλεσμα τον αποτελεσματικό σχεδιασμό της πρωτότυπης γεωμετρίας έγχυσης. Επιπλέον, οι υπολογιστικές προβλέψεις ήταν πολύ βοηθητικές προς την κατεύθυνση κατανόησης και ερμηνείας των πειραματικών αποτελεσμάτων, αφού παρείχαν την δυνατότητα συσχέτισης της προκύπτουσας κατανομής της θερμοκρασίας στο δάπεδο (endwall) με τις τοπικές τρισδιάστατες ροές. Πειραματικό Σκέλος (Experimental part): Για την πειραματική διερεύνηση της αποτελεσματικότητας της προτεινόμενης μεθόδου ψύξης, χρησιμοποιήθηκε μια νέα τεχνική η οποία αναπτύχθηκε ως μέρος της παρούσας εργασίας, υιοθετώντας θερμοκρασιακή κατανομή αντίστροφη από αυτήν που χρησιμοποιήθηκε για τις υπολογιστικές προβλέψεις, π.χ. Ψυχρή κύρια ροή (mainstream) - Θερμό πλαϊνό τοίχωμα (endwall) - Θερμότερος αέρας έγχυσης. Χρησιμοποιώντας την μέθοδο αυτή και με την χρήση υπέρυθρης θερμογραφίας (infrared thermography), ποσοτικοποιείται η αποτελεσματικότητα στην ψύξη του πλαϊνού τοιχώματος και προσδιορίζεται η περιοχή στην οποία η ψύξη είναι αποτελεσματική. Επιπλέον της ποιοτικής και ποσοτικής αποτίμησης της αποτελεσματικότητας της ψύξης, ήταν αναγκαίες αεροδυναμικές μετρήσεις για τον καθορισμό του αεροδυναμικού κόστους της προτεινόμενης μεθόδου ψύξης. Οι μετρήσεις αυτές, δίνουν μια ένδειξη του κατά πόσον η μέθοδος επηρεάζει την μεγέθυνση και ένταση των δευτερογενών ροών (π.χ. δίνη διακένου (passage vortex)) στην περιοχή κατάντη της ζώνης αλληλεπίδρασης του ψυκτικού τζετ με την τοπική τρισδιάστατη ροή. / The thermodynamic analysis of the Brayton cycle designates that the thermal efficiency and the specific work output of a Gas Turbine can be improved by increasing the Turbine Inlet Temperatures. Furthermore, increment of the turbine inlet temperatures also results into lower fuel consumption rates, while, if the gas turbine is meant for propulsion purposes, increment of the turbine inlet temperatures also results into increased engine thrust. Unfortunately, these high gas temperatures jeopardize the integrity of the high pressure turbine components and more particular, the turbine blades and the endwall on which the blades are attached. In modern turbines, the turbine inlet temperature may reach the level of 1900K, exceeding by far the melting temperature of the metal walls. As a result, the turbine components operate at much harsher environments than in the past. Maintaining adequate life in these high temperatures requires the development of new materials and manufacturing processes, as well as efficient cooling methods for the components of the turbine. In order to address and avoid the failure of the blades and endwall of a turbine cascade, the method of "film cooling" has been incorporated as part of the components design process. In the latter method, air is bled from the compressor stage, passed through internal chambers of the engine to the turbine components and is injected through discrete holes in the walls of the blades and the endwall, forming a thin protective layer film between the hot combustion gases and the metal surfaces of the parts. A critical region that is subjected into increased thermal stresses is the area around the leading edge - endwall juncture, which is inherently dominated by the presence of strong three dimensional secondary flows (also known as juncture flows) responsible for the increment of the local heat transfer rates to the order of 350%. Moreover, these flows, prevent the penetration of the fluid in the problematic area, displacing the coolant mass flux away from the surface of the endwall before providing adequate cooling. The subject of the current thesis, is the design, development and testing (both experimental and computational) of a prototype cooling scheme (with the method of film cooling), in order to effectively address the endwall overheating problem around the leading edge - endwall juncture, especially around the stagnation plane area. The main feature of the novel injection method is that the coolant air is ejected in such a way that the cooling effectiveness in the area is assisted by the rotational sense of local three-dimensional flows. The complexity of film cooling for the problematic area arises from two facts. Firstly, as mentioned previously the flow around the leading edge junction is characterized by complex three dimensional flows. Secondly, the problem is characterized by three temperatures: the temperature of the main flow, the endwall temperature and the temperature of the coolant air. In order to fully investigate the features and characteristics of the proposed cooling method, the work of the current thesis includes both, an experimental and a computational part: Computational part: The successful design of a possible cooling scheme for the particular region of the endwall requires the knowledge and understanding of local flow in which the coolant jet is to be entrained. Furthermore, it is important to understand the expected interaction between the coolant air and the local three-dimensional flow. For this purpose, the method of Computational Fluid Dynamics was employed for predicting the relevant three-dimensional flow field around the blade-endwall junction area. Simulations were made for both, the basic geometry in the absence of any coolant injection (which were used as a reference point) and simulations during the employment of the proposed coolant injection method which concerned the effect of various parameters on the cooling efficiency of the problematic area. For the simulations, the following temperature step was adopted: Warm endwall - Warmer main flow (mainstream) - Cold air injection, which is similar to that seen in real applications. The CFD predictions were very helpful towards understanding the relevant flow field in the junction area, both qualitatively and quantitatively in terms of the relative magnitudes of the flow structures and the aerodynamic characteristics of the flow in the region. Experimental part: For the experimental investigation regarding the effectiveness of the proposed cooling method, a new experimental technique was employed which was developed as part of the current thesis. In the latter technique, a reversed temperature step is adopted (when compared to the relevant temperature step adopted for the numerical simulations), e.g. Cold main flow (mainstream) - Warm endwall - Warmer air injection. Along with the use of infrared thermography, the endwall film cooling effectiveness is quantified and the region that the injection is effective is determined. In addition to the qualitative and quantitative evaluation of the cooling effectiveness, extensive aerodynamic measurements were necessary in order to evaluate the aerodynamic costs of the proposed cooling method. These measurements provide an indication of whether the cooling process affects the growth and intensity of secondary flows (e.g. passage vortex) in the region downstream of the coolant jet-local three-dimensional flow interaction.

