Physics informed neural networks for the analysis and optmization of structures

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 ro

Physics-informed neural networks for the

February 2023

Department of Architectural Engineering

Sejong University

Physics-informed neural networks for the

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

Major Advisor

Physics-informed neural networks for the analysis

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 model is first to analyze the stability of truss structures Different from most existing work, neural network (NN) 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 loss function without utilizing any incremental- iterative methods 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 fi­nite element analysis (FEA) and arc-length method are satisfied Furthermore, 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 physics- informed neural energy-force network (PINEFN) framework is first constructed

to directly solve the optimum design of truss structures that structural analy­sis is completely removed 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 con­sidered as design variables In this model, spatial coordinates of truss members are examined as input data, while corresponding cross-sectional areas and re­dundant forces unknown to the network are taken account of output Obtained outcomes indicated that it not only reduces the computational cost dramatically

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 stabil­ity, Hyperparameter optimization, Force method, Critical points, Com­plementary energy, Bayesian optimization, Truss optimization

ABSTRACT i

LIST OF TABLES vii

LIST OF FIGURES xi

1 INTRODUCTION 1

1.1 Structural nonlinear analysis 1

1.2 Size optimization 4

1.3 Physics-informed neural networks 5

1.4 Objective 7

1.5 Organization 9

2 PHYSICS-INFORMED NEURAL NETWORK FOR NON­ LIN EAR ANALYSIS OF TRUSS STRUCTURES 11

2.1 An overview 11

2.2 PINN for nonlinear analysis 12

2.2.1 Problem statement 12

2.2.2 Unsupervised learning-based approach framework 14

2.3 PINN for structural stability analysis 17

2.3.1 Problem statement 17

2.3.2 Direct instability-informed neural network framework 20

2.4 Numerical examples 23

2.4.1 Material and geometrical nonlinearities 23

2.4.2 Geometrical nonlinearity 31

2.4.3 Material nonlinearity 38

2.4.4 Structural stability 43

2.5 Conclusions 56

3 BAYESIAN DEEP NEUARL NETWORK-BASED PARAM­ ETERIZATION FRAMEWORK FOR OPTIMUM DESIGN OF NONLINEAR STRUCTURES 58

3.1 Introduction 58

3.2 Statement of structural optimization problem 60

3.3 BDNN-based parameterization framework 61

3.3.1 DNN-based parameterization model 63

3.3.2 Hyperparameter tuning 66

3.4 Numerical examples 70

3.4.1 25-bar space truss 71

3.4.2 52-bar dome truss 77

3.4.3 56-bar space truss 81

3.4.4 120-bar dome truss 85

3.5 Conclusions 89

4 PHYSICS-INFORMED NEURAL ENERGY FORCE NET­ WOR K FOR STRUCTURAL OPTIMIZATION 90

4.1 Introduction 90

4.2 Structural optimization based on energy-force methods 94

4.3 Physics-informed neural energy-force network 101

4.4 Numerical examples 105

4.4.1 Ten-bar truss 106

4.4.2 200-bar planar truss Ill 4.4.3 25-bar space truss 117

4.4.4 72-bar space truss 120

4.4.5 120-bar dome truss 126

4.5 Conclusions 130

5 CONCLUSIONS AND FUTURE WORK 131

5.1 Conclusions 131

5.2 Future work 133

REFERENCES 135

LIST OF PUBLICATIONS 150

ABSTRACT IN KOREAN 152

ACKNOWLEDGEMENTS 155

LIST OF TABLES

2.1 Type of material, network architecture, and epoch for different problems 252.2 Comparison of joint displacements and energy for 6-bars truss with different solution techniques and different materials 272.3 Comparison of member forces for 6-bars truss with different solu­tion techniques and different materials 272.4 Comparision of joint displacement results for the 31-bars truss with different algorithms and various materials 292.5 Comparision of member forces for the 31-bars truss with different algorithms and various materials 302.6 Loading conditions for the 25-bar space truss with geometrical nonlinearity 322.7 Comparison of displacements for 25-bars truss with various loading conditions 332.8 Comparison of member forces for 25-bars truss with various loading conditions 342.9 Comparison of error percentage of different algorithms with FEM for the 25-bar truss 35

