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Least-Squares Finite Element Formulation for Fluid-Structure Interaction

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Least-Squares Finite Element Formulation for Fluid-Structure Interaction Air Force Institute of Technology AFIT Scholar Theses and Dissertations Student Graduate Works 3-12-2009 Least-Squares Finite Element Formulation for Fluid-Structure Interaction Cody C. Rasmussen Follow this and additional works at: https://scholar.afit.edu/etd Part of the Aerodynamics and Fluid Mechanics Commons Recommended Citation Rasmussen, Cody C., "Least-Squares Finite Element Formulation for Fluid-Structure Interaction" (2009). Theses and Dissertations. 2386. https://scholar.afit.edu/etd/2386 This Dissertation is brought to you for free and open access by the Student Graduate Works at AFIT Scholar. It has been accepted for inclusion in Theses and Dissertations by an authorized administrator of AFIT Scholar. For more information, please contact [email protected]. LEAST-SQUARES FINITE ELEMENT FORMULATION FOR FLUID- STRUCTURE INTERACTION DISSERTATION Cody C Rasmussen, Major, USAF AFIT/DS/ENY/09-M16 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wright-Patterson Air Force Base, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED The views expressed in this thesis are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the United States Government. AFIT/DS/ENY/09-M16 LEAST-SQUARES FINITE ELEMENT FORMULATION FOR FLUID- STRUCTURE INTERACTION DISSERTATION Presented to the Faculty Department of Aeronautics and Astronautics Graduate School of Engineering and Management Air Force Institute of Technology Air University Air Education and Training Command In Partial Fulfillment of the Requirements for the Degree of Philosophy Cody C Rasmussen Major, USAF March 2009 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED AFIT/DS/ENY/09-M16 Abstract Fluid-structure interaction problems prove di¢ cult due to the coupling be- tween ‡uid and solid behavior. Typically, di¤erent theoretical formulations and numerical methods are used to solve ‡uid and structural problems separately. The least-squares …nite element method is capable of accurately solving both ‡uid and structural problems. This capability allows for a simultaneously coupled ‡uid struc- ture interaction formulation using a single variational approach to solve complex and nonlinear aeroelasticity problems. The least-squares …nite element method was com- pared to commonly used methods for both structures and ‡uids individually. The ‡uid analysis was compared to …nite volume methods and the structural analysis type compared to traditional Weak Galerkin …nite element methods. The simultaneous solution method was then applied to aeroelasticity problems with a known solution. Achieving these results required unique iterative methods to balance each domain’s or di¤erential equation’sweighting factor within the simultaneous solution scheme. The scheme required more computational time but it did provide the …rst hands-o¤ method capable of solving complex ‡uid-structure interaction problems using a si- multaneous least-squares formulation. A sequential scheme was also examined for coupled problems. iv Table of Contents Page Abstract.................................... iii Abstract.................................... iv ListofFigures ................................ xi ListofTables................................. xviii ListofSymbols................................ xx Acknowledgements . xxv I. Introduction ............................ 1-1 1.1 Motivation . 1-1 1.2 The Least-Squares Finite Element Method . 1-3 1.3 Simultaneously Coupled Fluid-Structure Interaction . 1-5 1.4 Research Goals . 1-6 1.5 Research Contributions . 1-6 1.6 Overview of Remaining Chapters . 1-7 II. Background............................. 2-1 2.1 Least-Squares Finite Elements . 2-1 2.2 Simultaneously Coupled Fluid-Structure Interaction . 2-3 2.3 Simultaneously Coupled Fluid-Structure Interaction Us- ing Least Squares Finite Elements . 2-6 2.4 Fluid Analysis . 2-6 2.5 Mesh Deformation . 2-7 2.6 Nonconformal Meshes . 2-8 v Page 2.7 Arbitrary Lagrangian-Eulerian (ALE) Schemes . 2-9 2.8 Mixed Finite Elements . 2-10 2.9 High Altitude Long Endurance Aircraft . 2-10 2.10 Aircraft Gust Loads and Response . 2-11 III. Finite Element Methods . 3-1 3.1 Basic Theory and Methodology of the Least Squares Fi- nite Element Method . 3-1 3.2 The p-Method and LSFEM . 3-11 3.2.1 Shape Functions . 3-13 3.2.2 Bubble Mode Condensation . 3-15 3.2.3 Isoparametric Curved Edge Elements . 3-16 3.2.4 Numerical Integration . 3-18 3.2.5 Boundary Condition Considerations When Us- ing the p-Method ................ 3-20 3.2.6 Angled Boundary Condition Considerations for Higher Order Elements . 3-21 3.3 Nonlinear Solution Methods . 3-24 3.4 Transient Discretization of LSFEM . 