Physics-informed neural networks for the
analysis and optimization of structures

MAI TIEN HAU

February 2023

Department of Architectural Engineering
The Graduate School
Sejong University


Physics-informed neural networks for the
analysis and optimization of structures

MAI TIEN HAU
A dissertation submitted to Faculty of Sejong University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Architectural Engineering

February 2023

Approved by
Professor Jaehong
Major Advisor

Lee


Physics-informed neural networks for the analysis
and optimization of structures

by
MAI TIEN HAU

Approved - ---------------------------------Professor Kihak Lee, Chair of the committee

Approved - ---------------------------------Professor Dongkyu Lee, Member of dissertation committee

Approved - ---------------------------------Professor Seunghye Lee, Member of dissertation committee

Approved - ---------------------------------Professor JongJae Lee, Member of dissertation committee

Approved - ---------------------------------Professor Jaehong Lee, Advisor



ABSTRACT

This thesis is concerned with nonlinear, stability analyses, and size optimization
of truss structures based on physics-informed neural networks (PINNs). For nonlinear analysis one, a robust and simple unsupervised neural network framework is
proposed to perform the geometrically nonlinear analysis of inelastic truss structures. To guide the training process, the loss function built via the total potential
energy principle under boundary conditions (BCs) is minimized in the suggested
NN model whose weights and biases are considered as design variables. And the
training data only contain the spatial coordinates of joints. In each training iteration, feedforward, physical laws, and back-propagation are applied for adjusting
the parameters of the network to minimize the loss function. Once the network
is properly trained, the mechanical responses of inelastic structures can be easily obtained without using any structural analysis as well as incremental-iterative
algorithms. Several benchmark examples regarding geometrical and material nonlinear analysis of truss structures are examined to demonstrate the effectiveness
and reliability of the proposed paradigm.

Subsequently, the proposed


first to analyze the stability of truss structures.
work,

neural network

(NN)

Different

from

most

model

is

existing

is designed to directly locate the critical point by

minimizing the loss function involving the residual load and property of the stiff-


ness matrix which they are established based on the outputs, loads, and BCs. It is
also significant because the first critical point will be located at the training end
corresponding to the minimum
iterative methods.


loss function without utilizing any incremental-

Additionally, this dissertation also develops a Bayesian deep

neural network-based parameterization framework to directly solve the optimum
design for geometrically nonlinear trusses for the first time. In this approach, the
parameters of the network are regarded as decision variables of the structural optimization problem, instead of the member’s

cross-sectional areas. Therein, the

loss function is constructed with the aim of minimizing the total structure weight
so that all constraints of the optimization problem

obtained by supporting fi-

nite element analysis (FEA)

are satisfied. Furthermore,

and are-length method

Bayesian optimization (BO) is applied to automatically tune the hyperparameters of the network. The effectiveness of this model

is demonstrated through a

series of numerical examples for geometrically nonlinear space trusses. And the
obtained results demonstrate that our framework can overcome the drawbacks of
applications of machine learning in computational mechanics. Finally, a physicsinformed neural energy-force network
to directly solve the optimum
sis is completely


removed

(PINEFN)

framework is first constructed

design of truss structures that structural analy-

from the implementation

of the global optimization

in this thesis. Herein, the loss function is designed based on the output values
and physics laws to guide the training. Now

only NN

is used in our scheme to

minimize the loss function wherein weights and biases of the network

are con-

sidered as design variables. In this model, spatial coordinates of truss members
are examined

as input

data, while corresponding


cross-sectional areas and re-

dundant forces unknown to the network are taken account of output. Obtained
outcomes indicated that it not only reduces the computational cost dramatically
ii


but also yields higher accuracy and faster convergence speed compared with recent literature. With the above outstanding features, it is promising to offer a
unified solver-free numerical simulation for solving complex issues in structural
optimization.

Keywords:

Physics-informed,

Geometric

nonlinear,

Structural

stabil-

ity, Hyperparameter optimization, Force method, Critical points, Complementary

energy,

Bayesian


optimization,

iii

Truss optimization


CONTENTS

ABSTRACT.