2.10 Comparison of deflection results for the 52-bar dome truss obtained

by different algorithms 372.11 Comparison of member forces for 52-bars truss obtained by differ­ent algorithms 372.12 Comparison of member forces for the 10-bar truss with different algorithms and various materials 402.13 Comparison of error percentage of different algorithms with ILM for the 10-bar truss 412.14 Comparison of member forces for the 21-bar truss with different algorithms and various materials 422.15 Comparison of error percentage of different algorithms with ILM for the 21-bar truss 432.16 Comparison results obtained for the two-bar truss in searching thefirst snap-through point with Ku=0 452.17 Comparison results obtained for the two-bar truss in searching thefirst snap-through point with varying Ku 482.18 Comparison of results of the triangular truss dome for searching the bifurcation point 502.19 Loading conditions for dome truss with 24 bars 532.20 Comparison results obtained for 24-bar dome truss in searching the first critical point 543.1 Configuration space for the hyper parameters of the network 703.2 Material properties, upper and lower bounds on design variables 723.3 Loading condition for the 25-bar space truss 72

3.4 Optimum hyperparameters of the network obtained using the BO

with different acquisition functions for the 25-bar space truss 73

3.5 Statistics of the optimal weight with different acquisition functions for the 25-bar space truss 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 problems 78

3.9 Statistics of the optimal weight with different problems 78

3.10 Comparison of optimal results for the 52-bar dome truss 79

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 106

4.2 Comparison of the obtained results for the 10-bar truss with the first loading condition 108

4.3 Error of the constraints for the 10-bar planar truss with the first loading condition 109

4.4 Comparison of the obtained results for 10-bar planar truss with the second loading condition 110

4.5 Error of the constraints for the 10-bar planar truss with the second loading condition 110

4.6 Design variables of the 200-bar planar truss 112

4.7 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 space 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 01) 122

4.16 Optimization results obtained for the 72-bar space truss (Case 02) 124 4.17 Error of the constraints for the 72-bar space truss (Case 02) 125

4.18 Optimization results obtained for the 120-bar dome truss 129

4.19 Error of the constraints for the 120-bar dome truss 130

LIST OF FIGURES

1.1 Flowchart depicting the data-driven approach for structural analysis 51.2 The schematic process of PINN for linear elasticity problem 62.1 Deformation of a space truss element 142.2 The whole process of an unsupervised learning-based frameworkfor geometrically nonlinear analysis of inelastic truss structures 152.3 Schematic representation of snap-through and bifurcation points 182.4 Indirect approach using bi-section method 192.5 The whole process of the direct instability-informed neural network framework 212.6 Stress—strain relationships considered in the analysis 242.7 A 6-bar planar truss structure 252.8 The convergence histories of the loss function for the 6-bar truss with the materials MAT3 and MAT4 262.9 A 31-bar planar truss structure 282.10 The convergence histories of the loss function for the 31-bar truss with different materials 312.11 Schematic of a 25-bar space truss 32

