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 PER
INTRODUCTION
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
2 are relevant to endure dynamic load and impact load due to their great energy absorption capacity
With the development of additive manufacturing (AM) technology, which is known as 3D printing technology, the application of complex structures as TPMS has become more practical However, deep analyses of the efficiency of these structures as a reinforcement component might remain limited The current method for analyzing the mechanical behavior of these components is mainly based on the finite element method (FEM) Nevertheless, this method requires the exact simulation for the structure which might be difficult due to the complex geometry of the TPMS In fact, the meshing grid might need to be relatively smooth, which results in more computational time and cost Therefore, it is necessary to conduct an investigation on the effect of parameters on this structure strength or build a surrogate model for this problem
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.
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
Simulating the plastic TPMS-reinforced cement beams under a three-point bending test in FEM software;
Investigating the influences of the number of TPMS layers and the plastic volume fraction on the beam behavior;
Conducting an ML-based surrogate model to predict the beam responses.
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
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.
Thesis structure
The contents of this thesis are presented in 05 chapters as follows:
Briefly demonstrating the topic, research objectives, and content; stating the object and scope of the research under the thesis; and evaluating the urgency and practical significance of the topic
Providing and summarizing related publications both domestically and internationally, and identifying the impact of the topic
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
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]
LITERATURE REVIEW
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]
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
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.
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
THEORETICAL BACKGROUND
Triply periodic minimal surface structures
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)
(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)
(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) a) b)
Figure 3.1 Classical minimal surfaces: a) Catenoids, and b) Helicoids (from https://wikipedia.org/)
A triply periodic minimal surface (TPMS) is a crystal structure formed by re- 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:
𝜙(𝑥, 𝑦, 𝑧) = cos(𝜔 𝑥) sin 𝜔 𝑦 + cos 𝜔 𝑦 sin(𝜔 𝑧) + cos(𝜔 𝑧) sin(𝜔 𝑥) (3 4)
I-graph and wrapped package-graph (IWP):
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
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
Figure 3.2 Typical TPMS structures: a) Primitive, b) Gyroid, c) I-graph and wrapped package-graph, and d) Diamond a) b) c)Figure 3.3 a) Network-based, b) skeletal-based, and c) sheet-based Primitive 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)
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-
14 to-weight and stiffness-to-weight ratios can be found and contribute to their wide- range applications, particularly in medical equipment and body implants
The high stiffness and stress-reducing properties of TPMS structures have attracted researchers studying their mechanical behavior Many studies have shown that metallic TPMS structures can be an alternative to the traditional lattice geometries of porous structures [10, 11, 32] Recently, 3D-printed TPMS shells made of recycled plastic have been used to reinforce concrete beams and plates, showing the potential to replace steel rebar in traditional reinforced concrete In these investigations, various analysis types have been adopted including static loads [18, 19], wave load [17], dynamic load, and impulsive load [15, 16, 20].
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
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
15 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
16 polymer fibers was used to investigate the ductility of the beam, with a fiber content of about 0.25% of the total volume of the beam [19]
The plastic component of the considering beam used in this study consists of the molds and the Primitive TPMS core These plastic parts are fabricated simultaneously by FDM printing method The cement material is casted into the beam after the plastic parts are created as in Figure 3.5 Hence, there is not any void inside the beam by vibrating after casting Due to the weakness in the printing direction mechanical responses, the beam’s side face is used to be the load-bearing surface Therefore, the effectiveness of the TPMS core on the beam can be investigated the more precisely Furthermore, along the load direction, there are not only the beam TPMS core and cement core but also two skin layers of the molds at the top and the bottom of the beam For this reason, the present beam can be denoted as a sandwich beam
Figure 3.5 The specimen of the plastic 3D printed TPMS-reinforced beam before and after filled with cement [20]
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,
17 the simulation was able to accurately capture the responses of the beams, hence verifying its reliability a) b)
Figure 3.6 The stress-strain curves of a) one-TPMS-layer reinforced beam and b) two-TPMS-layer reinforced beam [19]
Furthermore, according to Figure 3.7 , it can be seen that the key role of the TPMS core is reducing the cement brittleness This could be indicated by the beam's plastic behavior and increased energy absorption It is worth noting that the strain- softening regime of the beam is similar to that of a polymer fiber-reinforced beam [36] As a result, both the maximum load and maximum displacement of the beam have significantly increased in comparison to the unreinforced cement beam
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 a) b) Figure 3.8 The crack propagations of a) one, and b) two-layer TPMS
-reinforced beam with 25% and 100% bending load [19].
