Collaborative optimization design of process parameter and structural topology for laser additive manufacturing

Shoying LI, Hongki WEI, Shngqin YUAN, Jihong ZHU,b,*, Jing LI,Weihong ZHANG

a State IJR Center of Aerospace Design and Additive Manufacturing, Northwestern Polytechnical University, Xi’an 710072, China

b MIIT Lab of Metal Additive Manufacturing and Innovative Design, Northwestern Polytechnical University, Xi’an 710072, China

c Unmanned System Research Institute, Northwestern Polytechnical University, Xi’an 710072, China

KEYWORDS Back propagation neural network;Gradient algorithm;Laser additive manufacturing;Process-structure-property;Topology optimization

Abstract High-resolution laser additive manufacturing (LAM) significantly releases design freedom, promoting the development of topology optimization (TO) and advancing structural design methods. In order to fully take advantage of voxelated forming methods and establish the quantitative relationship between the mechanical properties of printing components and multiple process factors(laser-and process-parameters),the concurrent optimization design method based on LAM should cover the process-performance relationship. This study proposes a novel artificial intelligence-facilitated TO method for LAM to concurrently design microscale material property and macroscale structural topology of 3D components by adopting heuristic and gradient-based algorithms. The process-structure-property relationship of selective laser sintering is established by the back propagation neural network, and it is integrated into the TO algorithm for providing a systematic design scheme of structural topology and process parameter.Compared with the classical optimization method, numerical examples show that this method is able to improve the mechanical performance of the macrostructure significantly. In addition, the collaborative design method is able to be widely applied for complex functional part design and optimization, as well as case studies on artificial intelligence-facilitated product evaluation.

1. Introduction

With the rapid development of high-resolution laser additive manufacturing (LAM) technology, it has been widely applied to aerospace, electronics, medical and automotive industries.1Typical aerospace products, including fuel nozzle of General Electric and aircraft landing gear of Hindustan Aeronautics Limited, are fabricated by LAM, which reduces research and development cost and cycle and improves the performance of end-used products.2AM facilitates the application of complex components, and it dramatically releases design freedom of complex structures.3,4Therefore, design for additive manufacturing (DFAM)5is proposed to design upstream and terminal applications and optimize the AM system architecture,accelerating the industrialization of AM and providing general rules for designers and manufacturers6. Advanced structural design methods, such as topology optimization (TO)7,8and lightweight lattice structure design,9,10have been conceptualized in the DFAM framework. The designed TO and lattice structures usually possess complex geometry, and they are difficult to directly manufactured by traditional methods such as machining and casting.11Fortunately, AM is not sensitive to the geometric complexity of the printed parts, which supports advanced design methods to obtain the final high-performance products.12

Various combination of process parameters in AM affects the microstructure of printed materials and then influences the mechanical properties of printed macro-components.13Hossain, et al.14investigated the effects of slice parameters,building direction (BD), and temperature boundary on the bonding strength of fused deposition modeling (FDM)15,16printed parts. In addition, for Stereolithography Apparatus(SLA),17,18Selective Laser Sintering (SLS)19,20and Selective Laser melting (SLM),9,21laser power (LP), scanning strategy,and BD dominate the energy distribution and local melting pool, which further affect the dimensional accuracy and mechanical properties of fabricated specimens.22,23Due to the variety of AM technologies, equipment, and feedstock materials, the process-structure-property (PSP) relationship is uncertain and nonlinear, and it is difficult to establish analytical or physical models which incorporate the multidisciplinary factors related to the complicated system to predict the PSP relationship.24Hence, the approximate data-driven model based on statistical data methods is developed to address this issue.Artificial neural network(ANN)is a powerful tool for predicting nonlinear relationships as a computational method based on biological structures and functions,which has been widely applied in various areas such as intelligent control, pattern recognition, and nonlinear function fitting.25The simplicity and robustness of the back propagation (BP) neural network promote its application in AM field.26

