Reduced basis approximation and inverse analyses for dental implant problems

REDUCED BASIS APPROXIMATION AND INVERSE ANALYSES FOR DENTAL IMPLANT PROBLEMS HOANG KHAC CHI (B. Eng., Hochiminh University of Technology) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN COMPUTATIONAL ENGINEERING (CE) SINGAPORE–MIT ALLIANCE NATIONAL UNIVERSITY OF SINGAPORE 2012 DECLARATION I hereby and declare that this thesis is my original work it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. l{oang Khac Chi 2L May 2AL2 Acknowledgments This thesis would not have been possible without the help and contributions of many people. I would like to thank my thesis advisor Professor Anthony T. Patera. It was him who has taught me the way to research: how to understand complex mathematic theories, how to evaluate results exactly, how to improve theories to have better results. . . I admire his wide and deep knowledge, and I am really grateful for his constant support and guidance, his patience, understanding, humour, and the trust he showed me. I would like to thank my co-supervisors, Professor Liu Gui-Rong of the University of Cincinnati and Professor Khoo Boo Cheong of NUS, for their helpful comments, suggestions, encouragement and kindness throughout my study. My sincere thanks go to Professor Jaime Peraire and Professor Lim Kian Meng for serving on my thesis committee, and for their careful criticism and comments regarding this thesis. I am also grateful to my senior members of the “reduced basis” group: Dr. Nguyen Ngoc Cuong, Dr. Huynh Dinh Bao Phuong and Dr. David Knezevic for many interesting and helpful discussions during the time I was at MIT. I also would like to thank the Singapore–MIT Alliance (SMA) for giving me the most wonderful chance to connect with the “MIT world”. My special thanks go to all administration staffs of the SMA Office and Debra Blanchard during my stay at MIT. I would like to thank all of my friends in Singapore, who have helped and supported me through the bad times and shared the good times. The list is so long that I could not list all of them here! My deep thanks go to Prof. Nguyen Thien Tong from the Ho Chi Minh University of Technology in Vietnam. It was him who introduced the SMA to me and supported me to apply to this program as a PhD student. I surely could not pursue the PhD way without his help. Finally, I would like to express my love and gratitude to my family: my parents, Me Nga and Bo Vinh, my sister Minh Trang and my girlfriend, ii Xuan Hien – for their love, trust and support. Without them, I would not be able to pursue my dreams. This work is dedicated to them. iii Contents Acknowledgments ii Thesis summary viii List of Tables x List of Figures xi Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 The Dental Implant Problem with One-layered Interfacial Tissue . . . . . . . . . . . . . . . . . . . . The Dental Implant Problem with Three-layered Interfacial Tissue . . . . . . . . . . . . . . . . . . . . Computational Challenges . . . . . . . . . . . . . . 1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 1.1.3 1.2.1 Review of Methods to Assess Implant Stability . . . 1.2.2 Review of Finite Element Models in Dental Implant Research . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.3 Review of the Reduced Basis Method . . . . . . . . 17 1.2.4 Review of Computational Approaches in Inverse Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Purpose of the Thesis . . . . . . . . . . . . . . . . . . . . . 21 1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . 21 Preliminaries 23 2.1 Function Spaces . . . . . . . . . . . . . . . . . . . . . . . . 23 2.1.1 Linear Spaces . . . . . . . . . . . . . . . . . . . . . 23 2.1.2 Norm . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.3 Inner Product . . . . . . . . . . . . . . . . . . . . . 24 2.1.4 Spaces of Continuous Functions . . . . . . . . . . . 24 2.1.5 Lebesgue Spaces . . . . . . . . . . . . . . . . . . . 25 2.1.6 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . 26 2.1.7 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . 28 2.1.8 Dual Hilbert Spaces 28 . . . . . . . . . . . . . . . . . iv 2.2 Linear Functionals and Bilinear Forms . . . . . . . . . . . 29 2.2.1 Linear Functionals . . . . . . . . . . . . . . . . . . 29 2.2.2 Bilinear Forms . . . . . . . . . . . . . . . . . . . . 