A novel reduced basis method and application to inverse analyses

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 s

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

2.1.6 Parametric affine formulation using a single reference

2.2 Basic equations of heat conduction problem 27

2.2.2 Weak formulation for heat conduction problems 28

3.2.4 Properties of the Galerkin projection reduced-basis method 70

3.5 Numerical examples and results 82

Chapter 4 Rapid inverse parameter estimation using reduced-basis

Chapter 5 Rapid identification of elastic modulus of interfacial tissue

on dental implants surfaces using reduced-basis method and

5.3.2.1 Comparison of RBM outputs and experimental

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

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

G Bound gap between reduced-basis upper and lower bound

u 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 FEM displacement vector

Note: Temporal variables are not listed

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)

Γ for coarse mesh using the LC-PIM

Fig 2.8 Deflection along the boundary

B

Γ for fine mesh using the LC-PIM

Fig 2.9 Comparison of deflection along the boundary

B

solution vs the LC-PIM solutions

Fig 2.10 A thermal fin with 5

1

i

i=

Ω = ∪ Ω 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 μ( )i , 1,i= ,N

Fig 3.2 Distribution of reduced-basis sample set P obtained by adaptive N

sampling procedure using the greedy algorithm (N max =34)

Fig 3.3 Convergence of the averaged reduced-basis output of GP_RBM model

Fig 3.4 Comparison between the asymptotic error and the exact output error for

μ = (2.3936, 0.3681, 43.3353 )

Fig 3.5 Comparison between the asymptotic error and the exact output error for

μ = (1.6988, 0.6707, 33.8238 )

Fig 3.6 Illustration of the reference FEM output vs the GP_RBM,and SGP_RBM

outputs for three selected parameter points (D : fine mesh, 2 D : coarse 1

mesh)

Fig 3.7 Convergence of the solution of reduced-basis approximation for our

compliance output: average temperature at Γ root

Fig 3.8 Comparison between asymptotic and exact output error for a particular

Fig 4.1 A cantilever beam with a crack

Fig 4.2 Triangular mesh of quadratic finite element on the reference domain with

the crack in the middle

Fig 4.3 Distribution of reduced-basis sample set P obtained by adaptive N

sampling procedure using the greedy algorithm (N max =93)

Fig 4.4 Convergence of the solution of reduced-basis approximation for five

Fig 4.8 Comparison between the averaged asymptotic and exact output error for

output 4

Fig 4.9 Comparison between the averaged asymptotic and exact output error for

output 5

Fig 4.10 Flow chart of GA searching procedure

Fig 4.11 Sensitivity of function f test with respect to parameter b (with L=0.5,

Fig 5.1 Diagram of a dental implant-bone structure with 4 regions

Fig 5.2 Experimental setting for implant stability measurement

Fig 5.3 (a) Overall 3-D FEM model of a dental implant-bone structure

(b) Sectional view of the interface area

Fig 5.4 Distribution of reduced-basis sample set for RBM model obtained by

adaptive sampling procedure using the greedy algorithm

Fig 5.5 Comparison between the asymptotic error and the exact output error for

Fig 5.8 A NN model with two-hidden layers, n input inputs and n outputoutputs

Fig 5.9 A typical neuron in hidden layers and output layer of the neural network

Fig 5.10 Comparison of experimental, full FEM and RBM solutions

Fig 5.11 Effect of Young’s modulus of interface resin E on displacement

responses

List of Tables

Table 2.1 Subdomains for the cantilever beam

Table 2.2 Example-1: Compliance outputs of cantilever beam by the FEM and

LC-PIM and the reference output

Table 2.3 Example-2: Compliance outputs of cantilever beam by the FEM and

LC-PIM and the reference output

Table 3.1 Offline and online evaluations of GP_ and SGP_RBM

Table 3.2 Example-1: Compliance outputs by GP_RBM, reference FEM, and

SGP_RBM for the coarse mesh D and for the fine mesh 1 D 2

Table 3.3 Example-1: Bound gaps for two different background meshs, D and 1

2

D

Table 3.4 Example-1: Comparison of RBM online computational time and FEM

computational time

Table 3.5 Example-1: Time-saving by using RBM including GP_RBM and

SGP_RBM compared to FEM for N =34

Table 3.6 Example-2: Compliance outputs by GP_RBM, reference FEM, and

SGP_RBM for the coarse mesh D and for the fine mesh 1 D 2

Table 3.7 Example-2: Bound gaps for the two different background meshs, D1

and D 2

Table 3.8 Example-2: Comparison of RBM online computational time and FEM

computational time

Table 3.9 Example-2: Time-saving by using RBM including GP_RBM and

SGP_RBM compared to FEM for N =24

Table 4.1 Online computation time to calculate s N( )μ , , ( )

