Η συνεχής συζυγής μέθοδος για την αεροακουστική βελτιστοποίηση μορφής στην αυτοκινητοβιομηχανία

The present doctoral thesis deals with the mathematical formulation, programming and verification of the continuous adjoint method to the constituent parts of an existing noise prediction chain for automotive aeroacoustics. The proposed method is applied to optimize a generic vehicle, the SAE body, in order to reduce wind noise in its interior. When a car travels at high speeds, flow-induced noise is generated in the region near the side mirror and is radiated towards all directions, reaching also the side window. Its vibrational response to this acoustic load generates in turn sound waves that propagate into the cabin and are perceived by the passengers as noise. An existing aeroacoustic framework simulating these physical mechanisms consists of an Improved Delayed Detached Eddy Simulation (IDDES) of the Navier-Stokes equations to obtain the unsteady pressure distribution on the mirror, the Kirchhoff Integral method to compute the radiated acoustic pressure on the side window, the bending wave equation on the side window to compute its deflection and, finally, the wave equation in the interior to obtain the interior sound field.The continuous adjoint method in this thesis is based on the aforementioned framework which is split in two domains; the exterior domain that includes the flow-induced noise generation and radiation to the window and the interior domain that includes the vibroacoustic model for window vibration and interior wave propagation. These systems are firstly examined separately and, then, coupled and the contintuous adjoint chain for vehicle aeroacoustic optimization is proposed. Regarding the vibroacoustic model, the bending wave equation is solved on the car's side window, using as a source term the pressure load obtained by the exterior domain. The resulting window acceleration is then used as a boundary condition for the wave equation that is solved in the cabin to compute the interior sound field. The Sound Pressure Level at a point near the driver's ear is then defined as the objective function and the adjoint wave and bending wave equations are derived. These must be solved backwards in time and in the following order: the adjoint wave equation is solved first by considering a monopole source term at the location where the objective function is defined. The propagation of the adjoint interior pressure in the cabin is computed and used thereupon as a source term for the adjoint bending wave equation, solved at the window. The resulting adjoint deflection is then used in the expression of the sensitivity derivative term on the window. This term is used later to couple the interior and exterior domains and additional emphasis is laid upon its discretization. A hand-differentiated expression is proposed to ensure its accuracy. The developed method is verified against Finite Differences and, then, is applied to the cabin of the SAE body to minimize interior noise using synthetic pressure waves as a load on the window. The continuous adjoint method for the flow-induced sound radiation with the Kirchhoff Integral is proposed where the differentiated Kirchhoff Integral is used to compute the boundary condition of the adjoint velocity on the noise radiating (Kirchhoff) surface and, then, the unsteady adjoint Navier-Stokes equations are solved backwards in time. It should be noted that the time window where the simulation is performed and the one over which the objective function is evaluated do not coincide. This is reflected on the adjoint boundary conditions along the body and the time integration of the sensitivity derivatives. Furthermore, to ensure the consistency of the continuous adjoint-based gradients, grid sensitivities are taken into account which gives rise to the adjoint grid displacement equations along with an additional term in the sensitivity derivatives expression. The proposed method is verified against Finite Differences on a 3D turbulent flow around a