MODEL REDUCTION OF NONLINEAR MECHANICAL SYSTEMS VIA OPTIMAL PROJECTION AND TENSOR APPROXIMATION A DISSERTATION SUBMITTED TO THE DEPARTMENT OF AERONAUTICS & ASTRONAUTICS AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY Kevin Thomas Carlberg August 2011 © 2011 by Kevin Thomas Carlberg. All Rights Reserved. Re-distributed by Stanford University under license with the author. This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/ This dissertation is online at: http://purl.stanford.edu/td542hm2304 ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Charbel Farhat, Primary Adviser I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Juan Alonso I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy. Michael Saunders Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives. iii Abstract Despite the advent and maturation of high-performance computing, high-fidelity physics-based nu- merical simulations remain computationally intensive in many fields. As a result, such simulations areoftenimpracticalformanytime-criticalapplicationssuchasfast-turnarounddesign,‘inthefield’ analysis, control, and uncertainty quantification. The objective of this thesis is to enable rapid, ac- curate analysis of high-fidelity nonlinear models, thereby allowing them to be used in time-critical settings. Modelreductionpresentsapromisingapproachforrealizingthisgoal. Thisclassofmethodsgen- erates low-dimensional models that preserves key features of the high-fidelity model; such methods have been shown to generate fast, accurate solutions (often with error bounds and stability guaran- tees) when applied to specialized problems such as linear time-invariant systems. However, model reductionforhighlynonlinearsystemshasproventobesignificantlymoredifficult. Infact,nonlinear model reduction techniques have been limited primarily to methods based on the heuristic proper orthogonal decomposition (POD)–Galerkin approach. Unfortunately, these methods often generate inaccurate responses because 1) POD–Galerkin does not minimize any measure of the system error for general problems, and 2) the POD basis is not constructed to minimize output errors (i.e. it is notgoal-oriented). Furthermore,simulationtimesforthesemodelsusuallyremainlarge,asreducing the dimension of a nonlinear system does not necessarily reduce its computational complexity. Thisthesispresentstwomodelreductiontechniquesthataddressestheaccuracyandcomplexity shortcomings of the POD–Galerkin method. The first method is a ‘compact POD’ approach for computingthetrialbasis; thisapproachisapplicabletoparameterizedstaticsystems. Thecompact POD basis is constructed using a goal-oriented framework that allows sensitivity derivatives to be employed as snapshots. Numerical experiments on a parameterized aircraft wing illustrate the method’s ability to generate more accurate solutions than the typical POD basis. The second method is a Gauss–Newton with approximated tensors (GNAT) method applicable to nonlinear systems. Similar to other POD-based approaches, the GNAT method first executes high-fidelity simulations during a costly ‘offline’ stage; it computes a POD subspace that opti- mally represents the state as observed during these simulations. To compute fast, accurate ‘online’ solutions, the method introduces two approximations that satisfy optimality and consistency con- ditions. First, the method decreases the system dimension by searching for the solutions in the iv low-dimensional POD subspace. As opposed to performing a Galerkin projection, the method han- dles the resulting overdetermined system of equations arising at each time step by formulating a least-squares problem and solving it by the Gauss–Newton method; this ensures that a measure of the system error (i.e. the residual) is minimized. Second, the method decreases the model’s compu- tational complexity by approximating the residual and Jacobian using the ‘gappy POD’ technique; this requires computing only a few rows of the approximated quantities. For computational mechanics problems, the GNAT method leads to the concept of a sample mesh: the subset of the mesh needed to compute the selected rows of the residual and Jacobian. Because the reduced-order model uses only the sample mesh for computations, the online stage requires minimal computational resources. Results obtained for a compressible, turbulent fluid flow problem and a nonlinear structural dynamics problem highlight the ability of the method to reduce thecomputationalcostassociatedwithhigh-fidelitynonlinearmodelswhileretainingtheiraccuracy. v Acknowledgments First,Iamindebtedtomythesisadviser,Prof.CharbelFarhat. Ihavelearnedanincredibleamount from him, ranging from technical knowledge in computational mechanics to effective