2.12 52-bar dome space truss structure 36

2.13 The convergence history of the loss function for the 52-bar dome truss 38

2.14 A 10-bar planar truss structure 39

2.15 21-bar two-span continuous truss structure 41

2.16 Two-bar planar truss 44

2.17 The loss convergence history of the two-bar truss with Ku = 0 46

2.18 Load-deflection curve for the two-bar truss with Ku = 0 47

2.19 Effect of the spring stiffness on the structural stability 47

2.20 Triangular truss dome 49

2.21 Load-deflection curve and critical points of the triangular truss dome 51 2.22 Star dome truss with 24 bars 52

2.23 Case 01: Load-deflection curve and critical points of the 24-bar dome truss 55

2.24 Case 02: Load-deflection curve and critical points of the 24-bar dome truss 56

2.25 Case 03: Load-deflection curve and critical points of the 24-bar dome truss 57

3.1 Schematic of integration of BDNN-based parameterization frame­ work for structural optimization 62

3.2 An example of EI based Bayesian optimization of a one-dimensional minimization problem 69

3.3 A 25-bar space truss structure 71

3.4 The convergence histories of the HPO using BO for the 25-bar space truss structure 74

3.5 Iteration history of the SQP algorithm for different initial areas for the 25-bar space truss 763.6 The weight convergence histories of the optimal network and other studies for the 25-bar space truss 763.7 Schematic of a 52-bar dome truss structure 773.8 The convergence history of the HPO using BO for the 52-bar dome truss structure 803.9 The weight convergence histories of the optimal network and other works for the 52-bar dome truss 803.10 A 56-bar space truss structure 813.11 The convergence history of the HPO using BO for the 56-bar space truss structure 843.12 The weight convergence histories of the optimal network and FEA-

DE for the 56-bar space truss 843.13 120-bar dome space truss structure 853.14 The convergence histories of the HPO using BO for the 120-bar dome truss structure 883.15 The weight convergence histories of the optimal network and FEA-

DE for the 120-bar dome truss 884.1 Process of structural optimization, (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 91

4.2 Physics-informed neural energy-force networks framework for de­sign optimization 1034.3 A 10-bar planar truss structure 1064.4 The weight convergence histories of the 10-bar truss obtained using the PINEFN and DE for the first load case 1094.5 The weight convergence histories of the 10-bar truss obtained using the PINEFN and DE for the second load case Ill4.6 A 200-bar planar truss structure 1134.7 The weight convergence histories of the 200-bar truss obtainedusing the PINEFN and other algorithms 1164.8 A 25-bar space truss structure 1174.9 The weight convergence histories of the 25-bar truss obtained usingthe PINEFN and other algorithms 1204.10 A 72-bar space truss structure 1214.11 The weight convergence histories of the 72-bar truss obtained usingthe PINEFN and other algorithms for the first load case 1234.12 The weight convergence histories of the 72-bar truss obtained using the PINEFN and DE for the second load case 1254.13 A 120-bar dome truss structure 1274.14 The weight convergence histories of the 12-bar dome truss obtained using the PINEFN and DE 128

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 of the structure Owing to the salient advantages of this methodology, it has been widely and successfully applied in various nonlinear problems [6-8) Nevertheless, the implementation of this ap­proach still requires a large amount of numerical simulation works, depending

on controlling parameters of the nonlinear incremental-iterative techniques To circumvent this bottleneck, an alternative approach based on the minimum en­

ergy principle in combination with optimization schemes was developed to al­low the direct determination of nonlinear responses without the requirement of any incremental-iterative strategies as those implemented in FEA [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 [10], harmony search [11], and particle swarm optimization [12] etc Although these algorithms have the ability to search a near global optimum solution, they of­ten 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 (PEMP) or comple­mentary energy minimization problem (CEMP) In addition, an adaptive local search method (ALSM) 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 of critical points was delivered by Chan [24] And 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] Wrig- gers et al [32,33] were among the first authors to develop this method to find singular points A critical displacement framework is developed for predicting structural instability by Onate and Matias [34], Also, Battini et al [35] intro­duced an improved minimal augmentation approach 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 In this thesis, we also examine the size opti­mization of truss structure 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 and nonlinear anal­ysis technique was delivered by Saka and Ulker [39] to save the computational cost Schmit and Farshi [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 [44], particle swarm algorithm [45], chaotic coyote algorithm [46], big bang-big crunch [47], teaching-learning-based optimization (TLBO) [48], hybrid differential evolution and symbiotic organisms search [49], adaptive hybrid evolutionary firefly algorithm (AHEFA) [50], evo­lutionary symbiotic organisms search [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.

