Luận văn thạc sĩ Kỹ thuật xây dựng: Machine learning in predicting mechanical behavior of 3D printed beams with triply periodic minimal surface (TPMS) sandwich cores

VIETNAM NATIONAL UNIVERSITY HO CHI MINH CITY HO CHI MINH UNIVERSITY OF TECHNOLOGY

-

TRAN QUOC KIM

MACHINE LEARNING IN PREDICTING MECHANICAL BEHAVIOR OF 3D PRINTED BEAMS WITH TRIPLY

PERIODIC MINIMAL SURFACE (TPMS) SANDWICH CORES

Major: Civil Engineering Major ID: 8580201

MASTER THESIS

HO CHI MINH CITY, JANUARY 2023

TRAN QUOC KIM

MACHINE LEARNING IN PREDICTING MECHANICAL BEHAVIOR OF 3D PRINTED BEAMS WITH TRIPLY

PERIODIC MINIMAL SURFACE (TPMS) SANDWICH CORES

MÔ HÌNH MÁY HỌC TRONG DỰ ĐOÁN ỨNG XỬ CƠ HỌC CỦA DẦM IN 3D GIA CƯỜNG LÕI SANDWICH

BỀ MẶT CỰC TIỂU TAM TUẦN HOÀN

Major: Civil Engineering Major ID: 8580201

MASTER THESIS

HO CHI MINH CITY, January 2023

THIS THESIS IS ACCOMPLISHED AT

HO CHI MINH UNIVERSITY OF TECHNOLOGY – VNU HCMC Supervisors: Dr Nguyen Thi Bich Lieu Signature:

Assoc Prof Luong Van Hai Signature:

Examiner 1: Dr Thai Son Signature: Examiner 2: Dr Nguyen Phu Cuong Signature:

The master thesis is defended at Ho Chi Minh University of Technology – VNU HCMC on 13th January 2023

The thesis defense grading committee consists of: 1 Chairman: Assoc Prof Do Nguyen Van Vuong 2 Secretary: Dr Nguyen Thai Binh

3 Reviewer: Dr Thai Son

4 Reviewer: Dr Nguyen Phu Cuong

5 Council Member: Assoc Prof Luong Van Hai

Confirmations of the Chairman of thesis defense grading committee and the Dean of faculty of thesis major after the thesis has been corrected (if any)

CHARIMAN OF DEAN OF FACULTY

THESIS COMMITTEE FACULTY OF CIVIL ENGINEERING

Assoc Prof Do Nguyen Van Vuong

VIETNAM NATIONAL UNIVERSITY HCMC HO CHI MINH UNIVERSITY

OF TECHNOLOGY

SOCIALIST REPUBLIC OF VIETNAM Independence - Freedom - Happiness

MASTER THESIS ASSIGNMENTS

Full name: Tran Quoc Kim Student ID: 2170979 Date of birth: 30/07/1999 Place of birth: Can Tho Major: Civil Engineering Major ID: 8580201 I THESIS TITLE:

Machine learning in predicting mechanical behavior of 3D printed beams with triply periodic minimal surface (TPMS) sandwich cores

Mô hình máy học trong dự đoán ứng xử cơ học của dầm in 3D gia cường lõi sandwich bề mặt cực tiểu tam tuần hoàn

II THESIS ASSIGNMENTS AND CONTENTS:

1 Modeling the triply periodic minimal surface (TPMS) core reinforced beam, and comparing with experimental results;

2 Collecting beam’s behavior data based on simulations while changing beam’s geometric properties;

3 Creating a machine learning model to predict mechanical behavior of the beams

III DATE OF DELIVERING: 05/09/2022 IV DATE OF COMPLETION: 27/12/2022

V SUPERVISORS: Dr Nguyen Thi Bich Lieu Assoc Prof Luong Van Hai

Ho Chi Minh City, 6th March 2023 SUPERVISORS HEAD OF

ii

ACKNOWLEDGEMENT

Having had the opportunity to study a master's program at the Vietnam National University Ho Chi Minh City – Bach Khoa University, I would like to express my sincere appreciation to the school administrators and departments for creating favorable conditions for me to complete the study program I also would like to express my gratitude to the lecturers of the Faculty of Civil Engineering who have always been dedicated to teaching and imparting useful knowledge

In addition, I would like to express my deep acknowledgements to Professor Nguyen Xuan Hung and the CIRTech Institute of Technology for giving me the opportunity to work with the excellent lecturers and colleagues at here

I would not be able to complete this thesis without the guidance of my supervisors Doctor Nguyen Thi Bich Lieu and Associate Professor Luong Van Hai I would like to express my sincere gratitude to them Their orientations and suggestions are both the motivation and the objective to help me steady to complete the thesis

I am truly grateful to Vingroup Innovation Fund (VinIF) for the financial support during my research and implementation of the thesis under project code VINIF.2019.DA04

Moreover, I would like to thank my family and friends for always supporting and encouraging me throughout the study and research process

Finally, I would like to wish my teachers, colleagues, family and friends good health, successfulness and happiness

This thesis may have several shortcomings, so I would like to receive valuable comments from committee members and other students

ABSTRACT (Presented in English)

