- Email: [email protected]
Optimised lattice structure configuration for additive manufacturing
CIRP Annals - Manufacturing Technology 68 (2019) 117–120
Contents lists available at ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Optimised lattice structure configuration for additive manufacturing Nadhir Lebaal a, Yicha Zhang a,*, Frédéric Demoly a, Sébastien Roth a, Samuel Gomes a, Alain Bernard (1)b a b
Laboratoire Interdisciplinaire Carnot de Bourgogne, site UTBM, UMR 6303 CNRS, Univ. Bourgogne Franche-Comté, France Ecole Centrale de Nantes, LS2N, CNRS, UMR 6004, Nantes, France
A R T I C L E I N F O
Keywords: Design optimization Additive manufacturing Lattice structure
A B S T R A C T
Lattice structure is critical for designing light-weight or multi-function components in additive manufacturing. A lot of know-how had been accumulated via the benchmarking of cellular units. However, how to optimize the parameters and the topological distribution of lattice cells in a constrained design space to gain both mass and computation efficiency for structure design is still an open question. To answer it, this paper proposes a new optimization method using design of experiment and surrogate model to configure lattice structures in specified 3D hulls. A design case is presented to show the advantages of the proposed solution. © 2019 Published by Elsevier Ltd on behalf of CIRP.
1. Introduction
2. Related work
Additive Manufacturing (AM) brings more freedom to designers to enable extremely complex and light-weight design. The nonlinear relationship between the complexity and manufacturing cost is one of the main advantages of this new technology. Complexity is used as an indicator to guide the design for AM [1–3]. However, designers still have difficulties to apply lattice structures in their design since the current CAX tools have problems to represent and simulate complex freeform porous or periodic cellular structures [4]. Existing commercial tools can only generate lattice structure but lack of decision support on the selection of lattice patterns and its parameter set. In addition, the numerical simulation is too costly in design iteration. In the academic side, there is very little research to solve this problem although there were many works for the design and benchmarking of lattice pattern units. This means how to do the lattice design in the structure or component level is still an open question. To help designers realize their light-weight design via the use of lattice structure and fully harness the advantages of AM, this paper proposes a combined method that adopts knowledge engineering and mathematical optimization tools to provide decision support for lattice design configuration, including cell pattern selection and cell parameter optimization. The structure of the paper is organized as follows: the second section reviews key works related to the paper topic; the third section details the proposed method; the fourth section presents a case study for method demonstration and the last section concludes with discussions and perspectives.
Lattice structure and porous structure already exist in nature before AM. However, using these structures in design got more attention recently when AM becomes more mature. Basically, there are two main categories of lattice structure: stochastic and periodic. For the reason of ease control and better properties when compared with foam lattices [5], periodic cellular structures are more popular. The first lattice design framework, used for idea demonstration, was proposed by Ref. [6]. In this work, periodic lattices were controlled by parametric implicit modelling method to fill into 3D hull clusters. Then, only the strut diameter of lattices is optimized to change the density of lattice cells to meet design requirements. Numerical simulation is required for design validation in each iteration, which is costly. In addition, the modelling and analysis capability, e.g. maximum size or number of lattices, was not discussed and the computation cost of FEM (Finite Element Method) simulation in the design iteration was neither mentioned. However, this is the main barrier to remove for designers to adopt lattice structure since modelling capabilities and simulation efficiency are critical for qualified and high productive design. More recently, people tried to use TO (Topology Optimization) method for lattice structure design and configuration. In Refs. [7,8], TO is used to identify a variant material density distribution in a continuous way for a fixed design space with its constraints. Then, uniformed lattice cells with diverse strut diameters are used to map the continuous material distribution so as to approximate the mechanical properties. FEM is required for each lattice cell mapping/filling iteration. Similarly, computation cost would be high and the flexibility of design for AM is reduced since the original design space could not be changed. Another method was proposed in Ref. [9], where an initial design space was decomposed into different volumes (similar to the
* Corresponding author. E-mail address: [email protected] (Y. Zhang). https://doi.org/10.1016/j.cirp.2019.04.054 0007-8506/© 2019 Published by Elsevier Ltd on behalf of CIRP.