Data preparation DNN model Surrogate model

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 Helds 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 bẹ 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 mechan­ics [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 surro­gate 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

Backpropagation

Fig 1.2 The schematic process of FINN 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 The whole process of FINN 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 mechan­ics [60], structural analysis [61], and so on However, this paradigm also depends significantly on the resolution of samples, sampling techniques, the way of es­timating 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

time-consuming incremental-iterative algorithms as those done in standard FEA by minimizing the total potential energy under BCs Furthermore, its learning possibility is only relied upon the set of the nodal coordinates

It is a simple approach to the nonlinear analysis with the high efficiency and reliability The proposed model has been successfully tested for several numerical examples including geometric nonlinearity, material nonlinearity, and dual nonlinearities

(ii) A direct PINN is first developed to analyze the stability of truss struc­tures Instead of resolving the system of nonlinear equations consisting of the residual load and the test function as the conventional direct method, its residual as a loss function is minimized by our approach to indicate the optimal parameters of the network corresponding to the critical point The training data are just a set of spatial coordinates of the joints The effective­ness and robustness of the present method are verified via testing several benchmark examples to indicate the critical points including the limit and bifurcation points

(iii) A robust DNN-based parameterization framework is suggested to directly solve the optimum design for geometrically nonlinear trusses that the hy­perparameters of the network are automatically adjusted by BO Herein, the design variables of the structure are parameterized by weights and bi­ases of the network with the spatial coordinates of all joints as the train­ing data And the value of the loss function is established based on the predicted cross-sectional areas and deflection constraints obtained by sup­porting FEA This approach has been successfully applied to the optimum design problems and has proven its effectiveness in enhancing the accuracy

of optimal solution and escape from the local minima by hyperparameter optimization (HPO).

(iv) A PINEFN framework is first proposed to directly solve the optimum design

of truss structures that structural analysis is completely removed from the implementation of the global optimization Therein, the loss function is constructed to guide the training network based on physical laws and BCs The spatial coordinates of all truss members are treated as the training data which are easily gathered from the connectivity information of the structure The reliability, efficiency, and applicability of the proposed model are also demonstrated through several examples for the truss design under various constraints The obtained results showed that it not only saves a large computational cost but also yields higher accuracy, faster convergence speed, and not using any structural analyses

1.5 Organization

This dissertation includes 5 Chapters, and it is organized as follows:

Chapter 1 introduces an overview on the structural nonlinear, stability analyses, size optimization as well as deep neural network, and objective of this study

Chapter 2 presents the PINN approach to perform the geometrically nonlinear analysis of inelastic truss structures as well as structural stability analysis

Chapter 3 suggests the robust DNN-based parameterization framework to solve the optimum design for geometrically nonlinear trusses that BO is applied to self-adjusting hyperparameters of the network

Chapter 4 develops the PINEFN to tackle the optimum design of truss structures

without using any structural analyses.

Chapter 5 closes this thesis with crucial conclusions of this work And finally, several recommendations to future work are drawn

cated at the training ends by our network with the optimal network parameters The suggested methodology is also extremely simple to implement, while the un­labeled data is available, small in size, independent of sampling techniques, and without using any incremental-iterative algorithm and FEAs Finally, the present framework shows the effectiveness, reliability, high accuracy, and potential.

2.2 PINN for nonlinear analysis

2.2.1 Problem statement

The primary objective of structural analysis is to determine the responses under the action of external loads that the total potential energy (TPE) is minimized for the equilibrium configuration [8, 62], Let US consider a truss-like structure

that consists of n truss members and m nodes Then the TPE of the system np

including the strain energy u and external work w can be expressed as [62]

where Lfe, and Ak are the strain energy density, length, and cross-sectional

area of the fcth member; u is the vector of displacements at nodes; p denotes the vector of external forces; and ơk are strain and stress of the A; th member,

and Ơ (s') expresses the stress-strain relation for the material of member In order

to achieve the strain field, a space truss element with its initial configuration Lq

and current one Lc is considered as shown in Fig 2.1 It is easily to compute

L q and Lc of the element via coordinates of nodes (xị, yi, Zị, Xj, yj, Zj) and displacement field (uị, Vị, Wị, Uj, Vj, Wj) They are defined as follows

Lo = y(xj -Xi)2 + (yj -yi)2 + (zj - Zị)2 (2.5)

Alm ansi strain :

Lc—Lp

£ e L o ’

SG = LÌ-LỊ 2Ll ' SL

to be completely known for a given material type Hence, if the displacements

of member ends are determined, the strain and strain energy density of each member can be easily estimated by the integral Eq 2.4

Fig 2.1 Deformation of a space truss element.