Simulation model
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]
Eq (3.9) describes the relationships between the number of core layers, the size of a uniform TPMS unit, and the total number of TPMS units inside the beam when the size of the beam remains unchanged The illustration of these properties is provided in Figure 3.9
(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
20 where 𝑡 is the core’s shell thickness (see Figure 3.10)
𝑉𝐹 is the volume fraction of the plastic TPMS core,
𝐴 = 2.3526 × 𝑎 is the surface area of the TPMS isosurface with Primitive type, where 𝑡 = 0 [37]
Figure 3.10 Parameters of a Primitive sheet-based TPMS unit
Table 3.2 The core parameters of the investigating beams [1]
Table 3.2 shows the properties of the considering beams' core To evaluate the beam responses, the static bending load is employed in this study The three-point bending test is adopted where the load is at the center of the beam and the beam span is set as 200mm Figure 3.11 provides a representative simulation model
Figure 3.11 The three-layer TPMS-reinforced beam’s simulation under the three-point bending test [1]
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]
PVA fiber 0.25% of the total volume
Table 3.4 Mechanical characteristics of the cement material [19]
Young modulus, 𝐸 (MPa) 2500 Mass density, 𝜌 (kg/m ) 2200 Poisson’s ratio, 𝜈 0.2 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]
𝜎 (3 12) where 𝑑 , 𝜎 , 𝜎 are the compression damage, the nominal compressive stress and the compressive strength of the material, respectively
𝑑 , 𝜎 , 𝜎 are the tension damage, the nominal tensile stress and the tensile strength of the material, respectively
In this thesis, experimental SCDP model properties of the cement material in the previous study [19] are adopted as in Table 3.5
Table 3.5 The SCDP model parameters of the cement material [19]
Compressive behavior from experiment Tensile behavior (assumed)
The plastic parts of the beam including the molds and the TPMS core are fabricated with the ABS plastic material by 3D printing technology In the FEA simulation, the ideal elastic-plastic model has been used to demonstrate the behavior of this material This model parameters are provided in Table 3.6
Table 3.6 Mechanical characteristics of the ABS plastic material [19]
Young Modulus, 𝐸 (MPa) 2200 Mass Density, 𝜌 (g/m ) 1.05 Poisson’s Ratio, 𝜈 0.35 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
25 a) b) c) Figure 3.12 The simulations of the a) beam PC4, b) beam PC5, and c) beam PC6 under the three-point bending test [1]
The cement core of the beam is simulated using the four-node tetrahedral (C3D4) solid element, while the three-node triangular (S3/S3R) shell element is used
26 for the TPMS shell and beam molds in this study The demonstration and several key features of these elements are shown in Figure 3.13 and Table 3.7, respectively a) b)
Figure 3.13 The typical a) C3D4 and b) S3/S3R element
Table 3.7 The properties of C3D4 and S3/S3R elements
Description A tetrahedral element with 4 nodes and linear shape function (C 0 continuity interpolation functions);
A triangular element with 3 nodes which can be used in 3- dimensional space and have linear shape function (C 0 continuity interpolation functions);
Others + The number of Gauss points used for integrating is equal to 1 + The number of Gauss points used for integrating is equal to 1 + The transverse shear deformation can be taken into account with the First-order Shear Deformation Theory (FSDT)
Previous studies [14, 19] have shown that a mesh size of 4mm could be a relevant mesh grid for both accuracy and computational time, but it may not be appropriate for multi-layer beams As the number of core layers increases, the TPMS unit size is reduced For example, the one, two, and three-layer beams have the unit size of 50mm, 25mm, and 16.7mm, respectively With a constant mesh size, each TPMS unit is defined by less numbers of element In the three-layer beam, five elements are the number of elements to demonstrate a unit Therefore, a new mesh strategy is proposed in this thesis to address this issue In this meshing method, each TPMS unit is simulated with a specific number of mesh elements in order to maintain the zero-mean-curvature of the TPMS geometry Figure 3.14 shows three values of mesh-element numbers, that are 500, 1000, and 1500, investigated in this study
Besides, the full mesh of the one-layer beam is provided in Figure 3.15, where the mesh size of the molds and the cement core is 4mm
Figure 3.14 The meshing grids of a) 500, b) 1000, and c) 1500 triangular mesh elements for each TPMS unit [1] a) b)
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].
