 Original Article
 Open Access
 Published:
Robustness optimization for rapid prototyping of functional artifacts based on visualized computing digital twins
Visual Computing for Industry, Biomedicine, and Art volume 6, Article number: 4 (2023)
Abstract
This study presents a robustness optimization method for rapid prototyping (RP) of functional artifacts based on visualized computing digital twins (VCDT). A generalized multiobjective robustness optimization model for RP of scheme design prototype was first built, where thermal, structural, and multidisciplinary knowledge could be integrated for visualization. To implement visualized computing, the membership function of fuzzy decisionmaking was optimized using a genetic algorithm. Transient thermodynamic, structural statics, and flow field analyses were conducted, especially for glass fiber composite materials, which have the characteristics of high strength, corrosion resistance, temperature resistance, dimensional stability, and electrical insulation. An electrothermal experiment was performed by measuring the temperature and changes in temperature during RP. Infrared thermographs were obtained using thermal field measurements to determine the temperature distribution. A numerical analysis of a lightweight ribbed ergonomic artifact is presented to illustrate the VCDT. Moreover, manufacturability was verified based on a thermalsolid coupled finite element analysis. The physical experiment and practice proved that the proposed VCDT provided a robust design paradigm for a layered RP between the steady balance of electrothermal regulation and manufacturing efficacy under hybrid uncertainties.
Introduction
In recent years, in the field of rapid prototyping (RP), most composite materials have been formed layerbylayer via thermal energy fields without mold patterns. Therefore, the energy distribution of parts is very important for fabrication performance [1, 2]. The study of temperature distribution is helpful in reducing heat loss, improving processing efficiency, and saving energy.
The change in the RP process puts forward higher requirements for the applicability of materials. RP includes fused deposition modeling (FDM) and fused filament fabrication. Different processing technologies have diversified the requirements for processing environments, and different printing materials have different physical properties [3, 4]. It is necessary to consider the differences in the properties of these materials when designing the printing parameters. In particular, in a thermodynamic simulation, different materials exhibit different responses to the control signal. Therefore, the rapid response to the heating signal has become a key factor affecting the processing efficiency and environmental emissions of composite material RP [5, 6].
One of the key research points concerning the improvement in the RP process performance is the realization of the RP heating temperature. The accurate control of temperature provides the basis for the reconstruction of biological tissue structures with activity [7]. Researchers [8] have studied the effect of temperature on the emission rate of microparticles in the printing process of FDM and observed that the emission rate of particles was higher at higher working temperatures.
Carbon fiber composites have new applications in 3D printing. Kuncius et al. [9] improved the FDM production technology for continuous carbon fiber composites. Hou et al. [10] proposed a 3D printing technology for continuous fiberreinforced thermoplastic composites that controlled the content of printed fibers. Li et al. [11] replaced the traditional resistance heating method with microwave heating to achieve faster production of continuous carbonfiberreinforced plastic. Mohammadizadeh and Fidan [12] studied the effects of fiber parameters on the tensile properties of manufactured components. Kubota et al. [13] conducted tensile tests on samples with different stacking directions and studied the influence of the printing path.
In addition to the significant influence of the materials on the manufacturing process, the control scheme also determines the molding effect. An optimization algorithm that considers robustness can make the system operate under the interference of uncertain factors. Recently, some studies have been conducted on robust optimization in RP. The 3D printing parameters play a decisive role in printing quality. Different materials can be used as uncertainty factors to study the robustness of printing quality [14]. Material uncertainty is also used to evaluate the robustness of the optimal compliance design in additive manufacturing [15]. Naserifar et al. [16] studied the impact of 3D printing of stretchable aggregates on the robustness of wearable skin devices. Robustness is also an important factor considered in complex multiobjective optimization problems. Ehrgott et al. [17] applied robust optimization of multiobjective uncertainty to a practical field and studied the impact of weather on agricultural harvest. GasparCunha and Covas [18] used a multiobjective evolutionary algorithm to evaluate the robustness of research issues. Kotireddy et al. [19] used a genetic algorithm (GA) to improve the computational efficiency of multiobjective optimization considering uncertainty. The robustness of the design is guaranteed, and the calculation cost is reduced.
An increasing number of visualization technologies have been developed and applied for the visual presentation of digital twins (DTs), by which a physical object can be mapped into the real world and digitized in the form of software modeling. In particular, the development of DT technology makes realtime prediction, interaction, and visualization possible. Saiz et al. [20] optimized the robustness of visual defect segmentation using a generative adversarial network. Burch et al. [21] considered user interactions when implementing dynamic visualization of graphs. Fahd and Venkatraman [22] attempted to model unstructured data using visualization.