cylinder and, then, applied to the SAE body. Firstly, a sensitivity map analysis is conducted to investigate the influence the sensitivity derivative integration time window has on its computation but, also, to prove the importance of including the adjoint grid displacement equations. Finally, an optimization of the side mirror is performed, targeting at minimizing the radiated flow-induced sound at the vehicle's side window.After the formulation and verification of the continuous adjoint method for the systems of equations in the interior and exterior domains, their coupling is presented. Through the solution of the adjoint aeroacoustic chain, the sensitivity of the interior acoustic pressure with respect to a normal displacement of the mirror is computed to indicate the way the mirror shape should change, in order to improve the aeroacoustic performance of the vehicle. The method is applied to compute the adjoint aeroacoustic sensitivity map on the side mirror of the generic SAE vehicle and successfully perform several optimization cycles. In addition, the impact that optimizing for each individual step of the noise prediction chain has on interior noise is investigated. Finally, two approaches are proposed, in order to perform the aforementioned adjoint analysis, focusing however on a specific frequency range; the first approach uses an objective function which includes the Fourier Transform and is integrated over frequencies whereas the second one uses a signal processing filter that preserves only the necessary frequency components. The adjoint formulation, advantages and drawbacks of each approach are discussed. The adjoint aeroacoustic chain including the filtering process is finally used to compute sensitivity maps on the mirror for the frequency range 800Hz-4000Hz and also for each 1/3 Octave Band in this range.Σε αυτή τη διδακτορική διατριβή εφαρμόστηκε η συνεχής συζυγής μέθοδος σε ένα υπάρχον υπολογιστικό πλαίσιο για την πρόλεξη του αεροδυναμικού θορύβου στο εσωτερικό οχημάτων. Αυτό περιλαμβάνει την μη-μόνιμη πρόλεξη της ροής με μια IDDES προσομοίωση για τον υπολογισμό της υδροδυναμικής πίεσης στον καθρέφτη, την κυρίαρχη πηγή θορύβου, το ολοκλήρωμα Kirchhoff για τη διάδοση της ακουστικής πίεσης από τον καθρέφτη στο πλευρικό παράθυρο, την πρόλεξη της δομικής δόνησης του παραθύρου, και την προσομοίωση διαδόσης κυμάτων στο εσωτερικό του οχήματος, για τον τελικό υπολογισμό του επιπέδου θορύβου κοντά στο αυτί του οδηγού. Παρουσιάστηκε η συνεχής συζυγής διατύπωση για τις εξισώσεις που διέπουν τα προαναφερθέντα συστήματα και αναπτύχθηκε η συζυγής αλυσίδα για τον υπολογισμό του αεροακουστικού χάρτη ευαισθησίας στον καθρέφτη. Αυτός υποδεικνύει τις περιοχές, όπου η γεωμετρία πρέπει να παραμορφωθεί, είτε προς τα μέσα είτε προς τα έξω, με στόχο τη μείωση του θορύβου στο εσωτερικό. Κατά τη μαθηματική διατύπωση, έμφαση δόθηκε στα διαφορετικά παράθυρα της χρονικής ολοκλήρωσης των αντικειμενικών συναρτήσεων και της λύσης των πρωτεύοντων εξισώσεων. Επιπλέον, για να εξασφαλισθεί η ακρίβεια των παραγώγων ευαισθησίας, η ευαισθησία του πλέγματος μέσω της εξίσωσης μετατόπισης πλέγματος λήφθηκε υπόψη στην επαυξημένη συνάρτηση. Αυτό είχε ως αποτέλεσμα την εμφάνιση της συζυγούς εξίσωσης μετατόπισης πλέγματος καθώς και ενός επιπρόσθετου όρου στην έκφραση των παραγώγων ευαισθησίας. Η προτεινόμενη μέθοδος εφαρμόσθηκε για τη βελτιστοποίηση ενός οχήματος γενικευμένης μορφής, του SAE body, προκειμένου να μειωθεί ο εκ της ροής επαγόμενος θόρυβος στο εσωτερικό του. Εκτελέστηκαν επιτυχώς αρκετοί κύκλους βελτιστοποίησης του πλευρικού καθρέφτη και η πίεση σε ένα σημείο κοντά στο αυτί του οδηγού μειώθηκε κατά 14% . Ακόμα, διερευνήθηκε η επίδραση που έχει η βελτιστοποίηση κάθε επιμέρους βήματος της αεροακουστικής αλυσίδας στην ελαχιστοποίηση του εσωτερικού θορύβου και διαπιστώθηκε πως η μείωση μιας αντικειμενικής συνάρτησης εκφρασμένης σε ένα βήμα της αλυσίδας δεν συνεπάγεται αντίστοιχη μείωση στα βήματα που ακολουθούν

Η συνεχής συζυγής μέθοδος για την αεροακουστική βελτιστοποίηση μορφής στην αυτοκινητοβιομηχανία