Bioinspired porous structures are highly porous structures with an outstanding strength-to-weight ratio Their application has been applied in various fields such as aerospace and biomedical engineering, transportation, etc Recent research has indicated that 3D-printed plastic triply periodic minimal surfaces (TPMS) structure has tremendous impacts on cement beams, reducing maximum deflection, improving peak load, and enhancing ductility This study proposes a machine learning (ML) surrogate model to predict beam behaviors subjected to a static bending load To reinforce the considering beams, different combinations of core layer numbers and plastic volume fractions are adopted Their influences are investigated using the Finite Element Method (FEM) Consequently, the gathered data are used to develop the ML model through a three-phase assessment to achieve the most appropriate model for the present problem This assessment consists of model hyperparameter tuning, first performance assessment, and overfitting handling with Deep Learning (DL) techniques The results indicate a proportional relationship between the volume fraction and the beam peak load as well as the maximum deflection while increasing the number of TPMS layers enhances these properties nonlinearly Additionally, from the model predictions, there might be a limit value that each trait cannot achieve at a specific volume fraction with any number of layers The final model developed in this study is verified by the maximum deviations between FEM and predictions for peak loads and maximum deflections, that are 2.5% and 3.5%, respectively A new early stopping condition can maximize the final model performances on both train and test data, therefore verifying the model's reliability in handling noisy data from FEM

iv

TÓM TẮT LUẬN VĂN

(Trình bày bằng tiếng Việt)

Cấu trúc xốp được lấy cảm hứng từ sinh học là cấu trúc có độ xốp cao và tỉ lệ cường độ trên trọng lượng lớn Chúng được ứng dụng trong nhiều lĩnh vực như hàng không vũ trụ, kỹ thuật sinh học, giao thông vận tải, vv Nghiên cứu gần đây đã chỉ ra rằng cấu trúc bề mặt cực tiểu tam tuần hoàn (TPMS) bằng nhựa in 3D có tác động to lớn đến các dầm xi măng, giảm độ võng cực đại, cải thiện tải trọng giới hạn, và tăng cường độ dẻo dai Nghiên cứu này đề xuất một mô hình thay thế máy học (ML) để dự đoán hành vi của dầm chịu tải uốn tĩnh Để gia cố các dầm, nhiều kết hợp khác nhau của số lớp lõi và tỉ lệ thể tích nhựa được áp dụng Ảnh hưởng của chúng được khảo sát bằng phương pháp phần tử hữu hạn (FEM) Từ đó, dữ liệu được thu thập được sử dụng để phát triển mô hình ML thông qua phương pháp đánh giá ba bước để tìm ra được mô hình phù hợp nhất cho vấn đề hiện tại Đánh giá này bao gồm điều chỉnh siêu tham số của mô hình, đánh giá hiệu quả mô hình đầu và xử lý quá khớp bằng các kỹ thuật học sâu (DL) Kết quả cho thấy mối quan hệ tỉ lệ thuận giữa tỉ lệ thể tích và tải trọng giới hạn của dầm cũng như độ lệch tối đa trong khi tăng số lớp TPMS cải thiện các tính chất này một cách phi tuyến Ngoài ra, các dự đoán của mô hình chứng minh được tồn tại giá trị giới hạn mà mỗi đặc điểm không thể đạt được ở một tỉ lệ thể tích cụ thể với bất kỳ số lớp nào Mô hình cuối cùng được phát triển trong nghiên cứu này được kiểm chứng bằng các sai số tối đa giữa FEM và các dự đoán cho tải cực đại và độ lệch tối đa, lần lượt là 2,5% và 3,5% Một điều kiện dừng sớm mới có thể tối đa hóa hiệu quả mô hình trên cả hai tập dữ liệu huấn luyên và kiểm thử, qua đó chứng minh độ tin cậy của mô hình đối với các dữ liệu phức tạp từ FEM

COMMITMENT

I hereby declare that this thesis entitled "Machine learning in predicting mechanical behavior of 3D printed beams with triply periodic minimal surface (TPMS) sandwich cores” is my research work All sources referenced are properly and fully cited The research data and results in this thesis are guaranteed to be honest and have never been used to defend any other theses

I will take full responsibility for this statement

Ho Chi Minh City, 6th March 2023 GRADUATE STUDENT

TRAN QUOC KIM

LIST OF ABBREVIATIONS viii

LIST OF TABLES AND CHARTS ix

LIST OF FIGURES x

CHAPTER 1 INTRODUCTION 1

1.1 Research topic 1

1.2 Research objective and contents 2

1.3 Research object and scope 3

CHAPTER 3 THEORETICAL BACKGROUND 9

3.1 Triply periodic minimal surface structures 9

3.3.2 Materials 21

3.3.3 Finite element analysis simulation 23

3.4 Machine learning model 27

3.4.1 Introduction to machine learning 27

3.4.2 Artificial neural networks 28

3.4.3 Deep Learning 36

3.5 Thesis tasks 37

CHAPTER 4 RESULTS AND DISCUSSIONS 40

4.1 Finite element method process 40

4.1.1 Mesh convergence study 40

4.1.2 Impact of TPMS-core properties 41

4.2 Machine learning process 46

4.2.1 Model hyperparameter tuning 46

4.2.2 Model assessment 51

4.2.3 Handling overfitting 52

4.2.4 Best model predictions 57

CHAPTER 5 CONCLUSION AND RESEARCH DEVELOPMENT DIRECTION 63

viii

LIST OF ABBREVIATIONS

Abbreviation Meaning

MS Minimal Surface

TPMS Triply Periodic Minimal Surface

SCDP Simplified Cementitious Damage Plasticity AM Additive Manufacturing

FEM Finite Element Method FEA Finite Element Analysis

FSDT First-order Shear Deformation Theory AI Artificial Intelligence

ML Machine Learning DL Deep Learning

ANN Artificial Neural Networks

LIST OF TABLES AND CHARTS

Table 3.1 The investigating TPMS beams’ geometric descriptions and labels [1] 19

Table 3.2 The core parameters of the investigating beams [1] 20

Table 3.3 The mixture of the cementitious mortar [19] 21

Table 3.4 Mechanical characteristics of the cement material [19] 22

Table 3.5 The SCDP model parameters of the cement material [19] 23

Table 3.6 Mechanical characteristics of the ABS plastic material [19] 23

Table 3.7 The properties of C3D4 and S3/S3R elements 26

Table 3.8 The dataset adopted in the proposed three-phase process for conducting the final surrogate model [1] 39