With the aim of improving printing efficiency for mixedmaterial 3D printing, by considering previous works [23,24,25,26,27] on rapid manufacturing verification of product conceptual design entities, researchers continue to investigate an approach to robust RP. Accordingly, this study proposes a robustness design method for the RP of fiberreinforced composites based on visualized computing digital twins (VCDTs). The main difference between the wellknown DT and VCDT is that the latter can integrate thermal, structural, and multidisciplinary knowledge for computed visualization.
Methods
Experimental methods
Multiobjective robustness optimization for RP
In the process of robust optimization, the randomness of the genetic operators may cause the generated individuals to fail to meet the requirements of the control model. The generated parameter combination must be robustly optimized to eliminate individuals that do not meet the constraint conditions prior to calculating the fitness function. Robustness optimization considering the uncertainty model satisfies Eq. (1).
where the X vector can have 16 control variables, and ξ is the uncertain parameter, f and g are objective function and constraint function respectively.
In the robust optimization of RP, the uncertain factors are the temperatures of the printing materials and environment. Upon applying a GA to fuzzy theory, the iterative population becomes more stable and efficient after considering the robustness of the system.
Genetic operator and population iteration of membership function
In fuzzy theory, the membership function is the key factor in determining the input signal identification and processing of the control system. This maladaptive limitation can be overcome by using a GA that can iterate the membership function. The problem of system instability introduced by random parameters can be solved by applying the theory of robust optimization to the GA. After the controlled parameter variables are determined, the appropriate fitness function is selected as the evaluation index. Considering the error e of each measurement and relative error e_{c} of the two adjacent measurements in the control process, performance index J is defined as shown in Eq. (2).
where T is the sampling time, and n is the number of sampling points corresponding to the parameter group. The purpose of the algorithm is to optimize the performance index J using the genetic theory. The randomness of parameter selection may cause the performance index to have a magnitude difference; therefore, the fitness function \(F\) is more appropriate to represent the performance of each group of parameters. In contrast to the performance index \(J\), the fitness function \(F\) is positively correlated with the fitness of the parameter group. It is defined as follows:
where P is the scale coefficient, and J is the performance index.
After robust control is considered, the population is iteratively optimized. In previous research [25], the membership function was defined according to expert experience. It was defined as the membership rule of initial standardization. The fitness function of the response obtained by the control algorithm based on this rule is defined as a unit value, from which the value of the proportional coefficient P can be determined. According to the calculations, when the proportional coefficient P in Eq. (3) is obtained, the default fitness function value of the initial membership function model response can be obtained, and it is determined as the value of P. A population with high volumes of individuals is randomly generated as the initial population, and the fitness of each individual is calculated separately. The optimal schemes are compared and converged through the selection, crossover, and mutation steps. The best individual preservation and roulette methods can be used for the selection. The two individuals with the highest fitness in each generation are directly selected to enter the next generation, and a number of individuals with higher fitness are then randomly selected to cross or mutate using this genetic operator to obtain a new population. The overall flow of the algorithm is illustrated in Fig. 1.
Electrothermal regulating design of multimaterial RP
Thermodynamic equilibrium for conduction of RP
Regardless of the functional artifact M, axisaligned bounding boxes (AABBs) can be generated to define the scale of the manifold structure itself. The mechanical stroke lengths \({x}_{b}{,y}_{b},{z}_{b}\) of the AABBs along the \(x,y,z\) directions, respectively, can be calculated. Thus, any point \(Q\) can be represented in terms of its relative position \(ratio\) in the AABB.
The maximum print space for a printer along the \(x,y,z\) directions are \({x}_{p}{,y}_{p},{z}_{p}\), respectively. In the printing coordinate system, the normalized height \({h}_{n}\) of the i^{th} layer can be defined as follows:
Glass fiber composite materials for functional requirements
Glass fiber (GF) is an inorganic nonmetallic material with excellent performance, which has the advantages of good insulation, strong heat resistance, good corrosion resistance, and high mechanical strength. GF is usually used as reinforcement material, electrical insulation material, or thermal insulation material in composites utilized in various manufacturing fields. GF can improve the strength and rigidity of plastics and improve their heat resistance and thermal deformation temperature. It can also improve their dimensional stability, reduce shrinkage, and reduce material deformation.
Thermal conductivity refers to the heat transferred through an area of 1 m^{2} within a certain period by a 1 m thick material with a temperature difference of 1 K on both sides of the surface under stable heat transfer conditions. The thermal conductivity k_{x} (W/(m·K)) is calculated using the following expression:
where x is defined as the heat flow direction, and q_{x}" (W/m^{2}) is the heat flux in this direction. ∂T/ ∂x (K/m) denotes the temperature gradient in the assigned direction.
The thermal conductivity of GF is considerably low. The thermal conductivity of glass is 0.7–1.28 W/(m·K). However, after being drawn into a GF, its thermal conductivity is only 0.035 W/(m·K). The main reason for this phenomenon is that the gap between the fibers is large, the density is small, and the thermal conductivity of the air in the middle is low, which reduces the thermal conductivity of the entire material. The smaller the thermal conductivity, the better the thermal insulation performance. However, when the GF is damped, the thermal conductivity increases and the thermal insulation performance decreases.
The dielectric constant is a physical parameter that characterizes the dielectric or polarization properties of a dielectric material. It is a property of the material itself and measures the ability of a material to store charge. Relative permittivity is often used to characterize the dielectric properties of a material. Relative permittivity ε_{r} is calculated by measuring the capacitance of a thin plate material using the following expression:
where ε_{r} is the relative dielectric coefficient of the material, C(F) is the measured capacitance, d (nm) is the sample thickness, and S (m^{2}) is the sample area. ε_{0} is the vacuum dielectric constant (8.854 × 10^{–12} F/m).
The tensile strength σ_{b} (MPa) represents the resistance of the material to the maximum uniform plastic deformation and is the maximum stress that the material bears before breaking. Its value can be calculated using the following expression:
where F_{b} (N) is the maximum force applied when the specimen is broken, and S_{o} (mm^{2}) is the original crosssectional area of the specimen.
The tensile strength of GF is significantly higher than that of glass with the same composition. For example, the tensile strength of alkali glass is only 40–100 MPa, whereas that of the GF drawn from it can reach 2000 MPa, which is 2050 times higher. The tensile strength of GF can be even higher than that of highstrength alloy steel with the same diameter.
Temperature fuzzy decisionmaking and experimental control
GA of proportion integration differentiation fuzzy decisionmaking
The improved proportion integration differentiation (PID) algorithm based on fuzzy theory can more accurately control the temperature of the system. However, the membership function defined by experience has limitations, particularly for different control models. As an excellent global search algorithm, the GA can find a global optimal solution with high efficiency. The basis of the GA optimization of fuzzy decisionmaking is to select appropriate parameters as genes for combination.
Several parameters were selected for variable control to simplify the genetic model. As shown in Eqs. (9) and (10), 11 key nodes in the domain of the membership function were selected as variable parameters along with input coefficients (error e and relative error e_{c} for each signal acquisition) and output coefficients (adjustment coefficients K_{p0}, K_{i0}, K_{d0} for the three factors in PID control).
where the range of the independent variable x of the membership function is [6, 6].
Sixteen parameters were used as gene sequences in the GA. Figure 2 shows the output response of the step input and corresponding membership function when the parameters have different values.
Figure 3 shows the distribution of the intermediategeneration population. Individuals whose fitness function value is higher than the initial base level in the figure are selected for crossover and mutation to enter the nextgeneration population. The fitness values of the two individuals with the highest fitness function are 2.112 and 1.6706, respectively, which directly enter the nextgeneration population. New individuals are then generated by random coding until the number of individuals in the population reaches 400, and the fitness of the next generation is calculated and updated. After a limited number of iterations, the best individual fitness value is 2.112. This implies that the performance index J under the optimized membership rule is 41.1% of the initial value, significantly reducing the error and accelerating the temperature response.
Physical experiment of electrothermal regulating using programmable power
An actual heating experiment was conducted to further verify the effectiveness of the algorithm. The physical experiment was powered using a multichannel programmable DC linear power supply. This power supply unit can convert an input AC voltage of 220 V ± 10% at 50 Hz into DC linear power in each channel, which can be programmatically controlled independently under the mode of constant voltage, constant current, and constant resistance. Its voltage output range is 1224 V.
To verify the effectiveness of the temperature control strategy on the heating process of the GF PLA (polylactic acid) materials (PLAGF), temperature control experiments with variable loads were performed. The experimental setup is illustrated in Fig. 4. Doubletube heating has the unique advantages of a heating block and good thermal regulation.
Figure 5 shows the experimental results of the heating temperature control of the PLAGF material. Figures 5a and c show the experimental data of the two repeated tests. Each group includes curves before and after using the optimization algorithm under the same heating conditions. The PLAGF material was heated to the melting temperature (230 °C) using a temperature control algorithm before and after improvement. The improved temperaturecontrol algorithm exhibited faster heating and better stability under the same experimental conditions.
