Flow simulation and shape optimization for aircraft design

AbstractWithin the framework of the German aerospace research program, the CFD project MEGADESIGN was initiated. The main goal of the project is the development of efficient numerical methods for shape design and optimization. In order to meet the requirements of industrial implementations a co-operative effort has been set up which involves the German aircraft industry, the DLR, several universities and some small enterprises specialized in numerical optimization. This paper outlines the planned activities within MEGADESIGN, the status at the beginning of the project and it presents some early results achieved in the project

Simultaneous single-step one-shot optimization with unsteady PDEs

The single-step one-shot method has proven to be very efficient for PDE-constrained optimization where the partial differential equation (PDE) is solved by an iterative fixed point solver. In this approach, the simulation and optimization tasks are performed simultaneously in a single iteration. If the PDE is unsteady, finding an appropriate fixed point iteration is non-trivial. In this paper, we provide a framework that makes the single-step one-shot method applicable for unsteady PDEs that are solved by classical time-marching schemes. The one-shot method is applied to an optimal control problem with unsteady incompressible Navier-Stokes equations that are solved by an industry standard simulation code. With the Van-der-Pol oscillator as a generic model problem, the modified simulation scheme is further improved using adaptive time scales. Finally, numerical results for the advection-diffusion equation are presented. Keywords: Simultaneous optimization; One-shot method; PDE-constrained optimization; Unsteady PDE; Adaptive time scal

Towards Reduced-order Model Accelerated Optimization for Aerodynamic Design

The adoption of mathematically formal simulation-based optimization approaches within aerodynamic design depends upon a delicate balance of affordability and accessibility. Techniques are needed to accelerate the simulation-based optimization process, but they must remain approachable enough for the implementation time to not eliminate the cost savings or act as a barrier to adoption. This dissertation introduces a reduced-order model technique for accelerating fixed-point iterative solvers (e.g. such as those employed to solve primal equations, sensitivity equations, design equations, and their combination). The reduced-order model-based acceleration technique collects snapshots of early iteration (pre-convergent) solutions and residuals and then uses them to project to significantly more accurate solutions, i.e. smaller residual. The technique can be combined with other convergence schemes like multigrid and adaptive timestepping. The technique is generalizable and in this work is demonstrated to accelerate steady and unsteady flow solutions; continuous and discrete adjoint sensitivity solutions; and one-shot design optimization solutions. This final application, reduced-order model accelerated one-shot optimization approach, in particular represents a step towards more efficient aerodynamic design optimization. Through this series of applications, different basis vectors were considered and best practices for snapshot collection procedures were outlined. The major outcome of this dissertation is the development and demonstration of this reduced-order model acceleration technique. This work includes the first application of the reduced-order model-based acceleration method to an explicit one-shot iterative optimization process

Solving Optimal Control Problem of Monodomain Model Using Hybrid Conjugate Gradient Methods

We present the numerical solutions for the PDE-constrained optimization problem arising in cardiac electrophysiology, that is, the optimal control problem of monodomain model. The optimal control problem of monodomain model is a nonlinear optimization problem that is constrained by the monodomain model. The monodomain model consists of a parabolic partial differential equation coupled to a system of nonlinear ordinary differential equations, which has been widely used for simulating cardiac electrical activity. Our control objective is to dampen the excitation wavefront using optimal applied extracellular current. Two hybrid conjugate gradient methods are employed for computing the optimal applied extracellular current, namely, the Hestenes-Stiefel-Dai-Yuan (HS-DY) method and the Liu-Storey-Conjugate-Descent (LS-CD) method. Our experiment results show that the excitation wavefronts are successfully dampened out when these methods are used. Our experiment results also show that the hybrid conjugate gradient methods are superior to the classical conjugate gradient methods when Armijo line search is used

Combining Parameterizations, Sobolev Methods and Shape Hessian Approximations for Aerodynamic Design Optimization