To date,collaborative optimization of design and manufacturing is becoming a research hotspot. Science reported the holistic concept of material-structure-performance integrated AM (MSPI-AM) proposed by Gu, et al.27, which advocates’the right materials printed in the right positions and unique structures printed for unique functions’. The MSPI-AM methodology enables the parallelism of structural design and manufacturing processes and their mutual compatibility.Under the framework of MSPI-AM, the collaborative optimization design of structure and process is aims at the ’optimized macrostructure fabricated by proper process parameter’, matching stages of design and manufacturing,and fully exchange information of the two stages. Inspired by DFAM and MSPI-AM, TO integrated AM variables and constraints are proposed and developed,3including TO for self-supporting structural design,28gradient lattice structure design,29,30and the design model considering printing productivity.31-33Mirzendehdel, et al.34integrated the anisotropic property of FDM fabricated material into the TO model and realized the structural design considering customized strength failure criterion. The previous investigation established the multidisciplinary TO model incorporating the PSP relationship of AM.35However, the approximate functions are based on the traditional polynomial regression method, which strongly depends on the polynomial form, and greatly influences predicting results,causing the deviation of structural optimization and performance evaluation.Therefore,it is still challenging to select an appropriate fitting method to quantify the PSP relationship and solve the nonlinear TO problem coupling multidisciplinary variables and constraints.

In this work,the PSP relationship of SLS printed specimens is established by the BP neural network which is integrated into the TO model to provide a novel design method of structural topologies and process parameters. Firstly, a BP neural network is trained based on the experimental data to map the nonlinear PSP relationship from process parameters,including LP, scanning speed (SS), hatching space (HS), and cross angle(CA),to the mechanical properties(elastic modulus and Poisson’s ratio). Subsequently, the process parameter is optimized by the genetic algorithm(GA)for achieving the isotropy property,which means the customized mechanical property can be obtained by the BP neural network and heuristic optimization algorithms without any additional trial-anderror procedure. After that, the process parameters are adaptively optimized based on rudder and bracket to enhance macrostructural performance. Finally, the process parameter and structural topology are concurrently optimized by BP neural network and gradient-based algorithm,realizing the collaborative design of process and structure based on LAM.

2. Experiment and modeling

The proposed collaborative optimization workflow is demonstrated in Fig.1.Intelligent design includes structural and process design. In the process design, the reasonable range of process parameters is determined according to the existing literature and equipment suppliers. After setting the upper and lower limits of each process parameter, the design of experiment(DOE)36is utilized to obtain the process parameter combination, and the mechanical evaluation is performed based the transversely isotropic material model.The statistical experiment data is input to BP neural network for training the datadriven PSP relationship. Therefore, the material properties predicted by the BP neural network are integrated to the optimization process. In the structural design, the initial CAD model is discretized into a finite element(FE)model,and then the PSP relationship is integrated into the TO mathematical model through the solid anisotropic material with penalization(SAMP). After that, the sensitivity information is transferred into the TO algorithm to iteratively obtain the final structural topology and process parameters, realizing the integrated and intelligent design framework.

2.1. Mapping of PSP relationship

2.1.1. Experimental design

Fig. 1 Collaborative optimization design of process and structure based on LAM.

SLS technique was usually adopted to printed doggy-bone specimens owing to the flexible building position and low cost24,37. It has been proven that vital process parameters in SLS include LP,SS,HS,and CA38,and the central composite circumscribed(CCC)in response surface methodology was utilized to obtain experimental data statistically. Five levels of these parameters are listed in Table1, and the total 30 experimental runs in CCC design are shown in supplementary materials.