s

N M

Δ μ , and s μ( ) Table 4.2 GA settings used in the RBM-GA approach

Table 4.3 Parameter estimation using RBM-GA and simulated measurement with

2.4, 0.65, 25

test true

−

Table 4.4 Parameter estimation using RBM-GA and simulated measurement with

2.4, 0.6, 20

test true

−

Table 4.5 True parameters, estimated parameters by RBM-GA, number of

generation in the GA search and RBM calls (Noise level: 1 %)

Table 4.6 Comparison of computational time for inverse problem using FE and

RBM as forward solvers

Table 4.7 Time-saving by using RBM-GA compared to FEM-GA

Table 5.1 Material properties of dental implant-bone structure

Table 5.2 Comparison of CPU-time for a FEM and a RBM forward analysis

Table 5.3 Determination of Young’s modulus of resin using the RBM-NN and

experimental measurements

Table 5.4 Determination of Young’s modulus of resin using the RBM-NN and

simulated measurements with 0% and 5% noise contamination for

1

true

E −

Table 5.5 Determination of Young’s modulus of resin using the RBM-NN and

simulated measurements with 0% and 5% noise contamination for

2

true

E −

Table 5.6 Total forward analyses required in a RBM-NN inverse analysis

Table 5.7 Comparison of computational time for a NN model using FEM and

RBM as forward solvers

Chapter 1

Introduction

In practical engineering and sciences, mechanics of solids and structures, thermal

problems, and fluid mechanics problems are generally governed by different kinds of

partial differential equations Their outputs of interest are not the “full” field variables

but a certain kind of output, which can demonstrate the phenomenon of a given

system Important quantities (outputs) of interest usually include (i) displacement and stress fields in solids, (ii) flowrates, lift and drag forces in fluids, (iii) temperature and

heat flux in the thermal systems, etc These outputs are determined by certain system parameters called “inputs”, such as, geometry, material properties, boundary conditions and/or loading conditions of the system Therefore, outputs of interest can

be defined as the function of “input” parameters This leads to an “input-output” relationship which is a more general term representing the evaluation of the PDEs governing the physics of an engineering system by means of numerical simulation Numerical methods thus play an increasingly important role in evaluating outputs of engineering systems Hence, developing indispensable numerical methods becomes crucial for various aspects of computational simulation

In the last century, many numerical computation methods have been developed and are successfully used intensively in almost every field of science and engineering These methods include the finite element method (FEM), the finite difference method

(FDM), the boundary element method (BEM), meshfree methods and etc However, all the numerical approaches require an effectively dense approximation space in order to obtain reliable and accurate solutions It implies that classical numerical methods incur very long computational time in spite of CPU processing speed and very expensive computational cost

The main purpose of this thesis is to develop an efficient and reliable numerical

approach which can provide both upper and lower bounds to the exact solution This

development is inspired from the traditional finite element method (FEM), the standard reduced-basis method (RBM), and a smoothed Galerkin projection technique

in the area of meshfree methods Another goal of this thesis is to solve inverse problems efficiently and accurately as solving inverse problems is practically very important in areas of engineering and sciences, such as non-destructive evaluations, biology, computer imaging, and medical sciences

1.1 Review of numerical methods

Classical numerical approaches including the FEM (Bathe, 1996), the finite difference method (FDM) (Liszka and Orkisz, 1979), the finite volume method (FVM), the boundary element method (BEM) (Boresi et al., 2003) and Meshfree methods are powerful computational methods These numerical methods can be

classified into two categories: (i) techniques which rely on meshing and (ii)

techniques which do not rely on meshing The FEM, the FDM, the FVM, and the BEM are involved in the first category, and meshfree methods are in the second category of numerical methods In the first category, the most well-known and dominant numerical method is the traditional finite element method