Machine learning model
Machine learning (ML) refers to computer programs that generate predictions using input data through the train-test process It is a subset of artificial intelligence (AI) and has become common in many aspects of modern life since 21 st century, including smart devices like cellphones and computers The widespread use of the internet has resulted in an enormous amount of data being generated, which can be efficiently analyzed and exploited using ML Deep learning (DL) is a group of
28 techniques developed to improve the performance of ML models The relationship between AI, ML, and DL is shown in Figure 3.16
Figure 3.16 Artificial intelligence (AI), machine learning (ML), and deep learning (DL) (from www.usoft.com)
An artificial neural network (ANN) is a multi-layered ML algorithm that simulates the human nervous system [40] This algorithm can predict more accurately and quickly than traditional ML approaches like linear regression and decision trees ANN models have gained great attentions in numerous mechanical fields and be used as an optimization tool or to reduce computational time This calculation time can be reduced from hours to milliseconds with a well-trained model In these applications, the ANN models is used as a surrogate model to predict the mechanical behavior of structures instead of FEA simulations [22]
An ANN model has two primary processes: train and test During training, results are computed through feed-forward phase using the model parameters (weights and biases) and then adjusted via backpropagation using optimization algorithms The model performance is evaluated in the test process A well-trained model is denoted as a best-performance model that can be created Deep learning techniques can be used to minimize the error and increase the model performance, thus reducing the training time
The ANN architecture has three main components: the input layer (input data), the hidden layers (computation layers), and the output layer (output data)
The input layer is the first layer of the network (the 0 th layer), 𝐱 = 𝐡 (𝟎) It is the input data presented as a vector;
The model can have multiple hidden layers, each layer can have multiple nodes While the number of layers is denoted as 𝑛, the number of nodes in each layer (different from the 0 node) is denoted as 𝑑 ( ) with 𝑙 is the index of the layer;
Biases and weights demonstrate the connections from the 0 node and other nodes, respectively, of the previous layers (the (𝑙 − 1) layer) to all nodes of the current layer (the 𝑙 layer) These parameters are denoted as 𝐛 ( ) ∈ 𝐑 ( ) × and
The value of each node in a layer is determined by an activation function (𝑓), which takes into account the values of nodes in the previous layer and the parameters of the current layer, as shown in Eq (3.13);
For example, the value of node j in the 𝑙 layer is calculated by the following equation;
The output layer is the last layer of the network, 𝐲 = 𝐡 (𝐧 𝟏) The results of this layer are the predictions of the model
The architecture used in this study can be demonstrated as a diagram in Figure 3.17 with nodes as circles and parameters (weights and biases) as lines
Figure 3.17 The ANN architecture with a three-node input layer and a one-node output layer used this study
When designing an ANN model, it is important to specify the number of nodes in the input and output layers, as well as the number of hidden layers and their number of nodes In this work, three input variables including the displacement of the load roller, the number of TPMS layers and the plastic volume fraction are adopted The output layer, however, only contains one node which is the vertical reaction force of the load roller This force could be denoted as the applied load The number of parameters in the ANN model in this thesis can be calculated based on this setting In addition, as this number increases, the model can become more complex, making it more difficult to update parameters and requiring higher computational costs While a more complex model can achieve higher training accuracy, it can also lead to overfitting and lower accuracy on test data, as shown in Figure 3.18 Therefore, the model architecture should be chosen carefully The process of choosing this architecture is called hyperparameter tuning
Figure 3.18 The impact of the ANN model complexity on its performance [1] 3.4.2.2 Activation function
The activation function is responsible for deciding how much a node in a network can impact the node value in the next layer based on the sum products of the previous layer's parameters and node values (please see Eq (3.13) for details) As this sum can grow large, it can increase computational time and reduce model reliability To address these issues, activation functions are used Some commonly used activation functions are listed below, and their demonstrations are provided in Figure 3.19
Figure 3.19 Several usual activation functions
Using an appropriate activation function can improve reliability and provide higher-order nonlinearities, without increasing the number of parameters However, an unsuitable activation function can cause vanishing or exploding gradient problems when using gradient optimization algorithms Therefore, the activation function should be chosen carefully by the hyperparameter tuning process In this thesis, the model architecture and activation function are both determined through this process 3.4.2.3 Loss function