research- proposal writing. I am thankful for his mentorship throughout my graduate-student career, as heprovidedinvaluableguidancebeginningwithmyselectionofaresearchtopic(whichIhavegrown to really enjoy) and ending with my job search. IamalsogratefultothemembersofmyreadingcommitteeProf.MichaelSaundersandProf.Juan Alonso; their feedback has improved this thesis considerably. I also appreciate the financial support providedbytheNationalDefenseScienceandEngineeringGraduateFellowshipandNationalScience Foundation Graduate Research Fellowship. I have been fortunate to be in a research lab with many talented individuals. I am thankful for members of the “Cool Kids’ Corner” for not only being excellent research collaborators, but also for living up to the corner’s name: Julien Cortial, David Amsallem, Cl´ement Saint-Jalm, and honorary member Adam “The Weasel” Larat. I am grateful for the chance to work with model- reduction collaborators Charbel Bou-Mosleh, Matt Zahr, Kyle Washabaugh, and Wade Spurlock. I amalsothankfulfortheinsightfuldiscussionsprovidedbyotherlabmembers,includingPhilAvery, Irina Kalashnikova, S´ebastien Brogniez, Kevin Wang, and Meir Messingher Lang. I thank Robert Stephan for the opportunity to collaborate with the bright minds at the Archaeology Center. I am also indebted to Will Law and Godwin Zhang for their tireless technical support; they were instrumental in enabling me to obtain many of the numerical results presented in this thesis. IamalsogratefulformyrelationshipswithProf.DanielStrickland, BernieDaigle, IvanRoberto Canales Pineda II, Olivia Hatton, Jacob Mattingley, Pete Sommer, Todd and Alissa Murphy, Ken Franko, Alla Agafonov, Jason Campbell, Steve Imm, the “Breakfast Club” (Sandy Philipose, Lea Castro,EstherHong,DavidKlaus),androommates(CraigGoergen,AndrewSmith,JasonLewallen, Joe Johnson). They have made my years in graduate school some of the best in my life. Finally, I would like to thank my family. To my wonderful fianc´ee Carolyn: this past year would have been impossible without you. To my brother Brian and his fianc´ee Katelyn: I look forward to many years of great memories. I could not have asked for more love and support from my parents John and Kathy; thank you a million times over. This thesis is dedicated to all of you. vi Contents Abstract iv Acknowledgments vi Contents vii List of Tables x List of Figures xi 1 Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Strategy and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Thesis accomplishments and outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Time-critical analysis via surrogate modeling 4 2.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Objective: time-critical prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Surrogate models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.1 Data-fitting methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3.2 Lower-fidelity models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3.3 Model reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Model-reduction strategy 9 3.1 Static vs. dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Consistency and optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.3 Fully discrete computational framework . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Model reduction of static systems via compact POD 13 4.1 Previous work and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4.1.2 Compact POD method overview . . . . . . . . . . . . . . . . . . . . . . . . . 14 vii 4.2 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2.1 Dimension reduction via projection . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2.2 Goal-oriented strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2.3 Approximation error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.4 Optimal test subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.2.5 Consistent trial subspace and goal-oriented proper orthogonal decomposition 19 4.3 Compact POD basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.3.1 Taylor-expansion snapshots and weights . . . . . . . . . . . . . . . . . . . . . 23 4.3.2 Distance weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3.3 Inner-product choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.3.4 Computational efficiency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4.4 Application to time-critical problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5 Numerical examples: static system model reduction 31 5.1 Parameterized aeroelastic research wing . . . . . . . . . . . . . . . . . . . . . . . . . 31 5.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.3 Compact POD with Jacobian-induced inner product . . . . . . . . . . . . . . . . . . 