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N. Lebaal et al. / CIRP Annals - Manufacturing Technology 68 (2019) 117–120
method in Ref. [6]) to fill in with lattice or not. For the volumes to fill in, evolutionary TO is used to identify an optimal lattice strut diameter, to change the density, to respond to the desired mechanical properties. In this method, the original design space also could not be changed and the lattice parameter optimization is limited to strut diameter with too much computation since discrete evolutionary algorithm was adopted and FEM was called in each optimization iteration, which is quite costly. However, the computation performance was not discussed in their work. Different from those methods assigning uniformed cell type (although with no-uniformed thickness, strut diameter), in Ref. [10], a lattice configuration method based on element mechanics patterns was proposed. In this method, the original design space was decomposed into a set of conformal ‘ground structures cells’ where lattice cells mapped. Then, FEM analysis generated nodes were used to construct lattice cell joint nodes, and finally an optimization algorithm was applied to identify an optimal combination of predefined basic mechanics patterns to generate conformal lattice cells for each ‘ground structure’. Therefore, there is a need of large computation to deal with the overlapped struts created in the lattice generation process. Besides, the computation to find an optimal combination of lattice cells for each ‘ground structure’ is more expensive since the FEM should be engaged in each iteration when determining the strut diameter for each lattice cell. To solve the high computation cost problem of lattice configuration, more recently, there is a new method proposed by [4] almost avoiding the need of FEM in each design iteration. This method defined a relative density of lattice cells to the identical size volume of solid bounding box. Then it uses predefined cell patterns to map according to the stress distribution of the original specified design space. Variant strut diameters are used in the mapping to adjust the density of lattice cells so as to approaching the required mechanical performance. In addition, the representation of lattice structures in the configuration process is symbolic, which can further reduce the computation cost. This method has a much big advantage to other methods since it saves too much computation and make the wide use of lattice structure design for ordinary designers feasible. However, this method drives out unstable results some of which even exceeded 25% percent. This is not acceptable for the detail design stage where lattice configuration stays. The main reason is that they only use FEM simulation but no physical benchmarking of lattice cell’s properties to build a data base. The errors may also come from the omission of lattice orientation caused orthogonal properties and no FEM-based adjustment in the lattice configuration iteration. As reviewed above, the current research on lattice design configuration at the component level is limited. This problem should be well defined and further investigated to obtain wide application in light-weight design. Generally, the lattice configuration for component structure design has three main tasks: lattice pattern selection, lattice cell parameter set and lattice layout determination as described in Fig. 1. If the coupling aspect of these three tasks is considered, the configuration problem would become more complicated. It is clear that almost all existing
Fig. 1. Lattice configuration options for structure design.
methods use uniformed single pattern with only one parameter, thickness, for optimization but neglect many other configuration aspects. This only touches a small fraction of the original solution space of the configuration problem. To explore a wider solution space and provide more efficiency, this paper proposes to optimize more cell shape control parameters simultaneously in lattice configuration. Due to the complexity of this research problem, the proposed method limits its adaptability to the configuration of single lattice pattern with uniformed orientation and cell size. In addition, the lattice structure geometry modelling problem is also out of the scope of this paper. The proposed optimization method is detailed in the following section. 3. Proposed method Theoretically, for any specified design space, there is an infinite set of configuration solutions since the design space is continuous and full of design variables. Hence, to identify an optimal solution among the vast original solution space is already difficult. More seriously, during the solution searching, if the FEM is integrated for evaluating design alternative solutions, as did by some researchers in the literature, the computation cost would not be acceptable or even not feasible for median or big size component design. In [4], intensive benchmarking had been conducted and the result is that current CAD tools are only feasible to handle very small size lattice structures, and FEM tools are not feasible for big size structure analysis due to the high computation cost. To make lattice applicable for median or big size structure design with relative low computation cost, a new combined method is proposed as described in Fig. 2. The proposed method includes two blocks: the first block deals with the preselection of alternative lattice patterns and identification of significant variables of the selected lattice cell pattern that impact the concrete design problem; the second one adopts a statistical approximating method to explore the infinite solution space with an efficient optimization algorithm, which acts as a fine tuning of lattice key parameters.
Fig. 2. Lattice configuration optimization method.