The loss function measures the difference between the model-predicted output and the true output in the training data This helps to evaluate the model performance and identify solutions for improvement The mean square error (MSE) is used as the loss function in this study, along with other metrics such as root mean square error (RMSE) and mean absolute error (MAE) The formulas for these metrics are given below
𝑚 (3 19) where 𝐲, 𝐲 are model predictions and input data, respectively,
𝑚 is the number of input samples
To reduce the loss function, optimization algorithms (optimizers) such as gradient and non-gradient algorithms can be used Gradient-based optimizers are commonly adopted because the predicted value 𝒚 can be expressed mathematically in terms of input values 𝒙 using backpropagation These optimization algorithms use the gradient descent effect to find the solution, in which the gradient vector indicates the direction of decreasing the function value In this method, the learning rate is one of the key parameters Higher learning rates can lead to faster convergence, but this may not result in the global minimum solution, as provided in Figure 3.20
Figure 3.20 Influence of the learning rate on the optimizer efficiency [1] Various gradient optimization algorithms such as mini-batch gradient descent (GD), stochastic gradient descent (SGD), adaptive gradient algorithm (AdaGrad), root mean squared propagation (RMSprop), adaptive learning rate method (AdaDelta), adaptive moment estimation (Adam), etc have been developed Some of these algorithms have the prefix 'Ada', which indicates that they belong to the adaptive algorithms group The learning rate of these algorithms is adapted during training Additionally, to avoid getting trapped in local minima, most algorithms use a momentum factor as a solution
In an ANN model, an epoch indicates the point where the model has used all the available training data The maximum epoch is a hyperparameter that determines when the training process should stop Increasing the maximum epoch may improve the model's solution but also increase the training time The batch size is another important factor that determines how many data samples are used to update the model parameters each time Larger batch size can shorten the convergence time but may reduce the model generality, while a smaller batch size can update the model more
34 frequently but take more epochs to reach the best solution This study used various optimizers with different hyperparameters, such as learning rate, batch size, and epochs, to find the most suitable optimizer for the TPMS beam dataset
Data is a crucial factor in every ML model While a single set of input and output data is denoted as a data sample, the entire set of samples used in the model is called the dataset This dataset is usually split into three subsets: train dataset, validation dataset, and test dataset The size of the dataset depends on the complexity of the problem, thus increasing it can help solve underfitting Typically, a good model requires thousands of samples in the dataset Moreover, researchers are currently focusing on the problem of noisy data, which may consist of numerous meaningless and incorrect samples Various DL techniques have been introduced to deal with this type of problem These techniques focus on reducing both train loss and training times
Thesis tasks
In this thesis, all the tasks can be divided into two main processes, which are the FEM process and the ML process Figure 3.23 demonstrates the works in this thesis as a procedure flowchart
Figure 3.23 The two-process workflow in this thesis [1]
In the ML process, to create the most appropriate surrogate model, a three- phase process is introduced The first phase is the hyperparameter tuning for the basic ANN model Through the model assessment phase with six proposed performance metrics, the overfitting issue can be found due to the small FEA dataset of TPMS beams To address this issue, the last phase, handling overfitting with DL techniques, is adopted The illustration of this process can be seen in Figure 3.24 with details of the data and techniques used in each phase presented in Table 3.8 and Table 3.9, respectively
Figure 3.24 The three-phase process used in this thesis to create the final surrogate model [1]
This ML process were implemented on a personal computer with WindowOS,
4 processors, Intel Core i7U, CPU 2.7GHz, 16GB memory, Python 3.8, TensorFlow and scikit-learn libraries
Table 3.8 The dataset adopted in the proposed three-phase process for conducting the final surrogate model [1]
Train Validation Test Model hyperparameter tuning
PC1 + PC2 + PC4 + PC6 + PC7 + PC8
PC1 + PC2 + PC4 + PC6 + PC7 + PC8
Model assessment PC1 ÷ PC8 PC9 PC9 kfolds cross validation tuning PC1 ÷ PC8 PC9 PC9
Dropout tuning PC1 ÷ PC8 PC9 PC9
Modified early stopping condition tuning PC1 ÷ PC8 PC9 PC9
Table 3.9 The model properties that were used in this thesis ML process [1]
Stop condition Number of kfolds Dropout rate Model hyperparameter tuning min(𝑀𝑆𝐸 ) 4
Model assessment min(𝑀𝑆𝐸 ) 4 kfolds cross validation tuning min(𝑀𝑆𝐸 ) {2, 3, … , 10}
Modified early stopping condition tuning min(𝑀𝑆𝐸 ) & min 1 − 𝑀𝑆𝐸
RESULTS AND DISCUSSIONS
Finite element method process