1.1.1 Finite element method

The widely used traditional finite element method (Braess, 2001) is the most significant invention in 20th century, and it has been an indispensable technology in

solving most engineering and science problems with availability of many commercial software packages e.g ABACUS, ANSYS, etc The finite element method is well formulated by the variational principle or the principle of minimum potential energy The FEM algebraic equation appears in the form of integration over the entire problem domain

In the FEM routine, a real structure or component of an engineering system is first transformed into an appropriate physical domain or geometry The physical domain is

divided into a set of subdomains called finite elements which are connected by nodes

Therefore, the continuity between nodes is very important to obtain accurate and reliable solutions in the FEM Material properties of the system are then identified For example, Young’s modulus and Poisson ratio for solids, or thermal conductivity in thermal analyses are required to determine Different material properties are necessary

to identify for different sets of materials Neumann and/or Dirichilet boundary conditions and loading conditions are then imposed in the derivation of the finite element algebraic equation Detalied formulations of the finite element method for different problems can be found in earlier works of Oliveria (1968), Zienkiewicz and Taylor (1989; 1992) The FE discrete system equations are finally solved for the field variables, and/or required outputs of interest

Although the FEM (Reddy, 1993) is such a systematic approach, it is noted that the finite element method has encountered several difficulties due to the use of meshes Difficulties of the FEM are: in dealing with a class of problems such as large deformation problems, crack growth problems with arbitrary and complex paths

because of discontinuities between nodes (Liu and Quek, 2003), low accuracy in the derivatives of the field variables, difficulty in adaptive analysis and high computational cost of meshing for accurate solutions It is also noted that the FEM

produces a lower bound solution to the exact solution in energy norm because of its

over-stiff behavior Although the traditional FEM has such inherent shortcomings, there is no doubt that it has been influencing almost every branch of engineering and sciences Researchers thus have focused on improving the FEM

Recently, element-based smoothing domains used finite element method so-called

smoothed finite element method (SFEM) (Liu et al, 2006a; 2007a; 2007b) and edge-based smoothing domains used finite element method called the edge-smoothed finite element method (E-SFEM) (Liu et al., 2008) are developed, and remarkable achievements are obtained In the SFEM and the E-SFEM, it is observed that smoothed strain operations (Chen et al., 2001) incorporate with the standard finite element method More accurate and more stable solutions are achieved compared with the traditional finite element method

1.1.2 Meshfree methods

A class of numerical methods, say meshfree or meshless methods, have been developed to overcome mesh-related problems, and remarkable progresses have been obtained in recent years In meshfree methods, the problem domain is represented by

a set of scattered nodes Discrete system equation is developed from a set of nodes, and thus, no node-connectivity is necessary

Smoothed particle hydrodynamics (SPH) was firstly developed as a start of meshfree technologies (Gingold and Monaghan, 1977; Moraghan, 1988) The SPH (Liu and Liu, 2003) is based on the kernel approximation techniques, and it is

commonly used in fluid mechanics A variety of meshfree methods including the reproducing kernel particle method (RKPM) (Liu et al., 1995), the hp-clouds method (Duarte and Oden, 1996; Liszka et al., 1996), the partition of unity finite element method (PUFEM) (Melenk and Babuska, 1996), the finite point method (FPM) (Onate et al., 1996), radial point interpolation method (RPIM) (Liu, 2002; Liu et al, 2005c), and etc, have been developed with significant progresses