Due to the unique layer-by-layer deposition manner in LAM,the final printed components possess strong anisotropy.Literature39suggests that the transversely isotropic material model can describe the mechanical property of SLS printed materials,and the flexibility matrix in the local coordinate system (axis 3 is the building direction) is given by

where C(χ)mis the flexibility matrix and the subscript ‘m’denotes the local coordinate system,χ is the process parameter vector,E is elastic modulus,E1and E3are the elastic moduli in the axes 1 and 3, ν is Poisson’s ratio, and ν12and ν13are the Poisson’s ratios in 1-2 and 1-3 planes,and G is the shear modulus, G12and G13are the shear moduli in 1-2 and 1-3 planes,sym represents the symmetric part of the matrix.The flexibility matrix is able to be determined by the off-axis tensile experiment,and the doggy-bone specimens are printed by 4 building directions, as illustrated in Fig. 2(a). Tensile bars in every building direction contain 5 independent specimens. Each batch contains 20 tensile bars, and 600 parts of 30 experiment runs are fabricated and evaluated.Besides,tensile parts printed in different building directions are also independent. Therefore, the experimental data of each process combination is expanded by the cross-connection, as demonstrated in Fig. 2(b).There are 625 flexibility matrices for one specific combination of process parameters,and a total 18,750 sets of input and output are provided for training the BP neural network.

The properties of feedstock printed polyamide (PA) 12 powders are listed in Table 2,which are provided by the material datasheet from the supplier (Farsoon Technologies,Hunan). Commercial EP-P3850 (E-Plus-3D Company, Beijing, China) systems were utilized to fabricate involved specimens. Fig. 3 plots the printing temperature history. Firstly,the temperature rises to 120 °C, then the temperature is maintained during the initial 25 layers. In the heating chamber,the temperature gradually increases to 172 °C between the 25th and 50th layers. The temperature remains 172 °C until the printing is completed. The powder chamber holds 120 °C in the whole process.After the printing,the temperature is maintained for 2 hours to prevent significant warpage. The doggybone PA12 samples that follow the ASTM D638 standard were characterized, and the detailed data are listed in supplementary materials.degree of the connection between two neurons, and the input and output neurons are connected through hidden neurons and their weight properties. In the training process of the BP neural network, the inputs propagate forward to generate the output. And then, the error between the generated and actual outputs is calculated and backward propagated to the input layer in order to accordingly change the weight values and reduce the error. The adjusting process of the connection weights is performed iteratively until the error between the prediction and actual output is acceptable.

Table 1 Process parameters and their levels in CCC design.

Fig. 2 Experimental specimens in material test.

Table 2 Feedstock material properties of PA12.

Fig. 3 Temperature history in SLS process.

The implementation of a BP neural network consists of determining the number of neurons in input, hidden, and output layers and adjusting the weight values of connections.The numbers of input and output neurons are 4 and 5, which corresponds to four process parameters and five engineer constants, respectively. Reasonable number of hidden layer neurons could save the training time while ensuring the prediction accuracy. The number of hidden layer neurons can refer to the following formula40:

where f is the number of hidden neurons,m and q are the numbers of input and output neurons, respectively, and a is the constant between 1 and 10.The constant is varied to minimize the predicted error. The weight values are ceaselessly adjusted during the training process until the predicted errors are reduced below a default threshold. The training samples are selected by using the response surface methodology, which guarantees uniform dispersion and neat comparison.The sample value ranges are given in Table 3.

The training sample data has been given in the supplementary materials.In order to ensure the convergence of BP structure and TO mathematical model, the sample data is normalized with a formula

2.1.2. BP neural network for predicting PSP relationship

In this paper,the BP neural network will be applied to describe the PSP relationship of SLS printed PA12 and recommend appropriate parameter space38.The general BP neural network is demonstrated in Fig. 4, including the input layer, hidden layer,and output layer. A weight value represents the relation

where,xiand yiare the original and the normalized data of the i-th input data, respectively; xminand xmaxare the minimum and maximum of the original sample value range,respectively.The data is coded between 0 and 1 after the normalization process.

Table 3 Sample value ranges.

2.2. Optimization formulation

The established BP neural network integrates the PSP relationship into the optimization model. According to the literature,optimization problems of AM are mainly classified into three categories:

●Optimization of process parameters to obtain customized and desired material property.

●Optimization of process parameters based on a specific macrostructure aims to maximize structural performance.