29 2.2.3 Parametric Linear and Bilinear Forms . . . . . . . 30 2.2.4 Affine Parameter Dependence . . . . . . . . . . . . 31 2.3 Fundamental Inequalities . . . . . . . . . . . . . . . . . . . 32 2.3.1 Cauchy-Schwarz Inequality . . . . . . . . . . . . . . 32 2.3.2 H¨older Inequality . . . . . . . . . . . . . . . . . . . 32 2.3.3 Minkowski Inequality . . . . . . . . . . . . . . . . . 32 2.3.4 Friedrichs Inequality . . . . . . . . . . . . . . . . . 33 2.3.5 Poincar´e Inequality . . . . . . . . . . . . . . . . . . 33 Finite Element Method for Linear Elastodynamics 3.1 Review of Linear Elasticity in Time Domain . . . . . . . . 34 34 3.1.1 Strain-Displacement Relations . . . . . . . . . . . . 34 3.1.2 Constitutive Relations . . . . . . . . . . . . . . . . 35 3.1.3 Equation of Equilibrium/Motion . . . . . . . . . . 36 3.1.4 Initial and Boundary Conditions . . . . . . . . . . . 36 3.1.5 Weak Formulation . . . . . . . . . . . . . . . . . . 37 3.2 Finite Element Approximation . . . . . . . . . . . . . . . . 41 3.2.1 Weak Statement . . . . . . . . . . . . . . . . . . . 41 3.2.2 Time Discretization Scheme . . . . . . . . . . . . . 42 3.2.3 Space and Basis . . . . . . . . . . . . . . . . . . . . 43 3.2.4 Galerkin Projection . . . . . . . . . . . . . . . . . . 44 3.2.5 A Priori Convergence . . . . . . . . . . . . . . . . 46 3.2.6 Computational Complexity . . . . . . . . . . . . . . 46 3.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 50 3.3.1 Example – The Pure Normal Stress Problem . . . 51 3.3.2 Example – The Pure Shear Stress Problem . . . . 54 3.3.3 Remark . . . . . . . . . . . . . . . . . . . . . . . . 58 Reduced Basis Method for Linear Elastodynamics 4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . 59 60 4.1.1 Abstract Formulations . . . . . . . . . . . . . . . . 60 4.1.2 Impulse Response . . . . . . . . . . . . . . . . . . . 63 4.2 Reduced Basis Approximation . . . . . . . . . . . . . . . . 64 4.2.1 Dimension Reduction: Observation from Elliptic PDE 64 4.2.2 Approximation Formulation . . . . . . . . . . . . . 66 v 4.3 Sampling Procedure . . . . . . . . . . . . . . . . . . . . . 67 4.3.1 The Proper Orthogonal Decomposition (POD) Method 67 4.3.2 POD–Greedy Sampling Procedure . . . . . . . . . . 68 4.3.3 Offline-Online Computational Strategy . . . . . . . 70 A Posteriori Error Bound Estimation . . . . . . . . . . . . 72 4.4.1 Field Variable Error Bounds . . . . . . . . . . . . . 73 4.4.2 Output Error Bounds . . . . . . . . . . . . . . . . . 79 4.4.3 Offline-Online Computational Procedure . . . . . . 82 4.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 83 4.4 4.5.1 The Pure Normal Stress Problem . . . . . . . . . . 84 4.5.2 The Pure Shear Stress Problem . . . . . . . . . . . 86 4.5.3 Remark . . . . . . . . . . . . . . . . . . . . . . . . 88 Inverse Procedure 89 5.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . 89 5.1.1 Forward Problems . . . . . . . . . . . . . . . . . . 89 5.1.2 Inverse Problems . . . . . . . . . . . . . . . . . . . 91 5.2 Methods to Solve Inverse Problems . . . . . . . . . . . . . 92 5.2.1 The Gradient Descent Method . . . . . . . . . . . . 94 5.2.2 The Gauss–Newton Method . . . . . . . . . . . . . 95 5.2.3 The Levenberg–Marquardt Method . . . . . . . . . 97 5.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . 99 5.3.1 Equation of Motion with Damping Effects . . . . . 99 5.3.2 Equation of Motion without Damping Effects . . . 103 Dental Implant Problem with One-layered Interfacial Tissue 104 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6.2 Problem Description and Finite Element Approximation . 105 6.2.1 Models and Approximations . . . . . . . . . . . . . 105 6.2.2 Finite Element Approximation . . . . . . . . . . . . 109 6.3 Reduced Basis Approximation . . . . . . . . . . . . . . . . 113 6.3.1 Reduced Basis Method . . . . . . . . . . . . . . . . 114 6.3.2 POD–Greedy Sampling Procedure . . . . . . . . . . 115 6.3.3 Errors . . . . . . . . . . . . . . . . . . . . . . . . . 117 6.3.4 Offline-Online Computational Procedure . . . . . . 119 6.3.5 Numerical Results . . . . . . . . . . . . . . . . . . 121 6.4 Inverse Procedure . . . . . . . . . . . . . . . . . . . . . . . 127 6.4.1 The Levenberg–Marquardt–Fletcher Algorithm . . 127 vi 6.4.2 Numerical Results . . . . . . . . . . . . . . . . . . 