A novel reduced basis method and application to inverse analyses

Acknowledgements Acknowledgements I would like to express my deepest gratitude and most sincere appreciation to my thesis supervisor, Professor Liu Gui-Rong for his invaluable guidance, dedicated support and great patience, and genuine advises during my doctoral studies His continuous encouragement, passion and enthusiasm have strongly influenced me for my research as well as my future life I also would like to thank my co-supervisor, Dr Wang Yu Yong, for his helps and guidance in my research work I extend many thank to the National University of Singapore (NUS) for the financial support My special thanks go to Dr Huynh Dinh Bao Phuong for his fruitful discussions and helps throughout my research work I am very grateful to my fellow colleagues and friends in Center for ACES, Dr Dai Keyang, Dr Bernard Kee Buck Tong, Mr Song Chengxiang, Dr Li Zirui, Dr Zhang Guiyong, Dr Deng Bin, Dr Zhang Jian, Dr Chen Yuan, Mr Trung, and Mr George Xu I have enjoyed their invaluable discussion, suggestions, encouragement and support during my four years study in NUS I am proud of being a part of ACES research team I am also grateful to Daw Khin Khin Htar and Thura Win, for their friendship, comfort and support Finally, I wish to express my deepest love and gratitude to my parents U Kyaw Thein and Daw Myint Myint San, my parents-in-law, my sisters, my relatives, my uncles, my wife Mi Mi Oo and my daughter Khin Hnin Wai for their understanding, love, support and encouragement i Table of contents Table of contents Acknowledgements i Table of contents ii Summary vi Nomenclature viii List of Figures x List of Tables Chapter xiv Introduction 1.1 Review of numerical methods 1.1.1 Finite element method 1.1.2 Meshfree methods 1.2 Review of real-time computation technique 1.2.1 Reduced-basis method (RBM) 1.3 Inverse analysis 1.3.1 Review of genetic algorithms (GAs) 10 1.3.2 Review of neural network (NN) 12 1.4 Objectives 13 1.5 Organization of the thesis 15 Chapter Smoothed Galerkin projection 2.1 Basic equations of linear elasticity 18 19 2.1.1 Equilibrium equation 19 2.1.2 Constitutive relationship 20 2.1.3 Boundary conditions 21 2.1.4 Weak formulation for linear elasticity 22 2.1.5 Geometric affine mapping 23 2.1.6 Parametric affine formulation using a single reference domain 24 ii Table of contents 2.2 Basic equations of heat conduction problem 27 2.2.1 Governing equation 27 2.2.2 Weak formulation for heat conduction problems 28 2.2.3 Parametric weak formulation 29 2.3 Brief on the finite element method 2.3.1 Discrete equations 2.4 Smoothed Galerkin projection 2.4.1 30 31 34 Smoothed bilinear form 35 2.5 LC-PIM: Smoothed Galerkin projection (SGP) 37 2.5.1 Discrete equations of LC-PIM 37 2.5.2 Properties of the LC-PIM 39 2.6 Model problems 2.6.1 43 44 2.6.1.1 2.6.2 Example-1: Cantilever beam problem 44 Numerical results Example-2: Thermal fin problem 46 2.6.2.1 47 Numerical results 2.7 Remarks Chapter A novel reduced-basis method 3.1 Preliminaries 48 64 65 3.1.1 General problem statement 65 3.1.2 Numerical approximation 66 3.2 Reduced-basis method via Galerkin projection 66 3.2.1 Fundamental observation of the RBM 66 3.2.2 Galerkin projection reduced-basis method 67 3.2.3 Computational procedure of the GP_RBM 68 3.2.4 Properties of the Galerkin projection reduced-basis method 70 3.3 A novel reduced-basis method 71 3.3.1 Computational procedure of the SGP_RBM 72 3.3.2 Properties of the SGP_RBM 74 3.3.3 Output bound and the bound gap 77 3.4 Sample set construction 78 3.4.1 Asymptotic error estimation 78 3.4.2 Greedy adaptive procedure 79 iii Table of contents 3.5 Numerical examples and results 82 3.5.1 Example-1: Cantilever beam problem 82 3.5.2 Example-2: Thermal fin problem 85 3.6 Remarks Chapter 88 Rapid inverse parameter estimation using reduced-basis approximation with asymptotic error estimation 103 4.1 Introduction 103 4.2 Forward problem 105 4.2.1 Problem definition 105 4.2.2 Classical numerical approximation 107 4.2.3 Reduced-basis approximation 108 4.2.3.1 108 4.2.4 Sample set construction Numerical results for forward problem 4.3 Inverse procedure 109 110 4.3.1 Simulated measurements 111 4.3.2 Brief on GA inverse procedure 112 4.3.3 Sensitivity analysis 112 4.3.4 Numerical results for inverse analysis 114 4.4 Remarks Chapter 115 Rapid identification of elastic modulus of interfacial tissue on dental implants surfaces using reduced-basis method and a neural network 136 5.1 Introduction 136 5.2 Forward problem 138 5.2.1 Briefing on basic equations 138 5.2.2 Dental implant-bone problem 140 5.2.3 Experimental setting and numerical modeling 142 5.2.4 Finite element approximation 143 5.2.5 Reduced-basis method (RBM) 144 5.2.6 Numerical results 145 5.3 Inverse problem 146 5.3.1 Briefing