Table 3.9 The model properties that were used in this thesis ML process [1] 39

Table 4.1 The FEA results of the investigating TPMS-reinforced beams with both peak loads and maximum deflections [1] 46

Table 4.2 The ANN hyperparameters investigated in the model architecture tuning and the optimization tuning processes [1] 47

Table 4.3 The first three greatest ANN results for the validation loss with the ‘Adam’ optimizer [1] 48

Table 4.4 The first three greatest ANN results for the computational time along with a validation loss being less than 0.01 and ‘Adam’ optimizer [1] 48

Table 4.5 Validation losses and training times of various models with different stop patience values and optimizers [1] 50

Table 4.6 Assessment values of the hyperparameter tuned model [1] 52

Table 4.7 The assessment values of the final model [1] 57

x

LIST OF FIGURES

Figure 3.1 Classical minimal surfaces: a) Catenoids, and b) Helicoids (from https://wikipedia.org/) 10Figure 3.2 Typical TPMS structures: a) Primitive, b) Gyroid, c) I-graph and wrapped package-graph, and d) Diamond 12Figure 3.3 a) Network-based, b) skeletal-based, and c) sheet-based Primitive solids 12Figure 3.4 The nature-inspired TPMSs, a)-d) the butterfly wing with Gyroid geometry, e) the micro Fischer-Koch structure of nano-porous gold, and f) the sea urchin microstructure with the appearance of Primitive TPMS [31] 13Figure 3.5 The specimen of the plastic 3D printed TPMS-reinforced beam before and after filled with cement [20] 16Figure 3.6 The stress-strain curves of a) one-TPMS-layer reinforced beam and b) two-TPMS-layer reinforced beam [19] 17Figure 3.7 The stress-strain curves of various beam schemes including the plain cement beam, the ABS-mold cement beam, and the TPMS-reinforced cement beam [19] 17Figure 3.8 The crack propagations of a) one, and b) two-layer TPMS -reinforced beam with 25% and 100% bending load [19] 18Figure 3.9 The configuration of the three-layer TPMS-reinforced beam 19Figure 3.10 Parameters of a Primitive sheet-based TPMS unit 20Figure 3.11 The three-layer TPMS-reinforced beam’s simulation under the three-point bending test [1] 21Figure 3.12 The simulations of the a) beam PC4, b) beam PC5, and c) beam PC6 under the three-point bending test [1] 25Figure 3.13 The typical a) C3D4 and b) S3/S3R element 26Figure 3.14 The meshing grids of a) 500, b) 1000, and c) 1500 triangular mesh elements for each TPMS unit [1] 27

Figure 3.15 The illustration for the meshing grid of a) the plastic scaffolds with a meshed TPMS unit of 1000 elements and b) the cement core [1] 27Figure 3.16 Artificial intelligence (AI), machine learning (ML), and deep learning (DL) (from www.usoft.com) 28Figure 3.17 The ANN architecture with a three-node input layer and a one-node output layer used this study 30Figure 3.18 The impact of the ANN model complexity on its performance [1] 31Figure 3.19 Several usual activation functions 32Figure 3.20 Influence of the learning rate on the optimizer efficiency [1] 33Figure 3.21 Three types of the ANN model performances [1] 35Figure 3.22 The impact of the stop epoch on the ANN model effectiveness [1] 37Figure 3.23 The two-process workflow in this thesis [1] 37Figure 3.24 The three-phase process used in this thesis to create the final surrogate model [1] 38Figure 4.1 The load-displacement curves of PC1 beam with various meshing strategies for the TPMS core [1] 40Figure 4.2 The stress-strain curves of PC1 and PC2 beams from both experiments in the previous study [19] and FEA simulations in this thesis [1] 41Figure 4.3 The von-Mises stress distributions of plastic parts in a) beam PC4, b) beam PC2, c) beam PC5, d) beam PC8, and c) beam PC6 [1] 42Figure 4.4 The von-Mises stress distributions of cementitious cores in a) beam PC4, b) beam PC2, c) beam PC5, d) beam PC8, and c) beam PC6 [1] 43Figure 4.5 The force – displacement curves of the a) one-layer, b) two-layer, and c) three-layer TPMS reinforced beams [1] 45Figure 4.6 Loss values and training times of various models with the ‘Adam’ optimizer and stop patience of 30 epochs [1] 49Figure 4.7 Performances of various models with different optimizers and batch size values including a) the validation loss and b) the training time [1] 51Figure 4.8 The convergence history of the hyperparameter tuned model [1] 52

xii

Figure 4.9 The impacts of various numbers of folds in kfolds cross-validation model on the test loss 𝑀𝑆𝐸𝑡𝑒𝑠𝑡 and the training time [1] 53Figure 4.10 The impacts of various of the dropout rate of the dropout model on the test loss 𝑀𝑆𝐸𝑡𝑒𝑠𝑡 and the training time [1] 53Figure 4.11 The impacts of various of stop patience values of the modified early stopping condition model on the test loss 𝑀𝑆𝐸𝑡𝑒𝑠𝑡 and the training time [1] 54Figure 4.12 Comparisons of the efficiency of three DL techniques in handling overfitting including a) the average result and b) the best result for test loss value [1] 55Figure 4.13 The convergence history of the final model [1] 56Figure 4.14 The force-displacement curves of the a) 10%, b) 15% and c) 20% volume fraction reinforced beams from the ML model and the FEA simulations [1] 58Figure 4.15 The final surrogate model’s predictions for the beam peak load with a) the 3D view, the side view of b) the peak load – volume fraction plane and c) the peak load - number of core layers plane [1] 60Figure 4.16 The final surrogate model’s predictions for the beam maximum deflection with a) the 3D view, the side view of b) the maximum deflection – volume fraction plane and c) the maximum deflection - number of core layers plane [1] 61