129 6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 Dental Implant Problem with Three-layered Interfacial Tissue 135 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 135 7.2 Problem Description . . . . . . . . . . . . . . . . . . . . . 137 7.2.1 The In Vitro Model . . . . . . . . . . . . . . . . . . 137 7.2.2 The Simplified FEM Model . . . . . . . . . . . . . 137 7.3 Finite Element and Reduced Basis Approximation . . . . . 140 7.3.1 Finite Element and Reduced Basis Approximation . 140 7.3.2 Numerical Results . . . . . . . . . . . . . . . . . . 141 7.4 Inverse Procedure . . . . . . . . . . . . . . . . . . . . . . . 145 7.4.1 Sensitivity Analysis . . . . . . . . . . . . . . . . . . 145 7.4.2 The Levenberg–Marquardt–Fletcher Algorithm . . 150 7.4.3 Numerical Results . . . . . . . . . . . . . . . . . . 151 7.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Conclusions 154 8.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 8.2 Suggestions for Future Work . . . . . . . . . . . . . . . . . 156 Bibliography 158 A Calculation of the A Posteriori Error Bound (Chapter 4)170 B Calculation of the Dual Norm of the Residual (Chapter 6)175 vii Thesis summary Engineering and science nowadays require accurate, fast, reliable and efficient evaluations of several quantities of interest: displacement, stresses, strain, temperature or heat flux, etc. in engineering systems. In the field of dental implant research, the conditions of dental implant-bone interfacial tissues have received large interest from the research community. One of the popular ways to assess such conditions is the nondestructive evaluation, where one measures the displacement responses of a dental implant system when it is applied some stimulating forces in the time domain. In this work, we focus on the development of finite element approximations, reduced basis approximations and inverse techniques for material properties identification of implant-bone interfacial tissues in simulation dental implant systems. We first introduce our experimental work and characterize the main features of our in vitro model in order to approximate numerically that model. We then build the finite element approximation for the in vitro model taking into consideration that the model problem is governed by a second-order linear hyperbolic partial differential equation. We then establish reduced basis (RB) approximations using the Proper Orthogonal Decomposition (POD)–Greedy algorithm for “optimal” basis selection and the Galerkin projection for stable and fast convergence. This combination of RB–POD-Galerkin enables extremely fast and reliable computations of displacement responses for a large range of material properties parameters, leading to practically a real-time online model. In the inverse analysis, the reduced basis approximation for a dental implant-bone model is incorporated in the Levenberg–Marquardt–Fletcher (LMF) algorithm to enable rapid identification of unknown material properties. We integrate the RB– LMF computation strategy into two model problems of dental implants: the one-layered interfacial tissue problem and the three-layered interfacial tissue problem. We finally present numerical results and demonstrate that the RB–LMF strategy is extremely fast, efficient, reliable and robust viii against several levels of contaminated-noise. Keywords: reduced basis approximation, offline-online procedure, POD– Greeedy algorithm, dental implant, inverse analysis, Levenberg–Marquardt– Fletcher algorithm, second-order linear hyperbolic problem, finite element method, material characterization ix [43] M. A. Grepl and A. T. Patera. A posteriori error bounds for reducedbasis approximations of parametrized parabolic partial differential equations. ESAIM-Math. Model. Num., 39(1):157–181, 2005. [44] N. C. Nguyen, K. Veroy, and A. T. Patera. Certified real-time solution of parametrized partial differential equations. Handbook of Materials Modeling, pages 1529–1564, 2005. [45] M. A. Grepl, Y. Maday, N. C. Nguyen, and A. T. Patera. Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM-Math. Model Num., 41:575–605, May 2007. [46] G. Rozza and K. Veroy. On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Engrg., 196(7):1244–1260, 10 January 2007. [47] D. B. P. Huynh and A. T. Patera. Reduced basis approximation and a posteriori error estimation for stress intensity factors. Int. J. Numer. Meth. Eng., 72(10):1219–1259, December 2007. [48] G. Rozza, D. B. P. Huynh, and A. T. Patera. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations - application to transport and continuum mechanics. Arch. Comput. Method. Eng., 15(3):229– 275, 2008. [49] S. Sen. Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems. Numer. Heat Tr. B-Fund., 54(5):369–389, November 2008. [50] N. C. Nguyen, G. Rozza, and A. T. Patera. Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers’ equation. Calcolo, 46(3):157–185, 2009. [51] D. J. Knezevic, N. C. Nguyen, and A. T. Patera. Reduced basis approximation and a posteriori error estimation for the parametrized 163 unsteady Boussinesq equations. Accepted by the Math. Mod. Meth. Appl. Sci., August 2010. [52] L. Machiels, Y. Maday, I. B. Oliveira, A. T. Patera, and D. V. Rovas. Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris, Ser. 1, 331(2):153–158, 15 July 2000. [53] C. Prud’homme, D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. T. Patera, and G. Turinici. Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods. J. Fluid Eng., 124(1):70–80, March 2002. [54] A. Y. K. Tan. Reduced basis methods for 2nd order wave equation: Application to one dimensional seismic problem. MS thesis, Singapore-MIT Alliance, National University of Singapore, 2006. [55] B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM-Math. Model. Num., 42(2):277–302, 2008. [56] A. Buffa, Y. Maday, A. T. Patera, C. Prud’homme, and G. Turinici. A priori convergence of the greedy algorithm for the parametrized reduced basis. Submitted to the Math. Model. Numer. Anal., November 2009. [57] A. Quarteroni, G. Rozza, and A. Manzoni. Certified reduced basis approximation for parametrized partial differential equations and applications. Accepted by the Journal of Mathematics in Industry, June 2011. [58] D. B. P. Huynh. Reduced basis approximation and application to fracture problems. PhD thesis, Singapore-MIT Alliance, National University of Singapore, 2007. 164 [59] N. C. Nguyen. Reduced basis approximations and a posteriori error bounds for nonaffine and nonlinear partial differential equations: Application to inverse analysis. PhD thesis, Singapore-MIT Alliance, National University of Singapore, 2005. [60] Y. O. Solodukhov. Reduced-basis methods applied to locally nonaffine and locally non-linear partial differential equations. PhD thesis, Massachusetts Institute of Technology, June 2005. [61] G. R. Liu, K. Zaw, Y. Y. Wang, and B. Deng. A novel reduced-basis method with upper and lower bounds for real-time computation of linear elasticity problems. Comput. Method. Appl. M., 198(2):269– 279, December 2008. [62] K. Ito and S. S. Ravindran. A reduced-order method for simulation and control of fluid flows. J. Comput. Phys., 143(2):403–425, 1998. [63] G. R. Liu, J. H. Lee, A. T. Patera, Z. L. Yang, and K. Y. Lam. Inverse identification of thermal parameters using reduced-basis method. Comput. Method. Appl. M., 194(27–29):3090–3107, July 2005. [64] K. C. Hoang, B. C. Khoo, G. R. Liu, N. C. Nguyen, and A. T. Patera. Rapid identification of material properties of the interface tissue in dental implant systems using reduced basis method. Submitted to the Inverse Probl. Sci En., March 2011. [65] D. J. Knezevic and J. W. Peterson. A high-performance parallel implementation of the certified reduced basis method. Accepted by the Computer Methods in Applied Mechanics and Engineering, 2011. [66] D. B. P. Huynh, D. J. Knezevic, J. W. Peterson, and A. T. Patera. High-fidelity real-time simulation on deployed platforms. Comput. Fluids, 43(1):74–81, April 2011. [67] G. R. Liu, W. B. Ma, and X. Han. An inverse procedure for determination of material constants of composite laminates using elastic waves. Comput. Methods Appl. Mech. Engrg., 191:3543–3554, 2002. 