The present doctoral thesis deals with the mathematical formulation, programming and verification of the continuous adjoint method to the constituent parts of an existing noise prediction chain for automotive aeroacoustics. The proposed method is applied to optimize a generic vehicle, the SAE body, in order to reduce wind noise in its interior. When a car travels at high speeds, flow-induced noise is generated in the region near the side mirror and is radiated towards all directions, reaching also the side window. Its vibrational response to this acoustic load generates in turn sound waves that propagate into the cabin and are perceived by the passengers as noise. An existing aeroacoustic framework simulating these physical mechanisms consists of an Improved Delayed Detached Eddy Simulation (IDDES) of the NavierStokes equations to obtain the unsteady pressure distribution on the mirror, the Kirchhoff Integral method to compute the radiated acoustic pressure on the side window, the bending wave equation on the side window to compute its deflection and, finally, the wave equation in the interior to obtain the interior sound field. The continuous adjoint method in this thesis is based on the aforementioned framework which is split in two domains; the exterior domain that includes the flow-induced noise generation and radiation to the window and the interior domain that includes the vibroacoustic model for window vibration and interior wave propagation. These systems are firstly examined separately and, then, coupled and the contintuous adjoint chain for vehicle aeroacoustic optimization is proposed. Regarding the vibroacoustic model, the bending wave equation is solved on the car’s side window, using as a source term the pressure load obtained by the exterior domain. The resulting window acceleration is then used as a boundary condition for the wave equation that is solved in the cabin to compute the interior sound field. The Sound Pressure Level at a point near the driver’s ear is then defined as the objective function and the adjoint wave and bending wave equations are derived. These must be solved backwards in time and in the following order: the adjoint wave equation is solved first by considering a monopole source term at the location where the objective function is defined. The propagation of the adjoint interior pressure in the cabin is computed and used thereupon as a source term for the adjoint bending wave equation, solved at the window. The resulting adjoint deflection is then used in the expression of the sensitivity derivative term on the window. This term is used later to couple the interior and exterior domains and additional emphasis is laid upon its discretization. A hand-differentiated expression is proposed to ensure its accuracy. The developed method is verified against Finite Differences and, then, is applied to the cabin of the SAE body to minimize interior noise using synthetic pressure waves as a load on the window. The continuous adjoint method for the flow-induced sound radiation with the Kirchhoff Integral is proposed where the differentiated Kirchhoff Integral is used to compute the boundary condition of the adjoint velocity on the noise radiating (Kirchhoff) surface and, then, the unsteady adjoint Navier-Stokes equations are solved backwards in time. It should be noted that the time window where the simulation is performed and the one over which the objective function is evaluated do not coincide. This is reflected on the adjoint boundary conditions along the body and the time integration of the sensitivity derivatives. Furthermore, to ensure the consistency of the continuous adjoint-based gradients, grid sensitivities are taken into account which gives rise to the adjoint grid displacement equations along with an additional term in the sensitivity derivatives expression. The proposed method is verified against Finite Differences on a 3D turbulent flow around a cylinder and, then, applied to the SAE body. Firstly, a sensitivity map analysis is conducted to investigate the influence the sensitivity derivative integration time window has on its computation but, also, to prove the importance of including the adjoint grid displacement equations. Finally, an optimization of the side mirror is performed, targeting at minimizing the radiated flow-induced sound at the vehicle’s side window. After the formulation and verification of the continuous adjoint method for the systems of equations in the interior and exterior domains, their coupling is presented. Through the solution of the adjoint aeroacoustic chain, the sensitivity of the interior acoustic pressure with respect to a normal displacement of the mirror is computed to indicate the way the mirror shape should change, in order to improve the aeroacoustic performance of the vehicle. The method is applied to compute the adjoint aeroacoustic sensitivity map on the side mirror of the generic SAE vehicle and successfully perform several optimization cycles. In addition, the impact that optimizing for each individual step of the noise prediction chain has on interior noise is investigated. Finally, two approaches are proposed, in order to perform the aforementioned adjoint analysis, focusing however on a specific frequency range; the first approach uses an objective function which includes the Fourier Transform and is integrated over frequencies whereas the second one uses a signal processing filter that preserves only the necessary frequency components. The adjoint formulation, advantages and drawbacks of each approach are discussed. The adjoint aeroacoustic chain including the filtering process is finally used to compute sensitivity maps on the mirror for the frequency range 800Hz-4000Hz and also for each 1/3 Octave Band in this range

