Prediction of Mechanical Properties for Carbon fiber/PLA Composite Lattice Structures Using Mathematical and ANFIS Models

This study investigates the influence of design, relative density (RD), and carbon fiber (CF) incorporation parameters on mechanical characteristics, including compressive modulus (E), strength, and specific energy absorption (SEA) of triply periodic minimum surface (TPMS) lattice structures. The TPMS lattices were 3D-printed by fused filament fabrication (FFF) using polylactic acid (PLA) and carbon fiber-reinforced PLA(CFRPLA). The mechanical properties of the TPMS lattice structures were evaluated under uniaxial compression testing based on the design of experiments (DOE) approach, namely, full factorial design. Prediction modeling was conducted and compared using mathematical and intelligent modeling, namely, adaptive neuro-fuzzy inference systems (ANFIS). ANFIS modeling allowed the 3D printing imperfections (e.g., RD variations) to be taken into account by considering the actual RDs instead of the designed ones, as in the case of mathematical modeling. In this regard, this was the first time the ANFIS modeling utilized the actual RDs. The desirability approach was applied for multi-objective optimization. The mechanical properties were found to be significantly influenced by cell type, cell size, CF incorporation, and RD, as well as their combination. The findings demonstrated a variation in the E (0.144 GPa to 0.549 GPa), compressive strength (4.583 MPa to 15.768 MPa), and SEA (3.759 J/g to 15.591 J/g) due to the effect of the studied variables. The ANFIS models outperformed mathematical models in predicting all mechanical characteristics, including E, strength, and SEA. For instance, the maximum absolute percent deviation was 7.61% for ANFIS prediction, while it was 21.11% for mathematical prediction. The accuracy of mathematical predictions is highly influenced by the degree of RD deviation: a higher deviation in RD indicates a lower accuracy of predictions. The findings of this study provide a prior prediction of the mechanical behavior of PLA and CFRPLA TPMS structures, as well as a better understanding of their potential and limitations.


Introduction
Lattice structures are attracting attention in a wide range of industries, including aerospace, automotive, medical, and heat management, where lightweight and multifunctionality are required. Properties like thermal insulation, acoustic absorption, mechanical vibration damping, high stiffness-to-volume fraction ratio, and energy absorption are required within these structures [1]. They enable the enhancement of the performance-toweight ratio, the creation of highly controlled architectures, and the distribution of impact shock across an object.
In comparison to bulk materials, lattice structures have a high number of design variables, which increases their complexity and limits their ability to be fabricated using conventional manufacturing processes. However, the advent of additive manufacturing Primitive, with various cell sizes (8 mm and 12 mm), CF incorporation (0% and 15%), and RDs (30% and 44%) were considered. The analysis was conducted based on a full factorial design. The actual relative densities of the printed samples were measured to ensure no significant difference between the designed and printed RDs. Uniaxial compression testing was used to evaluate the mechanical properties of 3D-printed samples, including compression modulus, strength, and SEA. Moreover, ANFIS modeling was also employed for predicting the performance of the TPMS structures, and the results were compared with the mathematical models developed based on the DOE approach. By using the ANFIS modeling predictions, we could accurately predict mechanical characteristics considering the inherent imperfections in 3D printing, such as an RD variation, which cannot be avoided. Due to the freedom in the ANFIS modeling approach, the actual RDs were used instead of the intended (designed) RDs, leading to improved prediction accuracy. Finally, the best parameter settings for maximizing the TPMS lattices' performance were determined using multi-objective optimization through the desirability function.
The following section discusses the materials and methods used in this study. The results are presented in Section 3. Section 4 provides the discussions. Finally, Section 5 presents the conclusions.

Design and Relative Density
In the current research, three different TPMS cell topologies, including Gyroid (G), Diamond (D), and Primitive (P) were considered. Within the context of this study, the terms "cell topology" and "cell type" are used interchangeably. Figure 1a illustrates the G, D, and P cell topologies. The unit cell size (l) of all cell topologies was designed in a cubic unit cell with 8 mm and 12 mm edge lengths. The reason behind these particular lengths is to fit the whole lattice dimensions.
Relative density (RD) is one of the most factors that influence a lattice's performance. RD, also termed as volume fraction (V lattice /V overall ), is defined as the lattice structure volume (V lattice ) divided by the overall structure volume (V overall ) [34]. This study employed 30% and 44% RDs, taking into account the minimum feasible wall thickness to meet FDM resolution, nozzle diameter, and cell sizes. The selected range of the RDs also considered that the actual densities of the printed samples showed a close agreement with the designed RDs. For each TPMS cell topology, the needed RD is determined by the wall thickness and length of the cell. Hence, the intended RD was controlled by adjusting the ratio of the cell wall thickness parameter (t) to the cell size (t/l) [16]. It should be mentioned that every cell topology has its function in relation to the t/l ratio because of the differences in surface areas. The t values were determined based on the CAD predictions. Table 1 presents the values of the t parameter for each cell topology at the designed RD and cell lengths. areas. The t values were determined based on the CAD predictions. Table 1 presents the values of the t parameter for each cell topology at the designed RD and cell lengths.  The lattice structures were designed and printed in sizes of 24 × 24 × 48 mm 3 , see Figure 1b. The whole lattice dimensions were selected based on the aspect ratio (length to width ratio) of the ASTM D695-15 standard for compressive testing of rigid polymeric materials (), so that the length should be twice the sample width [17]. The number of cells to fit the selected dimensions depends on the cell size, whether 8 mm (3 × 3 × 6 cells) or 12 mm (2 × 2 × 4 cells). CREO 8.0 software was used to design the G, D, and P structures in the STL format. High-accuracy STL files were obtained for all investigated designs, with the number of generated triangles varying from 464,860 to 6,653,676 depending on the cell topology, cell size, and RD.