165 [68] Y. G. Xu and G. R. Liu. Detection of flaws in composites from scattered elastic-wave field using an improved µGA and a local optimizer. Comput. Methods Appl. Mech. Engrg, 191:3929–3946, 2002. [69] G. R. Liu, X. Han, and K. Y. Lam. A combined genetic algorithm and nonlinear least squares method for material characterization using elastic waves. Comput. Methods Appl. Mech. Engrg., 191:1909–1921, 2002. [70] G. R. Liu and S. C. Chen. Flaw detection in sandwich plates based on time-harmonic response using genetic algorithm. Comput. Methods Appl. Mech. Engrg., 190:5505–5514, 2001. [71] G. R. Liu and X. Han. Computational Inverse Techniques in Nondestructive Evaluation. CRC Press LLC, Boca Raton, Florida 33431, 2003. [72] A. Oishi, K. Yamada, S. Yoshimura, G. Yagawa, S. Nagai, and Y. Matsuda. Neural network-based inverse analysis for defect identification with laser ultrasonic. Res. Nondestr. Eval., 13(2):79–96, 2001. ¨ [73] B. Sawaf, M. N. Ozisik, and Y. Jarny. An inverse analysis to estimate linearly temperature dependent thermal conductivity components and heat capacity of an orthotropic medium. Int. J. Heat Mass Tran., 38(16):3005–3010, 1995. [74] D. W. Tang and N. Araki. An inverse analysis to estimate relaxation parameters and thermal diffusivity with a universal heat conduction equation. Int. J. Thermophys., 21(2):553–561, 2000. [75] C. Yang and S. Gao. An inverse analysis to estimate thermal conductivity components of an orthotropic medium. Third International Conference on Natural Computation (ICNC 2007), 3:246–251, 2007. 166 [76] D. S. Schnur and N. Zabaras. An inverse method for determining elastic material properties and a material interface. Int. J. Numer. Meth. Eng., 33(10):2039–2057, 1990. [77] K. M. Zhao and J. K. Lee. Inverse estimation of material properties for sheet metals. Commun. Numer. Meth. En., 20(2):105–118, 2004. [78] O. Ghouati and J. C. Gelin. Identification of material parameters directly from metal forming processes. J. Mater. Process Tech., 80– 81:560–564, August 1998. [79] J. C. Gelin and O. Ghouati. An inverse method for determining viscoplastic properties of aluminium alloys. J. Mater. Process. Technol., 45:435–440, 1994. [80] J. N. Reddy. Applied Functional Analysis and Variational Methods in Engineering. McGraw-Hill, New York, 1986. [81] W. J. T. Daniel. The subcycled Newmark algorithm. Comput. Mech., 20(3):272–281, 1997. [82] M. A. Grepl. Reduced-basis approximation and a posteriori error estimation for parabolic partial differential equations. PhD thesis, Massachusetts Institute of Technology, June 2005. [83] J. N. Reddy. An Introduction to the Finite Element Method. McGraw-Hill, New York, third edition, 2006. [84] K. J. Bathe. Finite Element Procedures. Prentice-Hall, New Jersey, 1996. [85] S. R. Wu. A priori error estimates for explicit finite element for linear elasto-dynamics by Galerkin method and central difference method. Comput. Method Appl. M., 192(51–52):5329–5353, December 2003. [86] J. D. Achenbach. Wave Propagation in Elastic Solids. Elsevier Science, The Netherlands, 1975. 167 [87] N. C. Nguyen. A multiscale reduced-basis method for parametrized elliptic partial differential equations with multiple scales. J. Comput. Phys., 227(23):9807–9822, December 2008. [88] D. R. Jones, C. D. Perttunen, and B. E. Stuckman. Lipschitzian optimization without the Lipschitz constant. J. Optimiz. Theory App., 79(1):157–181, 1993. [89] G. Venter. Review of optimization techniques. Encyclopedia of Aerospace and Engineering, 2010. [90] M. Hanke, A. Neubauer, and O. Scherzer. A convergence analysis of the Landweber iteration for nonlinear ill-posed problems. Numer. Math., 72:21–37, 1995. [91] B. Blaschke, A. Neubauer, and O. Scherzer. On convergence rates for the iteratively regularized Gauss-Newton method. IMA J. Numer. Anal., 17:421–436, 1997. [92] R. Fletcher. A modified Marquardt subroutine for nonlinear least squares. Technical Report Tech. Rep. AERE-R–6799, Atomic Energy Research Establishment, Harwell (England), January 1971. [93] Q. Jin. On a regularized Levenberg–Marquardt method for solving nonlinear inverse problems. Numer. Math., 115:229–259, 2010. [94] Anthony T. Patera. RB website (augustine) http://augustine.mit.edu/methodology.htm. [95] D. B. P. Huynh, D. J. Knezevic, and A. T. Patera. A Laplace transform certified reduced basis method; application to the heat equation and wave equation. C. R. Math., 349(7–8):401–405, April 2011. [96] M. Balda. An algorithm for nonlinear least squares. http://dsp.vscht.cz/konference matlab/MATLAB07/prispevky/balda m/balda m.pdf, 2010. 