●Collaborative optimization of process parameters and structural topology aims to maximize structural performance.

Three optimization schemes are proposed for solving different engineer problems. In the field of LAM, most research focus on the first problem, and the experiment and optimization are performed to search desired material property and corresponding process parameters based on the homogeneous solid material. However, it has been proven that the PSP relationship has a significant impact on the macrostructural performance and the polynomial function is utilized to optimize the structural performance and structural topology.41Therefore, optimization of process parameters is applied to plan the manufacturing process and the collaborative optimization is utilized to design the structural topology and physical manufacturing. In this section, the collaborative optimizations incorporating BP model driven PSP relationship are established and briefly explained.

2.2.1. Optimization for customized material property

The BP neural network maps SLS process parameters to material property, and customized mechanical property can be obtained by search a specific combination of process parameters. The isotropic property is acceptable since the inter-layer,and inner-layer material demonstrates substantially identical bonding strength. In this study, the anisotropy is mainly reflected in the difference between the elastic moduli of tensile parts of 0° and 90°. The material optimization model is described by where X is the vector of the process parameter,E0and E90are the elastic moduli of specimens printed in 0°and 90°,abs is the function of absolute value, and φ is the positive difference between them. Each parameter (LP, SS, HS, CA) is limited in their ranges. The subscript min and max are the lower and upper limits, respectively.

The elastic moduli are predicted by BP neural network,and the optimization formulation is nonlinear and implicit. Therefore, it is difficult to solve the problem using conventional algorithms such as the steepest descent and the conjugate descent methods.GA is a heuristic global optimization approach,and it is inspired by biological evolution. During the process,generate successive sets are generated,and the new population(solutions)can inherit properties from the best solutions of the precedent. The crossover and mutation operators are applied to promote the population diversity, and then the generated offspring will be compared and chosen based on the selection operator. GA has been adopted in various research fields for multi-objective optimization due to the capability of fast convergence.42In this study, GA is applied to search the parameter combination that leads to the isotropic material property based on the trained BP neural network.

2.2.2. Optimization for maximizing structural performance

The optimization for the customized material property is limited to the micro-scale. At the macro-scale, the macrostructure possesses complex topologies with multiple load cases, and the matching material property may not be isotropic.Besides,BD determines the principal direction of anisotropic property in the macrostructure, and it was optimized to enhance structural stiffness. The optimization models are briefly introduced, and the detailed derivation is given in the reference.41

The optimization formulation of process parameter is expressed as

where,α and β are the printing angles in the directional vector θ, which dominates the BD of parts, C is the structural strain energy; K, U, and F are the structural stiffness matrix, displacement,and load vectors,respectively,the constraint is that each process variable is between its upper and lower limits,which is similar to Eq. (4). The property parameters predicted by the BP neural network in Fig. 4 are calculated by Eq. (1),and the flexibility matrix are utilized to obtain the stiffness matrix K.

Further,the structural variable is added to the process optimization model in order to establish the collaborative optimization and concurrently design the process parameter and structural topology. For the isotropic solid material, SIMP(Solid Isotropic Material with Penalization) was usually utilized to achieve clear topologies43.PAMP(Porous Anisotropic Material with Penalization) was used in structures with microstructures in the macro-scale29. Similarly, the SLAprinted components could be optimized using SAMP35,41.The density-based approach and SAMP are adopted to build the optimization model.The global stiffness matrix is given by

where ρ is the pseudo-density vector of design domain Ω,Ωiis the domain of the i-th element, B, R and D are the geometric,rotation, and elastic matrices, Dmis the elastic matrix in the local coordinate system, n is the number of all the elements,P is the penalty factor and it is 4 in this study. The collaborative optimization of process parameters is formulated as

where V and V0are the structural volume before and after optimization, h is the preset volume fraction, ρiis the pseudo-density of the i-th element,ρminis 0.001 to avoid singularity of stiffness matrix.