3-26 3.5 Residual Weighting . 3-28 IV. The Structural Domain . 4-1 4.1 Linear Euler Bernoulli Beam . 4-1 4.2 Nonlinear Euler-Bernoulli Beam . 4-2 4.3 2D Elasticity . 4-3 4.4 Weak Galerkin Finite Element Method . 4-6 4.4.1 WGFEM Two-Dimensional Elasticity . 4-6 4.4.2 WGFEM Euler-Bernoulli Beam . 4-9 4.4.3 Transient Discretization of WGFEM . 4-10 vi Page 4.5 Example Elasticity Problems . 4-12 4.5.1 Serendipity vs. Full-Tensor Product Jacobi Shape Function..................... 4-12 4.5.2 Elliptic vs. Non-Elliptic Comparison . 4-14 4.5.3 Mixed Least-Squares FEM vs. Primal Weak Galerkin FEM Comparison . 4-20 4.6 Elasticity-Based Mesh Deformation . 4-23 4.7 Transient Formulation Study . 4-24 4.7.1 Transient Structural Problems . 4-30 V. TheFluidDomain ......................... 5-1 5.1 Stokes Linear Flow . 5-1 5.2 Navier-Stokes Incompressible Viscous Flow . 5-2 5.3 Compressible Euler . 5-5 5.3.1 Equation Development . 5-5 5.3.2 One-Dimensional Veri…cation . 5-10 5.3.3 Two-Dimensional Veri…cation . 5-11 5.3.4 Euler Time Stepping . 5-17 5.3.5 No Penetration Boundary Conditions and Their Respective Residual Weights . 5-19 5.3.6 Airfoil Problem Results . 5-20 5.4 Arbitrary Lagrangian-Eulerian Formulation . 5-23 5.4.1 Euler ALE . 5-23 5.4.2 Navier Stokes ALE . 5-24 5.4.3 Veri…cation of ALE . 5-25 VI. Fluid-Structure Interaction Theory and Methodology . 6-1 6.1 Simultaneous Coupling of Multiple Fields . 6-1 6.2 Simultaneous vs. Sequential Coupling . 6-3 vii Page 6.3 Mesh Movement Schemes . 6-6 6.3.1 Mesh Deformation Prediction . 6-7 6.4 Nonconformal Mesh Interaction . 6-7 6.4.1 Nonconformal Mesh Interaction Theory . 6-7 6.4.2 Examination of Nonconformal Mesh Interaction 6-8 VII. Steady-State Fluid-Structure Interaction . 7-1 7.1 Two-Field Simultaneously Coupled Problem –Driven Cav- ity Flow with Flexible Wall . 7-1 7.2 Three-Field Simultaneously Coupled Problem – Driven Cavity Flow with Flexible Wall . 7-2 7.3 Double Channel Flow Problem with Flexible Beam . 7-6 7.4 Residual Weighting Case Study . 7-9 7.5 Fluid-Structure Interaction Problem Created by Method of Manufactured Solutions (MMS) . 7-12 7.6 Collapsible Tube Problem . 7-24 7.7 Domain Weighting Sensitivity to h- and p-values . 7-29 7.8 Domain Weighting Sensitivity to Material Properties . 7-33 7.9 Comparison of LSFEM-LSFEM to LSFEM-WGFEM FSI Solutions ......................... 7-33 7.10 Examination of Nonconformal Mesh in FSI Solutions . 7-36 VIII. Transient Fluid-Structure Interaction . 8-1 8.1 Example Transient FSI Problem Using MMS . 8-1 8.1.1 Problem Properties . 8-1 8.1.2 Single Time Step Results (T=0.01 sec) . 8-2 8.1.3 Dependence on Error With Respect to Time Step Size ....................... 8-4 8.1.4 Multiple Time Step Results (T=0.25 sec) . 8-5 viii Page 8.1.5 Examination of Time Step Size Re…nement for Sequential Solution Accuracy . 8-8 8.1.6 Full Period Results (T=1.00 sec)....... 8-8 8.1.7 Dependence on Balanced Domain Residual Weights With Respect to Time Step Size . 8-11 8.2 Example Transient Collapsible Tube FSI Problem . 8-13 IX. Conclusions............................. 9-1 9.1 Research Goals . 9-1 9.2 Research Contributions . 9-1 9.3 Summary of Code Veri…cations Performed . 9-2 9.4 Summary of Comparisons Performed . 9-6 9.5 Key Abilities Demonstrated That Are Traceable to Tran- sient Gust Scenario . 9-10 9.6 Issues With LSFEM . 9-10 9.7 Future Work . 9-12 Appendix A. Code Structure . A-1 Appendix B. LSFEM Di¤erential Operators . B-1 B.1 Introduction . B-1 B.2 One-Dimensional Equations . B-1 B.2.1 Boundary Condition Application . B-1 B.2.2 Wave Equation Elliptic Form . B-2 B.2.3 Wave Equation Non-Elliptic Form . B-3 B.2.4 Axial Bar . B-3 B.2.5 Linear Euler-Bernoulli Beam . B-4 B.2.6 Nonlinear Euler-Bernoulli Beam . B-6 B.2.7 Compressible Inviscid Euler Flow . B-9 ix Page B.2.8 Pressure Equilibrium On a Beam With Top and Bottom Fluid Flow . B-10 B.2.9 Neumann-Type Stress Equilibrium Relationships B-10 B.2.10 Angled Velocity or Displacement Transformation B-12 B.2.11 Angled Stress Transformation . B-13 B.2.12 Stress To Displacement-Gradient Relationship B-14 B.2.13 Nonconformal Relationships . B-14 B.3 Two-Dimensional Equations . B-16 B.3.1 Poisson’sEquation . B-16 B.3.2 In-Plane Elasticity . B-16 B.3.3 Stokes Fluid Flow . B-22 B.3.4 Incompressible Navier-Stokes Fluid Flow . B-23 B.3.5 Compressible Inviscid Euler Fluid Flow . B-29 Appendix C. Synthetic Elasticity Problem Exact Solution . C-1 Appendix D. Method of Manufactured Solutions Applied to Fluid Struc- ture Interaction Problems . D-1 Bibliography ................................. BIB-1 Vita ...................................... VITA-1 x List of Figures Figure Page 1.1. Sample Joined-Wing Con…guration . 1-1 2.1. Sample Discrete and Nonlinear Transient Gust Response for Various Sample Frequencies .
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