Experimental Design
The influence of the material composition (CF incorporation), cell topology, cell size, and relative density on the mechanical properties, including compression modulus (E) and strength, and specific energy absorption (SEA), was evaluated by the DOE approach. Table 2 illustrates the four parameters and their respective levels. A full factorial design was used with 24 runs. Each run was repeated 3 times resulting in a total of 72 experiments. Analysis of variance (ANOVA) was performed with a 95% confidence interval to study the significant influence of the variables and their interactions on E, compressive strength, and SEA. p-Values less than 0.05 imply that model terms (main factors and interactions) are statistically significant. Mathematical relationships between the investigated parameters and each of the output responses were developed. The developed mathematical models were further considered for optimization of the considered parameters.  The lattice structures were designed and printed in sizes of 24 × 24 × 48 mm 3 , see Figure 1b. The whole lattice dimensions were selected based on the aspect ratio (length to width ratio) of the ASTM D695-15 standard for compressive testing of rigid polymeric materials (), so that the length should be twice the sample width [17]. The number of cells to fit the selected dimensions depends on the cell size, whether 8 mm (3 × 3 × 6 cells) or 12 mm (2 × 2 × 4 cells). CREO 8.0 software was used to design the G, D, and P structures in the STL format. High-accuracy STL files were obtained for all investigated designs, with the number of generated triangles varying from 464,860 to 6,653,676 depending on the cell topology, cell size, and RD.

Experimental Design
The influence of the material composition (CF incorporation), cell topology, cell size, and relative density on the mechanical properties, including compression modulus (E) and strength, and specific energy absorption (SEA), was evaluated by the DOE approach. Table 2 illustrates the four parameters and their respective levels. A full factorial design was used with 24 runs. Each run was repeated 3 times resulting in a total of 72 experiments. Analysis of variance (ANOVA) was performed with a 95% confidence interval to study the significant influence of the variables and their interactions on E, compressive strength, and SEA. p-Values less than 0.05 imply that model terms (main factors and interactions) are statistically significant. Mathematical relationships between the investigated parameters and each of the output responses were developed. The developed mathematical models were further considered for optimization of the considered parameters. The best settings of the considered parameters were optimized through the desirability approach to simultaneously (multi-objective optimization) achieve the maximum E, compressive strength, and SEA. Design-Expert 13 software was used to systematically analyze the influence of the investigated parameters, develop mathematical (prediction) models, and conduct multi-objective optimization.

FDM Printing of TPMS Lattices
An open-source FDM machine, Prusa FDM printer (Original Prusa i3 MK3S+, Czech Republic), equipped with a 0.4 mm nozzle, was used to additively manufacture the lattice structures. The STL files for the TPMS lattice structures were imported into slicer software (PrusaSlicer 2.4.2) in order to slice and generate "GCODE" files. Table 3 presents the printing parameters employed to 3D print the samples. All samples were printed without support structures. Examples of printed TPMS structures of different cell topologies are displayed in Figure 2. The best settings of the considered parameters were optimized through the desirability approach to simultaneously (multi-objective optimization) achieve the maximum E, compressive strength, and SEA. Design-Expert 13 software was used to systematically analyze the influence of the investigated parameters, develop mathematical (prediction) models, and conduct multi-objective optimization.

FDM Printing of TPMS Lattices
An open-source FDM machine, Prusa FDM printer (Original Prusa i3 MK3S+, Czech Republic), equipped with a 0.4 mm nozzle, was used to additively manufacture the lattice structures. The STL files for the TPMS lattice structures were imported into slicer software (PrusaSlicer 2.4.2) in order to slice and generate "GCODE" files. Table 3 presents the printing parameters employed to 3D print the samples. All samples were printed without support structures. Examples of printed TPMS structures of different cell topologies are displayed in Figure 2.

Metrological Characterization
The printed samples' actual dimensions (e.g., length, width, and thickness) were determined by a Lab Profile Projector (VOM-2515), as shown in Figure 3a. Three measurements were made for each dimension, and the averaged values were used. Density was obtained by utilizing Archimedes' method, a common method used for determining the density of both porous and solid structures. First, the sample' density was determined by Archimedes' method by weighing the sample in air and distilled water. The volume of the lattice (Vlattice) was then determined from the obtained density. Then, the ratio between

Metrological Characterization
The printed samples' actual dimensions (e.g., length, width, and thickness) were determined by a Lab Profile Projector (VOM-2515), as shown in Figure 3a. Three measurements were made for each dimension, and the averaged values were used. Density was obtained by utilizing Archimedes' method, a common method used for determining the density of both porous and solid structures. First, the sample' density was determined by Archimedes' method by weighing the sample in air and distilled water. The volume of the lattice (V lattice ) was then determined from the obtained density. Then, the ratio between (V lattice /V overall ), which represent the RD, was calculated. The overall volume (V overall ) was calculated from the actual sample' dimensions. Measurements were conducted on Shimadzu Analytical Balance (AUW220D, China) with a readability of 0.01 mg and a universal specific gravity kit (SGK-C, Mineralab, UK) as in [5]; see Figure 3b. (Vlattice/Voverall), which represent the RD, was calculated. The overall volume (Voverall) was calculated from the actual sample' dimensions. Measurements were conducted on Shimadzu Analytical Balance (AUW220D, China) with a readability of 0.01 mg and a universal specific gravity kit (SGK-C, Mineralab, UK) as in [5]; see Figure 3b.