The gradient-based algorithm (GCMMA) is applied to update design variables combined with the sensitivity analysis and filtering technique. The optimization is implemented through the ANSYS? with the corresponding APDL program. During the iterative process, the material property is predicted by the BP neural network, and the derivative of the process variable is obtained by the finite difference method and the step size is 0.001. As a result, the number of process parameters greatly affect the optimization efficiency. Adding one more process variable suggests an extra FE analysis process in ANSYS?. The number of finite elements determines the time cost of each FE analysis process. In addition, the filtering technique is adopted in the optimization to handle mesh dependency and checkerboard, ensuring optimization accuracy. The convergence criterion is given as below:

where Xkand Xk+1are the design vectors in the k-th and(k + 1)-th iteration, and ε is 1 × 10-5.

3. Results and discussion

3.1. PSP relationship

Fig. 5 Training process and predicted values of BP neural network.

The BP neural network is trained in the commercial software by the Bayesian regularization algorithm. The mean squared error (MSE) is demonstrated in Fig. 5 (a). The MSE reaches a minimum and remains almost constant after 35 iterations.The training process costs 18 seconds in the computing environment (Computer processor: Intel (R) Core (TM) i7-7700 CPU 3.60 GHz RAM 8 GB). Fig. 5(b) demonstrates the prediction performance by comparing the actual and the predicted outputs, and the maximum deviation is 3.74%. It is evident that the developed BP model exhibits a high accuracy for predicting the PSP relationship. Based on the trained BP model,the effects of the four input parameters on the mechanical properties can be investigated.In this section,the elastic modulus in the building plane is mainly studied and discussed,and the research method is the same for other properties such as Poisson’s ratio.

Fig. 6 plots the variations in different conditions. Unique combinations of process parameters are selected to investigate how the vital factors affect elastic modulus,which is labeled AE.From Fig.6(a),E0increases as the increase of LP for A,C,D, and E. This is because the higher the LP is, the more the energy density convert to the melting pool, resulting in better bonding strength of inter-layer materials and lower porosity.For combination B, E0slightly decreases with the increase LP since the higher LP with the lowest SS and HS induces concentrated energy density, which deepens the sintering depth and causes micro-defects inside the melting pool. In addition,the over-heat effect sinters extra PA12 powder, increasing the porosity of fabricated components. Similarly, the SS and HS influence the local energy density and the depth and width of the melting pool, resulting in the difference in sintering strength. Fig. 6(b) and (c) show the variation of E0with SS and HS. For A-D in both figures, E0decreases with the increase of SS and HS. The reason is that larger SS and HS lead to shorter local sintering times and smaller overlapping areas between neighboring scanning lines, strengthening printed material. The abnormal trends of combinations E0are also caused by the over-heat effect.

Fig. 6 Variations of E0 in different conditions.

As shown in Fig.6(d),it is noteworthy that CA also affects the mechanical properties of printed specimens. It is different from the prediction of the traditional ANOVA method,which suggests that the CA is not significant.41For combination C,the highest HS reduces the overlapping area of the scanning lines and increases the anisotropy in the printing plane. The number of layers with the horizontal scanning direction increases, enhancing the elastic modulus in the horizontal direction. For A, B, and E, the low HS improves the uniformity of the inner-layer printed materials, and the low CA increases the cycles of the inner-layer principal directions,which promotes the uniformity of the inter-layer printed materials. As a result, E0decreases slightly with the increase CA.

3.2. Optimization for isotropic material property

In this section,the AM process parameters are optimized from the point of view of mechanical performance. It is different from the previous work which mainly focus on the process feasibility.24The isotropic material property is obtained through GA algorithm which suggests that the any customized properties can be achieved by the data-driven approach and heuristic algorithm.

The BP neural network exhibits high reliability compared with polynomial function through the factor analysis in the previous section. Fig. 7 demonstrates the iterative history of fitness value using GA.Crossover probability is 0.8,and mutation probability is 0.01. The process parameters and corresponding mechanical properties are listed in Fig. 7. The difference between the inner-layer and inter-layer modulus converges in 60 iterations, and the difference of elastic and shear moduli are only 2.26% and 0.99%, respectively.