Design space exploration (DSE) is used to improve and understand engineering designs. Such designs must meet objectives and structural requirements. Design improvement is non-trivial and requires new DSE methods. Turbomachinery manufacturers must continue to improve existing engines to keep up with global demand. Two challenges of turbomachinery DSE are: the time required to evaluate designs, and knowing which designs to evaluate. This research addressed these challenges by developing novel surrogate and principal component analysis (PCA) based DSE methods. Node and PCA-based surrogates were created to allow faster DSE of turbomachinery blades. The surrogates provided static stress estimation within 10% error. Surrogate error was related to the number of sampled finite element (FE) models used to train the surrogate and the variables used to change the designs. Surrogates were able to provide structural evaluations three to five orders of magnitude faster than FEA evaluations. The PCA-based surrogates were then used to create a PCA-based design workflow to help designers know which designs to evaluate. The workflow used either two-point correlation or stress and geometry coupling to relate the design variables to principal component (PC) scores. These scores were projections of the FE models onto the PCs obtained from PCA. Analysis showed that this workflow could be used in DSE to better explore and improve designs. The surrogate methods were then applied to vibratory stress. A computationally simplified analysis workflow was developed to allow for enough fluid and structural analyses to create a surrogate model. The simplified analysis workflow introduced 10% error but decreased the computational cost by 90%. The surrogate methods could not directly be applied to emulation of vibration due to the large spikes which occur near resonance. A novel, indirect emulation method was developed to better estimate vibratory responses Surrogates were used to estimate the inputs to calculate the vibratory responses. During DSE these estimations were used to calculate the vibratory responses. This method reduced the error between the surrogate and FEA from 85% to 17%. Lastly, a PCA-based multi-fidelity surrogate method was developed. This assumed the PCs of the high and low-fidelities were similar. The high-fidelity FE models had tens of thousands of nodes and the low-fidelity FE models had a few hundred nodes. The computational cost to create the surrogate was decreased by 75% for the same errors. For the same computational cost, the error was reduced by 50%. Together, the methods developed in this research were shown to decrease the cost of evaluating the structural responses of turbomachinery blade designs. They also provided a method to help the designer understand which designs to explore. This research paves the way for better, and more thoroughly understood turbomachinery blade designs.

Design space exploration (DSE) is a tool used to evaluate and compare designs as part of the design selection process. While evaluating every possible design in a design space is infeasible, understanding design behavior and response throughout the design space may be accomplished by evaluating a subset of designs and interpolating between them using surrogate models. Surrogate modeling is a technique that uses low-cost calculations to approximate the outcome of more computationally expensive calculations or analyses, such as finite element analysis (FEA). While surrogates make quick predictions, accuracy is not guaranteed and must be considered. This research addressed the need to improve the accuracy of surrogate predictions in order to improve DSE of structural responses. This was accomplished by performing comparative analyses of linear and nonlinear dimensionality-reduction-based radial basis function (RBF) surrogate models for emulating various FEA nodal results. A total of four dimensionality reduction methods were investigated, namely principal component analysis (PCA), kernel principal component analysis (KPCA), isometric feature mapping (ISOMAP), and locally linear embedding (LLE). These methods were used in conjunction with surrogate modeling to predict nodal stresses and coordinates of a compressor blade. The research showed that using an ISOMAP-based dual-RBF surrogate model for predicting nodal stresses decreased the estimated mean error of the surrogate by 35.7% compared to PCA. Using nonlinear dimensionality-reduction-based surrogates did not reduce surrogate error for predicting nodal coordinates. A new metric, the manifold distance ratio (MDR), was introduced to measure the nonlinearity of the data manifolds. When applied to the stress and coordinate data, the stress space was found to be more nonlinear than the coordinate space for this application. The upfront training cost of the nonlinear dimensionality-reduction-based surrogates was larger than that of their linear counterparts but small enough to remain feasible. After training, all the dual-RBF surrogates were capable of making real-time predictions. This same process was repeated for a separate application involving the nodal displacements of mode shapes obtained from a FEA modal analysis. The modal assurance criterion (MAC) calculation was used to compare the predicted mode shapes, as well as their corresponding true mode shapes obtained from FEA, to a set of reference modes. The research showed that two nonlinear techniques, namely LLE and KPCA, resulted in lower surrogate error in the more complex design spaces. Using a RBF kernel, KPCA achieved the largest average reduction in error of 13.57%. The results also showed that surrogate error was greatly affected by mode shape reversal. Four different approaches of identifying reversed mode shapes were explored, all of which resulted in varying amounts of surrogate error. Together, the methods explored in this research were shown to decrease surrogate error when performing DSE of a turbomachine compressor blade. As surrogate accuracy increases, so does the ability to correctly make engineering decisions and judgements throughout the design process. Ultimately, this will help engineers design better turbomachines.

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