Based on formulations of meshfree methods, there are three categories of meshfree methods: (i) meshfree weakform methods, (ii) meshfree strong-form methods and (iii) meshfree weak-strong form methods Meshfree weakform methods include the diffuse element method (DEM) (Nayroles et al, 1992), the element free Galerkin method (EFG) (Belytschko et al., 1994), the meshless local Petrov-Galerkin method (MLPG) (Atluri and Zhu, 1998; 2000), the local radial point interpolation method (LRPIM) (Liu and Gu, 2001a; 2001b), the point interpolation method (PIM) (Liu and Gu, 2001c), etc Meshfree strongform methods include the finite point method (FPM) (Onate et al., 1996), the radial point collocation method (Kee et al., 2007a,b), the smoothed partical hydrodynamics (SPH), etc In weak-form and strong-form meshfree methods, it is observed that meshfree weakform methods give more stable solutions than meshfree strongform methods in solving Neumann boundary conditioned problems However, weakform techniques are more expensive computationally due to performing integration functions In contrast, no integration function is involved in meshfree strong-form methods Thus, meshfree strong-form methods are easy to implement, and are not as expensive as weakform techniques In order to use advantages of meshfree strongform and weakform methods, Liu and Gu (2003) proposed a meshfree weak-strong form method (MWS) In the MWS, local weak-form is used in the vicinity of the boundary on which Neumann boundary

condition is imposed Strong-form meshfree techniques are then employed for the rest part of the problem domain

In recent years, a new class of meshfree methods called a linearly conforming point interpolation method (LC-PIM) (Liu and Zhang, 2007) and a linearly conforming radial point interpolation method (LC-RPIM) (Liu et al., 2006b; 2006c) are established using the generalized smoothing operations (Liu, 2008a) with PIM shape functions (Wang and Liu, 2002; Liu and Gu, 2005) and RPIM shape functions (Liu et al., 2003; 2005c) It is found that the LC-PIM (Liu et al., 2005b) and LC-RPIM have a very important and attractive property: it gives upper bound solutions in strain energy norm for elasticity problems with homogeneous boundary conditions

It is obvious that classical numerical approaches and meshfree methods are very

important and useful in numerical simulations However, all these reported numerical methods require a large amount of CPU time even for engineering problems of normal-scale This situation becomes much more critical when one needs to explore intensively in the design space for optimal design, inverse identification of parameters or non-destructive evaluations (NDE) in which thousands times of forward evaluations are necessary Therefore, reducing computation time is of great importance in the field of numerical simulation for engineering analyses including forward and inverse analyses It is the primary concern in this study

1.2 Review of real-time computation technique

Development of fast computation techniques is necessary to improve computational efficiency in modern engineering One fast computation technique is called model order reduction (MOR) (Willcox et al., 2002; Bui-Thanh et al., 2007)

which has been used in many applications such as damage detection and flaws in structures (Banks et al., 2000; 2002) Another fast computational method called the reduced-basis method (RBM) is a systematic strategy consisting of dimension reduction, reduced-basis approximation, reduced-basis error estimation and offline-online computational decomposition, which will be discussed in detail in Chapter 3

1.2.1 Reduced-basis method (RBM)

The reduced-basis approximation method (RBM) was first developed in the late 1970s The reduced-basis method for nonlinear problems has been developed in 1980s (Fink and Rheinboldt, 1983; Noor and Peters, 1983; Porching, 1985; Peterson, 1989) The local approximation space was used in earlier works These studies mainly focused on the computational efficiency and accuracy of the reduced basis approach

In 1996, Balmes introduced a global reduced-basis approximation space which helps

to improve the computational efficiency and the reliability of reduced-basis solutions

In the last decade, Patera and coworkers (Machiels et al., 2000; 2001; Patera and Rønquist, 2007) have developed the reduced-basis method with several key contributions: 1) dimension reduction, 2) the global approximation, 3) offline-online computational decomposition and 4) two types of reduced-basis-approximation-error which are termed as the RBA-error estimations in this thesis for the convenience of

presentation One of the RBA-error estimation is a posteriori rigorous error estimation

stabilized by “inf-sup” condition Another type of RBA-error estimation is an asymptotic error estimation (Veroy, 2003)

Utilization of the reduced-basis approximation has been examined in solving different kinds of engineering problems underlying partial differential equations