Step 1: analyse the original design and identify the regions where lattice structure is possible to fill in and obtain the mechanical responses, e.g. stress and strain distribution, via FEM analysis according to the predefined loadings and constraints. Step 2: use the FEM analysis results to select potential lattice patterns and their orientations. Here, the method proposed in Ref. [3] is adopted. Relative density concept is used to roughly determine the lattice pattern, size and orientation to respond to the mechanical constraints. An allowance should be set for the selection so as to make the difference between the estimated performance and that of the real optimal solution locate at the positive direction to facilitate the later parameter optimization. A knowledge base storing the characteristics of lattice cell patterns and orientation impacts, which can be obtained either by numerical experiments as did in Ref. [4] or physical experiments as presented by Ref. [11,12], is required here. Step 3: analyse the selected lattice pattern and conduct DOE (design of experiment) to ascertain the significant lattice cell parameters for the specific design problem. It should be noted that the DOE here is numerical in order to save real experimental cost. Different lattice parameter combinations with designed value levels will form different lattice structures to the design. Then, FEM analysis is conducted to get their responses. In this step, the key
N. Lebaal et al. / CIRP Annals - Manufacturing Technology 68 (2019) 117–120
lattice parameters are identified and a set of representative lattice configurations’ performances or responses at the structure level are collected. Step 4: apply a statistical method to approximate the unknown solution space (build a surrogate model) of the design problem with the collected lattice structure responses in Step 3. If the collected DOE data is not enough to construct an approximating solution space, then more sample solutions are required. The number of required sample solutions depends on the method of building the surrogate model. Step 5: conduct multi-objective optimization for the design problem on the obtained surrogate model to identify the optimal parameter set for the selected lattice pattern. Step 6: use FEM to verify the obtained optimal solution. As shown in Fig. 2, there are two optimization iterations. In Iteration 1, there is no FEM analysis involved, In Iteration 2, FEM for DOE is required for each selected lattice pattern. But, Iteration 2 is not obligatory if a preselected lattice pattern and orientation can lead to an acceptable optimal result. Only when the preselected pattern does not work or there is a need to obtain better performance, then this iteration is launched. Therefore, the main computation cost actually is located at the numerical DOE via FEM. To further explain and demonstrate the proposed method, a design case study is presented in the next section. 4. Case study The design problem is redesign a scooter for AM to reduce mass but maintain acceptable strength. The original design is given below (Fig. 3a).
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Table 1 L9 Taguchi DOE with 4 factors in 3 levels. Hc/mm (central height)
Lb/mm (strut length)
Ab/mm (strut angle)
D/mm (strut diameter)
7 7 7 8 8 8 9 9 9
5 6 7 5 6 7 5 6 7
35 40 45 35 40 45 35 40 45
1.75 2.00 2.25 1.75 2.00 2.25 1.75 2.00 2.25
The operation of numerical DOE is realized in several technical steps. At first, a geometric model of the lattice cell and the deck 3D hull model with a fixed thickness of 1 mm are built with an ordinary CAD tool, CATIA; then, an FEM tool, COMSOL, is used to build an FEM model. In COMSOL, a lattice cell is represented as an FEM model with 36 beam elements (Fig. 5b), then this element model is copied and assembled in three directions to fully fill in the 3D hull. After that, the inner surfaces of the deck 3D hull are used to trimming the boundaries of lattice structure. The intersections between the trimmed beam elements and the 3D hull surfaces form a set of nodes on the hull’s surfaces. Then, these nodes are used to mesh the 3D hull into shell elements. Each FEM model has around 6000 beam and 35,000 shell elements with good connection (Fig. 5b). Due to the symmetric property of the deck geometry, only half of the deck is analysed to save computation. The DOE analysis shows that all of the 4 parameters are significant to this design problemwhen considering the strength and mass objectives (Fig. 6). Then, they should be all considered for optimization in the following fine tuning step. To be noted that different design parameters’ sensitivities depend on the specific design problem.
Fig. 3. Design problem.
Step 1: after analysis of the original design, the deck component (Fig. 3b), material of Aluminium T6-6060 (Young Modulus, 70 GPa; Tensile strength,160Map. Density, 2700 kg.m3) is selected to fill with lattice structures since it is solid and has many materials over the performance under the extreme loading of 1000 N of concentrated force on its middle top surface vertically. Its original weight is 0.6874 kg with a Von Mises stress of 160 MPa under loading. Step 2: according to the loading, the FEM analysis and an allowance of 50% more strength to the loading, two lattice patterns, Octet-truss with a minimum size of 20 mm (edge length of the cubic bounding box to host a lattice cell) and Hexa-truss with a minimum size of 15 mm are identified from an example lattice pattern base. The orientations of the lattice cells are their flat top bases in parallel to the deck top surface. To simplify the demonstration, only the hexa-truss cell is used for detail explanation in the following steps. Step 3: based on the selected lattice pattern, a numerical Taguchi DOE is conducted. Four shape control parameters (Fig. 4b) of this lattice cell are considered (Table 1).
Fig. 5. Numerical DOE FEM analysis.
Fig. 6. DOE impact analysis.