CHAPTER 1 INTRODUCTION 1.1 Research topic

Reinforced concrete materials have played an important role in the construction industry and the development of the world Structures made of reinforced concrete appear in almost every construction project from skyscrapers to bridges or canals The combination of concrete and reinforcement rebar is nearly perfect for load-bearing, specifically, the steel material has provided ductility to the concrete structures However, steel is a strongly corrosive material, so it may not be appropriate for underwater structures On the other hand, recycled plastics such as acrylonitrile butadiene styrene (ABS) or polylactic acid (PLA) with high corrosion resistance may be ideal materials for aquatic environments In addition, the specific weight of plastic is much smaller than that of steel and concrete, which is consistent with the current tendency of using lightweight materials Based on these analyzes, many structural components have been fabricated based on complex geometries that meet the architecture requirements but still provide acceptable load-bearing capacity Due to the differences in mechanical properties of plastic and steel, the method of using plastic to reinforce concrete structures can be one of the challenges Various potential reinforcement strategies have been proposed and verified An effective solution was using a plastic crystal-like geometry This geometry could split the solid structure into multiple spaces Depending on the reinforcement approach, these spaces could be filled with concrete, plastic, or even void The mechanical behavior of this structure was indicated to varied based on the topology which was adopted Several common geometries are the lattice shapes (i.e., cubic foam, octet lattice, etc.), honeycomb, and the emerging triply periodic minimal surface (TPMS) These TPMSs can be repeated eternally in three-dimensional space and can be changed into numerous solid structures The key characteristic of these complex geometries is that they are formed by zero-self-intersecting surfaces These surfaces do not create any sharp points which is the reason for stress concentration Therefore, these geometries

Artificial intelligence (AI) has had numerous applications in human life, AI is defined as computer intelligence that can independently provide the ability to perform tasks that require human intelligence and cognitive ability One of the most common ways to create an AI is by using machine learning (ML) algorithms, that describe the ability of a machine to learn and make predictions after learning In the field of mechanics, ML has been adopted to predict the behavior of structures Therefore, this ML model can be considered the most appropriate surrogate model for predicting the behavior of TPMS structures mentioned above Furthermore, this surrogate model could also be a design map for applying TPMSs in concrete beams The model prediction time is much lesser than the simulation and computational time of the FEM software, thereby shortening the time to conduct suitable TPMS geometric parameters for each specific application problem

1.2 Research objective and contents

Research objective: Creating a highly reliable machine learning model for predicting the mechanical behavior of plastic TPMS-reinforced cement beams with various TPMS core parameters

 Conducting an ML-based surrogate model to predict the beam responses

1.3 Research object and scope

Research object: Solid cement beam reinforced with 3D printed plastic Primitive TPMS core and molds under the three-point bending test

Research scope:

 The material is considered to be homogeneous, isotropic, and has elastoplastic behaviors The cement and plastic material properties are obtained from experiments of the previous study;

 The simply supported beam is subjected to a quasi-static load in the three-point bending test;

 The sheet-based TPMS structures are simulated with shell element which is only suitable for thin TPMS shells;

 The simulation results of three-layer TPMS beams are adopted without experimental reviews;

 Other hyperparameters that are not tuned in this study are set as default values in ML open-source libraries in Python programming language;

 The final surrogate model can produce excellent predictions for a small extension of the input domain due to the simulation limitations

4

 Chapter 2 Literature review

Providing and summarizing related publications both domestically and internationally, and identifying the impact of the topic

 Chapter 3 Theoretical background

Presenting the theoretical backgrounds of TPMS structure, TPMS reinforced beams, and simulation of three-point bending test in ABAQUS software;

Presenting the theoretical backgrounds of machine learning (ML) models, artificial neural networks (ANN), and deep learning (DL) techniques to ensure the accuracy of the model;

Presenting the research tasks and the thesis procedure as a flowchart of two main processes including the FEM process and the ML process;

Proposing a three-phase process to conduct the most suitable surrogate model for this thesis problem

 Chapter 4 Results and discussions

Demonstrating the results of the FEM process which consists of convergence study, comparison to experimental results, and influences of the plastic volume fraction and the number of core layers on the beam responses;

Displaying the results of each phase of the proposed process to create the final surrogate model and concluding on the necessity of this process;

Presenting the predictions of the achieved model and further discussions on the impacts of the investigating core parameters

 Chapter 5 Conclusions and research development direction Specifying several key conclusions achieved from the results; Indicating various future developments for the present topic

It should be noted that several figures and tables in this thesis were provided in the author’s publication in the Vietnam Journal of Mechanics [1]