Fig. 7 Iterative history of fitness value and final results.

Fig. 8 Rudder structure with ribs.

Fig. 9 Convergence curves of rudder based on three initializations.

The whole optimization process takes 1 minute and 18 seconds in the same computing environment (Computer processor:Intel(R)Core(TM)i7-7700 CPU 3.60 GHz RAM 8 GB).In the optimization process,GA directly call the BP neural network to output the predicted material properties, as a result,the training time (18 seconds) is saved to improve the optimization efficiency. The elastic surface in Fig. 7 suggests that the material property is close to isotropy, and this customized property can be obtained by an intelligent BP neural network and a global heuristic algorithm.

3.3. Process optimization

In the traditional process optimization, the feasible solutions are often a region and various combinations of process parameter can be selected to printed structures. In this section, the rudder structure and engine bracket with the particular load are utilized to optimize the AM process parameters. The matching process parameter maximize the structural stiffness of specific aircraft components and the systematic optimization formulation can be applied to assist manufactures to determine appropriate fabricating parameters.

3.3.1. Rudder structure

The rudder structure with ribs in Fig. 8 is selected to optimize process parameters combined with BP neural network. The volume of the rudder is 20,000 mm3, and the thickness of surface skin is 2 mm.In the FE model,the concentrated force F is 100 N, and it contains 447865 elements. The sensitivity information is obtained by the finite difference method since the model is implicit.Three initial values are selected,and the convergence curves and strain energies are shown in Fig. 9. In addition,the convergence results of rudder based on three initializations are listed in Table 4.The initial 1 and optimal 1 in Table 4 are the initial and final parameters of the curve 1 in Fig. 9, and it is the same for curve 2 or 3. Each iteration process takes 4 minutes and 45 seconds. The strain energy converged to 18.27 J at the 30th iteration. The strain energy of group 2 decreased by 20.35%, which remarkably demonstrated the effectiveness of the process optimization model to enhance the macrostructural performance.

Besides,the initial and final process parameters are input to the BP neural network in order to obtain the correspondingmaterial properties. As shown in Table 5, the final process parameters improve the stiffness of the printed material, and the optimized printing direction makes the principal stress direction perpendicular to the building direction. Therefore,the matching process parameters enhance the macrostructure with the specific boundary conditions.

Table 4 Convergence results of rudder based on three initializations.

Table 5 Initial and optimal material properties.

Fig. 10 Engine bracket with concentrated force.

Fig. 11 Convergence curves of the bracket based on three initializations.

3.3.2. Engine bracket

Engine bracket is an essential component of aircraft which links engine and wing.Excellent brackets can effectively transmit load from engine to wing and reduce the weight of the whole structure and create economic benefits. The process parameters are optimized based on the engine bracket in Fig.10 in order to promote structural stiffness.The FE model consists of 514,153 elements, and the volume is 35,000 mm3.The top of the bracket is fixed, and the vertical force is 1800 N.Each iteration process takes 5 minutes and 14 seconds.The optimized results are demonstrated in Fig. 11, and the convergence results of the bracket based on three initializations are listed in Table 6. The optimization target converges to 74.89 J at the 30th iteration, and the structural stiffness of group 2 increases 17.39%. It is noteworthy that the optimized building direction is not empirical. In addition, the material property is anisotropy, which suggests that the tunable material anisotropy combined with the appropriate building orientation can significantly promote the mechanical performance of printed complex macrostructures.

3.4. Collaborative optimization

In this section, the data-driven PSP relationship is integrated into the TO formulation to optimize process parameters and structural topology concurrently. Compared with the material and process optimization in Sections 3.2 and 3.3, the design variables are extended to the structural level which further matches the material property and structural topology,emphasizing the parallelism of process and structure design. In addition, it is the first report utilizing BP neural network and gradient algorithm to perform TO design. The collaborative optimization realizes the intelligent multidisciplinary design and it can be generally applied to other AM process.