Step 4: theoretically, the 9 computation results from the DOE in the former step are 9 representative alternative solutions in the infinite original design solution space. Therefore, they can be used to help building a surrogate model to approximate the original solution space. In this paper, the Kriging interpolation is adopted. For Kriging interpolation, the approximate functions can be expressed as follows. ~JðxÞ ¼ P T ðxÞa þ ZðxÞ
Fig. 4. Preselected lattice cell pattern.
where x is the design variable vector; a ¼ ½a1; a2; a3 . . . amT is the unknown parameters vector, ~J ðxÞ is the approximate function (objective or constraint functions) and Z(x) is the random fluctuation. To obtain a good accuracy of the surrogate model, 60 sample solutions (51 random combinations of lattice param-
N. Lebaal et al. / CIRP Annals - Manufacturing Technology 68 (2019) 117–120
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eters and the 9 collected DOE results) are used for this case. Specific design case may have different solution space which requires different number of sample solutions for interpolation. The determination of the minimum number of samples for interpolation depends on the interpolation validation test result. Due to limited space, the interpolation process and validation test of the surrogate model are not provided here. Readers are directed to authors’ other work on the modelling process [13,14]. Step 5: once the surrogate model is ready, an optimization algorithm can be applied to search the optimal solution on it. In this example, GA (genetic algorithm) is adopted due to its efficient stochastic search technique. The four parameters of the lattice cell pattern are set as design variables, and the minimization of relative global mass of the lattice deck structure with the decrease of the maximum value of relative Von Mises stress are set as two objectives (rM = f1(x) and rVM = f2(x)) under the constraints of material yield stress and variable value boundaries. 8 > > > > > > <
8 W > > < f 1 ðxÞ ¼ 1 W0 min s max > > : f 2 ðxÞ ¼
example, the average gain of mass is more than 75%. The following action is to select a non-dominated solution from the Pareto front according to design preferences. If required, new alternative lattice pattern can be selected to launch the second iteration of the proposed method to search for better result. The whole lattice modelling and optimization process is conducted in an ordinary PC with a configuration as: CPU E31535M [email protected] GHz, RAM, 32Go. However, due to the use of beam and shell element, the average calculation time for each FEM analysis is only about 11 min. Therefore, the total computation cost is about 660 min since the main contribution is the FEM analysis for the random sample solutions to build the surrogate model. This could not be imaged by other existing methods where FEM analysis is embedded into the optimization iteration. Another strong point of this proposed method is that the lattice pattern knowledge base can provide efficient pre-configuration. A good pre-configuration can greatly reduce the work of fine tuning on parameter optimization.
5. Conclusion
sy > > > > Such that : s max a:s y > > : With : xl x xu where W1 denotes the global structure weight; W0 is the original solid structure weight; s max is maximum VM (Von Mises) stress and s y is the yield stress. x is the design variable of the four selected lattice parameters. More details about GA coding can be found in author’s work [15]. The optimization result, a set of nondominated solutions on the Pareto front, is presented in Fig. 7.
Via the use of benchmarking knowledge and surrogate model based optimization, lattice configuration optimization can be realized for median or large size structure design with acceptable computation cost. Reliable design solution can be obtained by using current design and simulation tools, which implies that lattice structure would be possibly adopted widely by ordinary designers for light weight design. It is the main contribution of this paper. However, this work is limited to uniformed lattice configuration. More work should be done in the future to deal with conformal or gradient lattice configuration and optimization.
References
Fig. 7. GA based optimization result.
Step 6: Since the obtained optimal solutions come from the optimization on the surrogate model, there is a need to validate via the FEM analysis. Three solutions (marked by blue, orange and red colour respectively as shown in Fig. 7) are selected for calculation validation. Table 2 shows the validation result. Table 2 Optimization result analysis (demi-deck). Design solutions
Hc /mm
Lb /mm
Ab /mm
D /mm
Mass /kg
VM /MPa
Surrogate-B FEM-B Surrogate-O FEM-O Surrogate-R FEM-R Original solid
7.45 7.45 8.93 8.93 8.99 8.99 –
6.99 6.99 6.99 6.99 6.99 6.99 –
43 43 37.74 37.74 44.42 44.42 –
1.89 1.89 1.75 1.75 1.81 1.81 –
0.155 0.154 0.116 0.113 0.115 0.110 0.687
82.2 90 139.6 159.5 149 166 160
Observation: Average error of surrogate model is 5.23% for mass & VM. (Note: Bblue, O-orange, R-red).