Mechanical Properties
The influence of the considered parameters on the mechanical characteristics of the TPMS structures was conducted under a uniaxial compression test. The compression tests were carried out following the ASTM D695-15 standard. Compression tests were performed along the build direction at a constant crosshead speed of 1.6 mm/min up to 60% strain on a Zwick Z100 testing machine equipped with a 100 KN load cell. During the tests, forces and displacements were recorded using testXpert II software.
Mechanical characteristics, including compressive modulus (E), compressive strength, and SEA, were gathered from the force-displacement curves. First, engineering compressive stress was used and calculated by dividing the recorded forces (F) by the measured original cross-sectional area. The strain was computed by dividing the recorded displacements (δ) during the tests by the original length of the sample. Then, compression modulus and strength were determined. Compression modulus was calculated by the slope of the tangent line at the linear portion of the stress-strain curves by testXpert II software. Compression strength (σpeak) was considered as the peak strength, the maximum stress-value of the first peak (i.e., first local maximum) in the stress-strain curve [17,35].
Specific energy absorption (SEA) is a useful indicator for measuring a structure's capability to absorb energy per unit weight. The SEA of a lattice structure is represented by the area under the force-displacement curve divided by the structure's weight. The SEA was calculated as in Equation (1) [14]. The area under the force-displacement curve was calculated up to 55% strain (theoretical strain densification strain) using MATLAB R2022a.
where F is the force, δ is the displacement, and m is the structure's weight.

Mechanical Properties
The influence of the considered parameters on the mechanical characteristics of the TPMS structures was conducted under a uniaxial compression test. The compression tests were carried out following the ASTM D695-15 standard. Compression tests were performed along the build direction at a constant crosshead speed of 1.6 mm/min up to 60% strain on a Zwick Z100 testing machine equipped with a 100 KN load cell. During the tests, forces and displacements were recorded using testXpert II software.
Mechanical characteristics, including compressive modulus (E), compressive strength, and SEA, were gathered from the force-displacement curves. First, engineering compressive stress was used and calculated by dividing the recorded forces (F) by the measured original cross-sectional area. The strain was computed by dividing the recorded displacements (δ) during the tests by the original length of the sample. Then, compression modulus and strength were determined. Compression modulus was calculated by the slope of the tangent line at the linear portion of the stress-strain curves by testXpert II software. Compression strength (σ peak ) was considered as the peak strength, the maximum stress-value of the first peak (i.e., first local maximum) in the stress-strain curve [17,35].
Specific energy absorption (SEA) is a useful indicator for measuring a structure's capability to absorb energy per unit weight. The SEA of a lattice structure is represented by the area under the force-displacement curve divided by the structure's weight. The SEA was calculated as in Equation (1) [14]. The area under the force-displacement curve was calculated up to 55% strain (theoretical strain densification strain) using MATLAB R2022a.
where F is the force, δ is the displacement, and m is the structure's weight.

Adaptive Neuro-Fuzzy Inference System (ANFIS) Model
ANFIS is a hybrid neuro-fuzzy method for modeling complex systems. It integrates the best learning abilities of the artificial neural network (ANN) and inference capabilities of the fuzzy inference system (FIS) [36,37]. ANFIS accomplishes sample-based learning using the train data set to develop an efficient ANFIS structure for solving the associated problem. The developed ANFIS structure is being evaluated for its validity through a test data set. ANFIS uses a five-layer, feed-forward propagation structure [38]. Figure 4 shows an illustration of the ANFIS structure with two inputs, three membership functions (MFs), and one output. The layer explanation is described as follows [38][39][40]:   Layer 1: input membership functions (MFs). In this layer, also called the fuzzification layer, the fuzzy membership value of each input, i.e., µA i (x) and µB i (y) is calculated by a proper membership function, e.g., trapezoidal, Gaussian, and triangular.
Layer 2: fuzzy rules. The firing strength ( w i ), the weight for each rule's output is calculated in this layer. Each node's output is the product of all its input signals, which can be calculated using Equation (2).
Layer 3: normalization. This layer represents the normalization of the firing strength (w ι ), as computed by Equation (3).
Layer 4: defuzzification. The output of each node in this layer is calculated based on the function given in Equation (4).
where p i , q i , and r i called a consequent parameter set. Layer 5: output layer. This layer has only one node to calculate the system output, as in Equation (5).
The ANFIS modeling was used for predicting the performance of the TPMS structures in terms of E, σpeak, and SEA. Using the ANFIS modeling predictions, we could accurately predict mechanical characteristics considering imperfections in 3D printing, such as an RD variation. This way, the actual RDs were used instead of the intended RDs, leading to improved prediction accuracy. Figure 5 depicts the methodology adopted in this study.    Table 4 shows the average values of the actual RDs, E, σ peak , and SEA of the 24 runs according to the full factorial DOE. The variabilty in the three repeated measured results is indicated by the standard deviation (SD). The results, including E, σ peak , and SEA, were derived from the force-displacement curves. Typical stress-strain curves under the uniaxial compression test of the 24 runs listed in Table 4 are depicted in Figure 6. As can be seen in Figure 6, the mechanical behavior of the lattice structures is susceptible to changes in the CF incorporation, design, and RD factors. This influence is shown by how much the stressstrain curves vary in terms of either the stress range or the shape of the curves. For instance, the influence of the cell type in terms of the stress range and the deformation behavior is shown between Diamond-based structures (Figure 6a  Deformation mechanisms are graphically illustrated in Figure 7. A bending-torsion coupled failure is evident for the Diamond-based structures; see Figure 7a. Figure 7b shows that the Gyroid-based structure exhibits bending and buckling mechanisms of failure. Figure 7c illustrates the deformation of the Primitive-based structure showing a layer-by-layer deformation mechanism as also reflected by the stress-strain curves; see Figure 6e,f.

