A novel reduced basis method and application to inverse analyses

Acknowledgements
i
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.

Table of contents
ii
Table of contents
Acknowledgements i
Table of contents ii
Summary vi
Nomenclature viii
List of Figures x

List of Tables xiv
Chapter 1 Introduction 1
1.1 Review of numerical methods 2
1.1.1 Finite element method 3
1.1.2 Meshfree methods 4
1.2 Review of real-time computation technique 6
1.2.1 Reduced-basis method (RBM) 7
1.3 Inverse analysis 9
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 2 Smoothed Galerkin projection
18
2.1 Basic equations of linear elasticity 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
Table of contents
iii
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 30
2.3.1 Discrete equations 31
2.4 Smoothed Galerkin projection 34
2.4.1 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 43
2.6.1 Example-1: Cantilever beam problem 44
2.6.1.1 Numerical results 44
2.6.2 Example-2: Thermal fin problem 46
2.6.2.1 Numerical results 47
2.7 Remarks 48
Chapter 3 A novel reduced-basis method 64
3.1 Preliminaries 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
Table of contents
iv
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 88
Chapter 4 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 Sample set construction 108
4.2.4 Numerical results for forward problem 109
4.3 Inverse procedure 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 115
Chapter 5 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
Table of contents
v
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 152
Chapter 6 Conclusions and future works 165
6.1 Summary 165
6.2 Recommendations of future works 168
References 171
Publications arising from the thesis 186



Summary

vi
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.
Summary
vii
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.
Nomenclature
viii
Nomenclature
()
,a
Standard bilinear form
()
,

D
a
Smoothed bilinear form
D
A parameter domain
f

Linear functional
F


FEM force vector
N
F


GP_RBM force vector
F
LC-PIM force vector
N
F
SGP_RBM force vector
D
G
Bound gap between reduced-basis upper and lower bound
K


FEM stiffness matrix
N

K


GP_RBM stiffness matrix
N
K
SGP_RBM stiffness matrix
K
LC-PIM stiffness matrix

Linear functional
M
Low dimensional manifold
ℵ
Dimension of a FEM mesh
N
P ,
M
P
reduce-basis sample sets
e
s
Exact output
s


FEM output
N
s


,
M
s


GP_RBM outputs
N
s ,
M
s
SGP_RBM outputs
e
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
N
u

,
M
u


GP_RBM solutions in terms of field variables
N

u ,
M
u
SGP_RBM solutions in terms of field variables
Nomenclature
ix
u


FEM displacement vector
N
u


GP_RBM displacement vector
u
LC-PIM displacement vector
N
u
SGP_RBM displacement vector
N

W
,
M

W

Reduced-basis spaces
Ω

Problem domain
μ

A parameter point in the parameter set D
κ

Thermal conductivity
()
q
Θ
μ

Parametric coefficient functions
i
ϕ

a nodal basis (or shape) function constructed based on elements
n
Ω
smoothing domains/cells
Γ
Boundary of problem domain
∇
the Laplacian operator
,
s
NM
Δ
Asymptotic error
,

s
N exact
Δ
Exact error
,,
s
NMavg
Δ

Averaged asymptotic error
,,
s
Nexactavg
Δ
Averaged exact error
α

CPU-time saving factor
Note: Temporal variables are not listed.



List of Figures
x
List of Figures
Fig 2.1
Division of problem domain Ω into smoothing domains Ω
n.

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
B
Γ
for reference mesh (
ref
ℵ ) using the
FEM.

Fig 2.7
Deflection along the boundary
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
Γ
: Reference FEM
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 (
760,4
=
ℵ
) 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
new
μ
by a linear
combination of predetermined solutions
(
)
,
i
u

μ
1, ,iN=  .
List of Figures
xi

Fig 3.2
Distribution of reduced-basis sample set
N
P obtained by adaptive
sampling procedure using the greedy algorithm ( 34

=
max
N ).

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 (
2
D : fine mesh,
1
D : coarse
mesh).

Fig 3.7 Convergence of the solution of reduced-basis approximation for our
compliance output: average temperature at

root
Γ
.

Fig 3.8 Comparison between asymptotic and exact output error for a particular
configuration
1test
−
μ .

Fig 3.9 Comparison between asymptotic and exact output error for a particular
configuration
2test
−
μ .

Fig 3.10 Comparison between asymptotic and exact output error for a particular
configuration
3test
−
μ .

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
N

P obtained by adaptive
sampling procedure using the greedy algorithm. ( 93
=
max
N ).

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

Fig 4.5 Comparison between the averaged asymptotic and exact output error for
output 1.

Fig 4.6 Comparison between the averaged asymptotic and exact output error for
output 2.

Fig 4.7 Comparison between the averaged asymptotic and exact output error for
output 3.
List of Figures
xii

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

test
f with respect to parameter b (with 0.5L
=
,
and 45
θ
=

).

Fig 4.12
Sensitivity of function
test
f with respect to parameter L (with 2.0b
=
,
and
45
θ
=

).

Fig 4.13
Sensitivity of function
test
f with respect to parameter
θ
(with
2.0b

=
and 0.5L = ).

Fig 4.14
Sensitivity of function
test
f with respect to parameters b and L (with
45
θ
=

)

Fig 4.15
Sensitivity of function
test
f with respect to parameters b and
θ
(with
0.5L
=
).

Fig 4.16
Sensitivity of function
test
f with respect to parameters L and
θ
(with
2.0b

=
).

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
()
9
1
2.05 10 ,2500
test
Pa Hz
−
=×μ .

Fig 5.6 Comparison between the asymptotic error and the exact output error for
()
9
2
1.7 10 ,1800
test
Pa Hz

−
=×μ .

Fig 5.7 Comparison between the averaged asymptotic error and averaged exact
error.

List of Figures
xiii
Fig 5.8
A NN model with two-hidden layers,
input
n inputs and
output
n outputs.

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
xiv
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
1
D and for the fine mesh
2
D .

Table 3.3
Example-1: Bound gaps for two different background meshs,
1
D and
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
34N
=
.

Table 3.6 Example-2: Compliance outputs by GP_RBM,
reference FEM, and
SGP_RBM for the coarse mesh
1
D and for the fine mesh
2
D .

Table 3.7
Example-2: Bound gaps for the two different background meshs,
1
D
and
2
D .

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 24N
=
.

Table 4.1
Online computation time to calculate
(
)
N
s

μ
,
()
,
s
NM
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
different noise levels of contamination:
()
1
2.4,0.65,25
test

true
−
≡=

μμ
.
List of Tables
xv

Table 4.4 Parameter estimation using RBM-GA and simulated measurement with
different noise levels of contamination:
()
2
2.4,0.6,20
test
true
−
≡=

μμ .

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
1true
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
2true
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

1
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

Chapter 1 Introduction

2
(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.

Chapter 1 Introduction

3
1.1.1 Finite element method
The widely used traditional finite element method (Braess, 2001) is the most
significant invention in 20
th
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
Chapter 1 Introduction

4
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
Chapter 1 Introduction

5
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
Chapter 1 Introduction

6
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)
Chapter 1 Introduction

7

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
Chapter 1 Introduction

8
(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
.
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
Chapter 1 Introduction

9
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
Chapter 1 Introduction

10
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