on neural network (NN) 147 5.3.2 Numerical analysis 148 iv Table of contents 5.3.2.1 Comparison of RBM outputs and experimental measurements 148 5.3.2.2 Inputs of NN model 148 5.3.2.3 Training of NN model 149 5.3.2.4 Simulated and experimental measurements 149 5.3.2.5 Normalization 150 5.3.2.6 Application of NN and results 151 5.4 Remarks Chapter Conclusions and future works 152 165 6.1 Summary 165 6.2 Recommendations of future works 168 References 171 Publications arising from the thesis 186 v Summary Summary In practice, quantities of interest including stresses or strains, temperature or heat flux, and solution outputs in the form of energy norm describe behaviors of engineering systems These outputs of interest are often used in inverse parameter identifications including non-destructive evaluations, non-evasive evaluations, and material characterizations Thus, it is very important to develop efficient computational techniques for both forward and inverse analyses in modern engineering and sciences The main purposes of this thesis are twofold In the first part, a real-time computation method called a smoothed Galerkin projection reduced-basis method (SGP_RBM) has been developed based on the standard reduced-basis method and a smoothed Galerkin projection An upper bound to the exact solution in the form of energy norm is generated by the newly developed SGP_RBM, while a lower bound (in energy norm) to the exact solution is obtainable from the standard reduced-basis method The properties of the SGP_RBM have been studied theoretically and numerically in details A linear elasticity solid mechanic problem and a heat conduction problem are conducted using the developed method Both theoretical studies and numerical results show good features of the proposed reduced-basis method including upper and lower bounds to the exact solution, computational efficiency, accuracy and very fast convergent rate Therefore, numerical solutions which lie within our proposed solution bounds can be certified as a reliable solution although the exact solution is practically impossible to obtain for general science and engineering problems The proposed reduced-basis method is, thus, invaluable to gauge reliability of numerical solutions in the form of energy norm vi Summary In the second part, emphasis is given to inverse problems Two different inverse parameter identification procedures have been developed In the first inverse approach, a rapid and reliable inverse searching approach called a RBM-GA (combination of reduced-basis method and the Genetic Algorithm) approach for non-destructive evaluations has been developed to solve inverse problems of parameter estimation for structural systems In the RBM-GA approach, a reduced-basis model is developed and used for fast computation of solving forward problems, and a genetic algorithm is then used in the inverse searching procedure for parameter estimation An inverse problem is performed to estimate the crack location, length and orientation insider the cantilever beam In the second inverse searching approach, a rapid inverse analysis approach for non-evasive evaluations has also been established based on the reduced-basis method (RBM) and a neural network (NN) to identify the “unknown” elastic modulus of interfacial tissue between a dental implant and surrounding bones An RBM model is first built to compute displacement responses of dental implant-bone structures subjected to a harmonic loading for a set of assumed elastic moduli The RBM model is then used to train a NN model that is used for inverse identifications Actual experimental measurements of displacement responses are then fed into the trained NN model to inversely determine the “true” elastic modulus of interfacial tissue An example of a 3D dental implant-bone structure is built and inverse analysis is conducted to verify the present RBM-NN approach Based on numerical results of the RBM-GA and the RBM-NN, it is confirmed that the numerical results are very accurate and reliable Significant computational saving has been demonstrated The proposed RBM-GA and RBM-NN approaches are thus found very robust and efficient for inverse problems vii Nomenclature Nomenclature a (,) Standard bilinear form aD ( , ) Smoothed bilinear form D A parameter domain f Linear functional F FN FEM force vector GP_RBM force vector F LC-PIM force vector FN SGP_RBM force vector GD Bound gap between reduced-basis upper and lower bound K KN FEM stiffness matrix GP_RBM stiffness matrix KN SGP_RBM stiffness matrix K LC-PIM stiffness matrix Linear functional M Low dimensional manifold ℵ Dimension of a FEM mesh PN , PM reduce-basis sample sets se Exact output s FEM output s N , sM GP_RBM outputs s N , sM SGP_RBM outputs ue Exact solution in terms of field variables u FEM solution