20.

ce

LIST

OF

TABLES:

LIST

OR

FIGURES:

1

2


ee

2.3:

i cee eees

1 6c

se

6 csi ciinue ee

INTRODUCTION...

ed

ewe

we

6 6 ecto

1...

ee

eee

ee


ewe

vii

8 ee

otis

xi

ew
ee

ee

ee

ee

1

Structural nonlinear analysis

1.2

Sizexoptintization:

ot oF BRE om Om we


4

1.3

Physics-informed neural networks...............

5

lợi

(QJQCHDVỀ

OOS

7

1.5

Organization

ee

9

. « « «sss wewin

2 eek

EER


©...

EMMONS

soe Rw

SELER

eH

EEE

OF

2.2...

NEURAL

NETWORK

TRUSS

STRUCTURES

21

Anoverview..

2.2


PINN for nonlinear analysis...

2.3

oo

EEE

HE ERO

2...

PHYSICS-INFORMED
ANALYSIS

2...

i

1.1

LINEAR

. 2...

ee

FOR

1


NON-

........

0.
2...

ee

11

ee

11

20.20...
00. eee eee

12

2.2.1

Problem siatemenL.........
.

13

2.2.2


Unsupervised learning-based approach amework........

14

PINN for structural stability analysis.................

17

2.3.1

Problem statemeniL.........
.

17

2.3.2

Direct instability-informed neural network framework

20

iv

...


2.4

2.6


Numerical examples

...........

0.0000.

2.4.1.

Material and geometrical nonlinearlles

2.4.2

Geometrical nonlinearity...

2.4.3

Material nonlinearity...

2.4.4

Structural stability...

Comebusiovis

BAYESIAN

. vv

DEEP


ETERIZATION

oe

eww

NEUARL

2.2...
2...
0...

waras

FOR

0...

2.2.02...

00000004

ee

ee

8 8 ee

ee


Be

PARAM-

OPTIMUM

DESIGN

STRUCTURES..................

Imroduelon......

3.2

Statement of structural optimization problem

3.3.

BDNN-based parameterization framework

3.5

000000004

2.0.2...
0.0.00 0004

3.1

3.4


eee

...........

NETWORK-BASED

FRAMEWORK

OEFNONLINEAR

we

eee

.

ch
hy
vẻ

ky vở

............

..............

3.3.1

DNN-based parameterization model


3.3.2

Hyperparameter tuning

.............

.................00.

Numericalexamples..........
. .
0000 pee ee eee
34,1

25-bar space truss

«sees

3.42

52-bardometruss

...

2.0.0.0...
..... 0.0000.

3.4.3

56-bar space truss


2...

0...

344

[20-bar dome truss

Conclusions

..

FOR

ơn...

«oe

a aE

EER EE

EH

0.0... ee
ee

eee


ee

EMM

ee

OBS

eee

ee

2...

PHYSICS-INFORMED
WORK

a0.

oe

ee
NEURAL

STRUCTURAL

ENERGY

FORCE


OPTIMIZATION

NET-

.........

ỘỚIẶẶÁa....

4.2

Structural optimization based on energy-foree methods........

4.3

Physics-informed neural energy-force network

4.4

Numericalexamples

............

........
20.200. 00: eee eee eee


4.5
5

AA


fRendbar truest

4.4.2

200-bar planar truss

4.4.3

25-barsDpacelTUSS

.......

444

72-bar space truss

.....

44.5

em

opm oO we Me

1...

2.

ee


106

120

120-bardometruss...................000.

126

....

ec

ce

AND
<2

2 4 4

Q2
0.0...

peewee
FUTURE
eee

EHH

va


0.002

ey

TẾ TP SE

130

WORK..............

131

ENN

EE

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EEE

EER


EHS

ee
2...

osiciccie

ee

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6 ci

131
133

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PUBLICATIONS

ABSTRACT

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Comelusions


OF

RM

117

REFERENCES.
LIST

eR

và

CONCLUSIONS

`.

se

111

2.

..

wv

ee


Comwebusiotig

Sil

« «sso www

ee

ee ens
eee

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wom

135
150

IN KOREAN

..

1...

ee
ee

152

ACKNOWLEDGEMENTS


..

1...

eee
ee ee

155

vi


LIST

21

Type

of material,

network

problems..........
222

OF

.


TABLES

architecture,

and epoch

.

xa

Comparison of joint displacements and energy for 6-bars truss with
different solution techniques and different materials.