To determine the most suitable number of mesh elements for each TPMS unit, all three mesh grids, introduced in the previous section, are used in the simulations The convergence study for beam PC1 in Figure 4.1 showed that a mesh grid with at least 1000 elements can produce good results The force-displacement curves of beam PC1 and PC2 are provided in Figure 4.2, where the deviations between the present simulation and the experiments in a previous study [19] for these beams are 9.6% and 10.7%, respectively Therefore, this mesh of 1000 elements is used for further inves- tigation to reduce computational time
Figure 4.1 The load-displacement curves of PC1 beam with various meshing strategies for the TPMS core [1]
Figure 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]
4.1.2 Impact of TPMS-core properties
This section provides the results of the FEM simulations for several beams using the above meshing method The stress distribution in both the plastic and cement components are presented in Figure 4.3 and Figure 4.4, respectively The results are consistent with previous studies and show that both higher numbers of TPMS layers and thicker shells result in more evenly distributed stress Because the confinement effect is smaller at the middle of odd-layer beams, stress tends to concentrate at this place
Although the von Misses stress may not be appropriate to investigate the behav- ior of the cement material, for comparison with the previous study [19], this stress criteria is still adopted It is the fact that the stress at the load interaction lines of beam PC2 in this study are similar to those obtained in a previous one The results provide a maximum stress of 76MPa (compared to 82MPa) and an average stress of 44MPa (compared to 45MPa) Deviations in stress values may be due to differences in mesh- ing strategies used The study suggests that increasing the number of TPMS layers and the volume fraction can improve the von Mises stress of cement components However, increasing the number of TPMS layers may not generate an impact on max- imum cement stress as significant as a 5% volume fraction increase in a two-layer
42 beam Consequently, increasing TPMS shell thickness may lead to greater confine- ment of the cement In addition, according to previous study [19], the destructive state of the TPMS-reinforced beams occurs when the crack in cement extends from the bottom to the top of the beam At this point, the cement core of beam is split into two parts However, this problem is not in the scope of this study due to the requirement of using the extended finite element method (XFEM) to investigate a) b) c) d) e)
Figure 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]
Figure 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]
The force-displacement relationship of the investigated beams is shown in Figure 4.5 By increasing the number of layers, the smoothness of these curves has
44 been reduced The load oscillates around the true value with a higher distance when the vertical displacement increases These values are considered noise data in this thesis Table 4.1 presents the peak loads and maximum deflections of the beams, with mean values of unstable curves given in parentheses In this study, the beam peak load is described as the maximum applied load in the force-displacement curve The midpoint displacement which provides the peak load is considered as the maximum deflection Beam PC9 generated the maximum peak load among all nine cases with a value of 8.295kN The results indicate that a higher volume fraction can increase ductility and bending stiffness, which is consistent with a previous report [17] For instance, the peak loads of one-layer beams with 15% and 20% volume fractions increased by about 21% and 45%, respectively, while the peak loads of two-layer and three-layer beams increased by 14% and 29%, and 13% and 28%, respectively Therefore, a higher number of TPMS layers provides more stability in peak load value, and increasing the volume fraction is less effective with a higher number of layers due to the unique geometric property of TPMSs Previous studies have shown that the true mechanical (micromechanical) properties of the TPMS structure can only be achieved with a specific number of layers, typically at least four with the Primitive type [41] A similar trend is observed in the maximum midpoint displacement responses, where increasing the number of layers only results in a slight increase in displacement, even with double the thickness
45 a) b) c) Figure 4.5 The force – displacement curves of the a) one-layer, b) two-layer, and c) three-layer TPMS reinforced beams [1]
Table 4.1 The FEA results of the investigating TPMS-reinforced beams with both peak loads and maximum deflections [1]
Increasing the number of TPMS layers results in improvements in both peak load and maximum deflection of the beam The average increment in peak load for the three-layer beam with all volume fractions is 37%, while the average increment in midpoint displacement is 98% This indicates that the effect of TPMS layers on the maximum load is smaller than on deflection A more detailed analysis of this trend is included in the next section, with predictions for non-computed schemes from the fine-tune surrogate model Additionally, the force-displacement curves of three-layer beams exhibit nonlinearity due to the material's behavior, resulting in unsmooth curves The mean values from fit curves need to be adopted to determine the peak load and maximum displacement In this case, the ML surrogate model may be a reliable approximation not only for one particular case but for all training cases as well The results of the following subsection can confirm this assumption.