(PDEs) for evaluating different kinds of outputs Machiels et al., (2000, 2001) presented the reduced-basis method and its output bounds for elliptic partial differential equations Sen et al., (2006) reported “natural norm” of RBA-error estimation for coercive and noncoercive linear elliptic partial differential equations Furthermore, applications of the RBM and RBA-error estimations for parabolic partial differential equations were proposed by Veroy et al., (2005), Grepl (2005), and Grepl and Patera (2005) Also, Nguyen (2005) found effective implementations of the

reduced-basis method and the rigorous RBA-error estimation for nonaffine and

nonlinear partial differential equations He also introduced the use of the reduced-basis method in solving inverse parameter estimation problems Moreover, the reduced-basis method and the RBA-error estimation found its applications in solving a broader class of equations/problems including: viscous Burger equations (Veroy , 2003), steady state incompressible Navier Strokes equations (Rozza and Veroy, 2007), heat conduction problems (Prud’homme et al., 2002), Boltzmann equation (Patera and Rønquist, 2007) and stress intensity factor analysis (Huynh et al.,

2006 and Huynh, 2007) The basic methodology and developments of reduced-basis technology can also be found at http://augustine.mit.edu

The above studies revealed that the RBA-error bound given in these works is actually the error bound for the RBM solution with respect to the solution of a very

fine FEM model It is not an error bound to the exact solution of the original problem

because the FEM model surely contains errors Therefore, solution bounds in the RBM models are in reduced-basis approximation errors (RBA-errors) which are very expensive to obtain No work has been done on developing the reduced-basis method

for solution bounds with respect to the exact solution Hence, there is still a room for

developing a reduced-basis method which produces upper and lower bounds to the

exact solution as the exact solution bounds are important in practical engineering, at

least for safety reason It is also found that the reduced-basis method is computationally very efficient due to the global approximation and offline-online computational decomposition The reduced-basis solution is very close to the traditional FEM solution because of very fast convergence rate Thus, the RBM is very suitable for inverse problems due to its computational efficiency, and robustness

1.3 Inverse analysis

In all the areas of engineering and sciences, inverse problems such as identification of unknown parameters in solids and structural systems, characterization of unknown material properties (Han, 2000; Han et al., 2002), etc., based on measurements or observations play a very important role In general, there are four main steps to solve an inverse problem In the first step, a computational forward model for a defined problem/system is developed to generate outputs of interest In the second step, sensitivity analysis is performed to ensure that outputs of the forward model are highly sensitive to the system parameters to be determined inversely Required modifications to the forward model and to the choice of inputs for inverse searching techniques need to be made based on the sensitivity analysis In the third stage, experiments should be undertaken to measure responses of the system subjected to the external loading or excitation In the final stage, optimization techniques such as direct search algorithm, gradient-base algorithms, neural network, Genetic Algorithms (GAs) need to be used as an inverse search methodologies; required system parameters can be predicted from the knowledge of experimental measurements

In inverse searching procedures, genetic algorithm (GA) stands one of the most

efficient and robust inverse searching techniques for complex and nonlinear inverse problems due to global searching nature and discrete formulation Additionally, a neural network (NN) is popular for its unique computing feature so that the NN can

be employed to identify structural parameters which are complexly and nonlinearly related to the dynamic responses of a given structure In this work, the GA and the NN are employed as inverse searching procedures due to their unique features

1.3.1 Review of genetic algorithms (GAs)

From Darwin’s genetic evolution theory, genetic algorithms (GAs) were established by Holland (1975) The GAs (Goldberg, 1989) is stochastic (random) searching technique to minimize an error function or objective function that is generally nonlinear and implicit function of parameters to be identified First, an initial population or generation of chromosomes (individuals) is randomly created The fitness value (objective function value) of each individual is then evaluated Next, the GA finds good individuals which possess the best fitness value (minimum objective function value) in current generation The population of the next generation

is produced from good individuals of the past generations and newly (random) selected individuals Note that individuals of current generation are created by three main operations of the GA to ensure a “healthy” evolution with proper notions Finally the best individual can be found and the GA searching is terminated