168 [97] Y. G. Xu and G. R. Liu. Detection of flaws in composites from scattered elastic-wave field using an improved µGA and a local optimizer. Comput. Method. Appl. M., 191(36):3929–3946, August 2002. [98] J. E. Jackson. A User’s guide to Principal Components. John Wiley & Sons, Inc., Hoboken, NJ, USA, 1991. [99] C. Le Niliot and F. Lefevre. A parameter estimation approach to solve the inverse problem of point heat sources identification. Int. J. Heat Mass Tran., 47(4):827–841, February 2004. [100] K. J. Dowding and B. F. Blackwell. Joint experimen- tal/computational techniques to measure thermal properties of solids. Meas. Sci. Technol., 9:877–887, June 1998. [101] J. Simunek, O. Wendroth, and M. T. V. Genuchten. Parameter estimation analysis of the evaporation method for determining soil hydraulic properties. Soil Sci. Soc. Am. J., 62:894–905, 1998. [102] N. Sun, N. Z. Sun, M. Elimelech, and J. N. Ryan. Sensitivity analysis and parameter identifiability for colloid transport in geochemically heterogeneous porous media. Water Resour. Res., 37(2):209–222, February 2001. [103] E. P. Scott, P. S. Robinson, and T. E. Diller. Development of methodologies for the estimation of blood perfusion using a minimally invasive thermal probe. Meas. Sci. Technol., 9(6):888–897, June 1998. 169 Appendix A Calculation of the A Posteriori Error Bound (Chapter 4) In this section, we present in detail the calculation of the a posteriori error bounds for the field variable and output described in Section 4.4. We recall the error bounds for the field variable and output are derived respectively as follows ((4.47) and (4.67)) √ k ∆N (µ, t ) = ( ( ′ max E(µ, tk ) [1:k] αLB Y ′ E(µ, tk −1 ) + ( ∑ ′ ′ k−1 E(µ, tk +1 ) − E(µ, tk −1 ) + ∆t 2∆t k′ =1 ) Y )2 ) 21 , (A.1) Y and ( ∆sN (µ, t k+ 12 )≡ ∥ℓ1 ∥2Y ′ ∥ℓ2 ∥2X + αLB σLB ) 12 ∆N (µ, tk ), ∀µ ∈ D, ≤ k ≤ K−1. (A.2) In these equations, E(µ, tk ) Y is the dual norm of the residual that is defined as in (4.43) 170 R(·; µ, tk ) Y ′ ≡ sup v∈Y R(v; µ, tk ) , ∥v∥Y = E(µ, t ) k Y , (A.3) ≤ k ≤ K − 1, where E(µ, tk ) ∈ Y is given by the Riesz representation ( E(µ, tk ), v ) Y = R(v; µ, tk ), ∀v ∈ Y, ≤ k ≤ K − 1. (A.4) The residual mentioned here relates to the discrete RB equation (4.24) ( ) that is discretized by the Newmark’s scheme1 ψ = 21 , φ = 12 ( uN (µ, tk+1 ) − 2uN (µ, tk ) + uN (µ, tk−1 ) R(v; µ, t ) = geq (t )f (v)−m , v; µ ∆t2 ( ) uN (µ, tk+1 ) + uN (µ, tk−1 ) −a , v; µ , ∀v ∈ Y, ∀µ ∈ D, ≤ k ≤ K −1. k ) k (A.5) It is obvious that (A.2) is readily followed from (A.1), we thus need to compute all terms in (A.1) only. The right-hand side of (A.1) is divided into two parts: the first part which comprises the terms E(µ, tk ) Y , ≤ k ≤ E(µ, tk+1 ) − E(µ, tk−1 ) K, and the second part which comprises the terms ,1≤ 2∆t Y k ≤ K − 1. Let’s expand these second terms in more details We need to emphasize that all error bounds formulations and derivations presented in Chapter and Appendix A are valid only with this Newmark’s scheme. In Appendix B, we shall present the calculation of the dual ( norm of the) residual for the discrete RB equation using another Newmark’s scheme ψ = 14 , φ = 12 . 171 E(µ, tk+1 ) − E(µ, tk−1 ) 2∆t Y √( ) k+1 k−1 E(µ, t ) − E(µ, t ) E(µ, tk+1 ) − E(µ, tk−1 ) , = 2∆t 2∆t Y √ = ∥E(µ, tk+1 )∥2Y − (E(µ, tk+1 ), E(µ, tk−1 ))Y + ∥E(µ, tk−1 )∥2Y , 2∆t Sk = ≤ k ≤ K − 1. (A.6) In summary, in order to find the error bounds in (A.1) and (A.2), we need to compute two kinds of terms which are E(µ, tk ) ( ) and E(µ, tk+1 ), E(µ, tk−1 ) Y , ≤ k ≤ K − 1. Y , ≤ k ≤ K, Substituting (A.4) into (A.5) and noting the affine parameter dependence, we have ( E(µ, tk ), v ) Y = geq (tk )f (v) + Qm N ∑ ∑ Θqm (µ) n=1 q=1 + Qa N ∑ ∑ Θqa (µ) n=1 q=1 −uN n (µ, tk+1 ) + 2uN n (µ, tk ) − uN n (µ, tk−1 ) q m (ζn , v) ∆t2 −uN n (µ, tk+1 ) − uN n (µ, tk−1 ) q a (ζn , v), ∀v ∈ Y, ≤ k ≤ K. (A.7) It is clear from linear superposition that we can express E(µ, tk ) as E(µ, t ) = geq (t )F + k k Qm N ∑ ∑ Θqm (µ)λm,n (µ, tk )Mq,n n=1 q=1 + Qa N ∑ ∑ Θqa (µ)λa,n (µ, tk )Aq,n , ≤ k ≤ K − 1, n=1 q=1 (A.8) 172 where −uN n (µ, tk+1 ) + 2uN n (µ, tk ) − uN n (µ, tk−1 ) ∆t2 −uN n (µ, tk+1 ) − uN n (µ, tk−1 ) λa,n (µ, tk ) = ; λm,n (µ, tk ) = (A.9) and the parameter independent terms F, M and A are calculated as F ∈Y from (F, v)Y = f (v), ∀v ∈ Y, Mq,n ∈ Y from (Mq,n , v)Y = mq (ζn , v), ∀v ∈ Y for ≤ n ≤ N, ≤ q ≤ Qm Aq,n ∈ Y from (Aq,n , v)Y = aq (ζn , v), ∀v ∈ Y for ≤ n ≤ N, ≤ q ≤ Qa . (A.10) It is followed from (A.8) that E(µ, tk ) Y ( ) = E(µ, tk ), E(µ, tk ) Y = geq (tk )geq (tk )Λf f {Q } Qa N m ∑ ∑ ∑ +2 geq (tk ) Θqm (µ)λm,n (µ, tk )Λfqnm + Θqa (µ)λa,n (µ, tk )Λfqna n=1 + N ∑ n,n′ =1 { q=1 ∑ Qm q=1 ′ Θqm (µ)Θqm (µ)λm,n (µ, tk )λm,n′ (µ, tk )Λmm qnq ′ n′ q,q ′ =1 +2 Qm Qa ∑ ∑ ′ Θqm (µ)Θqa (µ)λm,n (µ, tk )λa,n′ (µ, tk )Λma qnq ′ n′ q=1 q ′ =1 + Qa ∑ } q′ Θqa (µ)Θa (µ)λa,n (µ, tk )λa,n′ (µ, tk )Λaa qnq ′ n′ , q,q ′ =1 ≤ k ≤ K, (A.11) where 173 Λf f = (F, F)Y , Λfqnm = (F, Mq,n )Y , ≤ n ≤ N, ≤ q ≤ Qm Λfqna = (F, Aq,n )Y , ≤ n ≤ N, ≤ q ≤ Qa , ′ ′ Λmm qnq ′ n′ = (Mq,n , Mq ′ ,n′ )Y , ≤ n, n ≤ N, ≤ q, q ≤ Qm , Λma qnq ′ n′ = (Mq ′ ,n′ , Aq,n )Y , ≤ n, n′ ≤ N, ≤ q ≤ Qa , ≤ q ′ ≤ Qm , Λaa qnq ′ n′ = (Aq,n , Aq ′ ,n′ )Y , ≤ n, n′ ≤ N, ≤ q, q ′ ≤ Qa . (A.12) Finally, it remains to compute the inner product term in (A.6); it is followed from (A.8) that ( E(µ, tk+1 ), E(µ, tk−1 ) + N ∑ { n=1 Qm ∑ + + N ∑ n,n′ =1 { ∑ = geq (tk+1 )geq (tk−1 )Λf f Y ( Θqm (µ) q=1 Qa ) geq (t k+1 )λm,n (µ, t k−1 ) + geq (t k−1 )λm,n (µ, t k+1 ) ) Λfqnm ( ) Θqa (µ) geq (tk+1 )λa,n (µ, tk−1 ) + geq (tk−1 )λa,n (µ, tk+1 ) Λfqna } q=1 Qm ∑ q,q ′ =1 Qm Qa + ′ Θqm (µ)Θqm (µ)λm,n (µ, tk+1 )λm,n′ (µ, tk−1 )Λmm qnq ′ n′ ∑∑ ( ′ Θqm (µ)Θqa (µ) λm,n (µ, t k+1 )λ a,n′ (µ, t k−1 )+λ m,n′ (µ, t k−1 )λa,n (µ, t q=1 q ′ =1 } Qa ∑ ′ + Θqa (µ)Θqa (µ)λa,n (µ, tk+1 )λa,n′ (µ, tk−1 )Λaa qnq ′ n′ , ′ q,q =1 ≤ k ≤ K − 1. (A.13) 174 k+1 ) ) Λam qnq ′ n′ Appendix B Calculation of the Dual Norm of the Residual (Chapter 6) In this section, we present explicitly the calculation of the dual norm of the residual associated with the discrete RB equation (6.15) discretized by ( ) the Newmark’s scheme ψ = 41 , φ = 12 . We consider the residual defined in (6.20) and its dual norm given in (6.21). The dual norm of the residual can be computed alternatively as ∥R(v; µ, tk )∥Y ′ ≡ sup v∈Y R(v; µ, tk ) , ∥v∥Y = ∥ˆ e(µ, tk )∥Y , (B.1) ≤ k ≤ K − 1, where eˆ(µ, tk ) ∈ Y is given by the Riesz representation: (ˆ e(µ, tk ), v)Y = R(v; µ, tk ), ∀v ∈ Y, ≤ k ≤ K − 1. (B.2) From (6.20), (6.24) and the affine parameter dependence (6.8) it thus follows that eˆ(µ, tk ) satisfies 175 (ˆ e(µ, tk ), v)Y = g eq (tk )f (v) + N ∑ −uN n (µ, tk+1 ) + 2uN n (µ, tk ) − uN n (µ, tk−1 ) ∆t2 n=1 + Qc N ∑ ∑ Θqc (µ) −uN n (µ, tk+1 ) + uN n (µ, tk−1 ) q c (ζn , v) 2∆t Θqa (µ) −uN n (µ, tk+1 ) − 2uN n (µ, tk ) − uN n (µ, tk−1 ) q a (ζn , v). n=1 q=1 + Qa N ∑ ∑ m(ζn , v) n=1 q=1 (B.3) It is clear from linear superposition that we can express eˆ(µ, tk ) as eˆ(µ, tk ) = g eq (tk )F + N ∑ λm,n (µ, tk )Mn n=1 + Qc N ∑ ∑ n=1 q=1 Θqc (µ)λc,n (µ, tk )Cq,n + Qa N ∑ ∑ Θqa (µ)λa,n (µ, tk )Aq,n , n=1 q=1 (B.4) where −uN n (µ, tk+1 ) + 2uN n (µ, tk ) − uN n (µ, tk−1 ) , ∆t2 −uN n (µ, tk+1 ) + uN n (µ, tk−1 ) λc,n (µ, tk ) = , 2∆t −uN n (µ, tk+1 ) − 2uN n (µ, tk ) − uN n (µ, tk−1 ) λa,n (µ, tk ) = ; λm,n (µ, tk ) = (B.5) and the parameter independence terms F, M, C, A are calculated from 176 F ∈Y from (F, v)Y = f (v), ∀v ∈ Y, Mn ∈ Y from (Mn , v)Y = m(ζn , v), ∀v ∈ Y for ≤ n ≤ N, Cq,n ∈ Y from (Cq,n , v)Y = cq (ζn , v), ∀v ∈ Y for ≤ q ≤ Qc , ≤ n ≤ N, Aq,n ∈ Y from (Aq,n , v)Y = aq (ζn , v), ∀v ∈ Y for ≤ q ≤ Qa , ≤ n ≤ N. (B.6) From (B.1) and (B.4) it follows that ∥ˆ e(µ, tk )∥2Y +2 N ∑ = (ˆ e(µ, tk ), eˆ(µ, tk ))Y = g eq (tk )g eq (tk )Λf f { eq k g (t ) λm,n (µ, t k )Λfnm n=1 + Θqc (µ)λc,n (µ, tk )Λfqnc } q=1 + N ∑ + Qc ∑ { Qa ∑ Θqa (µ)λa,n (µ, tk )Λfqna q=1 k λm,n (µ, t )λm,n′ (µ, t k )Λmm nn′ n,n′ =1 +2 Qc ∑ Θqc (µ)λc,n (µ, tk )λm,n′ (µ, tk )Λmc qnn′ q=1 ∑ Qa +2 Θqa (µ)λa,n (µ, tk )λm,n′ (µ, tk )Λma qnn′ q=1 Qc + ∑ q,q ′ =1 Qc Qa +2 ′ Θqc (µ)Θqc (µ)λc,n (µ, tk )λc,n′ (µ, tk )Λcc qnq ′ n′ ∑∑ ′ Θqc (µ)λc,n (µ, tk )Θqa (µ)λa,n′ (µ, tk )Λca qnq ′ n′ q=1 q ′ =1 } Qa ∑ ′ + Θqa (µ)λa,n (µ, tk )Θqa (µ)λa,n′ (µ, tk )Λaa qnq ′ n′ , q,q ′ =1 (B.7) where the parameter-independent quantities Λ are defined as follows 177 Λf f = (F, F)Y , Λfnm = (F, Mn )Y , ≤ n ≤ N, Λfqnc = (F, Cq,n )Y , ≤ q ≤ Qc , ≤ n ≤ N, Λfqna = (F, Aq,n )Y , ≤ q ≤ Qa , ≤ n ≤ N, Λmm nn′ = (Mn , Mn′ )Y , ≤ n, n′ ≤ N, Λmc qnn′ = (Mn′ , Cq,n )Y , ≤ q ≤ Qc , ≤ n, n′ ≤ N, Λma qnn′ = (Mn′ , Aq,n )Y , ≤ q ≤ Qa , ≤ n, n′ ≤ N, Λcc qnq ′ n′ = (Cq,n , Cq ′ ,n′ )Y , ≤ q, q ′ ≤ Qc , ≤ n, n′ ≤ N, Λca qnq ′ n′ = (Cq,n , Aq ′ ,n′ )Y , ≤ q ≤ Qc , ≤ q ′ ≤ Qa , ≤ n, n′ ≤ N, ′ ′ Λaa qnq ′ n′ = (Aq,n , Aq ′ ,n′ )Y , ≤ q, q ≤ Qa , ≤ n, n ≤ N. (B.8) 178 [...]