ANOVA Analysis
The effect of the considered variables on the TPMS structure performances was studied statistically using ANOVA. Reduced ANOVA tables were utilized so that nonsignificant terms were eliminated using backward/forward methods to enhance the model accuracy without sacrificing the model fit. It is worth mentioning that the normality assumption was satisfied (see Figure 8). The R 2 of E, σ peak , and SEA are 0.995, 0.999, and 0.97, respectively, which indicates an excellent representation of the variability of the data by the model terms. Table 5 shows the reduced ANOVA table of the compressive modulus, and p-values less than 0.05 indicate that model terms are significant. From the ANOVA table (Table 5), all considered factors, including CF incorporation, RD, cell type, and cell size, significantly influence the E. Two-source interactions, including the cell type and RD, and cell type and cell size, show a significant influence on the E. Furthermore, three-source interactions, namely RD, cell size, and cell type, have a significant influence on the E. The most significant effects are caused by changing the RD variable (58.95%), followed by cell type (31.32), and then the CF incorporation (3.51%). Figures 9 and 10 also provide a visual representation of the results, showing the influence direction of the main factors ( Figure 9) and their interactions ( Figure 10) on the compressive modulus. The Diamond cell type demonstrates the highest compressive modulus, followed by the Gyroid cell type, and finally, the Primitive cell type, which demonstrated the lowest compressive modulus performance. The other parameters-CF, RD, and cell size-have a proportional influence, meaning that a change from a low level to a high level of any of them can increase in the E.  Table 4, classified based on the cell type and CF %: (a,b) Diamond-based samples at 0% CF and 15% CF, respectively; (c,d) Gyroid-based samples at 0% CF and 15% CF, respectively; (e,f) Primitive-based samples at 0% CF and 15% CF, respectively.

ANOVA Analysis
The effect of the considered variables on the TPMS structure performances was studied statistically using ANOVA. Reduced ANOVA tables were utilized so that nonsignificant terms were eliminated using backward/forward methods to enhance the model accuracy without sacrificing the model fit. It is worth mentioning that the normality assumption was satisfied (see Figure 8). The of E, σpeak, and SEA are 0.995, 0.999, and 0.97, respectively, which indicates an excellent representation of the variability of the data by the model terms. Table 5 shows the reduced ANOVA table of the compressive modulus, and p-values less than 0.05 indicate that model terms are significant. From the ANOVA table (Table 5), all considered factors, including CF incorporation, RD, cell type, and cell size, significantly influence the E. Two-source interactions, including the cell type and RD, and cell type and cell size, show a significant influence on the E. Furthermore, three-source interactions, namely RD, cell size, and cell type, have a significant influence on the E. The most significant effects are caused by changing the RD variable (58.95%), followed by cell type (31.32), and then the CF incorporation (3.51%).       Similarly, the reduced ANOVA table for the σpeak, peak strength is presented in Table  6. According to the σpeak ANOVA table, the factors that significantly influence σpeak are CF incorporation, RD, and cell type. The two-source interaction terms, including RD and cell type, and cell size and cell type, have a significant influence on the σpeak. Furthermore, RD and cell size interact significantly with the other two parameters, CF and cell type. The most significant effects are caused by changing the RD variable (71.68%), followed by cell type (23.87%). Figure 11 demonstrates the directional influence of the main components, whereas Figure 12 shows the two-and three-source interaction terms. Figure 11 shows that the TPMS structures with Diamond-cell type exhibit the highest σpeak, whereas the Primitive-based cell type structures show the lowest. Compressive strength is proportionally influenced by RD; increasing RD results in an increase in σpeak ( Figure 11). Regarding CF, Figure 11 demonstrates that σpeak decreases as CF increases. The interaction between cell type and size is illustrated in Figure 12b, which demonstrates that as cell size increases, the σpeak of the Diamond and Primitive structures improve while the σpeak of the Gyroid structures decreases. Figure 12d shows how the interaction of the RD, cell size, and cell type affects the σpeak. For instance, the σpeak of the Diamond-based structures improved at high RD (44%) but decreased at low RD (30%) as the cell size increased.   Similarly, the reduced ANOVA table for the σpeak, peak strength is presented in Table  6. According to the σpeak ANOVA table, the factors that significantly influence σpeak are CF incorporation, RD, and cell type. The two-source interaction terms, including RD and cell type, and cell size and cell type, have a significant influence on the σpeak. Furthermore, RD and cell size interact significantly with the other two parameters, CF and cell type. The most significant effects are caused by changing the RD variable (71.68%), followed by cell type (23.87%). Figure 11 demonstrates the directional influence of the main components, whereas Figure 12 shows the two-and three-source interaction terms. Figure 11 shows that the TPMS structures with Diamond-cell type exhibit the highest σpeak, whereas the Primitive-based cell type structures show the lowest. Compressive strength is proportionally influenced by RD; increasing RD results in an increase in σpeak ( Figure 11). Regarding CF, Figure 11 demonstrates that σpeak decreases as CF increases. The interaction between cell type and size is illustrated in Figure 12b, which demonstrates that as cell size increases, the σpeak of the Diamond and Primitive structures improve while the σpeak of the Gyroid structures decreases. Figure 12d shows how the interaction of the RD, cell size, and cell type affects the σpeak. For instance, the σpeak of the Diamond-based structures improved at high RD (44%) but decreased at low RD (30%) as the cell size increased.  Similarly, the reduced ANOVA table for the σ peak , peak strength is presented in Table 6. According to the σ peak ANOVA table, the factors that significantly influence σ peak are CF incorporation, RD, and cell type. The two-source interaction terms, including RD and cell type, and cell size and cell type, have a significant influence on the σ peak . Furthermore, RD and cell size interact significantly with the other two parameters, CF and cell type. The most significant effects are caused by changing the RD variable (71.68%), followed by cell type (23.87%). Figure 11 demonstrates the directional influence of the main components, whereas Figure 12 shows the two-and three-source interaction terms. Figure 11 shows that the TPMS structures with Diamond-cell type exhibit the highest σ peak , whereas the Primitive-based cell type structures show the lowest. Compressive strength is proportionally influenced by RD; increasing RD results in an increase in σ peak (Figure 11). Regarding CF, Figure 11 demonstrates that σ peak decreases as CF increases. The interaction between cell type and size is illustrated in Figure 12b, which demonstrates that as cell size increases, the σ peak of the Diamond and Primitive structures improve while the σ peak of the Gyroid structures decreases. Figure 12d shows how the interaction of the RD, cell size, and cell type affects the σ peak . For instance, the σ peak of the Diamond-based structures improved at high RD (44%) but decreased at low RD (30%) as the cell size increased.  The reduced ANOVA table for SEA is presented in Table 7. The results indicate that the cell type, CF incorporation, and RD have a significant influence on the SEA. In addition, the two-source interaction between the cell type and CF has a significant influence on the SEA. The most important effects are contributed by the cell type (70.51%), followed by RD (22.44%), and then the CF and cell type interaction (2.66%). The influence of the main factors is also graphically presented in Figure 13. Similar to the compressive modulus and strength, Diamond-based cell types exhibit the best SEA. SEA is proportionally influenced by RD, i.e., an increase in RD results in an upsurge in SEA. Figure 13 demonstrates that SEA decreases as CF increases. This influence, however, varies depending on the cell type, as illustrated in Figure 14. For instance, Diamond-based TPMS structures show an enhancement in SEA with an increase in CF (Figure 14a,b), whereas Primitive-based TPMS structures show a decrease in SEA (Figure 14a,d). Furthermore, Table 7 demonstrates the significant influence of the interaction between the CF and cell type (shown in Figure 14a) on SEA, providing further evidence that CF's influence on SEA is dependent on the cell topology. The reduced ANOVA table for SEA is presented in Table 7. The results indicate that the cell type, CF incorporation, and RD have a significant influence on the SEA. In addition, the two-source interaction between the cell type and CF has a significant influence on the SEA. The most important effects are contributed by the cell type (70.51%), followed by RD (22.44%), and then the CF and cell type interaction (2.66%). The influence of the main factors is also graphically presented in Figure 13. Similar to the compressive modulus and strength, Diamond-based cell types exhibit the best SEA. SEA is proportionally influenced by RD, i.e., an increase in RD results in an upsurge in SEA. Figure 13 demonstrates that SEA decreases as CF increases. This influence, however, varies depending on  based TPMS structures show a decrease in SEA (Figure 14a,d). Furthermore, Table 7 demonstrates the significant influence of the interaction between the CF and cell type (shown in Figure 14a) on SEA, providing further evidence that CF's influence on SEA is dependent on the cell topology.