CHAPTER 2 LITERATURE REVIEW 2.1 International research

The 3D printing technology has become popular and been industrialized Numerous studies on this fabrication method have been conducted for the past decade Recently, researchers have focused on the use of cement printing materials, an example is the study of Liu et al [2] The influences of printing parameters including material injection speed, nozzle size, printing layer height, printing direction, etc., are top research interests Several representative results could be found in studies [3, 4] Nguyen-Van et al [5] have also studied the influence of these printing parameters on cement triply periodic minimal surface (TPMS) structures The results showed that the printing speed was the most influential factor in the load-bearing capacity of the structure The high printing speed might result in the reduced strength of cement material and then damage the bottom layers Besides, the destructive patterns of two TPMS types that were Primitive and Gyroid were also demonstrated in this study

The following publications concentrated on investigating the mechanical properties of TPMS structures including the anisotropy indexes, elastic modulus E, Poisson's ratio 𝜈, bulk modulus K, shear modulus G, shear and compression strength, The relationships between the volume fraction and these properties of a typical TPMS structure have been conducted by Lee et al [6] Besides, both thermal and electrical responses of various TPMS types were revealed in the previous work [7] Moreover, by adopting the selective laser melting (SLM) and selective laser sintering (SLS) printing methods in the research [8] and [9], the great energy absorption capacity of these porous structures has been verified The experimental results in research [10] have indicated the more robust behavior of sheet-based TPMS solids compared with skeletal-based ones Finite element analysis (FEA) simulations have been proposed to demonstrate the efficiency of the metallic TPMS structures in the publication of Yang et al [11]

6

As the emerging trend of reinforcing concrete materials with recycled plastic, various strategies have been proposed The plastic lattice structures have shown increments in the structure’s overall ductility [12] Xu et al [13] later investigated the influence of the volume-occupation ratio of plastic material in this lattice-core reinforcement Nguyen-Van et al [14] replaced the above lattice structure with Primitive and Gyroid TPMS structures and concluded on their efficiencies For instance, the TPMS structure might have better uniform-distributed stress and improved compression strength compared to lattice structures due to its zero-intersection-surface trait The plastic core in the lattice structure might not be fully mobilized for load-bearing, this could be changed by using the TPMS geometries instead

Based on previous studies on the energy absorption capacity of TPMSs, the great potential of using these structures under dynamic loads was revealed Sandwich plates with TPMS cores were later investigated with impact loads [15] and explosive loads [16] Recycled plastic was considered a suitable material to use in aquatic environments Therefore, the plastic TPMS-reinforced cementitious block has been studied and adopted by Dang et al [17] as a breakwater solution Simulations showed an effective reduction of wave effects toward 50% in porous Gyroid TPMS blocks In addition to the offshore-construction application, TPMSs could be applied to reduce the structure weight and thus reduce the self-weight load of the building An architecture component that has load-bearing capacity was one of the application goals of these structures A porous sandwich beam fabricated entirely by 3D printing technology was indicated as one of the bright candidates for this role [18]

Plastic TPMS-reinforced cement structure was also an effective alternative solution for traditional reinforcement concrete beams Nguyen-Van et al [19] evaluated the mechanical behavior of this reinforcement method with static loads The results showed that the load-bearing capacity of the beam has increased significantly The TPMS core has also created cement confinement and therefore the maximum deformation of beams was increased remarkably Another study by the same group of authors investigated the impact of dynamic loads on these beams [20]

Both simulations and experiments showed excellent results about the beam’s energy absorption capacity, which was the most important factor when studying the dynamic load in general In fact, while the energy absorption capacity of the normal cement beam was about 15J, the TPMS core reinforced beam was capable of absorbing up to 20J, which could be considered a valuable increment

Machine learning (ML) models have been increasingly applied in numerous fields of mechanics and structural engineering The development of these models was indicated to base on artificial neural networks (ANN) Author Adeli [21] summarized these applications in the field of civil engineering from 1989 to 2000 Moreover, Lee et al [22] have also presented a detailed overview of deep learning (DL) techniques that were applied in structural applications and their robustness in alleviating the overfitting problem

The major applications of ML in construction consist of optimization and behavior prediction Studies on prediction have been conducted for various structures such as reinforced concrete beams [23], steel-concrete connections [24], etc A recent study by Lieu et al [25] has included the reliability of the structure for truss problems On the other hand, the study of Nguyen et al [26] has demonstrated the typical optimization application of ML algorithms for truss problems By using the gradient-based optimizer of the ML model, the optimization time was significantly reduced compared to other metaheuristic algorithms Along with this result, various activation functions and optimization methods have been employed for comparisons Currently, the ML model was adopted to solve governing equations as an alternative method for classical solutions including the weighted residual method, Garlerkin, Rayleigh-Ritz, finite element method (FEM), etc Research by Samaniego et al [27] used ANN as an approximation function to solve the problem of minimizing the total potential energy of the structure Tremendous following studies based on this idea have been published that contributed to a new direction for computational mechanics

8

2.2 Domestic research

It can be noted that the applications of TPMS structures are fairly new Besides, with the essentials of using 3D printing technology for fabrication, the equipment requirement might also be a limitation when studying this structure domestically For these reasons, domestic papers on these TPMSs and their application are now inconsiderable

2.3 Summary

The effectiveness of the TPMSs in the mechanical fields has been verified by the above studies The results from these studies showed remarkable mechanical behaviors of these TPMS structures Moreover, as a reinforcement component, plastic TPMS core could increase the ductility of concrete materials Therefore, both the ultimate load and maximum deflection of the cement beam could be improved However, either an FEA simulation or an experiment might need to be conducted for specifying the result The geometric complexity of the TPMS structure might lead to difficulties in both methods While the meshing grid in simulation should be reasonably fine, the fabrication method should only be additive manufacturing (AM) For certain TPMS parameters, both the FEA computation time and the experimental time might be excessive and uneconomical This could be a barrier to applying this type of beam in real-world problems Consequently, the influence of the TPMS core properties on the cement beam should be achieved