According to the principle of minimum potential energy, the nodal displacements vector u are considered as unknown solutions and are determined by minimizing the TPE with BCs Once the displacement field is found, the other structural responses can be completely determined by the constitutive equations

2.2.2 Unsupervised learning-based approach framework

A flowchart of the overall proposed algorithm is illustrated in Fig 2.2 where the

parameters 0 including weights and biases of the network are design variables

Note that the framework presented here relies on unsupervised learning, so the output data is not given here In other words, the responses of the structure including displacement, strain, stress, member force, etc are not included in

the training data It means that we only have input data including a set of coordinates (xị, yi, Zi) of all structural nodes for the learning process Hence, the training data is easily obtained from the geometry of the structure and completely independent of sampling techniques Additionally, its size is small (m X 2) and (m X 3) for planar and space truss structures, respectively Here m denotes the number of nodes, while 2 or 3 implies the number of spatial coordinates for 2- or 3-dimensional truss.

Total potential energy

According to this scheme, a fully connected NN with 3 layers is set first up with the initial weight and bias values according to the normal distribution criterion Therein, the input layer is known as the first layer which consists of three neurons corresponding with the spatial coordinates (x,i/,z), the second layer as hidden layer consists of mfr neurons which depends on the complexity of the application, and the last layer is called the output layer with three neurons which corresponds

to the predicted displacements (ủ, u, w) And the neurons of the present layer are connected to all units in the previous layer via the parameters of the NN which

consists of weights and biases b^\ respectively.

Firstly, a mapping from input to output nodes can be expressed as ỉ: R3 R3

by FF process When the data is transmitted from the first layer to the last layer

by the transformations And the relation between the input and output of each layer is expressed as

input layer : o° = [x, y, z] € R3,

hidden layer: Ô1 = /1 (w1o° + b1) e Rm\

output layer: Ô2 = /2 (w2ô1 + b2) = [Ú, V, w] e R3,

(2-8)

where /1(.) and /2C) are in turn the activation function for hidden and output layers, which supports the network to learn the nonlinear relationship between input and output, and make accurate predictions Several common choices include Linear, ReLU, LeakyReLU, Sigmoid, Softmax, and Tanh Ô1 and Ô2 denote the output of the hidden and output layers, respectively

Next, the TPE is establish as a loss function by a sum of the strain energy and external work based on the outputs, loads, and BCs Its mathematical expression

is given as follows

£(0) =Ỉ7(Ũ(0)) +T7(Ũ(0)) , (2.9)where Ũ is the predicted displacement vector; 0 is the parameter vector including weights and biases of the network At the same time, it is easily seen that instead

of solving the nonlinear analysis problem by using incremental-iterative methods

as conventional methods, we now turn to minimize the loss function by training

to find the optimal parameters Ỡ* of the NN.

Ỡ* = arg min (c (0) ) (2.10)

e

To do this, BP is required to automatically calculate the gradient information of the loss with respect to the weights and biases And then, they are updated in the direction of the gradient descent by optimizer Adam And the above operations are performed and repeated many times until reaches the minimum loss function value which is called the training process Once the network is properly trained, the results in the context of nonlinear structural analysis can be found with minimum energy