Extensive theoretical developments of the GA (Haupt, 1998) such as simple GA, micro-GA (Krishnakumar, 1989), intergeneration projection Genetic Algorithm (IP-GA) (Liu and Han, 2003) have been done Applications of the GAs have been found in inverse parameter identifications, design and optimization fields with great achievements For example, Liu et al., (2002) presented an inverse procedure using a

micro genetic algorithm and non linear least square method of composite laminate plates Yang et al., (2002) also reported an inverse approach for detection of crack in laminates by using micro-GA and internal strain of optical fibers Similarly, Wu et al., (2002) proposed a nondestructive evaluations (NDE) procedure to detect the location and length of the crack in an isotropic plate using elastic waves governed by the differential equation of wave propagation, and the uniform micro-GA was employed for inverse searching Besides, Liu et al., (2005) articulated an inverse search technique, for material characterization of composite plates, based on real-micro GA which treated the dynamic response on the composite plate surface as an input A material characterization technique of laminated cylinder shell using uniform micro-GA was also articulated by Han el al., (2002) An intergeneration-projection Genetic Algorithm (IP-GA) was proposed for the inverse parameter characterization

of heat convection constants (Liu et al., 2005a) In addition, the genetic algorithm (GA) was also implemented to perform flaw detection in sandwich plates (Liu and Chen, 2001) In the study, the time-harmonic response of sandwich plate was evaluated by the FEM and used as an input of GA From the earlier studies, it has been found that the genetic algorithms (GAs) are very effective in many inverse applications It is because the genetic algorithm (GA) requires only evaluating the objective function on top of its simplicity and robustness, while many inverse searching procedures require evaluating of objective error function as well as additional information such as the derivatives of the objective function that can be very complex and nonlinear in nature For these reasons, the GA is often preferred to

be used alone in order to keep all these good features Hence, the Genetic Algorithm

is chosen as an inverse search technique in this study On the other hand, the GA usually requires a large times of forward analyses Hence, fast forward solvers are

needed

1.3.2 Review of neural network (NN)

Another searching procedure called a neural network (NN) technique, originated

by McCulloch and Pitts (1943), is a useful information processing tool The parallelism of NN makes possible to solve many problems which cannot be handled analytically A typical NN consists of an input layers, several hidden layers and an output layer Each NN layer is organized by a set of neurons which are information processing units in NN models

The NN technique (Lippmann, 1987) is usually employed in inverse analyses (Sribar , 1994), material characterizations (Huber and Tsakmakis, 1999) and design problems (Sumpter and Noid, 1996) More applications of NN in detection of 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., 2003a; 2003b; Liu et al., 2001a; 2001b; 2002) have developed a progressive neural network for material characterization of functionally graded material (FGM) based on dynamic displacement responses Beyond engineering problems, Deng et al (2004; 2008a; 2008b; 2008c) have articulated a material characterization of implant-bone structure based on the neural network technique and the finite element method in dentistry From earlier works, it is noted that the NN possesses unique computing features for identification of structural parameters which are nonlinearly related to dynamic responses of the structure in a complicated manner, and the NN searching procedure offers high processing speed compared with other inverse searching techniques

However, inverse procedures including the GA and the NN normally require a fast

forward solver as the CPU-time of forward solvers has a lot of influence over computational efficiency in inverse analyses In this thesis, the RBM is implemented

as a “teacher” to train an NN and as an efficient forward solver in GA to achieve high computational efficiency in inverse analyses

1.4 Objectives

The main objectives of this study are two fold The first is to develop an efficient and reliable reduced-basis method which provides upper and lower solution bounds to

the exact solution in the form of energy norm The second is to establish efficient

inverse searching techniques using the standard Galerkin projection RBM and inverse techniques for inverse parameter identifications in both engineering and science problems

As discussed in the above sections, it is recognized that the original Galerkin projection reduced-basis method (RBM) provides solution bounds with respect to a FEM solution output, and the errors in the solution introduced by the use of RBM need to be properly quantified for solution bounds No research development has been investigated on developing the reduced-basis method for solution bounds with respect

to the exact solution Thus, developing a novel reduced-basis method for both upper and lower bounds to the exact solution (in energy norm) has become a major objective

in this thesis A novel reduced-basis method is generated based on the theory of standard reduced-basis method and a smoothed Galerkin projection used LC-PIM In the smoothed Galerkin projection, smoothed strain operation provides softening effects to the discretized model governed by PDEs These effects will make the discretized numerical model weaker than the original As a result, the solution of smoothed Galerkin projection becomes an upper bound solution in strain energy norm

with respect to the FEM solution as well as the exact solution This is a very

important property which leads to the development of a general procedure of a novel

reduced-basis method to obtain an upper bound to the exact solution while the standard RBM provides a lower bound to the exact solution Therefore, upper and

lower bounds in the form of energy norm or compliance output can be obtained using the proposed RBM approach regardless of reduced-basis approximation error estimation The reliability and computational efficiency are promising