The continuous adjoint method for periodic flows. Application in the optimal flow control of vortex shedding around a cylinder

σ.Σκοπός της παρούσας διπλωματικής εργασίας είναι η εφαρμογή της συνεχούς συζυγούς μεθόδου (continuous adjoint method) για τον ενεργητικό έλεγχο (active flow control) περιοδικής, στρωτής ροής ασυμπίεστου ρευστού γύρω από κύλινδρο, με στόχο την ελαχιστοποίηση των δυνάμεων που ασκούνται σε αυτόν, εξαιτίας της εμφάνισης των στροβίλων von Karman. Οι συζυγείς μέθοδοι χρησιμοποιούνται για την εύρεση της κλίσης της αντικειμενικής συνάρτησης, όπου η τελευταία εκφράζει τη χρονικά μέση τιμή του τετραγώνου των δυνάμεων, ως προς τις μεταβλητές σχεδιασμού. Ο ενεργητικός έλεγχος της ροής επιτυγχάνεται με δέσμες έγχυσης ή αναρρόφησης ρευστού (jets), οι οποίες τοποθε- τούνται σε ισαπέχουσες θέσεις σε όλη την περιφέρεια του κυλίνδρου. Οι παράμετροι της ταχύτητας κάθε δέσμης, δηλαδή το πλάτος ταλάντωσης, η φάση και η συχνότητα, αποτελούν και τις μεταβλητές σχεδιασμού του προβλήματος βελτιστοποίησης. Οι συνεχείς συζυγείς εξισώσεις, οι οριακές συνθήκες και οι παράγωγοι ευαισθησίας προκύπτουν από την παραγώγιση της συνάρτησης κόστους επαυξημένης με το χωρικό, σε όλο το πεδίο, και το χρονικό, σε μια περίοδο του φαινομένου, ολοκλήρωμα του γινομένου των εξισώσεων κατάστασης (Navier-Stokes) και των συζυγών μεταβλητών. Οι συζυγείς εξισώσεις διακριτοποιούνται και επιλύονται με τις αντίστοιχες οριακές συνθήκες και έτσι προκύπτει το πεδίο τω συζυγών μεταβλητών. Έπειτα υπολογίζεται η παράγωγος ευαισθησίας για κάθε μεταβλητή σχεδιασμού, με βάση την οποία αυτή ανανεώνεται, σύμφωνα με τη μέθοδο της απότομης καθόδου. Στην εργασία αυτή, χρησιμοποιούνται δύο συναρτήσεις κόστους, που αφορούν στη μέση τιμή, σε εύρος μιας περιόδου του φαινομένου, του τετραγώνου κάθε δύναμης, της άνωσης ή της οπισθέλκουσας κατά περίπτωση. Για κάθε συνάρτηση κόστους μελετώνται δύο περιπτώσεις συνδιασμών μεταβλητών σχεδιασμού, όπου στην πρώτη μεταβάλλονται τα πλάτη ταλάντωσης και στη δεύτερη τα πλάτη ταλάντωσης και οι φάσεις των δεσμών ρευστού. Με το πέρας των κύκλων βελτιστοποίησης εξάγονται συμπεράσματα για την επίδραση που έχει το πλάτος ταλάντωσης και η φάση κάθε δέσμης στο βέλτιστο έλεγχο της ροής. Κατά τη διατύπωση της συνεχούς συζυγούς μεθόδου εμφανίζονται όροι οι οποίοι εξαρτώνται της χρονικής στιγμής αφετηρίας υπολογισμού του χρονικού ολοκληρώματος της αντικειμενικής συνάρτησης, κάτι που οφείεται στο ότι με την επίδραση των δεσμών ρευστού η περίοδος του φαινομένου ταυτίζεται με την περίοδο των jets που αποτελεί μεταβλητή σχεδιασμού. Έτσι,λοιπόν, εξετάζεται η εξάρτηση που έχει η παράγωγος ευαισθησίας ως προς τη συχνότητα, μέσω της εφαρμογής της συνεχούς συζυγούς μεθόδου σε μονοδιάστατο πρόβλημα υποθετικής περιοδικής ροής με αναλυτική λύση, όπου η περίοδός της αποτελεί και αυτή μεταβλητή σχεδιασμού. Aπό αυτήν την ανάλυση προκύπτουν συμπεράσματα για τη συμπεριφορά της παραγώγου ευαισθησίας ως προς τη συχνότητα για περιπτώσεις περιοδικών ροών, που η περίοδος αποτελεί μεταβλητή σχεδιασμού.This diploma thesis aims at the adaptation and use of continuous adjoint methods for the active flow control of a periodic, laminar and incompressible flow around a cylinder. The purpose of flow control is the minimization of the forces that act on the cylinder, caused by the induced vortices, also known as, the von Karman vortices. The adjoint method is used to compute the derivatives of the objective function which, in this