In this thesis, the beam’s responses were revealed with various combinations of numbers of TPMS core layers and volume fraction values by FEA simulations From the received data, a surrogate model based on ML was created This model was evaluated by a three-process assessment strategy that both hyperparameter tuning and overfitting handling were included The final model predictions were adopted to generate a deeper analysis of the impact of these TPMS parameters on the beam behaviors

CHAPTER 3 THEORETICAL BACKGROUND 3.1 Triply periodic minimal surface structures

3.1.1 Minimal surface

A minimal surface (MS) can be defined as a surface where the local area at any point is the smallest In the two-dimensional space, the MS is also the entire space; which means that any closed boundary on this space always creates a minimal area inside On the other hand, an MS in the three-dimensional space is determined when the mean curvature of any point on the surface is vanished

Soap film is a liquid surface on which the air pressure on both the inside and outside surfaces is equal and therefore the mean curvature at any point is zero From this phenomenon, the very first minimal surfaces can be found based on soap film experiments One of the most classical MS to be discovered is the catenoids surface This surface can be created by rotating a catenary curve about an axis The catenoids surface can be described in space by Eq (3.1)

⎧𝑥 = 𝑐 cosh 𝑣

𝑐 cos(𝑢)𝑦 = 𝑐 cosh 𝑣

𝑐 cos(𝑢)𝑧 = 𝑣

(3 1)

where 𝑢 ∈ [−𝜋, 𝜋), 𝑣 ∈ 𝑹 and 𝑐 is a real number differing from 0

By splitting and bending the catenoids surface without stretching it, a new MS may be obtained which is called the helicoids This new surface can be created due by changing the shape of an MS but maintaining its zero-curvature Similar to the catenoids, the helicoids surface can be described by Eq (3.2)

𝑥 = cos(𝜃) sinh(𝑣) cos(𝑢) + sin(𝜃) cosh(𝑣) cos(𝑢)𝑦 = − cos(𝜃) sinh(𝑣) cos(𝑢) + sin(𝜃) cosh(𝑣) cos(𝑢)𝑧 = 𝑢 cos(𝜃) + 𝑣 sin(𝜃)

(3 2)

where (𝑢, 𝑣) ∈ (−𝜋, 𝜋] × (−∞, ∞), with the bending angle −𝜋 < 𝜃 ≤ 𝜋 These two minimal surfaces are typical examples of helicoids – catenoids minimal surface family Both of them were first discovered by using soap film experiments as shown in Figure 3.1 Moreover, other MS families were discovered

10

in later studies For instance, several famous MS families are the Schwarz surface family, the Riemann's surface family, the Ennerper surface family, etc The following contents of this thesis focus on the Schwarz surface family, which is well known in the name of triply periodic minimal surface (TPMS)

Figure 3.1 Classical minimal surfaces: a) Catenoids, and b) Helicoids (from https://wikipedia.org/)

3.1.2 Triply periodic minimal surface

A triply periodic minimal surface (TPMS) is a crystal structure formed by peating minimal surface units in all three perpendicular dimensions Minimal surfaces are non-self-intersecting surfaces that reduce stress concentration, making TPMSs useful for various applications [28] Since the discovery of the first TPMS, many new types have been introduced, including Primitive, Diamond, Hexagonal, Neovius, and the famous bioinspired geometry, Gyroid This Gyroid type was developed based on skeleton graphs of crystals by Alan Schoen [29] Various methods can be used to create TPMS geometries, the implicit function might be the most common one [30] Several implicit functions of typical TPMS geometries are:

 I-graph and wrapped package-graph (IWP):

𝜙(𝑥, 𝑦, 𝑧) = 2 𝑐𝑜𝑠(𝜔 𝑥) 𝑐𝑜𝑠 𝜔 𝑦 + 𝑐𝑜𝑠 𝜔 𝑦 𝑐𝑜𝑠(𝜔 𝑧) + 𝑐𝑜𝑠(𝜔 𝑧) 𝑐𝑜𝑠(𝜔 𝑥)− 𝑐𝑜𝑠(2𝜔 𝑥) + 𝑐𝑜𝑠 2𝜔 𝑦 + 𝑐𝑜𝑠(2𝜔 𝑧) (3 5)  Diamond (D):

𝜙(𝑥, 𝑦, 𝑧) = sin(𝜔 𝑥) sin 𝜔 𝑦 sin(𝜔 𝑧) + sin(𝜔 𝑥) cos 𝜔 𝑦 cos(𝜔 𝑧) +cos(𝜔 𝑥) sin 𝜔 𝑦 cos(𝜔 𝑧) + cos(𝜔 𝑥) cos 𝜔 𝑦 sin(𝜔 𝑧) (3 6)with

𝜔 = 2𝜋𝑛

𝐿 | 𝑖 = 𝑥, 𝑦, 𝑧 (3 7) where 𝑥, 𝑦, 𝑧 are the spatial directions in Cartesian coordinate system,

𝜔 are periodicities of the TPMS structure in three directions, 𝐿 are lengths of a TPMS unit in three directions,

𝑛 are numbers of TPMS unit in three directions

The study focuses on using uniform TPMSs, where the properties are similar in all three dimensions (𝜔 = 𝜔) As a result, the size of a uniform TPMS is defined as 𝑎 = 𝐿 /𝑛 The geometries of several typical uniform TPMS types are demonstrated in Figure 3.2

Most TPMS structures should split their enclosing cube into two equally volumetric parts, resulting in a volume fraction of 50% between the volume of the solid TPMS and its cube However, a variety of solid shapes can be formed from a single TPMS type For example, Figure 3.3 shows three types of Primitive TPMS solids, including skeletal-based, network-based, and sheet-based solids

These solid types are typical demonstrations of the controllability of TPMS structures The implicit function modification technique is adopted for these solids By introducing the control parameter (𝑡) in the implicit functions, the volume fraction of the structure can be changed This study uses the sheet-based solid of the Primitive type to reinforce the beam structure The general implicit function of the sheet-based structures is provided in Eq (3.8).