Machine learning process
During this phase, it is important to note that the performance of the ANN model is strongly influenced by its hyperparameters, which can be divided into two main groups: the architecture group and the optimizer group To investigate the impact of the model architecture, various combinations of the number of layers, nodes, and activation functions are adopted, as shown in Table 4.2 Two optimizers
47 and three different stop patience values are also used It should be noted that the output layer's activation function is the ‘SoftPlus’ function for all cases to ensure a smooth output range of [0, +∞)
Table 4.2 The ANN hyperparameters investigated in the model architecture tuning and the optimization tuning processes [1]
Number of nodes-per-layer {50; 100; 150} 150
Optimizer {′Adam′; ′SGD′} { Adam ; SGD ;
The optimizer parameters are also important, especially the learning rate, to the performance of an ANN model In this study, the default optimizer parameters in the Keras library are used except for the momentum value of the ‘SGD’ optimizer, which is modified to 0.9 The data samples of one and two-layer beams are used to train the model using a kfolds cross-validation of four In this phase, the stop condition is based on the minimum validation loss function To have the smallest loss value, the model weights and bias are concluded at the stop epoch The above hyperparameter-tuning method results in approximately 650 models being evaluated The evaluation metrics for these models include the number of parameters, validation loss (𝑀𝑆𝐸 ), and training time To provide an accurate evaluation, the average metrics of three separate trains are used The reason is that the performance of a model may vary due to the random initialization of its weights and biases However, by evaluating the metrics across multiple runs, it is possible to assess the influence of a given hyperparameter
Table 4.3 The first three greatest ANN results for the validation loss with the
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]
The study findings suggest that the 'Adam' optimizer is more effective than 'SGD' in terms of performance, which is consistent with other publications The three best models, based on a minimum loss value of 0.01, are presented in Table 4.3 and Table 4.4, respectively The study also finds that a stop patience value of 30 epochs can lead to better model solutions The impacts of different optimizers and stop patience values on model performance are discussed in the next step Figure 4.6 shows the validation loss and training time for proposed architectures using the 'Adam' optimizer and stop patience of 30 epochs, along with a linear fit curve for each activation function data
In Figure 4.6, it can be observed that the 'ReLU' activation function is the most effective in this model Its output range of (0, +∞) and step point at 0 can reduce the loss function value with fewer epochs in gradient descent optimization algorithms Additionally, the metrics of the top three models in Table 4.3 indicate that increasing the model complexity can lead to higher accuracy during training, which is demonstrated by the linear fit curves in the figure It can be noted that a model with higher total parameters can achieve better results in less computational time with the help of the early stopping condition Therefore, the selected model for further investigation is model number #643, which has the minimum loss value in the architecture tuning process and a reasonable training time of 41.7s
Figure 4.6 Loss values and training times of various models with the ‘Adam’ optimizer and stop patience of 30 epochs [1]
In the following step, the optimization hyperparameters are fine-tuned using the best architecture found in the previous step, and different optimizers are evaluated The same methodology is used as in the previous tuning process, where the average of 10 training runs is used to assess the results The tunable parameters include the optimizer type, learning rate, batch size, and stop patience The specific values of these parameters are presented in Table 4.2
The effect of stop patience on validation loss and training time is presented in Table 4.5 The results show that a higher stop patience can lead to lower loss values but with a longer training time With the increase in stop patience from 30 to 50 epochs, the ‘Adam’ optimizer shows a 20% reduction in loss values for both learning rates with an increase in training time by 50% and 37%, respectively A minimum stop patience value of 30 epochs is recommended, as models trained with lower stop patience values exhibit significantly higher loss values The ‘SGD’ optimizer is more sensitive to stop patience compared to other optimizers due to its non-adaptive learning rate In general, the ‘Adam’ and ‘RMSprop’ optimizers require less time to produce good results Moreover, with a stop patience of 30 epochs, the ‘Adam’,