In this thesis, attempts also made to propose efficient inverse searching procedures for non-destructive evaluations (NDE) From summarizing the past works, it is observed that a wide variety of computational procedures have been developed for inverse problems However, almost all the inverse techniques are very expensive

because of (i) very long computation time, and (ii) lack of fast forward solvers These

key issues are resolved proposing two rapid computational inverse procedures The first proposed inverse procedure is developed combining the reduced-basis method (RBM) and Genetic Algorithm, so-called RBM-GA approach In the RBM-GA inverse approach, time-consuming problem is addressed by using the standard RBM approach as a fast forward solver since the RBM could provide the low marginal cost

of repeated forward evaluations which are the essential requirement of inverse problems To avoid a very complex and non-linear nature of optimization techniques, the GA is utilized as an inverse operator controlling the forward solver because of the simplicity of the GA (see Section 1.3.1) Another inverse procedure is, then, established using the RBM and neural network (NN) technique, and it is termed as the RBM-NN approach Again, the computational efficiency is increased due to the used

of the RBM Due to the unique information processing feature discussed in Section 1.3.2, the neural network (NN) is employed for the purpose of material

characterization in implant dentistry in which dental-implant structural behaviors subjected to harmonic loading are nonlinearly and complexly related to the material properties The proposed inverse procedures ensure to confirm that the applicability of the reduced-basis method in inverse analyses of nondestructive evaluations in engineering as well as noninvasive evaluations in dentistry

In our RBM-GA and RBM-NN approaches (discussed in Chapter 4 and Chapter 5), we use standard RBM method, but not the novel RBM method It is because (i) outputs of forward solver used in our inverse analyses are not compliance outputs, (ii) outputs bounds of newly proposed novel reduced-basis method are only promising for compliance outputs at the current stage, (iii) the standard reduced-basis method has been analyzed for different types of outputs including compliance output (energy norm) in the past, (iv) the standard RBM solution approaches to the traditional FEM solution when the RBM space is refined, and (v) our work focuses to ensure that the proposed inverse procedures are workable and stable Therefore, it is possible to assume that the proposed inverse procedures are applicable for any reliable forward solutions without any problems Further note that only the reduced-basis method is considered in this thesis and another real-time computation method called the model order reduction (MOR) is beyond the scope of this study

1.5 Organization of the thesis

The organization of this thesis is mainly divided into two parts Chapter 2-3 are

involved in the first part which is concerned with the development of reduced-basis

method for an upper bound to the exact solution The second part deals with the

inverse parameter estimation problems discussed in Chapter 4-5, and Chapter 6 is

the conclusions

The smoothed Galerkin projection used the linear conforming point interpolation method (LC-PIM) is reviewed before proceeding with the reduced-basis method In

Chapter 2, fundamental equations and parametric weakforms for linear elasticity

solid mechanic problems and heat conduction problems are firstly introduced The

standard Galerkin weakform of the finite element method (FEM) and its important

properties are briefed Then, a smoothed Galerkin projection or linearly conforming point interpolation method (LC-PIM) is introduced Properties of the LC-PIM are examined through the numerical examples of a cantilever beam problem and a thermal fin problem Numerical results are then presented to demonstrate accuracy, reliability and properties of the smoothed Galerkin projection (LC-PIM)

In Chapter 3, the basic concept of standard reduced-basis approximation method

(RBM) is introduced firstly with its special properties including: lower bound property and monotonic convergence property A novel reduced-basis method is then proposed which is the focus of this thesis The detail formulation and the properties of the proposed method are presented theoretically Two numerical examples: (1) a cantilever beam with an oblique crack and (2) a thermal fin problems are conducted to verify the properties of proposed reduced-basis method

Computational inverse searching techniques are established in the second part of

this thesis In Chapter 4, a fast inverse parameter estimation technique called the

RBM-GA is established In the RBM-GA, the RBM is employed as a fast forward solver, and the GA is used as an inverse searching methodology A numerical example

of detection of crack in a structural component is conducted for our RBM-GA approach in order to demonstrate substantial accuracy, robustness and significant computational saving