The result shows that the average error between surrogate model-based optimization result and the FEM simulation result is about 5%, which is acceptable for engineering design, especially when a relative large security coefficient value is set. For this
[1] Zhang Y, Bernard A, Gupta RK, Harik R (2014) Evaluating the Design for Additive Manufacturing: a Process Planning Perspective. Procedia CIRP 21:144–150. [2] Thompson MK, Moroni G, Vaneker T, Fadel G, Campbell RI, Gibson I, Martina F (2016) Design for Additive Manufacturing: Trends, Opportunities, Considerations, and Constraints. CIRP Annals Manufacturing Technology 65(2):737–760. [3] Zhang Y, Bernard A (2018) A KBE CAPP Framework for Qualified Additive Manufacturing. CIRP Annals Manufacturing Technology 67(1):467–470. [4] Azman AH (2017) Media Epidemics: Method for Integration of Lattice Structures in Design for Additive Manufacturing(Doctoral dissertation) . [5] Ashby MF, Evans A, Fleck NA, et al (2001) Metal Foams: a Design Guide. Applied Mechanics Reviews 54:105. [6] Rosen DW (2007) Computer-aided Design for Additive Manufacturing of Cellular Structures. Computer-aided Design and Applications 4(5):585–594. [7] Zhang P, Toman J, Yu Y, et al (2015) Efficient Design-Optimization of Variabledensity Hexagonal Cellular Structure by Additive Manufacturing: Theory and Validation. Journal of Manufacturing Science and Engineering 137(2):021004. [8] Cheng Lin, Zhang P, Biyikli E, et al (2017) Efficient Design Optimization of Variable-Density Cellular Structures for Additive Manufacturing: Theory and Experimental Validation. Rapid Prototyping Journal 23(4):660–677. [9] Tang Y, Kurtz A, Zhao YF (2015) Bidirectional Evolutionary Structural Optimization (BESO) Based Design Method for Lattice Structure to be Fabricated by Additive Manufacturing. Computer-Aided Design 69:91–101. [10] Nguyen J, Park SI, Rosen DW, Folgar L, Williams J (2012) Conformal Lattice Structure Design and Fabrication. Solid Freeform Fabrication Symposium (SFF), 6–8. Austin, TX. [11] Chu J, Engelbrecht S, Graf G, Rosen DW (2010) A Comparison of Synthesis Methods for Cellular Structures with Application to Additive Manufacturing. Rapid Prototyping Journal 16(4):275–283. [12] Yang L, Harrysson O, West H, Cormier D (2013) Modeling of Uniaxial Compression in a 3D Periodic Re-entrant Lattice Structure. Journal of Materials Science 48(4):1413–1422. [13] Chebbah MS, Lebaal N (2018) Tube Hydroforming Optimization Using a Surrogate Modeling Approach and Genetic Algorithm. Mechanics of Advanced Materials and Structures. http://dx.doi.org/10.1080/15376494.2018.1482578. [14] Azaouzi M, Lebaal N, Rauchs G, Belouettar S (2012) Optimal Design of Multistep Stamping Tools Based on Response Surface Method. Simulation Modelling Practice and Theory 24:1–14. [15] Zhang Y, Bernard A, Harik R, Karunakaran KP (2017) Build Orientation Optimization for Multi-Part Production in Additive Manufacturing. Journal of Intelligent Manufacturing 28(6):1393–1407.
Contents lists available at ScienceDirect
CIRP Annals - Manufacturing Technology jou rnal homep age : ht t p: // ees .e lse vi er . com /ci r p/ def a ult . asp
Optimised lattice structure configuration for additive manufacturing Nadhir Lebaal a, Yicha Zhang a,*, Frédéric Demoly a, Sébastien Roth a, Samuel Gomes a, Alain Bernard (1)b a b
Laboratoire Interdisciplinaire Carnot de Bourgogne, site UTBM, UMR 6303 CNRS, Univ. Bourgogne Franche-Comté, France Ecole Centrale de Nantes, LS2N, CNRS, UMR 6004, Nantes, France
A R T I C L E I N F O
Keywords: Design optimization Additive manufacturing Lattice structure
A B S T R A C T
Lattice structure is critical for designing light-weight or multi-function components in additive manufacturing. A lot of know-how had been accumulated via the benchmarking of cellular units. However, how to optimize the parameters and the topological distribution of lattice cells in a constrained design space to gain both mass and computation efficiency for structure design is still an open question. To answer it, this paper proposes a new optimization method using design of experiment and surrogate model to configure lattice structures in specified 3D hulls. A design case is presented to show the advantages of the proposed solution. © 2019 Published by Elsevier Ltd on behalf of CIRP.
