Model Order Reduction for Aerodynamic Lifting Surfaces Gonçalo da Cunha Laboreiro Mendonça Thesis to obtain the Master of Science Degree in Aerospace Engineering Supervisors: Prof. Fernando José Parracho Lau Dr. Frederico José Prata Rente Reis Afonso Examination Committee Chairperson: Prof. Filipe Szolnoky Ramos Pinto Cunha Supervisor: Prof. Fernando José Parracho Lau Member of the Committee: Prof. Afzal Suleman November 2017 ii Dedicated to my family and friends, who were always there for me. iii iv Acknowledgments I would like to thank dearly my supervisors Prof. Lau and Dr. Afonso who gave me constant support throughout this thesis up until the very end. Their insights on aerodynamics and CFD models guided me in my research of model order reduction methods and allowed me to better understand the models with which I had to work. Their demands for rigor and quality also pushed me to better my work, and in the end write a better thesis. v vi Resumo Nesta tese o tema de reduc¸ao˜ de modelos e a sua aplicac¸ao˜ a Mecanicaˆ de Fluidos Computacional sao˜ abordados. E´ mostrada a necessidade da industria´ aeroespacial, seja nacional ou Europeia, de modelos mais rapidos´ mas fieis´ a` realidade. Isto e´ devido ao elevado tempo de calculo´ associado aos modelos de alta-fidelidade. Estes mostram-se pouco viaveis´ para aplicac¸oes˜ do tipo Optimizac¸ao˜ Multi- disciplinar, como a plataforma de optimizac¸ao˜ NOVEMOR. Tendo por objectivo testar e aplicar reduc¸ao˜ de modelos a modelos CFD de superf´ıcies sustentadoras, uma pesquisa bibliografica´ abrangendo a reduc¸ao˜ de modelos nao-lineares,˜ dinamicosˆ e ou estaticos´ foi feita. Esta demonstrou a predominanciaˆ dos metodos´ de Projecc¸ao˜ de Galerkin e Reduc¸ao˜ por M´ınimos Quadrados, que funcionam atraves´ da Decomposic¸ao˜ Propria´ Ortogonal de soluc¸oes˜ obtidas do modelo a reduzir. Tecnicas´ complementares de amostragem e reduc¸ao˜ h´ıbrida tambem´ sao˜ apresentadas e discutidas. Os metodos´ de Projecc¸ao˜ e M´ınimos Quadrados sao˜ testados em modelos-referenciaˆ presentes na literatura, e o seu desem- penho em tempo e erro suplementar e´ analisado. Modelos dinamicos,ˆ estaticos,´ nao-lineares˜ e mul- tiparametricos´ foram reduzidos, com as versoes˜ mais simples dos metodos´ referidos a apresentar um desempenho superior. Estes metodos´ foram depois aplicados a problemas de fluidos uni-parametricos,´ nomeadamente a` cavidade a tampa movel´ com Navier-Stokes incompress´ıvel e Reynolds variavel,´ e ao perfil RAE-2822 com Euler compress´ıvel e anguloˆ de ataque variavel.´ No final da tese o desem- penho dos metodos´ de reduc¸ao˜ e´ analisado, demonstrando-se o tempo de calculo´ menor obtido e os problemas com modelos multi-parametricos´ ou com singularidades locais. Palavras-chave: Reduc¸ao˜ de Modelos,Projecc¸ao˜ de Galerkin, Reduc¸ao˜ por M´ınimos Quadra- dos, Decomposic¸ao˜ Propria´ Ortogonal , Mecanicaˆ de Flu´ıdos Computacional vii viii Abstract In this thesis the subject of model reduction and its application to Computational Fluid Dynamics is ex- plored. We show the need of the aerospace industry, at national or European level, of faster yet reliable models. This need is linked to the time cost of high-fidelity models, rendering them inefficient for ap- plications like Multi-Disciplinary Optimization, implemented in the NOVEMOR framework. With the goal of testing and applying model reduction to CFD models applicable to lifting surfaces, a bibliographical research covering reduction of non-linear, dynamic or static models was done. This established the prevalence of Projection and Least Mean Squares methods, which rely on solutions of the original high- fidelity model and their Proper Orthogonal Decomposition to work. Other complementary techniques such as adaptive sampling, greedy sampling and hybrid models are also presented and discussed. These Projection and Least Mean Squares methods are then tested on simple and documented bench- marks, to test the error and speed-up of the reduced order models thus generated. Dynamic, static, non-linear and multi-parametric problems were reduced, with the simplest version of these methods showing the most promise. These methods were later applied to single parameter problems, namely the lid-driven cavity with Incompressible Navier-Stokes and varying Reynolds, and the RAE-2822 airfoil at varying angles of atack for compressible Euler flow. An analysis of the performance of these methods is given at the end of this thesis, highlighting the computational speed-up obtained with these techniques, and the challenges presented by multi-parametric problems and problems showing point singularities in their domain. Keywords: Model Order Reduction, Galerkin Projection, Least Mean Squares Reduction, Proper Orthogonal Decomposition, Computational Fluid Dynamics ix x Contents Acknowledgments...........................................v Resumo................................................. vii Abstract................................................. ix List of Tables.............................................. xiii List of Figures............................................. xv Nomenclature.............................................. xix Glossary................................................ xxi 1 Introduction 1 1.1 Motivation.............................................1 1.2 Topic Overview..........................................2 1.3 Objectives.............................................4 1.4 Thesis Outline..........................................4 2 Model reduction methods7 2.1 Model Reduction methods....................................7 2.1.1 A posteriori methods...................................7 2.1.2 A priori methods - the PGD............................... 11 2.2 Basis creation........................................... 12 2.2.1 Proper Orthogonal Decomposition........................... 13 2.2.2 Compact POD...................................... 16 2.2.3 Error Estimation and control............................... 17 2.3 Sampling............................................. 18 2.3.1 Quad-Tree exploration with leave-one-out error cross validation........... 19 2.3.2 Kriging-based greedy search.............................. 20 2.3.3 Fast low-rank modifications of the SVD........................ 21 2.3.4 Argument error estimation................................ 21 2.4 Precision............................................. 22 2.4.1 Stability preservation................................... 22 2.4.2 Hybrid Methods..................................... 23 xi 3 Reduction of Benchmarks 25 3.1 Introduction............................................ 25 3.2 Advection - dynamical...................................... 26 3.2.1 Methods Applied..................................... 27 3.2.2 Results and Discussion................................. 27 3.3 Oscillatory............................................. 29 3.3.1 Methods Applied..................................... 31 3.3.2 Results and Discussion................................. 31 3.4 Advection - static......................................... 34 3.4.1 Methods Applied..................................... 35 3.4.2 Results and Discussion................................. 35 3.5 Diffusion - static......................................... 40 3.5.1 Methods Applied..................................... 41 3.5.2 Results and Discussion................................. 41 3.6 Conclusion............................................ 43 4 CFD Analysis and Reduction 45 4.1 Introduction............................................ 45 4.2 Choice of Solver......................................... 45 4.3 Incompressible Navier Stokes.................................. 48 4.3.1 Verification........................................ 49 4.3.2 Validation......................................... 52 4.3.3 Reduction......................................... 57 4.4 Compressible Euler....................................... 65 4.4.1 Verification........................................ 66 4.4.2 Validation......................................... 66 4.4.3 Reduction......................................... 70 5 Conclusions 77 5.1 Achievements........................................... 78 5.2 Future Work............................................ 78 Bibliography 79 xii List of Tables 3.1 Relative errors for ROM of order 5 for different POD methods and different ∆S for the Dynamic Advection problem................................... 29 3.2 Effect of the order of the integration method on the error for ROM of order 5, ∆S = 5 for the Dynamic Advection problem. We give the difference relative to the case of rectangle integration, in other words, we present ei−e0 , where i is the order of the integration method. 29 e0 3.3 Wall Clock times for the HDM and ROM for the Dynamic Advection problem. The time does not depend on the method for basis generation, being the same for whatever snap- shot set or POD used. Values taken are approximate averages................ 29 3.4 Relative errors for the different snapshot sets for a ROM of order 6 for the Oscillatory Problem.............................................. 32 3.5 Wall clock time for ROM and HDM solves (left) and SVD computation time for different snapshot sets (right) for Oscillatory problem.......................... 33 3.6 Relative errors for a ROM of order 6 for the Static Advection Problem. The errors for the System Induced POD (not shown), are all superior to 10.000 %............... 36 3.7 Wall clock times for the HDM, SVD computation and the several ROMs for the Static Advection problem.