Mathematical and ANFIS Prediction Models
Mathematical relationships between the responses, including E, σpeak, and SEA, and the investigated variables were developed based on the reduced models in the ANOVA analysis, e.g., Tables 5-7. The developed mathematical models are presented in Table 8. show an enhancement in SEA with an increase in CF (Figure 14a,b), whereas Primitivebased TPMS structures show a decrease in SEA (Figure 14a,d). Furthermore, Table 7 demonstrates the significant influence of the interaction between the CF and cell type (shown in Figure 14a) on SEA, providing further evidence that CF's influence on SEA is dependent on the cell topology.

Mathematical and ANFIS Prediction Models
Mathematical relationships between the responses, including E, σpeak, and SEA, and the investigated variables were developed based on the reduced models in the ANOVA analysis, e.g., Tables 5-7. The developed mathematical models are presented in Table 8.

Mathematical and ANFIS Prediction Models
Mathematical relationships between the responses, including E, σ peak , and SEA, and the investigated variables were developed based on the reduced models in the ANOVA analysis, e.g., Tables 5-7. The developed mathematical models are presented in Table 8.
Similarly, the ANFIS models were developed to predict the performance of the TPMS structures in terms of E, σ peak , and SEA. An advantage of this type of modeling is that it eliminates the requirement to strictly adhere to predetermined settings for any variable in the DOE, where any change in that setting will most likely impact the results. For instance, the ANOVA analysis results (Section 3.2) proved that RD contributed the most to the mechanical properties, including E, σ peak , and SEA. Thus, a variation in the actual RD from the designed RD will certainly lead to a variation in the mechanical properties. In this regard, actual RD should be used instead of using the designed one, particularly when there is a high variation, which cannot be avoided. Using the actual RDs makes the analysis and modeling more accurate. Variations between the actual and designed RDs are attributed to a number of reasons depending on the AM process. For instance, deviation in wall thickness and the presence of voids and cracks are reasons to deviate RD in the FDM process [5,41]. Excess porosity is an example of the source RD deviation in the SLS process, as elaborated by [42]. In laser-powder bed fusion (L-PBF), unmelted powder particles that have no pathway to get out of the structure, closed pores due to powder clog [43], deviation in wall thickness, voids and cracks, and surface roughness [16] are reasons for RD deviation. Therefore, a modeling approach such as ANFIS will be more accurate as it can consider the measured RDs instead of the designed ones. Table 8. Mathematical prediction models of E, σ peak , and SEA.