Figures 5b and d show the curves of heating the PLAGF material to the unmelted temperature to demonstrate the change in the output voltage duty cycle (the proportion of high level in one pulse cycle) in fuzzy decisionmaking, which is realized by pulse width modulation (PWM) of the voltage. They correspond to the optimized heating curves in Figs. 5a and c respectively. From the experimental results, it can be observed that the PWM of the output voltage is large when the difference between the collected data of the sensor and target temperature (180 °C) is large. The PWM of the output voltage begins to decrease when the temperature gradually increases, thus avoiding a control overshoot. When approaching the target temperature, the PWM fluctuates significantly with the temperature difference and its change value. When the temperature is stable, the PWM also tends to be stable and fluctuates within a small range when heating is required. The experimental outcomes were in accordance with the optimization objective of fuzzy decisionmaking.
Infrared thermographs of thermal field measurements obtained using fuzzy logic are shown in Fig. 6. A higher surface temperature indicates remarkable characteristics of the composite material.
Results and discussion
Results
D functional artifact to be fabricated
The proposed VCDT was implemented on a platform coded in Python 3.7, and all the numerical tests were performed on a PC operating on Windows 10 64bit.
A slender, thinwalled functional half handle (Fig. 7) was used as a calculation example to verify the previously stated theoretical method. By default, the length unit hereinafter is millimeter (mm).
The overall dimension (\({x}_{b}{,y}_{b},{z}_{b}\)) of the AABB is equal to (150.9527, 148.9764, 46.5349) with a ratio of 3.2439:3.2014:1. The percentages of the AABB to the center of gravity are (50.4337%, 58.1379%, 42.8853%). The minimum bounding sphere was placed at (148.9994, 33.0316, 8.5189) with a spherical radius of 102.4477. The total surface area \({S}_{object}\) is 46497.5272, the total volume of the enclosed manifold \({V}_{object}\) is 77661.2048, and the specific surface area is 0.7838. The net mass was calculated as 81.5133 g when Acrylonitrile Butadiene Styrene(ABS) was used.
Figure 8 shows the layered surface and volume along with the specific surface area. Table 1 lists the values of the important parameters of the specific surface area and slope corresponding to Figs. 8b and d.
Virtual printing via layered orthogonal projection areas using visual computing digital twins
Figure 9 presents the comparison of the layered orthogonal projection areas of various supports using stacked bars rather than a grouped style. Table 2 lists the corresponding calculation outcomes under different conditions.
Figure 10 indicates that visualized virtual 3D printing can be utilized to evaluate the thermoplastic adhesion process.
Finite element analysis via transient thermal structure coupling
Finite element analysis was conducted based on transient thermal structure coupling theories using a quasilinear solution with 8node 3D thermal solid element. The structural analysis was implemented using the mesh domain decomposition method. Table 3 lists the common physical and chemical properties of ABS and the carbon fiber and GF mixed materials. These parameters were used for the finite element analysis.
The simulation results of the temperature distribution, total deformation, and directional deformation are shown in Fig. 11. The detailed data are presented in Table 4.
The convergence outcomes of the cumulative iteration calculations are shown in Fig. 12. The elements were sliced into 52 layers according to their height, and the heat load was applied when the elements of a new layer were active.
Experimental test of VCDT
The stereolithography equipment was an extrusionbased 3D printer operated at an ambient temperature of 25 °C and 55% relative humidity, as depicted in Fig. 13. Digital twinning here can be further divided into visual twinning, functional twinning and mechanism twinning. The layer thickness can be varied from 0.02 mm to 0.3 mm. The resolution precision size in the X/Y direction was 0.1 mm. The fabricated heavyduty half handles are shown in Fig. 14.
Conclusions
(1) A robustness optimization method for RP of functional artifacts based on VCDT was proposed
A generalized multiobjective robustness optimization for RP was first built, where thermal, structural, and multidisciplinary knowledge can be integrated for visualization. The wellknown DT was extended to VCDT with a more intuitive computing visualization ability owing to the integration of intelligent multidisciplinary algorithms. This provides a robust design paradigm for a layered RP between the steady balance of electrothermal regulation and manufacturing efficacy under hybrid uncertainties.
(2) A GA was employed to improve the fuzzy decisionmaking scheme, which was fed back to the visualization process to obtain a better strategy
A fuzzy decisionmaking model of RP was established, and the optimal membership function rules were obtained through a GA iteration, which realized a more accurate control scheme. Therefore, the fuzzy decisionmaking model can be adapted to different material and form requirements and can be customized according to diverse application scenarios.
(3) A physical experiment regarding layered RP and infrared thermographs was conducted
Infrared thermographs were obtained using thermal field measurements to determine the temperature distribution, which was highly consistent with the DT outcomes. The physical experiment and practice proved that the proposed VCDT provided a robust design paradigm for a layered RP between the steady balance of electrothermal regulation and manufacturing efficacy under complex RP uncertainties.
Future work involves the application of VCDT to more asymmetric artifacts with composite materials via fuzzy heating systems to enhance the universality of the theory in RP for product conceptual design.
Availability of data and materials
The metadata about robustness optimization of the paper can be shared for availability, in case of publication.
Abbreviations
 RP:

Rapid prototyping
 VCDT:

Visualized computing digital twins
 FDM:

Fused deposition modeling
 GA:

Genetic algorithm
 DT:

Digital twin
 AABB:

Axisaligned bounding box
 GF:

Glass fiber
 PID:

Proportion integration differentiation
 PWM:

Pulse width modulation
 ABS:

Acrylonitrile Butadiene Styrene
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Funding
The work is funded by the National Natural Science Foundation of China, Nos. 51935009 and 51821093; National key research and development project of China, No. 2022YFB3303303; Zhejiang University president special fund financed by Zhejiang province, No. 2021XZZX008; Zhejiang provincial key research and development project of China, Nos. 2023C01060, LZY22E060002 and LZ22E050008; The Ng Teng Fong Charitable Foundation in the form of ZJUSUTD IDEA Grant, No. 188170–11102.
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JX initiated the essential innovations and finalized the article; KL carried out the fuzzy mathematical programming; LW conducted FEA of composite material; HG carried out manifold optimization; JZ conducted manufacturing process; XL participated in article checking; SZ is PI (Principal Investigator) of research team; JT provided guidance to the team as Chief Scientist. The author(s) read and approved the final manuscript.
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Xu, J., Liu, K., Wang, L. et al. Robustness optimization for rapid prototyping of functional artifacts based on visualized computing digital twins. Vis. Comput. Ind. Biomed. Art 6, 4 (2023). https://doi.org/10.1186/s4249202300131w
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DOI: https://doi.org/10.1186/s4249202300131w
Keywords
 Robustness optimization design
 Rapid prototyping
 Functional artifacts
 Fuzzy decisionmaking
 Infrared thermographs
 Visualized computing digital twins