Aerodynamic design optimization, considered in this thesis, is a large and complex area spanning different disciplines from mathematics to engineering. To perform optimizations on industrially relevant test cases, various algorithms and techniques have been proposed throughout the literature, including the Sobolev smoothing of gradients. This thesis combines the Sobolev methodology for PDE constrained flow problems with the parameterization of the computational grid and interprets the resulting matrix as an approximation of the reduced shape Hessian. Traditionally, Sobolev gradient methods help prevent a loss of regularity and reduce high-frequency noise in the derivative calculation. Such a reinterpretation of the gradient in a different Hilbert space can be seen as a shape Hessian approximation. In the past, such approaches have been formulated in a non-parametric setting, while industrially relevant applications usually have a parameterized setting. In this thesis, the presence of a design parameterization for the shape description is explicitly considered. This research aims to demonstrate how a combination of Sobolev methods and parameterization can be done successfully, using a novel mathematical result based on the generalized Faà di Bruno formula. Such a formulation can yield benefits even if a smooth parameterization is already used. The results obtained allow for the formulation of an efficient and flexible optimization strategy, which can incorporate the Sobolev smoothing procedure for test cases where a parameterization describes the shape, e.g., a CAD model, and where additional constraints on the geometry and the flow are to be considered. Furthermore, the algorithm is also extended to One Shot optimization methods. One Shot algorithms are a tool for simultaneous analysis and design when dealing with inexact flow and adjoint solutions in a PDE constrained optimization. The proposed parameterized Sobolev smoothing approach is especially beneficial in such a setting to ensure a fast and robust convergence towards an optimal design. Key features of the implementation of the algorithms developed herein are pointed out, including the construction of the Laplace-Beltrami operator via finite elements and an efficient evaluation of the parameterization Jacobian using algorithmic differentiation. The newly derived algorithms are applied to relevant test cases featuring drag minimization problems, particularly for three-dimensional flows with turbulent RANS equations. These problems include additional constraints on the flow, e.g., constant lift, and the geometry, e.g., minimal thickness. The Sobolev smoothing combined with the parameterization is applied in classical and One Shot optimization settings and is compared to other traditional optimization algorithms. The numerical results show a performance improvement in runtime for the new combined algorithm over a classical Quasi-Newton scheme

PDE-constrained Models with Neural Network Terms: Optimization and Global Convergence

Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, and the neural network parameters are calibrated to available data. Calibration of the embedded neural network can be performed by optimizing over the PDE. Motivated by these applications, we rigorously study the optimization of a class of linear elliptic PDEs with neural network terms. The neural network parameters in the PDE are optimized using gradient descent, where the gradient is evaluated using an adjoint PDE. As the number of parameters become large, the PDE and adjoint PDE converge to a non-local PDE system. Using this limit PDE system, we are able to prove convergence of the neural network-PDE to a global minimum during the optimization. The limit PDE system contains a non-local linear operator whose eigenvalues are positive but become arbitrarily small. The lack of a spectral gap for the eigenvalues poses the main challenge for the global convergence proof. Careful analysis of the spectral decomposition of the coupled PDE and adjoint PDE system is required. Finally, we use this adjoint method to train a neural network model for an application in fluid mechanics, in which the neural network functions as a closure model for the Reynolds-averaged Navier-Stokes (RANS) equations. The RANS neural network model is trained on several datasets for turbulent channel flow and is evaluated out-of-sample at different Reynolds numbers

Online adjoint methods for optimization of PDEs

We present and mathematically analyze an online adjoint algorithm for the optimization of partial differential equations (PDEs). Traditional adjoint algorithms would typically solve a new adjoint PDE at each optimization iteration, which can be computationally costly. In contrast, an online adjoint algorithm updates the design variables in continuous-time and thus constantly makes progress towards minimizing the objective function. The online adjoint algorithm we consider is similar in spirit to the the pseudo-time-stepping, one-shot method which has been previously proposed. Motivated by the application of such methods to engineering problems, we mathematically study the convergence of the online adjoint algorithm. The online adjoint algorithm relies upon a time-relaxed adjoint PDE which provides an estimate of the direction of steepest descent. The algorithm updates this estimate continuously in time, and it asymptotically converges to the exact direction of steepest descent as →∞. We rigorously prove that the online adjoint algorithm converges to a critical point of the objective function for optimizing the PDE. Under appropriate technical conditions, we also prove a convergence rate for the algorithm. A crucial step in the convergence proof is a multi-scale analysis of the coupled system for the forward PDE, adjoint PDE, and the gradient descent ODE for the design variables.First author draf

Assessment of the One-Shot Strategy for the Calibration ofMarine Ecosystem Models