‘SGD’, and ‘RMSprop’ optimizers reduce loss values by 73%, 66%, and 81%, respectively
Table 4.5 Validation losses and training times of various models with different stop patience values and optimizers [1]
The effect of batch size on the model is provided in Figure 4.7 using different optimizers and learning rates The results suggest that for the current problem, the 'Adam' optimizer with a learning rate of 0.005 is the most suitable solution Increasing the batch size can improve model accuracy, but may also increase training time, particularly for adaptive optimizers like 'Adam' and 'RMSprop' A higher batch size can also result in a faster completion time for each epoch, but the model may require more epochs to achieve convergence The learning rate of 0.005 generally outperforms the learning rate of 0.01 in terms of loss value, but the computational time for the higher learning rate is typically shorter To find a model that produces both good results and a short training time, the 'Adam' optimizer with a learning rate of 0.005 and a batch size of 32 is adopted for further investigation
51 a) b) Figure 4.7 Performances of various models with different optimizers and batch size values including a) the validation loss and b) the training time [1]
The best hyperparameters found in the previous step were used to create a new model which includes architecture #643 and trained with the 'Adam' optimizer and other properties as concluded in the previous section The values of six performance metrics of the present model are provided in Table 4.6 The test loss obtained is five times higher than the train loss, with a correlation metric of 4.8553, indicating that the model is overfitting This affirmation is further demonstrated by the convergence history shown in Figure 4.8 for a typical fold in four kfolds
Table 4.6 Assessment values of the hyperparameter tuned model [1]
Figure 4.8 The convergence history of the hyperparameter tuned model [1]
This thesis employs three techniques to handle overfitting and investigates the impact of their key hyperparameters on the model performance, using a model architecture and optimizer from a previous tuning process The average results of ten times running the technique tuning are shown in Figure 4.9, Figure 4.10, and Figure 4.11 for kfolds, dropout and modified early stopping conditions techniques, respectively A steady test loss value of 0.05 can be found when the number of folds is greater than five However, with the increment in this number, the training time increases proportionally This is similar to the relationship between stop patience value and training time The dropout rate has a smaller effect on training time, and increasing it from 0.1 to 0.3 can slightly reduce the test loss value Model performance is slowly enhanced with increasing stop patience, as shown in Figure 4.11
Figure 4.9 The impacts of various numbers of folds in kfolds cross-validation model on the test loss 𝑀𝑆𝐸 and the training time [1]
Figure 4.10 The impacts of various of the dropout rate of the dropout model on the test loss 𝑀𝑆𝐸 and the training time [1]
Figure 4.11 The impacts of various of stop patience values of the modified early stop- ping condition model on the test loss 𝑀𝑆𝐸 and the training time [1]
In order to compare the performance of different methods, the best model based on the loss value for each method is selected Specifically, these models include the eight-fold model, the 0.1-dropout-rate model, and the 300-stop-patience modified early stopping model Six performance metrics proposed in the previous are normal- ized with the smallest value among the three techniques as a reference point to give a clear comparison as follows:
𝑚𝑖𝑛(𝑚𝑒𝑡𝑟𝑖𝑐 ) (4 1) where 𝐴 are the normalized performance values,
Figure 4.12 Comparisons of the efficiency of three DL techniques in handling overfit- ting including a) the average result and b) the best result for test loss value [1] The modified early stopping technique shows the best performance on test loss and metrics in Figure 4.12, and has a comparable train-test loss correlation to the dropout technique While the dropout model performs well in terms of time and test loss, its train loss value is much smaller compared with other models The kfolds model has the best result in train loss, but may not predict new data properly In
56 contrast, the early stopping model achieved 79% of the maximum train loss performance value (𝐴 ) which is a 94% ratio of the best-test-loss models Figure 4.8 shows that several epochs have comparable loss for both train and test data, but the previous model cannot stop training at those epochs This section’s results suggest that the stop condition of loss correlation can prevent the model from overfitting effectively Therefore, the modified early stopping model with a stop patience of 300 epochs is the most efficient model for this problem