2.3

for different

........

Comparison of member forces for 6-bars truss with different solution techniques and diferent materials.................

2.4

Comparision of joint displacement results for the 31-bars truss with
điÑferent algorithms and various materials...............

2.5

Comparision of member forces for the 31-bars truss with different
algorithms and various materials................


2.6

Loading

conditions

nonlÌinearltY........
3ã

.

geometrical

. cvkg

2...

kg

.-.aAdaa

. ..

. aÁẶẶ..

Comparison of member forces for 25-bars truss with various loading
@OMdvIONS. . vv

2.9


space truss with

Comparison of displacements for 25-bars truss with various loading
conditions...

2.8

for the 25-bar

ve

ee

Rw

aa

woe

ee

ee

ee

ee

ee


Re

Comparison of error percentage of different algorithms with FEM
forthe

25-bar truss:

« sem

ees

vii

BRR

RE

EERE

ERE

EM

MRS


2.10

Comparison of deflection results for the 52-bar dome truss obtained
by different algorithms.


2.11

...........

Comparison

.

of member

. Q0

2...

Comparison

©...)

of member

ee

eee

ee

2...

©...


2.

2...

ee

ee

ee

Ha

Comparison results obtained for the two-bar truss in searching the
..............00.

Comparison results obtained for the two-bar truss in searching the
first snap-through point with varying Ky.

2.18

ee

forces for the 21-bar truss with different

first snap-through point with Ky=0.
2.17

2.


Comparison of error percentage of different algorithms with ILM
for the 21-bar truss).

2.16

0...

2

algorithms and various materials...
2.15

va

Comparison of error percentage of different algorithms with ILM
for the 10-bar truss.

2.14

Q cv

forces for the 10-bar truss with different

algorithms and various materials...
2.13

eee

Comparison of member forces for 52-bars truss obtained by different algorithms...........


2.12

0.000000

Comparison

..............

of results of the triangular truss dome

for searching

the bifuireation pOÏnE.........ee
2.19

Loading conditions for dome truss with 24 bars...............

2.20

Comparison

results obtained for 24-bar dome

the first critical pointy

truss in searching

gssyeees
22a RE REE EEE EROS OS


3.1

Configuration space for the hyperparameters of the network

3.2

Material properties, upper and lower bounds on design variables.

3.3

Loading condition for the 25-bar space truss...............

viii

...

.

70
Tổ


3.4

Optimum

hyperparameters of the network obtained using the BO

with different acquisition functions for the 25-bar space truss.
3.5


...

73

Statistics of the optimal weight with different acquisition functions
for the 25-bar space truss.

. 2...

ee

73

3.6

Comparison of optimal results for the 25-bar space truss.

.....

75

3.7

The displacement constraints of the 25-bar space truss... .....

75

3.8


Optimum hyperparameters obtained by using the BO for different
PEODIGMIS:

sg

gk

kde

EMMSEH

MS

GRR

RE

EERE

LEAH

EMBED

78

..... .

78

.....


79

3.9

Statistics of the optimal weight with different problems.

3.10

Comparison of optimal results for the 52-bar dome truss.

3.11

The displacement constraints of the 52-bar dome truss........

79

3.12

Comparison of optimal results for the 56-bar space truss.

.....

83

3.13

The displacement constraints of the 56-bar space truss...

.....


83

3.14

Comparison of optimal results of the 120 dome truss... ......

87

3.15

The displacement constraints of the 120-bar dome truss. ......

87

4.1

Hyperparameters for the benchmarks tested in this study.

4.2

Comparison of the obtained results for the 10-bar truss with the
first loading condition.

4.3

2...

Comparison


eseaw meas

eee

108

sa

ee

ee

ee

ee

Be we

me

109

...

2...

..

20.2...


0000084

110

Error of the constraints for the 10-bar planar truss with the second
loading condition...

4.6

ee

of the obtained results for 10-bar planar truss with

the second loading condition.
4.5

. 0...

106

Error of the constraints for the 10-bar planar truss with the first
loading condition: . 6

4.4

..

.....

2...


ee

ee 110

Design variables of the 200-bar planar truss... ...........

ix

112


47

Optimization results obtained for the 200-bar planar truss.

...