3.4.1. Cantilever beam

The cantilever beam is plotted in Fig. 12, and the force is 1800 N. The corresponding FE model contains 76,800 elements,and the volume fraction is 30%.The classical TO based on given process parameters is also performed for comparison.The classical and collaborative optimization results are demonstrated in Fig. 13, respectively. The optimization process converges at the 200th iteration,and the optimized results show the obvious difference. It costs 34 seconds and 1 minute and 27 seconds for each iteration in classical and collaborative optimization,respectively.The extra computing time is caused by the sensitivity analysis. For the collaborative optimization,material moves to the fixed region, and the main beam is printed in the building plane. Besides, the high LP and low HS increase the elastic modulus of inner layer materials.Hence, the structural stiffness of collaborative optimization promotes 19.43% compared with the classical optimization,exhibiting the superiority of the optimization model.

3.4.2. Cube

The cube as a test scenario is illustrated in Fig. 14, and eight corner points are fixed. The size of elements is 1 mm, and the cube consists of 125000 elements. Random load cases are generated with random position, direction, and magnitude.Both the classical and collaborative optimizations are based on the same FE model. The total volume fraction is 15%.

Table 6 Convergence results of the bracket based on three initializations.

Fig. 12 Initial design domain of cantilever beam.

Fig. 13 Convergence curves and results of concurrent and classical TO.

Fig. 14 Initial design domain of cube.

Fig. 15 demonstrates the classical and collaborative optimization results.Similarly,the optimization process converges before the 200th iteration, and the optimized topologies and parameters show the apparent difference.About the computational efficiency, it costs 1 minute and 38 seconds and 5 minutes and 16 seconds for each iteration in classical and collaborative optimization, respectively. The marked regions in Fig. 15 suggest that material aggregates to different positions of the cube in the different building directions. Besides,the optimized process parameters enhance the elastic modulus of the microscale materials.As a result,the structural stiffness of collaborative optimization increases by 9.1% compared with the classical optimization.

Fig. 15 Convergence curves and results of concurrent and classical TO.

Fig. 16 Optimized structural topologies.

In order to compare with the TO results without considering the LAM, the isotropic material property is integrated in the classical optimization to obtain the structural topology.The elastic modulus is 1200 MPa and Poisson’s ratio is 0.4.The elastic constants are input to Eq.(1)to obtain the flexibility matrix and perform optimization algorithm.The structural topologies are demonstrated in Fig.16,and the structure based on the isotropic properties is plotted in Fig.16(a).As shown in Fig.16(d)and(e),the difference between the collaborative TO is obvious, which indicates that the material property significantly affects the final structure design. Hence, it is necessary to consider the manufacturing process in the TO algorithm.

4. Conclusions

In this study, a collaborative optimization framework integrated BP neural network-driven PSP relationship is proposed to concurrently design microscale material property and macroscale structural topology of 3D parts. Total 18,750 sets of input and output parameters are utilized for training the BP neural network, and it exhibits the excellent ability of factor analysis and property prediction.The BP neural network is easily combined with the heuristic algorithm (GA) and gradient-based algorithm to obtain customized material property and high-stiffness structure. The maximum difference of the elastic moduli in a different direction is only 2.26%, and the structural performance is promoted 20.35% and 17.39%by simply adjusting process parameters for rudder structure and engine bracket, respectively. Finally, the structure and parameters are concurrently optimized, and the cantilever beam is extra enhanced by 19.43%compared with the classical TO result,demonstrating the great potential in industry application of the collaborative optimization. Besides, the collaborative design method can be widely applied for complex component design and optimization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by National Natural Science Foundation of China (U1930207), Key Project of National Natural Science Foundation of China (51790171), National Natural Science Foundation of China for Excellent Young Scholars(11722219), National Natural Science Foundation of China for Young Scholars (51905439), and 111 Project (B21013).