... to establish efficient inverse procedures that combine the RB with optimization techniques for parameter inverse identification in science and engineering problems These inverse procedures will be applied for two parameter identification inverse problems in dental implant research Both problems require the estimation of material properties of the interfacial tissues in the dental implant- bone systems,... performed in the frequency domain for elastostatic model problems Thus, a fast and reliable inverse analysis method in the time domain for elastodynamic model problems is of challenge 1.1.1 The Dental Implant Problem with One-layered Interfacial Tissue Figure 1-1: The 3D simplified FEM model and its sectional view of a dental implant- bone system with one-layered interfacial tissue 3 We consider a dental. .. 6.1 Material properties of the dental implant- bone structure 108 6.2 Comparison of the CPU-time for a FEM, RB and ABAQUS forward analysis 125 6.3 Total number of forward analyses required in a RB–LMF inverse analysis (for one particular µmeasure ) 134 6.4 Comparison of computational time for a LMF model using FEM and RB as forward solvers (for one particular µmeasure )... Assess Implant Stability Fig.1-3 illustrates a typical real tooth structure and a corresponding dental implant structure In general, a dental implant structure consists of host bone, an implant, an abutment and a single artificial crown Host bone consists of two major parts: cortical bone and cancellous bone Cortical 7 Figure 1-3: A typical real tooth structure (left-half) and a corresponding dental implant. .. on dental implant FEA can be found in [29] or [25]; this section mentions very briefly some aspects of dental implant FEA Due to the high complexity of real dental implant structures, some simplifications and assumptions were made on FE models for possible ap14 proximations and calculations Assumptions were made on the following aspects: geometry, material properties, boundary conditions and boneimplant... × 10−5 ), noise-free and pe = 1% noise level 152 7.5 Identification results for µm = (8 × 106 Pa, 12 × 106 Pa, 3 × 106 Pa, 1 × 10−5 ), pe = 5% and pe = 7% noise level 152 7.6 Total number of forward analyses required in a RB–LMF inverse analysis (for one particular µm ) 152 7.7 Comparison of computation time for a LMF model using FEM and RB as forward solvers (for one particular µm... shorter for teeth (implants) where the attenuating ability of their periodontal zone 12 is greater – less mobile and thus stiffer Since the contact time is clinically meaningless, one transforms it into a “Periotest value” (PTV) and uses this PTV to evaluate teeth (implants) stability Currently, though some positive claims for Periotest, the accuracy of PTV for implant stability has been criticized for. .. principle and method as described in Section 1.1.1 above The work still consists of two stages: the forward and inverse problems with constitutive equations and relations (1.1)–(1.7) In particular, the forward problem shall provide the fast and reliable input-output relationship (1.6) that is based on solving the governing equation (1.1); while the inverse analysis incorporates that relationship into an inverse. .. Review of the Reduced Basis Method Reduced basis discretizations were first introduced in the late of 1970s for single-parameter problems in nonlinear structure analysis [35, 36] The method was then extended for multi-parameter problems [37], as well as certain classes of ODEs/PDEs [38, 39] The main focus of much work at this period was on the efficiency and accuracy of the method through local approximation. .. directly from metal forming processes This identification scheme was then applied for two problems: first, the determination of aluminum alloy behavior from a tensile test, and second, the 3D cross deep drawing test [78] 20 1.3 Purpose of the Thesis This thesis has two main goals The first goal is to develop efficient and reliable reduced basis approximations and associated error estimators for the parametrically . REDUCED BASIS APPROXIMATION AND INVERSE ANALYSES FOR DENTAL IMPLANT PROBLEMS HOANG KHAC CHI (B. Eng., Hochiminh University of Technology) A THESIS SUBMITTED FOR THE DEGREE OF. Finite Element and Reduced Basis Approximation . . . . . 140 7.3.1 Finite Element and Reduced Basis Approximation . 140 7.3.2 Numerical Results . . . . . . . . . . . . . . . . . . 141 7.4 Inverse Procedure. a dental implant system when it is applied some stimulating forces in the time domain. In this work, we focus on the development of finite element approximations, reduced basis approximations and