. 115

4.8

Error of the constraints for the 200-bar planar truss. ........

116

4.9

Loading conditions for the 25-bar space truss (Kips).


........

118

4.10

Stress limitation for the 25-bar space truss................

118

4.11

Optimization results obtained for the 25-bar spaece truss........

119

4.12

Error of the constraints for the 25-bar space truss.

.........

119

4.13

Loading conditions for the 72-bar space truss (kips).

........


121

4.14

Optimization results obtained for the 72-bar space truss (Case 01).

122

4.15

Error of the constraints for the 72-bar planar truss (Case Ol).

4.16

Optimization results obtained for the 72-bar space truss (Case 02).

4.17

Error of the constraints for the 72-bar space truss (Case 02).

4.18

Optimization results obtained for the 120-bar dome truss.

4.19

Error of the constraints for the 120-bar dome truss..........

. . . 122
124


. . . 125

..... 129
130


LIST

OF

FIGURES

1.1

Flowchart depicting the data-driven approach for structural analysis.

12

The schematic process of PINN for linear elasticity problem...

Ded

Deformation oŸ a space truss elemen...................

2.2

The whole process of an unsupervised

..


learning-based framework

for geometrically nonlinear analysis of inelastic truss structures.

. .

2.3

Schematic representation of snap-through and bifurcation points.

2.4

Indirect approach using bi-section method.

2.5

The whole process of the direct instability-informed neural network
framework.

.

.............

2.2...

ee

2.6


Stress—strain relationships considered in the analysis. .......

Đã

A 6-bar planar truss structure.

2.8

The convergence histories of the loss function for the 6-bar truss
with the materials MAT3

2...

2...

ee

and MATA..................

2.9

A 31-bar planar truss structure...)

2.10

The convergence histories of the loss function for the 31-bar truss
wilhalifieventomteriale.

2.11


ww

2...

aan

Schematic of a 25-bar space truss.

xi

14

a ee

ee

2...

2...

ee

ew
2...

ee

ee

ee


ee

we

ee an

000000004

15
18


2.12

52-bar dome space truss structure.

2.13

The convergence history of the loss function for the 52-bar dome
WUE

s oe

Ree

EE

ER


...

2...

ee

36

EMRE
ER REE EE ES Sw we

38

2.14

A 10-bar planar truss structure... 2...

ee

39

2.15

21-bar two-span continuous truss siructure................

41

2.16

Tworbar planar truss.


44

2.17

The loss convergence history of the two-bar truss with K, =0.

2.18

Load-deflection curve for the two-bar truss with Kk, =Q.......

47

2.19

Effect of the spring stiffness on the structural stability........

47

2.20

Triangular truss dome............
. .
ee
ee

49

2221


Load-deflection curve and critical points of the triangular truss dome

51

2.22

Star dome truss with 24 bars........

52

2.23

Case O1: Load-defection
Choe WOUBE.

2.24

mm

ge

ee

SE

.

BE

BS


BE

BỤ BoNn SỈ HỆ Re 3
..

ee

46

curve and critieal points of the 24-bar
eo

OO aE

Oe

ee

EM

MUO

OE

...

ác Các CÁC CÁC HE HE HP HP

RRR


RE

EEE

ER

EROS

. Q Q Q Q Q ng

55

56

y

a

57

Schematic of integration of BDNN-based parameterization framework for structural optimization

3.2

c

Case 03: Load-deflection curve and critical points of the 24-bar
đome fFUSS............


3.1

mm

3a

Case 02: Load-deflection curve and critical points of the 24-bar
KHONG URE:

2.25

cc

:s « «

...............0008.

62

Anexample of EI based Bayesian optimization of a one-dimensional
minimization problem. ............
0.00 eee eee eee

69

3.3

A 25-bar space truss structure.

71


3.4

The convergence

2...

histories of the HPO

space truss structure...

6.

ee

xii

2...

ee

eee

using BO

for the 25-bar

. . .eắe he...

74



3.5

Iteration history of the SQP
for the 25-bar space truss.

3.6

algorithm

for different initial areas

©...)

ee

The weight convergence histories of the optimal network and other
studies for the 25-bar space truss...

2...

0...

02..0.20.
0005

3.7

Schematic of a 52-bar dome †russ sfrueture................