Figure 4.13 The convergence history of the final model [1]
The convergence history of the model is shown in Figure 4.13 with the min- imum loss values for both the train and test data occurring at epoch 99 Although a smaller train loss could be obtained during training, the test loss may be slightly higher The performance metrics of the model are presented in Table 4.7 showing a 10.3 times reduction in correlation index and a significant decrease (three times smaller) in test loss compared to the previous model The high stop patience also results in longer training time, but the model can generate good approximations for the response of beam PC9, which is not included in train data A low average absolute error of 0.1kN and root mean square error of 0.15kN can be found in the test metrics
Table 4.7 The assessment values of the final model [1]
The well-trained model concluded in the previous subsection is used to generate force-displacement curves for nine beams in Figure 4.14 While the calculation time of FEA simulations for each beam varied from 266s to 7600s, 0.806s is the total computational time for all beams with this surrogate model Besides, the
CONCLUSION AND RESEARCH DEVELOPMENT
Conclusion
In this thesis, the plastic reinforced cement beam with the Primitive TPMS type has been studied by FEA simulations These beams’ behaviors have been adopted to conduct a surrogate model based on ML algorithms Some key conclusions can be denoted as follows:
A novel meshing strategy for the complicated geometry of TPMS structures is introduced The reliability of this number-of-element-based mesh has been verified by the experimental results of the reference study;
Nine TPMS reinforced beam scenarios including 10%, 15%, and 20% volume fractions along with one, two, and three core layers are investigated
The relationships between the beam’s peak load and its volume fraction could be denoted as straight lines Similar relations could also be indicated with the impact of this volume fraction on the maximum midpoint displacement However, these linear relations have different inclination angles for each number of layers;
The influences of the number of layers on both maximum deflection and maximum load are nonlinear curves By increasing the number of layers, the beam's mechanical properties tend to attain ceiling values;
The beam strength might mainly depend on the strength of the cement core The most important contribution of the TPMS core may be confinement creation;
The heaviest proposed architecture, which is the three 150-node layers, along with the ‘ReLU’ activation function and ‘Adam’ optimization algorithm can produce good predictions on the training dataset despite the noisy FEA data;
Different deep learning techniques, namely kfolds cross-validation, dropout, and modified early stopping conditions are adopted to alleviate the overfitting in this work;
The model which used a new stopping condition of loss correlation could be denoted as the most appropriate model for this thesis problem;
The final model concluded from the proposed three-phase process can produce excellent predictions for various reinforcement cases without inefficient FEA simulations;
Two key mechanical properties, that are the peak load and the maximum displacement, are illustrated as surfaces to indicate the influence of both volume fraction and the number of layers
In brief, the surrogate model has been achieved based on a well-trained ANN model with an effective DL technique This result can expand the potential of using the TPMS-reinforced beam in numerous practical applications.
Research development direction
Based on the product of this thesis and its limitation, some potential research areas should be noted as follows:
Employing different TPMS types as reinforcement strategies;
Studying other behaviors of the reinforced structures including crack pattern, dynamic response, responses under various boundary conditions, etc.;
Investigating the theoretical solution of this beam type;
Using the TPMS to reinforce other types of concrete structures such as slabs, columns, etc.;
Creating an overall surrogate model that can predict the beam behavior with any configuration
Enhancing the surrogate model with different ML algorithms such as support vector machine (SVM), random forest regression, extreme gradient boosting (XGBoost), and unsupervised or reinforce learning algorithms;
Applying the latest physics-informed neural networks (PINN) to deal with the present problem without FEA simulations
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