−𝑡 ≤ 𝜙(𝑥, 𝑦, 𝑧) ≤ 𝑡 (3 8) 3.1.3 Applications of TPMS

Bioinspired structures have demonstrated superior performance in various applications due to their evolutionary optimization over time Among all, TPMS structures can be observed in numerous natural objects, including living creatures such as butterflies, sea urchins, and humans and non-living things such as gold, as illustrated in Figure 3.4

Figure 3.4 The nature-inspired TPMSs, a)-d) the butterfly wing with Gyroid geometry, e) the micro Fischer-Koch structure of nano-porous gold, and f) the sea urchin

microstructure with the appearance of Primitive TPMS [31]

An outstanding characteristic of TPMSs is their optimized surface area and volume, leading to minimal self-mass In order to maintain the organism's mobility, these structures also need great strength and stiffness Therefore, the high strength-

19], wave load [17], dynamic load, and impulsive load [15, 16, 20]

3.2 TPMS-reinforced beam

The 3D printing technology has provided the fabrication feasibility of these complex TPMS geometry Scientists have applied this technology to investigate the effectiveness of these geometries on aforementioned structures under various loading conditions This thesis focuses on investigating the solid cement beams reinforced with recycled plastic TPMS cores, which have been denoted as a significant reinforcement strategy in load-bearing capacity and energy absorption

3.2.1 Additive manufacturing

Additive manufacturing (AM) technology, also known as 3D printing, has been widely developed and applied in various research fields and industries The basis of this fabrication method is to create layers of material using equivalent printing devices These layers are then superimposed on top of one another to form a 3D specimen This approach differs from traditional fabrication technology, which is subtractive manufacturing Identifying the object to be fabricated and layering it appropriately are critical tasks in the implementation of 3D printing These tasks are supported by modern software and spatial-model-design formats, such as computer-aided design (CAD) and standard triangle language (STL), making 3D printing technology more accessible for practical applications It is necessary to carefully

consider suitable printing equipment for specific objects and materials [5, 33] The following are the most commonly used 3D printing methods, as described in [34]:

 Fused deposition modeling (FDM): A process in which layers are created by melting thermoplastic materials and then allowing them to solidify;

 Inkjet printing: A process that forms melted ceramic powder into droplets and solidifies them into layers;

 Powder bed fusion (PDF): A process in which ultra-fine metallic powder is fused together using a laser beam or binder to create layers;

 Stereolithography (SLA): A process that treats plastic materials as liquid and uses an ultraviolet laser to harden them

The AM technology has been adopted to fabricate bioinspired structures, which are difficult to create using traditional manufacturing methods, due to its ability to produce intricate geometries with high precision [35] 3D printing technology also allows the use of various materials, such as metal, plastic, ceramic, cement, etc This is the main reason for its attractiveness in scientific research Recently, recycled plas-tic materials have gained enormous attention and have been investigated for 3D print-ing applications Several thermoplastics, including acrylonitrile butadiene styrene (ABS), polycarbonate (PC), polylactic acid (PLA), etc., have been used for this pur-pose Specifically, this study investigates the TPMS structures fabricated with ABS plastic by the FDM printing method

3.2.2 Cement beam with 3D printed TPMS core

Recent research has demonstrated that the TPMS core can provide confinement to cement beams, leading to increased compressive strength and ultimate strain The use of TPMS cores made from lightweight materials like ABS or PLA can also reduce the total weight of the structure However, the volume fraction of the TPMS core needs to be considered attentively As the ABS material volume ratio reaches 19.2%, the compressive strength of the concrete may decrease by 22% [12] With low-volume fractions, a sheet-based TPMS solid can be approximated by giving a thickness to the isosurface, which has the control parameter 𝑡 = 0 This approach is adopted in the present study Additionally, a lightweight cement reinforced with

Figure 3.5 The specimen of the plastic 3D printed TPMS-reinforced beam before and after filled with cement [20]

3.2.3 Effectiveness of TPMS core

Nguyen-van et al [19], conducted a study in which they validated the impact of the TPMS core on beams using both experimental and Finite Element Analysis (FEA) simulation methods The results showed that the standard error between the simulation and experimental results for one-layer and two-layer beams are 3% and 7%, respectively Moreover, Figure 3.6 illustrates the good agreement of both simulations and experiments through the stress-strain curve of the beams Therefore,

the simulation was able to accurately capture the responses of the beams, hence verifying its reliability

Figure 3.7 The stress-strain curves of various beam schemes including the plain cement beam, the ABS-mold cement beam, and the TPMS-reinforced cement beam [19]

Another finding was that increasing the number of TPMS layers could lead to greater improvements in the beam's response However, as the number of layers increases, the size of the units decreases, and the manufacturing process might