3.8

The convergence history of the HPO using BO for the 52-bar dome
truss siTU€EUF©.......

3.9

.

hư

vờ

Tờ

ky

The weight convergence histories of the optimal network and other
works for the 52-bar dome truss.

.............-...004

3.10

A 56-bar space truss øfUEEUF€.

3.11

The convergence history of the HPO using BO for the 56-bar space

truss siTU€EUF©.......

3.12

. . . . .

HQ

ee

.

hư

vờ

ee

Tờ

ky

The weight convergence histories of the optimal network and FEADE for the 56-bar space truss... 2... ee

3.13

120-bar dome space truss structure.

3.14


The convergence histories of the HPO
dome truss structure...

3.15

..........000.

using BO

for the 120-bar

2...

ee

The weight convergence histories of the optimal network and FEADE

4.1

©...

for the 120-bar dome truss.

..

Process of structural optimization.

2...

0.000.200.0200

00045

(a) Conventional approach in-

cluding optimizer algorithm and structural analysis. (b) Framework
combines between the deep neural network and structural analysis.
(c) Physics-informed neural network without using any structural
analyses...

Q

Q Q kh

kh

ga

ga

xii

ga

k k kg

kg

xi k v và

ko



4.2

Physics-informed neural energy-force networks framework for design optimization...

2...

ee

ee

0.2

ee

4.3

A 10-bar planar truss structure...)

4.4

The weight convergence histories of the 10-bar truss obtained using
the PINEF'N and DE for the first load case.

4.5

2.

ee


2

The weight convergence histories of the 10-bar truss obtained using
the PINEEN

and DE for the second load case.

4.6

A 200-bar planar truss structure.

47

The weight

convergence

using the PINEFN

©...

histories

2...

of the 200-bar

..


2...

2

2...

ee

ee

truss obtained

and other algorthms...............

4.8

A 25-bar space truss structure.

4.9

The weight convergence histories of the 25-bar truss obtained using
the PINEFN

2...

and other algorthms

2...
ee ee


..................

4.10

A 72-bar space truss structure.

4.11

The weight convergence histories of the 72-bar truss obtained using
the PINEFN

4.12

2...

2...

ee

ee

and other algorithms for the first load case.

ee

.....

The weight convergence histories of the 72-bar truss obtained using
the PINEEN


and DE for the second load case.

2...

2.2...

4.13

A 120-bar dome truss structure...

4.14

The weight convergence histories of the 12-bar dome truss obtained
using the PINEEN

2...

2 ee

ee

and DH...................

xiv


CHAPTER

1


INTRODUCTION

1.1.

Structural

nonlinear

analysis

Most mechanical behavior of structures is nonlinear in one way or another [1, 2].
Due to the nonlinear changes of geometrical and material properties, designing
structures under such responses becomes more complex and is required in analysis
to obtain more accurate behavior [3,4]. A variety of algorithms for solving nonlinear structural problems have been proposed during the last few decades, and
they are generally classified into two groups [5]. In the first one, the stiffness-based
approach derived from the FEA

employs

incremental-iterative procedures such

as incremental load method (ILM), Newton-Raphson,

Quasi-Newton, Arc-length

techniques, and so on, to update the stiffness matrix concerning the changes in
the stress-strain relation and geometry
advantages of this methodology,
various nonlinear problems


Owing

to the salient

it has been widely and successfully applied in

[6-8]. Nevertheless, the implementation

proach still requires a large amount
on controlling parameters

of the structure.

of numerical simulation works,

of this apdepending

of the nonlinear incremental-iterative techniques.

circumvent this bottleneck, an alternative approach based on the minimum

To
en-


ergy principle in combination with optimization schemes

was developed to al-

low the direct determination of nonlinear responses without the requirement

any incremental-iterative strategies as those implemented in FEA

of

[9]. According

that core idea, several metaheuristic algorithms known as gradient-free methods
have been successfully applied for the above issue, such as genetic algorithm
harmony

search

[11], and particle swarm

optimization

[10],

[12] etc. Although these

algorithms have the ability to search a near global optimum

solution, they of

ten require a larger number of evaluation functions and are of a relatively slow
convergence speed. Besides, the gradient-based optimization algorithms have also
been successfully applied in this context. For instance, Ohkubo et al. [13] developed a modified sequential quadratic programming algorithm for the structural
analysis by solving potential energy minimization problem
mentary


energy minimization

search method

(ALSM)

problem

(CEMP).