In Chapter 5, another rapid inverse computational searching technique, using the

RBM for a forward solver and neural network (NN) for inverse searching procedure,

is articulated The approach is termed as the RBM-NN approach Identification of elastic constants of interface tissues between dental implant surfaces is performed using frequency responses of the implant bone structure High computational efficiency, and stable inverse solutions are demonstrated

presented

Chapter 2

Smoothed Galerkin projection

In practice, engineering problems governed by different kinds of PDEs are generally impossible to solve in analytical means but in numerical means In numerical methods, the conventional finite element method (FEM) is one of the most well-known numerical methods with its important properties: lower bound (in energy norm) property, monotonic convergence property and reproducibility property (Liu, 2008a; 2008b) In recent years, Liu et al.(2006a; 2006b; 2006c; 2007a; 2007b) developed a class of numerical methods, with upper bound properties, which have been established by using a smoothed Galerkin projection Therefore, both upper and

lower bounds for the exact solution in energy norm are able to determine using the

standard Galerkin projection (FEM) and the smoothed Galerkin projection

A smoothed Galerkin weakform is formulated using the smoothed strain operation (Liu, 2008a) It appears to confirm three major innovations of softening effect, monotonic convergence property and an upper bound (in energy norm) property

Based on the smoothed Galerkin weakform, numerical methods including the

smoothed finite element method (SFEM), the linearly conforming point interpolation method (LC-PIM) and linearly conforming radial point interpolation method (LC-RPIM) (Liu et al., 2006b; 2006c), and the edges-smoothed finite element method where edge-based smoothing domain are used (Liu et al., 2008) have been developed

In this work, the LC-PIM in the area of meshfree method is employed in order to develop an upper bound reduced-basis method which will be discussed in next chapter The linearly conforming point interpolation method (LC-PIM) has been articulated using the smoothing operation and shape functions of the point interpolation method (PIM) In this chapter, the smoothed Galerkin projection (SGP) via LC-PIM is presented

Before discussing the LC-PIM, fundamental equations of linear elasticity solids mechanics problems and heat conduction problems are first presented including the

standard weak statement, and the parametric weak statement The standard Galerkin

projection (GP) via the finite element method and its important properties are then briefly discussed The smoothed Galerkin projection via the LC-PIM is formulated, and the properties of LC-PIM are also presented Two numerical examples of a linear elasticity cantilever beam problem and a thermal fin problem are conducted to verify the properties of the smoothed Galerkin projection (SGP)

2.1 Basic equations of linear elasticity

Basic equations for solid mechanics problem of linear elasticity are established in both mathematical community and mechanics community

0 0

0ij i 0,

j

b x

The relationship between the stress tensor σij0 and the strain tensor εij0 is given

by the Generalized Hook’s Law:

ij C ijkl kl

where C is elasticity tensor of material property constants that are symmetrical: ijkl

ijkl jikl ijlk klij

u stands for the prescribed

value for the displacement component Let Γ denote a part of 0N Γ , on which 0

Neumann boundary condition is satisfied,

2.1.4 Weak formulation for linear elasticity

The weakform formulation for linear elasticity solid mechanics problems (Prud'homme et al., 2002) is stated here To begin, we introduce a functional space

This is the standard weakform for linear elasticity models

The general statement of Eq (2.16) for linear elasticity PDEs is now defined A

(μ1, ,μP)∈D

μ = is an input of a given problem, such as geometry of the model or

material properties For a given μ∈D, the solution u μ0( ) satisfies

( )

where a and f are - parameterized μ bilinear form and linear functional

We now assume that the parameter μ∈D describes the physical properties or geometry of a given model The a(.,.;μ) and f ( ).;μ can be formulated to be affine using geometric affine mappings theory which is very important in formulation

of the reduced-basis method Note that the geometric affine mapping allows us to formulate a(.,.;μ) and f ( ).;μ for all the possible μ∈D based on a single reference domainΩ

2.1.5 Geometric affine mapping

To begin geometric affine mapping theory, a physical domain Ω is divided into 0

and the components of G μ r( ) are denoted as r( )

reference domain, for any μ∈D, can be formulated as