case, is the mean square of the forces acting on the cylinder, with respect to the design variables. Active flow control is implemented via pulsating jets, which are equidistributed across the whole surface of the cylinder. The jet velocity parameters, which are the amplitude, phase and frequency, constitute the design variables. The continuous adjoint equations, the corresponding boundary conditions and the sensitivity derivatives are derived with the use of the objective function, augmented with the field and time integral of the product of the state equations (Navier-Stokes) with the time dependent adjoint variables, during a period of the phenomenon. The adjoint equations are discretized and solved together with the corresponding boundary conditions to determine the field of the adjoint variables and, through them, the sensitivity derivatives of the problem. Then, the value of each design variable is updated with the use of its sensitivity derivative, accordingly to the steepest decent method. The objective functions used in this diploma thesis are the mean square of lift or drag acting on the cylinder, during a period of the phenomenon. For each force and, thus, objective function, two cases are studied. In the first one, only the amplitude of each jet is free to change while, in the second case, both amplitude and phase of each jet are considered as design variables. After the completion of the optimization cycles, conclusions can be drawn about the effect that amplitude and phase of each jet have in the optimal flow control. When the jets act on the cylinder, the period of the phenomenon becomes equal to the period of the jets. In the continuous adjoint formulation, this leads to the derivation of terms that depend on the limits of the time integral of the objective function. This dependency is studied by virtue of the adaptation of the continuous adjoint method to a hypothetical, one-dimensional periodic flow problem, where the period is also a design variable. Useful conclusions are drawn about the behaviour of the sensitivity derivative with respect to the period.Χρήστος Σ. Καπέλλο

A Continuous Adjoint Approach for Vehicle Interior Noise Reduction

<p>In this paper the continuous adjoint method is developed for a vibroacoustic model that predicts the interior noise of a vehicle induced by the airflow. The model simulates the front side window vibration, excited by the acoustic and hydrodynamic pressure load, and the<br> resulting sound wave propagation into the cabin. Targeting interior noise reduction, the continuous adjoint formulation is used to derive the adjoint to the state equations, namely the bending wave and the wave equations, whilst taking into consideration their coupling. The developed method is applied to a generic vehicle test body for the minimisation of the interior noise.</p

Using Parametric Effectiveness for Efficient CAD-Based Adjoint Optimization

Parametric effectiveness is a measure of the ability of the parameters defining a CAD model to be used for optimization. It compares the optimum change in performance that can be achieved using a CAD model’s parameterization, to the maximum performance improvement that could be obtained if the model is free to move. In this work, the approach is applied to the shape optimization of an S-Bend duct for minimizing the power-loss and an automotive car mirror for minimizing the noise perceived by the driver of the car