(PEMP)

or comple

In addition, an adaptive local

was also released by Toklu [14]. Despite the fact that

their convergence speed is fairly high, these methods always demand the compulsory calculation of derivative information of the energy function with respect to
the displacements. This performance is one of the main difficulties in solving the
problem, even impossible in many cases.
Nonlinear stability analysis is an integral part and plays a central role in the
structural design process [15,16]. According to investigate the instability, critical
points must be identified along the equilibrium path, which is a major challenge
for FEA [17]. In general, many different algorithms were developed by researchers
for solving structural stability problems. And therein, they were commonly classified into two main groups called indirect and direct methods. The first one relies
on a detecting parameter that will be evaluated during each incremental load step


to gain equilibrium [18-20]. For instance, Riks [21] has tried to locate the limit

point

by using Newton’s

method.

Simo

et al. [22] introduced a scaling vector

to control the constraints for tracking limit points. To save the computational
effort, Shi and Crisfield [23] developed a semi-direct approach to identify singular points. Besides, an exact technique for the determination
was delivered by Chan

[24]. And

of critical points

more recently, several alternatives have been

developed to estimate the singular points

[25-27]. In spite of its success, some

variants still face challenges, such as increased computational cost due to intermediate solutions and its dependency on incremental-iterative techniques as well
as control parameters. To circumvent these drawbacks, the direct approach has
been introduced, in which a system of nonlinear equations is derived to directly
detect the critical points. During the last decades, it has received much interest
from researchers and achieved the remarkable success in this field [28-31]. Wriggers et al. [32,33] were among the first authors to develop this method to find
singular points.


A critical displacement

framework

structural instability by Ofiate and Matias
duced an improved

minimal

augmentation

is developed

for predicting

[34]. Also, Battini et al. [35] introapproach

for the elastic stability of

beam structures. Despite their remarkable success in the stability analysis, they
still have certain limitations. First, as indicated by Shi [17], it required a good
starting vector, sufficiently close to the singular point, and gradient information
to obtain convergence. Moreover, criterion for selecting the starting point highly
depends on expert experience, as well as the incremental-iterative methods.


1.2

Size optimization


Over the past decade,

simultaneous

topology,

shape,

and size optimization

of

structures has received considerable attention from many researchers in the computational mechanics community.
mization of truss structure.

In this thesis, we also examine the size opti-

Its objective is to minimize the structural weight

while satisfying all constraints. In general, it often requires structural analysis
and optimization algorithms to find the solution. And they can be divided into
two main classes. In the first one, the gradient-based algorithms have been successfully applied for searching optimal solutions. For instance, an algorithm based

on the optimality criterion (OC) is developed by Khot et al. [36] and Rizzi [37]
to find the optimal weight of the truss structure. Hrinda et al. [38] has proposed
a new algorithm by combining the design-variable update scheme and arc-length
method.

Besides, a coupling methodology based on the OC


ysis technique was delivered by Saka and Ulker
cost. Schmit and Farshi

and nonlinear anal-

[39] to save the computational

[40] developed a sequence of linear programs to sizing

structural systems. However, this approach cannot deal with the lack of gradient
information from the objective and constraint

functions. The other one is the

gradient-free algorithms which rely on evolutionary and population genetics to
address the optimal design of truss structures, such as firefly algorithm (FA) [41],
harmony

search

[42, 43], genetic algorithms

chaotic coyote algorithm
optimization (TLBO)
search

[44], particle swarm

[46], big bang—-big crunch


algorithm

[45],

[47], teaching-learning-based

[48], hybrid differential evolution and symbiotic organisms

[49], adaptive hybrid evolutionary

lutionary symbiotic organisms

search

firefly algorithm

(AHEFA)

[50], evo-

[51], and so on. Despite these algorithms

have achieved certain success, they require many evaluation functions, slow con-


vergence rate, high computational

cost due to large-scale problems,


and many

turning parameters.
ee a i ai

Data preparation

im

lata

amie

“————————kL—=—=—=—=—=—=————~

|

e Normalization

Structural analysis using

® Dividing data

FEA

up into:

|

+ Training

+ Validation

Output data Y
(displacement, stress ,..)
`

`

cL

wasn

ee EMRE

model

Surrogate model

Sitar ene

Sree cities

\

!