18

become more difficult Additionally, the arrangement of the units could also affect the behavior of the beam, as demonstrated in Figure 3.8 For example, the one-layer beam exhibited an inclined crack path between the TPMS units, rather than a straight path in the center of the beam In contrast, due to the absence of TPMS units at the middle of the beam, the two-layer beam produced a straight crack path which was similar to that of a non-reinforced beam

During the investigation, the dimensions of the 50mm × 50mm × 250mm beam remain constant, but the thickness of the core’s shell (𝑡 ) is changed depending on the volume fraction of the plastic material The thickness of the molds (𝑡 ) is constant at 2𝑚𝑚 for all cases Along with the volume fraction, the number of TPMS layers is also varied This thesis uses different combinations of these features to strengthen the beam and investigates how the number of TPMS layers and the volume fraction of the plastic core affect the beam performance Specifically, the present study use TPMS cores with one, two, and three layers, and plastic core volume fractions ranging from 10% to 20% Table 3.1 presents the labels and parameters of each beam

Table 3.1 The investigating TPMS beams’ geometric descriptions and labels [1]

Volume fraction

𝑎 =50𝑚𝑚𝑛𝑙𝑎𝑦𝑒𝑟

𝑛𝑢𝑛𝑖𝑡= 5 × 𝑛𝑙𝑎𝑦𝑒𝑟3

(3 9)

where 𝑎 is the size of a uniform TPMS unit,

𝑛 is the number of TPMS core layers inside the beam, 𝑛 is the total number of TPMS units inside the beam

Figure 3.9 The configuration of the three-layer TPMS-reinforced beam

As stated in the previous section 3.2.2, the present study treats the sheet-based TPMS solids as the thicken-based ones As a result, the equation provided below can be used to determine the volume fraction of the TPMS core through the average thickness of the sheet-based structures

𝑡 = 𝑉𝐹 ×𝑎

𝐴 = 𝑉𝐹 ×𝑎

2.3526 (3 10)

Figure 3.10 Parameters of a Primitive sheet-based TPMS unit

Table 3.2 The core parameters of the investigating beams [1]

Figure 3.11 The three-layer TPMS-reinforced beam’s simulation under the three-point bending test [1]

3.3.2 Materials

3.3.2.1 Cement material

The polymer fiber cement used in this study is made using the mix proportions listed in Table 3.3.Previous research used the same cement mixture to determine the mechanical properties of cementitious mortar, using a 50mm × 50mm × 50mm cubical specimen This specimen was cast at the same time as the beam After being vibrated and cured in a humid environment for 28 days, the test specimen could eliminate all the air inside and be used to conduct the mechanical properties of the cement material which are provided in Table 3.4 Hence, there is not any void inside the beam either by similar vibrating

Table 3.3 The mixture of the cementitious mortar [19]

22

Table 3.4 Mechanical characteristics of the cement material [19]

Young modulus, 𝐸 (MPa) 2500 Mass density, 𝜌 (kg/m ) 2200

When the stress in the brittle material reaches the yield state, the failure occurs and hence cracks initialize The simplified concrete damage plasticity (SCDP) model has been verified as an appropriate theory to investigate the cement material This model mentions both damage theory and plasticity theory for cement The uniaxial damage behavior of the cement material can be described by two damage factors in compression and tension as described in Eq (3.11) and Eq (3.12), respectively [38]

Table 3.5 The SCDP model parameters of the cement material [19]

Dilation Angle

Eccentricity Initial Biaxial/Uniaxial Ratio, 𝜎 /𝜎

Damage parameter, 𝑑

Stress (MPa)

Cracking strain

Damage parameter, 𝑑

Table 3.6 Mechanical characteristics of the ABS plastic material [19]

Young Modulus, 𝐸 (MPa) 2200 Mass Density, 𝜌 (g/m ) 1.05

Yield stress, 𝜎 (MPa) 56

3.3.3 Finite element analysis simulation

To investigate the beam behaviors, the three-dimensional bending simulation has been applied with both nonlinear geometry and nonlinear responses of materials These simulations are implemented in a finite element method (FEM) commercial

24

software which is Abaqus The explicit dynamic analysis in this software has been used to reduce the computation time of the simulation By using this algorithm, the most important factor is the stable incremental time which has a great impact on the accuracy of the solution In this study, the incremental time is automatically computed by the program without mass-scaling

The three-point bending test model used in this study includes three rigid semi-cylindrical rollers: one load transmission device and two test supports along with the beam Illustrations for these models are shown in Figure 3.12 The interactions between the beam and supports are considered "hard" contacts To prevent out-of-plane sliding, a 0.15 friction coefficient is also adopted A similar general contact but with a friction coefficient of one is adopted for the contact between the beam molds and the cementitious core Due to the assumption that these two components are completely tied together, no sliding effect occurs on the interaction surfaces Therefore, the shear force on these interfaces vanishes [14] It should be noted that the elastic moduli of cement and plastic materials are 2500MPa and 2000MPa respectively This similarity can lead to a perfect bond between these two components, which means that the bond-slip effect can be neglected [39] As the result, the embedded method in Abaqus software has been used to simulation the TPMS core inside the beam’s cementitious core Additionally, the TPMS shell's zero-intersecting-surface characteristics can lead to confinement of the cementitious core, resulting in a reduction of deviatoric stress and less sliding on the contact surface between the TPMS and the cement cores

To simplify the simulation process and reduce computational time, the vertical displacement of the roller at the beam mid-span is adopted as an alternative for the load During the simulation, the value of the load is calculated based on the reactions of the support rollers