1. Introduction
2. Related work
Additive Manufacturing (AM) brings more freedom to designers to enable extremely complex and light-weight design. The nonlinear relationship between the complexity and manufacturing cost is one of the main advantages of this new technology. Complexity is used as an indicator to guide the design for AM [1–3]. However, designers still have difficulties to apply lattice structures in their design since the current CAX tools have problems to represent and simulate complex freeform porous or periodic cellular structures [4]. Existing commercial tools can only generate lattice structure but lack of decision support on the selection of lattice patterns and its parameter set. In addition, the numerical simulation is too costly in design iteration. In the academic side, there is very little research to solve this problem although there were many works for the design and benchmarking of lattice pattern units. This means how to do the lattice design in the structure or component level is still an open question. To help designers realize their light-weight design via the use of lattice structure and fully harness the advantages of AM, this paper proposes a combined method that adopts knowledge engineering and mathematical optimization tools to provide decision support for lattice design configuration, including cell pattern selection and cell parameter optimization. The structure of the paper is organized as follows: the second section reviews key works related to the paper topic; the third section details the proposed method; the fourth section presents a case study for method demonstration and the last section concludes with discussions and perspectives.
Lattice structure and porous structure already exist in nature before AM. However, using these structures in design got more attention recently when AM becomes more mature. Basically, there are two main categories of lattice structure: stochastic and periodic. For the reason of ease control and better properties when compared with foam lattices [5], periodic cellular structures are more popular. The first lattice design framework, used for idea demonstration, was proposed by Ref. [6]. In this work, periodic lattices were controlled by parametric implicit modelling method to fill into 3D hull clusters. Then, only the strut diameter of lattices is optimized to change the density of lattice cells to meet design requirements. Numerical simulation is required for design validation in each iteration, which is costly. In addition, the modelling and analysis capability, e.g. maximum size or number of lattices, was not discussed and the computation cost of FEM (Finite Element Method) simulation in the design iteration was neither mentioned. However, this is the main barrier to remove for designers to adopt lattice structure since modelling capabilities and simulation efficiency are critical for qualified and high productive design. More recently, people tried to use TO (Topology Optimization) method for lattice structure design and configuration. In Refs. [7,8], TO is used to identify a variant material density distribution in a continuous way for a fixed design space with its constraints. Then, uniformed lattice cells with diverse strut diameters are used to map the continuous material distribution so as to approximate the mechanical properties. FEM is required for each lattice cell mapping/filling iteration. Similarly, computation cost would be high and the flexibility of design for AM is reduced since the original design space could not be changed. Another method was proposed in Ref. [9], where an initial design space was decomposed into different volumes (similar to the
* Corresponding author. E-mail address: [email protected] (Y. Zhang). https://doi.org/10.1016/j.cirp.2019.04.054 0007-8506/© 2019 Published by Elsevier Ltd on behalf of CIRP.
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N. Lebaal et al. / CIRP Annals - Manufacturing Technology 68 (2019) 117–120
method in Ref. [6]) to fill in with lattice or not. For the volumes to fill in, evolutionary TO is used to identify an optimal lattice strut diameter, to change the density, to respond to the desired mechanical properties. In this method, the original design space also could not be changed and the lattice parameter optimization is limited to strut diameter with too much computation since discrete evolutionary algorithm was adopted and FEM was called in each optimization iteration, which is quite costly. However, the computation performance was not discussed in their work. Different from those methods assigning uniformed cell type (although with no-uniformed thickness, strut diameter), in Ref. [10], a lattice configuration method based on element mechanics patterns was proposed. In this method, the original design space was decomposed into a set of conformal ‘ground structures cells’ where lattice cells mapped. Then, FEM analysis generated nodes were used to construct lattice cell joint nodes, and finally an optimization algorithm was applied to identify an optimal combination of predefined basic mechanics patterns to generate conformal lattice cells for each ‘ground structure’. Therefore, there is a need of large computation to deal with the overlapped struts created in the lattice generation process. Besides, the computation to find an optimal combination of lattice cells for each ‘ground structure’ is more expensive since the FEM should be engaged in each iteration when determining the strut diameter for each lattice cell. To solve the high computation cost problem of lattice configuration, more recently, there is a new method proposed by [4] almost avoiding the need of FEM in each design iteration. This method defined a relative density of lattice cells to the identical size volume of solid bounding box. Then it uses predefined cell patterns to map according to the stress distribution of the original specified design space. Variant strut diameters are used in the mapping to adjust the density of lattice cells so as to approaching the required mechanical performance. In addition, the representation of lattice structures in the configuration process is symbolic, which can further reduce the computation cost. This method has a much big advantage to other methods since it saves too much computation and make the wide use of lattice structure design for ordinary designers feasible. However, this method drives out unstable results some of which even exceeded 25% percent. This is not acceptable for the detail design stage where lattice configuration stays. The main reason is that they only use FEM simulation but no physical benchmarking of lattice cell’s properties to build a data base. The errors may also come from the omission of lattice orientation caused orthogonal properties and no FEM-based adjustment in the lattice configuration iteration. As reviewed above, the current research on lattice design configuration at the component level is limited. This problem should be well defined and further investigated to obtain wide application in light-weight design. Generally, the lattice configuration for component structure design has three main tasks: lattice pattern selection, lattice cell parameter set and lattice layout determination as described in Fig. 1. If the coupling aspect of these three tasks is considered, the configuration problem would become more complicated. It is clear that almost all existing
Fig. 1. Lattice configuration options for structure design.