2.3 PINN for structural stability analysis

2.3.1 Problem statement

The primary objective of structural stability analysis is to determine the position

of critical points And their outstanding characteristic is a singular tangent stiff­ness matrix Ky [25] When the structural system may lose its stability in dealing with various geometrical or material properties or both changes However, this work only considers the effect of geometrical changes on the instability arising Based on the phenomena of structural instability, there are two types of singular points: limit (snap-through) and bifurcation points, as shown in Fig 2.3 Note that the sign of the determinant of the stiffness matrix Ky changes in the vicinity

of the limit point, whereas it does not vary with the bifurcation point [17,64] A standard criterion for the distinction between the limit and bifurcation points is given as follows [32,33]

Limit points

Bifurcation points

where <Ị> is the critical eigenvector, and q denotes the reference load vector

Fig 2.3 Schematic representation of snap-through and bifurcation points

For the indirect method, its underlying principle relies on the combination of a test function and numerical methods to estimate the singular point [23,65,66] And this test function allows evaluating the positive definiteness of Kt such as the determinant, smallest pivot, smallest eigenvalue, etc This scheme is illustrated

in Fig 2.4 In this way, incremental-iterative procedures are used to trace the nonlinear behaviors of structure, and almost at the same time, the value of the test function T is determined Normally, its value at the stable equilibrium state

is positive, and vice versa is unstable Hence, when the sign of T changes between

two consecutive increments (1 and 2), there occurs a critical point in this interval

To locate the singular point (5), the numerical method, such as the Bisection method, is applied to reduce the amount of uncertainly between them to the user-defined threshold

Fig 2.4 Indirect approach using bi-section method

In the second tactic, the direct procedure, which is known as one of the most powerful techniques, was developed and widely applied in structural stability analysis to enhance computational efficiency [28-30] As noted in this procedure, the critical points are indicated directly by solving a system of nonlinear equations consisting of the equilibrium and test functions [67] According to Wriggers and

Simo [32], it is defined as

f (u) - Aq

w-1

where u denotes the displacement field; A represents the load factor; f refers

to the internal force vector, and Ộ indicates the critical eigenvector There are

(2n +1) design variables where n is the number of acting degrees of freedom It

is worth mentioning that the iterative method is used only to obtain the singular point, and there would not trace the behavior of the structure

2.3.2 Direct instability-informed neural network framework

A direct instability-informed neural network (DIINN) framework is first presented

in this section to detect the point of structural instability Herein, the NN acts

as an identification model to detect the critical point by minimizing the loss function, and its whole process is shown in Fig 2.5 Overall, this framework is similar to the previous approach from the implementation point of view However, there are significant differences between them Firstly, the output layer consists

of three neurons corresponding to the unknown displacements and load factor Next, the initial parameters of the network were set to zeroes, which corresponds

to the unloading state of the structure And finally, the loss function involving the residual load and property of the stiffness matrix is established to minimize by the training process As soon as the training process completed, the first critical point will be located by our model corresponding to the minimum loss function Therefore, we mainly focus on a detailed description of the building loss function

Fig 2.5 The whole process of the direct instability-informed neural network framework.

The first one provides briefly the derivation of internal force vector and tangent stiffness matrix Let US consider a space truss element in its initial and current configurations shown in Fig 2.1 The interested readers may refer to Ref [68] for

a complete treatment of the problem As a consequence, the internal force vector

of the eth truss member is expressed as

where E, and A are the elastic modulus and the cross-section area, respectively;The matrix B is defined as follows

B = 1

^0 Lex Ley Lcz Lex Ley Lez (2.15)

in which Lcx — Xj Xị 4- Uj Uịy Ley — yj yị T Vj Lcz — Zj Zị 4- Wj yji'i

Lo, L c and £ are the initial, final lengths, and strain of the eth element, and they easily obtained from Eqs 2.5-2.7, respectively;

The tangent stiffness matrix of the eth truss member is obtained by differentiating the internal force vector with respect to the nodal displacements [68].

2

g (u, a) + min {diag(JD2) (2.17)

where ù (x, y, z, Ổ) is the displacement field with the fulfilled boundary conditions;

0 denotes the parameter vector of the network including weights and biases; X, y, and z are the coordinate vectors of the joints; Â represents the load parameter which is computed by a sum of the load factor components Aị, as follows