in terms of field variables u LC-PIM solution in terms of field variables u N , uM GP_RBM solutions in terms of field variables u N , uM SGP_RBM solutions in terms of field variables viii Nomenclature u uN FEM displacement vector GP_RBM displacement vector u LC-PIM displacement vector uN SGP_RBM displacement vector WN , WM Reduced-basis spaces Ω μ Problem domain κ Thermal conductivity Θq ( μ ) Parametric coefficient functions ϕi a nodal basis (or shape) function constructed based on elements Ωn smoothing domains/cells Γ Boundary of problem domain A parameter point in the parameter set ∇ the Laplacian operator Δ D Asymptotic error s N ,M Δ s ,exact N Exact error Δ s , M , avg N Averaged asymptotic error Δ s ,exact , avg N Averaged exact error α CPU-time saving factor Note: Temporal variables are not listed ix List of Figures List of Figures Fig 2.1 Division of problem domain Ω into smoothing domains Ωn Fig 2.2 Illustration of background triangular mesh and the smoothing domain created by sequentially connecting the centroids with the mid-edge-points of the surrounding triangles of a node Fig 2.3 A cantilever beam with a crack Fig 2.4 Subdomain divisions of a cantilever beam: (a) Reference domain and (b) Original domain Fig 2.5 Triangular mesh of finite element on the reference domain with the crack in the middle Fig 2.6 Deflection along the boundary ΓB for reference mesh ( ℵref ) using the FEM Fig 2.7 Deflection along the boundary ΓB for coarse mesh using the LC-PIM Fig 2.8 Deflection along the boundary ΓB for fine mesh using the LC-PIM Fig 2.9 Comparison of deflection along the boundary ΓB : Reference FEM solution vs the LC-PIM solutions Fig 2.10 A thermal fin with Ω = ∪5 Ωi i= Fig 2.11 Two dimensional illustration of the thermal fin Fig 2.12 Triangular mesh ( ℵ = 4,760 ) of finite element on the 2-D thermal fin problem Fig 2.13 Temperature distribution graph for the FEM reference solution Fig 2.14 Temperature distribution graph for coarse mesh using the LC-PIM Fig 2.15 Temperature distribution graph for fine mesh using the LC-PIM Fig 3.1 (a) Low dimensional manifold spanned by the field variables (b) The approximate solution for a parameter μ new by a linear ( ) combination of predetermined solutions u μi , i = 1, ,N x References Deng B, Han X, Liu GR, Tan KBC (2004) Prediction of elastic properties of the maxillary bone Proceeding 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(1989) The Finite Element Method McGraw-Hill: New York, Vol.1 Zienkiewicz OC, Taylor RL (1992) The Finite Element Method McGraw-Hill: New York, Vol.2 185 Publications arising from the thesis Publications arising from the thesis Journal Paper [1] Liu GR, Khin Zaw, Wang YY (2008) Rapid inverse parameter estimation using reduced-basis approximation with asymptotic error estimation Comput Methods Appl Mech Engrg., doi:10.1016/j.cma.2008.03.012 [2] Liu GR, Khin Zaw, Wang YY, Deng B (2008) A novel reduced-basis method with upper and lower bounds for real-time computation of solids mechanics problem, Comput Methods Appl Mech Engrg (Accepted) [3] Khin Zaw, Liu GR, Deng B, Tan KBC (2008) Rapid identification of elastic modulus of the interface tissue on dental implants surfaces using reduced-basis method and neural network (submitted) [4] Deng B, Tan KBC, Liu GR, Khin Zaw (2008) Inverse identification of elastic constant of dental implant-bone interfacial tissue using neural network and FEA model Inverse Problems in Science and Engineering (revised) Conference Paper [1] Liu GR and Khin Zaw, A real-time computation technique: A reduced-basis method with asymptotic error estimation, International Conference on Computational Methods, Hiroshima, Japan, April 2007 [2] Liu GR, Khin Zaw, Han X, A novel reduced-basis method for certified real-time solution to solid mechanics with exact upper and lower bounds, International Conference on Computational & Experimental Engineering and Sciences, Honolulu, Hawaii, USA, March, 2008 186 Publications arising from the thesis [3] Liu GR, G Y Zhang, Khin Zaw, LC-PIM Method for Certified Solution with Bounds, Adaptive Analysis and Real-time computation, APCOM’07 in conjunction with EPMESC XI, December 3-6, 2007, Kyoto, JAPAN 187 ... procedures have been developed In the first inverse approach, a rapid and reliable inverse searching approach called a RBM-GA (combination of reduced- basis method and the Genetic Algorithm) approach... structural damages can also be found (Wu et al., 1992; Zhoa et al., 1998; Zgnoc and Achenbach, 1996; Luo and Hanagud, 1997) Additionally, Liu and coworkers (Han and Liu, 2003; Han et al., 200 3a; 2003b;... orientation insider the cantilever beam In the second inverse searching approach, a rapid inverse analysis approach for non-evasive evaluations has also been established based on the reduced- basis