Parameter optimization is an important and highly topical task in all kinds of cli- mate models or models that simulate parts of the climate system. These models are to provide reliable projections on the future climate. Computed model variables are fit to measurements or to output from other models by correcting model parameters. This is a mathematical minimization problem with constraints in the form of nonlinear ordinary or partial differential equations. Since analytical solutions often cannot be provided, usually an iterative method is applied to solve the model equations. Clas- sical optimization strategies perform this iterative method (or fixed point solver) for each function evaluation. The fixed point solver may need extensive computing time lasting hours or even days. Depending on the optimization approach, each optimization run requires numerous function evaluations and if necessary derivative information to adjust parameters. In this work, for the first time the application of the One-shot optimization strategy according to Hamdi and Griewank is investigated for the calibration of two illustrative marine ecosystem models. The One-shot method corrects parameters in each step of the iterative process applied for solving the model equations. The fixed point iteration is augmented by an update of the adjoint state and the correction of the parameters. It aims at computing a feasible state and optimal parameters with only bounded re- tardation compared to the underlying fixed point iteration with fixed parameters. This work examines the applicability of the One-shot strategy in the calibration of ma- rine ecosystem models with respect to the theoretical assumptions of the convergence theory, its implementation, the quality of computed results and the efficiency of the algorithm. The application to a model with unsteady (here annually periodic) PDEs is of great value. Instructions for the implementation, simplifications and suitable ad- justments are presented. The numerical results identify the costs, the advantages and the disadvantages of the One-shot optimization technique in the calibration of marine ecosystem models.Das Optimieren von Modellparametern ist eine a ̈ußerst wichtige und hochaktuelle Auf- gabe in der Entwicklung von Klimamodellen oder Modellen, die Teile des Klimasys- tems simulieren. Diese Modelle sollen verla ̈ssliche Aussagen u ̈ber das zuku ̈nftige Klima liefern. Dafu ̈r werden berechnete Modellvariablen an vorhandene Messdaten oder an Ausgaben anderer Modelle ausgerichtet, indem Modellparameter geeignet korrigiert werden. Dies entspricht einem mathematischen Minimierungsproblem mit Nebenbe- dingungen in Form von partiellen oder gewo ̈hnlichen Differentialgleichungen. Ha ̈ufig liegt der Lo ̈sung der Modellgleichungen ein iterativer Prozess zugrunde, da analytische Lo ̈sungen oft nicht gegeben sind. Konventionelle Optimierungsalgorithmen fu ̈hren fu ̈r jede Funktionsauswertung diesen iterativen Prozess (oder Fixpunktlo ̈ser) durch, der allein schon erheblichen Rechenaufwand und Rechenzeit von Stunden bis zu Tagen beno ̈tigen kann. Im gesamten Optimierungsprozess werden je nach Methode zahlreiche Funktionsauswertungen und gegebenenfalls Ableitungsinformationen beno ̈tigt, um Pa- rameter geeignet zu adjustieren. In dieser Arbeit wird die One-shot Optimierungsmethode nach Hamdi und Griewank erstmalsfu ̈rdieKalibrierungzweierillustrativermarinerO ̈kosystemmodelleuntersucht. Die One-shot Methode korrigiert Parameter bereits in jedem Schritt des Fixpunktlo ̈sers, der fu ̈r die Berechnung der Lo ̈sung der Modellgleichungen benutzt wird. Dabei wird der Fixpunktlo ̈ser durch eine Aufdatierung des adjungierten Zustandes und der Pa- rameterkorrektur erweitert und soll schließlich wa ̈hrend einer Fixpunktiteration (mit nur wenig Verzo ̈gerung) eine zula ̈ssige Lo ̈sung der Modellgleichungen als auch optimale Parameter liefern. In der vorliegenden Arbeit wird gepru ̈ft, ob die One-shot Methode fu ̈r Modelle des mari- nenO ̈kosystemsanwendbaristinBezugauftheoretischeVoraussetzungenderKonver- genztheorie, Implementierbarkeit und ihrer Gu ̈te bezu ̈glich der berechneten Lo ̈sungen und der Effizienz des Algorithmus’. Großen Wert hat die Beschreibung der Anwen- dung auf ein Modell mit instationa ̈rer (hier ja ̈hrlich periodischer) Lo ̈sung. Es wird eine Anleitung zur Implementierung erstellt und mo ̈gliche Vereinfachungen und geeignete Einstellmo ̈glichkeiten empfohlen. Anhand der vorgestellten Ergebnisse ko ̈nnen der Aufwand, die Vorteile und die Nachteile der One-shot Methode in Zusammenhang mit marinen O ̈kosystemmodellen klar auf- gezeigt werden