Response
Topology In this study, the ANFIS models were developed during the training phase based on the full factorial results (as training data) presented in Table 4. The ANFIS technique allows for the utilization of input data even if it does not comply with a specific DOE. It should be noted that the actual RDs were used instead of the designed RDs when developing ANFIS models in both the training and testing phases. Thus, more realistic models could be developed, resulting in improved prediction accuracy. The accuracy of the developed models was evaluated with the testing data set presented in Table 9. For each TPMS performance (e.g., E, σ peak , and SEA), the accuracy of the developed ANFIS models was evaluated using the root mean square error (RMSE), which was calculated using Equation (6).
where n is the number of testing data while Exp i and Pred i are the experimental and predicted results of the ith testing experiment, respectively. Furthermore, for each TPMS performance, RMSE was used during the training phase for tuning and selecting the fuzzy inference parameters to minimize the RMSE. The fuzzy inference parameters and their settings that resulted in the lowest RMSE are presented in Table 10. An illustration of the ANFIS structure of the σ peak response is provided in Figure 15.  Figure 15. ANFIS structure network for modeling the σ peak with "2 2 2 3" number of MFs.
In Figure 16, the predicted values for E, σ peak , and SEA based on the mathematical and ANFIS models are shown alongside the experimental results. Figure 16a,c,e depict the experimental and predicted findings for the training data, demonstrating that experimental and predicted results obtained from both methods are comparable. Figure 17a-c show evidence of the close agreement between predicted results obtained by both models and the experimental results in terms of R 2 for E, σ peak , and SEA. Similarly, Figure 16b,d,f depict the experimental and predicted findings for the testing data, showing that the predicted results obtained from ANFIS methods are much close to the experimental results. Furthermore, an enhancement is clearly shown in ANFIS predicting results for E, σ peak , and SEA compared with mathematical results. For instance, the R 2 of ANFIS prediction for σ peak is 0.977, which is significantly higher than that of mathematical prediction, which is 0.763. Table 11 compares the RMSE performance of the mathematical and ANFIS models with respect to testing experiments, revealing that ANFIS models have outperformed mathematical models in all responses. Furthermore, a comparison between mathematical and ANFIS results for each test experiment in terms of absolute percent deviation Table 12. From Table 12, the maximum deviation in the ANFIS' prediction results was 7.61% (the 3rd test experiment, σ peak ), while a deviation of 21.11% (the 5th test experiment, σ peak ) was found for the mathematical model. It should be noted that the maximum deviation in the mathematical prediction results (21.11%) occurred in the 5th testing experiment, which showed a high RD deviation (actual and designed RDs are 37% and 34.91%, respectively). Furthermore, mathematical models also displayed some deviations exceeding 10%, contrary to ANFIS deviation values.
ANFIS' prediction results was 7.61% (the 3rd test experiment, σpeak), while a deviation of 21.11% (the 5th test experiment, σpeak) was found for the mathematical model. It should be noted that the maximum deviation in the mathematical prediction results (21.11%) occurred in the 5th testing experiment, which showed a high RD deviation (actual and designed RDs are 37% and 34.91%, respectively). Furthermore, mathematical models also displayed some deviations exceeding 10%, contrary to ANFIS deviation values.  The results shown in Figures 16 and 17 and Tables 11 and 12 highlight the significance of employing AI models such as the ANFIS model for predicting the performances of TPMS lattice structures while accounting for the issues associated with 3D printing, such as RD deviation. Table 12 shows that whenever the actual RD is close to the designed RD, both models provide accurate predictions. However, for a high variation in RD, the ANFIS model is preferable. This indicates the ability of both modeling approaches to predict various mechanical properties of TPMS structures while considering different variables, including material composition, geometry, and RD.  The results shown in Figures 16 and 17 and Tables 11 and 12 highlight the significance of employing AI models such as the ANFIS model for predicting the performances of TPMS lattice structures while accounting for the issues associated with 3D printing, such

Multi-Objective Optimization
Desirability analysis was used to select the best settings of the CF incorporation, relative density, cell type, and cell size RD that led to maximizing the σ peak , E, and SEA. Table 13 shows the optimal combination values of the considered variables for multi-objective optimization. Diamond topology, 12 mm cell size, 15% CF, and 44% RD should be used to achieve an overall desirability of 97.8%. Nevertheless, if, for whatever reason, either Gyroid or Primitive cell topologies are selected, the optimal combinations as well as the overall desirability values for both designs are illustrated in Table 14. Tables 13 and 14 present the significance of carefully selecting the cell topology, cell size, and CF incorporation, as different combinations of these factors result in varying performances.  Table 15 shows the validation experiments related to the multi-objective optimization findings reported in Tables 13 and 14, demonstrating a good agreement between the multi-objective optimization and experimental results.