methods use uniformed single pattern with only one parameter, thickness, for optimization but neglect many other configuration aspects. This only touches a small fraction of the original solution space of the configuration problem. To explore a wider solution space and provide more efficiency, this paper proposes to optimize more cell shape control parameters simultaneously in lattice configuration. Due to the complexity of this research problem, the proposed method limits its adaptability to the configuration of single lattice pattern with uniformed orientation and cell size. In addition, the lattice structure geometry modelling problem is also out of the scope of this paper. The proposed optimization method is detailed in the following section. 3. Proposed method Theoretically, for any specified design space, there is an infinite set of configuration solutions since the design space is continuous and full of design variables. Hence, to identify an optimal solution among the vast original solution space is already difficult. More seriously, during the solution searching, if the FEM is integrated for evaluating design alternative solutions, as did by some researchers in the literature, the computation cost would not be acceptable or even not feasible for median or big size component design. In [4], intensive benchmarking had been conducted and the result is that current CAD tools are only feasible to handle very small size lattice structures, and FEM tools are not feasible for big size structure analysis due to the high computation cost. To make lattice applicable for median or big size structure design with relative low computation cost, a new combined method is proposed as described in Fig. 2. The proposed method includes two blocks: the first block deals with the preselection of alternative lattice patterns and identification of significant variables of the selected lattice cell pattern that impact the concrete design problem; the second one adopts a statistical approximating method to explore the infinite solution space with an efficient optimization algorithm, which acts as a fine tuning of lattice key parameters.
Fig. 2. Lattice configuration optimization method.
Step 1: analyse the original design and identify the regions where lattice structure is possible to fill in and obtain the mechanical responses, e.g. stress and strain distribution, via FEM analysis according to the predefined loadings and constraints. Step 2: use the FEM analysis results to select potential lattice patterns and their orientations. Here, the method proposed in Ref. [3] is adopted. Relative density concept is used to roughly determine the lattice pattern, size and orientation to respond to the mechanical constraints. An allowance should be set for the selection so as to make the difference between the estimated performance and that of the real optimal solution locate at the positive direction to facilitate the later parameter optimization. A knowledge base storing the characteristics of lattice cell patterns and orientation impacts, which can be obtained either by numerical experiments as did in Ref. [4] or physical experiments as presented by Ref. [11,12], is required here. Step 3: analyse the selected lattice pattern and conduct DOE (design of experiment) to ascertain the significant lattice cell parameters for the specific design problem. It should be noted that the DOE here is numerical in order to save real experimental cost. Different lattice parameter combinations with designed value levels will form different lattice structures to the design. Then, FEM analysis is conducted to get their responses. In this step, the key
N. Lebaal et al. / CIRP Annals - Manufacturing Technology 68 (2019) 117–120
lattice parameters are identified and a set of representative lattice configurations’ performances or responses at the structure level are collected. Step 4: apply a statistical method to approximate the unknown solution space (build a surrogate model) of the design problem with the collected lattice structure responses in Step 3. If the collected DOE data is not enough to construct an approximating solution space, then more sample solutions are required. The number of required sample solutions depends on the method of building the surrogate model. Step 5: conduct multi-objective optimization for the design problem on the obtained surrogate model to identify the optimal parameter set for the selected lattice pattern. Step 6: use FEM to verify the obtained optimal solution. As shown in Fig. 2, there are two optimization iterations. In Iteration 1, there is no FEM analysis involved, In Iteration 2, FEM for DOE is required for each selected lattice pattern. But, Iteration 2 is not obligatory if a preselected lattice pattern and orientation can lead to an acceptable optimal result. Only when the preselected pattern does not work or there is a need to obtain better performance, then this iteration is launched. Therefore, the main computation cost actually is located at the numerical DOE via FEM. To further explain and demonstrate the proposed method, a design case study is presented in the next section. 4. Case study The design problem is redesign a scooter for AM to reduce mass but maintain acceptable strength. The original design is given below (Fig. 3a).