33 5.4 Compact POD with output-induced inner product . . . . . . . . . . . . . . . . . . . 34 6 Model reduction of nonlinear dynamical systems 40 6.1 Previous work and objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 6.1.2 Computational bottleneck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.1.3 GNAT method overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2 Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.2.1 Dimension reduction via optimal projection . . . . . . . . . . . . . . . . . . . 44 6.2.2 Consistent trial subspace and proper orthogonal decomposition . . . . . . . . 44 6.2.3 Optimal test subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 6.2.4 Computational complexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 6.3 System approximation via tensor approximation . . . . . . . . . . . . . . . . . . . . 47 6.3.1 Approximation of the reduced state equations . . . . . . . . . . . . . . . . . . 48 6.3.2 Gappy POD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 6.3.3 Consistency-driven construction of Φ and Φ . . . . . . . . . . . . . . . . . 51 R J 6.3.4 Partial computation of the state . . . . . . . . . . . . . . . . . . . . . . . . . 53 6.3.5 Offline–online splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 7 Sample mesh 56 7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 7.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.3 Sample node selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 7.4 Output computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 viii 8 Numerical examples: dynamical system model reduction 62 8.1 Nonlinear structural dynamics: finite-element formulation . . . . . . . . . . . . . . . 62 8.1.1 Experiment 1: Illustrate sufficient conditions for consistency. . . . . . . . . . 64 8.1.2 Experiment 2: Compare performance with truncation . . . . . . . . . . . . . 64 8.2 Nonlinear fluid dynamics: finite-volume formulation . . . . . . . . . . . . . . . . . . 70 8.2.1 Ahmed body: coarse mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 8.2.2 Ahmed body: fine mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 8.2.3 Burgers equation: predictive study . . . . . . . . . . . . . . . . . . . . . . . . 92 9 Conclusions and future work 96 A System approximation by Broyden’s method 98 A.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 A.2 Broyden’s method for system approximation. . . . . . . . . . . . . . . . . . . . . . . 98 A.3 Modified Broyden’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 A.4 Numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 B POD computation methods 104 C Proofs 107 Bibliography 110 ix List of Tables 5.1 Parameterized ARW-2 wing: physical shape parameters p , i=1,...,33 . . . . . . . 31 i 5.2 Parameterized ARW-2 wing: performance of various static system model reduction methods. These results employ a Jacobian-induced inner product Θ=K(µ¯). . . . . 34 5.3 Parameterized ARW-2 wing: performance of various static system model reduction methods. These results employ an output-induced inner product Θ=HTH.. . . . . 35 6.1 GNAT snapshot collection procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.1 Truss structure: GNAT performance for various snapshot collection procedures. All experiments employ n =32, n =n =42, and η =1.0. . . . . . . . . . . . . . . . 66 x R J 8.2 Truss structure: effect of increasing the number of sample indices on GNAT perfor- mance. The number of residual and Jacobian basis vectors n = n = 42 are held R J fixed. All experiments employ n =32 . . . . . . . . . . . . . . . . . . . . . . . . . . 69 x 8.3 Trussstructure: effectofincreasingthesampleindexfactorη onGNATperformance. The number of sample indices n are held fixed. . . . . . . . . . . . . . . . . . . . . . 70 i 8.4 Ahmed body coarse mesh: simulation times for the full-order model using different numbers of cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.5 Ahmed body fine mesh: GNAT performance for 378 sample nodes using different snapshot procedures. Simulation times are reported using 4 cores. . . . . . . . . . . 86 8.6 Ahmed body fine mesh: sample mesh attributes for different numbers of sample nodes 89 8.7 Ahmed body fine mesh: GNAT performance with snapshot procedure 1 for different numbers of sample nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.8 Ahmed body fine mesh: simulation times for GNAT with snapshot procedure 1 and 378 sample nodes using different numbers of cores . . . . . . . . . . . . . . . . . . . 91 A.1 Ahmed body medium-sized mesh: effect of skipping period n on the performance of s ModelIIIbasedonBroyden’smethod. Allsimulationsemployn =400basisvectors x for the state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 x