Discussion
Stress-strain curves depicted in Figure 6 show evident variations in the mechanical response during the compression testing. Variations are clearly detected in terms of stress, compressive modulus, and deformation patterns. Primitive structures showed a wave pattern of deformation (Figure 6e,f (e.g., run #17-24)), while Diamond structures (Figure 6a,b) and Gyroid structures (Figure 6c,d) seemed to deform uniformly, giving them the advantage of accumulating the load-bearing capacity (e.g., run #1-8 for Diamond and run #9-16 for Gyroid. Compared with D ( Figure 6a) and G (Figure 6c), samples with 0% CF, which exhibit a sharp reduction in stress following elastic deformation, D ( Figure 6b) and G ( Figure 6d) samples with 15% CF demonstrate more plastic deformation. However, Figure 6f shows that incorporating CF into the Primitive structures, particularly with samples of 12 mm cell sizes (run#22 and run#22 24), makes the deformation more wavy than in samples with 0% CF (Figure 6e).
From the results presented in Table 4, Diamond-based TPMS lattice structures showed the best mechanical properties, including compressive modulus, σ peak , and SEA. For instance, the maximum compressive modulus (0.549 GPa), σ peak (15.768 MPa), and SEA (15.591 J/g) were observed with Diamond, 15% CF, 44% RD, and 12 mm cell size (run #8). On the other hand, the worst mechanical responses were obtained by Primitive-based celltype TPMS structures. The minimum compressive modulus (0.144 GPa) was obtained with Primitive, 0% CF, 30% RD, and 8 mm cell size (run #17), while the lowest σ peak (4.583 MPa) was observed with Primitive, 15% CF, 30% RD, and 8 mm cell size (run #21). Similarly, the minimum SEA (3.759 J/g) was observed with Primitive, 15% CF, 30% RD, and 12 mm cell size (run #22). The walls' orientation and better material distribution within the geometry of the Diamond structures improve wall contact and reduce the empty spaces, making it less susceptible to fracture initiation [5,24]. For illustration, Figure 18 shows cross-sections of lattice structures with 12 mm cell size and 44% RD. It is evident from the cross sections that empty spaces between walls in the Diamond structure (Figure 18a) are less than those in Gyroid and Primitive structures; Figure 18b,c, respectively. larly, the minimum SEA (3.759 J/g) was observed with Primitive, 15% CF, 30% RD, and 12 mm cell size (run #22). The walls' orientation and better material distribution within the geometry of the Diamond structures improve wall contact and reduce the empty spaces, making it less susceptible to fracture initiation [5,24]. For illustration, Figure 18 shows cross-sections of lattice structures with 12 mm cell size and 44% RD. It is evident from the cross sections that empty spaces between walls in the Diamond structure (Figure 18a) are less than those in Gyroid and Primitive structures; Figure 18b,c, respectively. The influence of all considered parameters, including cell topology and size, CF incorporation, and RD, as well as their different combinations on the compressive response of the TPMS structures, was statistically significant. Results showed that RD had the greatest impact on mechanical response among the four studied variables, followed by cell type and CF incorporation. Cell size had the least impact. The authors of [44] stated that RD is the key factor in mechanical performance, including elastic modulus and strength of a given lattice structure. This is in line with the statistical findings, which show that RD has a high influence on both the E and σpeak (58.95% and 71.68%, respectively). The statistical analysis confirms that cell type and RD influence the compressive modulus, σpeak, and SEA of the TPMS lattice structures. This finding is in line with previous studies that investigated the effect of TPMS cell topologies and RDs on E, σpeak, and SEA characteristics, such that these properties were enhanced as RD increased [16]. Diamond structures showed the best performance, while Primitive structures were the worst [24]. In this regard, when statistically investigating the influence of design parameters, such as strut/wall thickness and cell type and size, RD has to be controlled; otherwise, the results could be misleading.
CF incorporation was found to be a statistically significant influence on the mechanical properties and SEA. These findings are consistent with [3]: CF-reinforced PLA lattices showed enhanced compressive modulus and energy absorption. According to results in [3], the shear forces acting on the polymer melt during extrusion cause fibers to align along the printing direction, enhancing the structures' mechanical stability. Furthermore, results The influence of all considered parameters, including cell topology and size, CF incorporation, and RD, as well as their different combinations on the compressive response of the TPMS structures, was statistically significant. Results showed that RD had the greatest impact on mechanical response among the four studied variables, followed by cell type and CF incorporation. Cell size had the least impact. The authors of [44] stated that RD is the key factor in mechanical performance, including elastic modulus and strength of a given lattice structure. This is in line with the statistical findings, which show that RD has a high influence on both the E and σ peak (58.95% and 71.68%, respectively). The statistical analysis confirms that cell type and RD influence the compressive modulus, σ peak , and SEA of the TPMS lattice structures. This finding is in line with previous studies that investigated the effect of TPMS cell topologies and RDs on E, σ peak , and SEA characteristics, such that these properties were enhanced as RD increased [16]. Diamond structures showed the best performance, while Primitive structures were the worst [24]. In this regard, when statistically investigating the influence of design parameters, such as strut/wall thickness and cell type and size, RD has to be controlled; otherwise, the results could be misleading.
CF incorporation was found to be a statistically significant influence on the mechanical properties and SEA. These findings are consistent with [3]: CF-reinforced PLA lattices showed enhanced compressive modulus and energy absorption. According to results in [3], the shear forces acting on the polymer melt during extrusion cause fibers to align along the printing direction, enhancing the structures' mechanical stability. Furthermore, results reported in [4] stated that the tensile modulus and energy absorption at the break of chiral structures were significantly enhanced (by two times) when incorporating CF into PLA. The interaction of CF incorporation with the cell topology significantly influenced the SEA. This finding agrees with [5] in that the CF incorporation evidently enhanced the energy absorption in the octahedral lattices, while a slight influence was found on the octet lattices. This study confirms these previous findings by demonstrating that CF increases the SEA of Diamond-based structures while (Figure 14b) decreasing the SEA of Primitive-based structures (Figure 14d). This influence of CF and cell type interaction on the SEA is also depicted in Figure 19 using ANFIS 3D surface plot.
Cell size had the smallest effect, and the statistical analysis confirmed its influence only on the compressive modulus. However, its interactions with cell type and RD significantly influenced the compressive modulus and strength, as seen in Tables 5 and 6. Moreover, the influence of the combination of cell size, CF incorporation, and RD on the σ peak was significant. Even though the combination of the three factors had no statistical influence on the SEA contrary to [28] (Truncated octahedron lattices), the negative influence on SEA was consistent with [28]. Findings in [41] stated that certain combinations of wall thickness and cell size (which control the structure RD) of Gyroid structures outperformed others, suggesting the need for predicting the optimal combination of wall thickness and cell size. This observation is noteworthy since it implies that different combinations of design, RD, and material composition (e.g., CF incorporation) factors could be used to attain the desired performance.
The interaction of CF incorporation with the cell topology significantly influenced the SEA. This finding agrees with [5] in that the CF incorporation evidently enhanced the energy absorption in the octahedral lattices, while a slight influence was found on the octet lattices. This study confirms these previous findings by demonstrating that CF increases the SEA of Diamond-based structures while (Figure 14b) decreasing the SEA of Primitivebased structures (Figure 14d). This influence of CF and cell type interaction on the SEA is also depicted in Figure 19 using ANFIS 3D surface plot. Figure 19. ANFIS 3D surface plot of SEA variation with the CF and cell type interaction at a cell size of 12 mm and RD of 44%; 1, 2, and 3 denote Diamond, Gyroid, and Primitive, respectively.
Cell size had the smallest effect, and the statistical analysis confirmed its influence only on the compressive modulus. However, its interactions with cell type and RD significantly influenced the compressive modulus and strength, as seen in Tables 5 and 6. Moreover, the influence of the combination of cell size, CF incorporation, and RD on the σpeak was significant. Even though the combination of the three factors had no statistical influence on the SEA contrary to [28] (Truncated octahedron lattices), the negative influence on SEA was consistent with [28]. Findings in [41] stated that certain combinations of wall thickness and cell size (which control the structure RD) of Gyroid structures outperformed others, suggesting the need for predicting the optimal combination of wall thickness and cell size. This observation is noteworthy since it implies that different combinations of design, RD, and material composition (e.g., CF incorporation) factors could be used to attain the desired performance.
For the training data set, factorial design experiments presented in Table 4, both prediction models, mathematical and ANFIS models, performed well in predicting E, σpeak, and SEA, as illustrated in Figures 16 and 17. However, regarding the testing data set presented in Table 9, ANFIS models clearly outperformed the mathematical models in terms of MAPE (Table 11) and the absolute percent deviation (Table 12). Furthermore, Table 12 and Table 16 demonstrate that mathematical modeling predictions deviated more from experimental results compared with ANFIS predictions, notably for experiments having high RD deviation. This implies a connection between mathematical prediction performance and RD deviation. From the ANOVA analysis, Section 3.1, the RD influenced E, σpeak, and SEA with a contribution of 58.95%, 71.68%, and 22.44%, respectively.
A correlation was observed between the contribution percentage of the RD influence on the TPMS structure performance (e.g., E, σpeak, and SEA) reported in ANOVA analysis (Tables 5-7) and the performance of the mathematical predictions. In other words, a high contribution percentage of RD influence on a TPMS structure's performance (e.g., E, σpeak, or SEA) indicates a high error in mathematical modeling predictions for experiments having a high RD deviation. For instance, the higher mathematical modeling prediction Figure 19. ANFIS 3D surface plot of SEA variation with the CF and cell type interaction at a cell size of 12 mm and RD of 44%; 1, 2, and 3 denote Diamond, Gyroid, and Primitive, respectively.
For the training data set, factorial design experiments presented in Table 4, both prediction models, mathematical and ANFIS models, performed well in predicting E, σ peak , and SEA, as illustrated in Figures 16 and 17. However, regarding the testing data set presented in Table 9, ANFIS models clearly outperformed the mathematical models in terms of RMSE (Table 11) and the absolute percent deviation (Table 12). Furthermore, Tables 12 and 16 demonstrate that mathematical modeling predictions deviated more from experimental results compared with ANFIS predictions, notably for experiments having high RD deviation. This implies a connection between mathematical prediction performance and RD deviation. From the ANOVA analysis, Section 3.1, the RD influenced E, σ peak , and SEA with a contribution of 58.95%, 71.68%, and 22.44%, respectively. A correlation was observed between the contribution percentage of the RD influence on the TPMS structure performance (e.g., E, σ peak , and SEA) reported in ANOVA analysis (Tables 5-7) and the performance of the mathematical predictions. In other words, a high contribution percentage of RD influence on a TPMS structure's performance (e.g., E, σ peak , or SEA) indicates a high error in mathematical modeling predictions for experiments having a high RD deviation. For instance, the higher mathematical modeling prediction deviations for σ peak (16.67%, 16.23%, 21.11%, and 8.30%) were observed for the testing experiments (2, 3, 5, and 6, respectively), with relatively high RD deviations; see Table 16. The fourth testing experiment (i.e., regarding the mathematical prediction deviations for the E and SEA) was an exceptional case where the aforementioned phenomenon was not valid. Furthermore, Figure 20 depicts the relationship between the RD deviation and the mathematical prediction deviation for E, σ peak , and SEA. A high correlation between the RD deviation and mathematical prediction deviation in the case of σ peak (R 2 = 0.802) can be observed, while a low correlation is in the case of SEA. Table 17 presents the results of the Spearman Rho correlation test, indicating a significant correlation (0.943) between the RD deviation and mathematical prediction performance in the case of σ peak , which is highly influenced by the RD variable (71.68%).
prediction absolute percent deviation.