Create input data X
(Loads, BCs, materials)

.


DNN

ena,

;

ae

“+

#

1

1

Build model training
(Layers, nodes,...)

1
1
1
1
1

1

1 |
1

1
1

#

Loss function based on

data MSE(Y,

1

1
1
1

5

Y)

i

|

|

Predicted responses

weights, biases

(displacement, stress,...)


1

{

Deep neural networkbased surrogate model

Training and adjusting

I
1

:

1

`
“Soo

cocoee eo eee

1
1
1
1
1
1

|


|

1

606UC( tll

1

New input data (Loads,
BCs, materials)

NO

ee

¢

id

1
1
1
1
1
1
1
1
1
1
{


Fig.

1.1. Flowchart depicting the data-driven approach for structural analysis.

1.3.

Physics-informed

neural networks

Machine learning (ML) is a computational model inspired by architecture of biological NNs that mimics the way of human brain activities. In recent years, it has
attained remarkable

success in many

fields to help decision-making,

e.g. speed

recognition, industrial automation, medical diagnoses, material informatics, etc.
There are a variety of ML models such as convolutional neural network (CNN),

long short-term memory (LSTM), recurrent neural network (RNN), and so on.
Among

ML

models, NNs have attracted attention in computational mechanics.


As indicated by Li et al. [52], NN-based approaches can be categorized as purely
data-driven and physics-informed ones. Firstly, the data-driven methodology is
often used to build surrogate models, as shown in Fig. 1.1. Note that the training data contains both input and expected output data for this approach. And
the output

data are often obtained through numerical methods,

such as FEA.

In recent times, the data-driven approach has been successfully applied to solve


complex structural analysis problems

[53], materials sciences [54], fluid mechan-

ics [55], structural optimization [56], structural healthy monitoring [57], fracture
mechanics

[58], and so on. Despite the remarkable success achieved in this area,

it is worth mentioning that most of the above studies use a NNs

as the surro-

gate model and employ supervised learning to construct the model. The above
approaches have several disadvantages, making them inefficient such as:
(i) They require a time-consuming effort due to the numerical simulation for
collecting data of training process.


(ii) It is difficult to estimate the suitable training data size.
(iii) They depend strongly on the quality and quantity of data to build the
high-accuracy data-driven predictable model.
Deep Neural network

Training
Loss function

`...

|

Đ(c:)

Update rule

Automatic
differentiation

Backpropagation

Fig. 1.2. The schematic process of PINN for linear elasticity problem.
Hence, physics-informed neural networks have been developed to circumvent these
challenges. PINNs are algorithms from deep learning leveraging physical laws by
including partial differential equations (PDEs) together with a respective set of
boundary

and initial conditions as penalty terms into their loss function.

whole process of PINN


The

is illustrated in Fig. 1.2 for linear elasticity problem. It


has received much attention in computational mechanics fields. In comparison to
conventional numerical solvers, it has the following advantages:
(i) It can easily handle the problems with irregular domains as well as completely avoids a discretization like FEA.
(ii) The training data are easily collected from the known

design information

of the structure without any numerical simulation, for example, boundary
conditions, geometry, properties of materials, etc.
(iii) One of its outstanding characteristic is that the sensitivity can be quickly
and easily calculated with a back-propagation algorithm of the network.
PINNs have recently been proven effective for solving PDEs

[59], fluid mechan-

ics [60], structural analysis [61], and so on. However, this paradigm also depends
significantly on the resolution of samples,

sampling techniques,

the way of es-

timating a suitable training data size, tuning hyperparameters


as well as the

local minimum.

Furthermore, it has still not been yet utilized for geometrically

nonlinear analysis of inelastic truss structures, structural stability analysis, and
structural optimization design thus far.

1.4

Objective

The mainly focus of this dissertation is on developing PINN

framework for the

analysis and optimization of structures. The following specific aims are addressed
in this thesis:

(i) A robust PINN framework is proposed to perform the geometrically nonlinear analysis of inelastic truss structures. In this approach, NN is employed
to directly estimate

nonlinear

structural responses without

utilizing any