Conclusions
In this study, the influence of different TPMS-based cell topologies with varying unit cell sizes, relative densities, and CF incorporation on the mechanical and specific energy absorption was statistically investigated. The FDM 3D-printed lattices were tested using a uniaxial compression testing, and their E, σ peak , and SEA were evaluated. Prediction models were developed using ANFIS and mathematical modeling. By using the ANFIS models, we were able to predict mechanical characteristics considering imperfections in 3D printing, such as an RD variation. This was achieved by using the actual RDs instead of the designed RDs. Multi-objective optimization was conducted using the desirability approach with the objective of maximizing E, σ peak , and SEA. The following inferences can be made from this study's findings:

•
The findings demonstrated a change in the E (0.144 GPa to 0.549 GPa), σ peak (4.583 MPa to 15.768 MPa), and SEA (3.759 J/g to 15.591 J/g) due to the impact of the considered variables. • RD had a significant influence on both E and σ peak , with a contribution of 58.95% and 71.68%, respectively. The cell type had the highest impact on the SEA, contributing to 70.51% of the total influence. In general, RD had the highest influence on mechanical responses among the four studied variables, followed by cell type, CF incorporation, and finally, cell size having the least impact.

•
The findings revealed the importance of statistically evaluating the influence of the design, RD, and material composition (e.g., CF incorporation) parameters and their combination to attain the desired TPMS lattice structure performance. • For the training data set (factorial design experiments), both mathematical and ANFIS models predicted E, σ peak , and SEA well. However, when it comes to the testing data set (validation experiments), ANFIS models clearly outperformed mathematical models in terms of RMSE and absolute percent deviation in predicting all mechanical characteristics. For instance, the maximum absolute percent deviation was 7.61% for ANFIS prediction, while it was 21.11% for mathematical prediction.

•
The accuracy of mathematical predictions is highly influenced by the degree of RD deviation; a higher deviation in RD results in lower accuracy of predictions. Furthermore, a correlation between the mathematical models' prediction accuracy and the RD deviation was found when the RD influence contribution on a TPMS performance was high. For instance, when RD accounted for 71.68% of the variation in σ peak , there was a significant correlation (94.3%) between the accuracy of mathematical predictions and RD deviation. Therefore, for a high variation in RD, ANFIS models are preferable. • Whenever the actual RD is close to the designed RD, both models provide accurate prediction models. This also indicates the ability of both models to predict different mechanical properties of TPMS structures, taking into account different variables, including material composition, geometry, and the RD.

•
This study provides a better understanding of PLA and CFRPLA TPMS structures, as well as an ability to better predict their mechanical behavior. For instance, based on the multi-objective optimization using the desirability approach, the Diamond topology, 12 mm cell size, 15% CF, and 44% RD combination are the best settings that achieved an overall desirability of 97.8% for E, σ peak , and SEA.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.