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Table 1 L9 Taguchi DOE with 4 factors in 3 levels. Hc/mm (central height)
Lb/mm (strut length)
Ab/mm (strut angle)
D/mm (strut diameter)
7 7 7 8 8 8 9 9 9
5 6 7 5 6 7 5 6 7
35 40 45 35 40 45 35 40 45
1.75 2.00 2.25 1.75 2.00 2.25 1.75 2.00 2.25
The operation of numerical DOE is realized in several technical steps. At first, a geometric model of the lattice cell and the deck 3D hull model with a fixed thickness of 1 mm are built with an ordinary CAD tool, CATIA; then, an FEM tool, COMSOL, is used to build an FEM model. In COMSOL, a lattice cell is represented as an FEM model with 36 beam elements (Fig. 5b), then this element model is copied and assembled in three directions to fully fill in the 3D hull. After that, the inner surfaces of the deck 3D hull are used to trimming the boundaries of lattice structure. The intersections between the trimmed beam elements and the 3D hull surfaces form a set of nodes on the hull’s surfaces. Then, these nodes are used to mesh the 3D hull into shell elements. Each FEM model has around 6000 beam and 35,000 shell elements with good connection (Fig. 5b). Due to the symmetric property of the deck geometry, only half of the deck is analysed to save computation. The DOE analysis shows that all of the 4 parameters are significant to this design problemwhen considering the strength and mass objectives (Fig. 6). Then, they should be all considered for optimization in the following fine tuning step. To be noted that different design parameters’ sensitivities depend on the specific design problem.
Fig. 3. Design problem.
Step 1: after analysis of the original design, the deck component (Fig. 3b), material of Aluminium T6-6060 (Young Modulus, 70 GPa; Tensile strength,160Map. Density, 2700 kg.m3) is selected to fill with lattice structures since it is solid and has many materials over the performance under the extreme loading of 1000 N of concentrated force on its middle top surface vertically. Its original weight is 0.6874 kg with a Von Mises stress of 160 MPa under loading. Step 2: according to the loading, the FEM analysis and an allowance of 50% more strength to the loading, two lattice patterns, Octet-truss with a minimum size of 20 mm (edge length of the cubic bounding box to host a lattice cell) and Hexa-truss with a minimum size of 15 mm are identified from an example lattice pattern base. The orientations of the lattice cells are their flat top bases in parallel to the deck top surface. To simplify the demonstration, only the hexa-truss cell is used for detail explanation in the following steps. Step 3: based on the selected lattice pattern, a numerical Taguchi DOE is conducted. Four shape control parameters (Fig. 4b) of this lattice cell are considered (Table 1).
Fig. 5. Numerical DOE FEM analysis.
Fig. 6. DOE impact analysis.
Step 4: theoretically, the 9 computation results from the DOE in the former step are 9 representative alternative solutions in the infinite original design solution space. Therefore, they can be used to help building a surrogate model to approximate the original solution space. In this paper, the Kriging interpolation is adopted. For Kriging interpolation, the approximate functions can be expressed as follows. ~JðxÞ ¼ P T ðxÞa þ ZðxÞ
Fig. 4. Preselected lattice cell pattern.
where x is the design variable vector; a ¼ ½a1; a2; a3 . . . amT is the unknown parameters vector, ~J ðxÞ is the approximate function (objective or constraint functions) and Z(x) is the random fluctuation. To obtain a good accuracy of the surrogate model, 60 sample solutions (51 random combinations of lattice param-
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eters and the 9 collected DOE results) are used for this case. Specific design case may have different solution space which requires different number of sample solutions for interpolation. The determination of the minimum number of samples for interpolation depends on the interpolation validation test result. Due to limited space, the interpolation process and validation test of the surrogate model are not provided here. Readers are directed to authors’ other work on the modelling process [13,14]. Step 5: once the surrogate model is ready, an optimization algorithm can be applied to search the optimal solution on it. In this example, GA (genetic algorithm) is adopted due to its efficient stochastic search technique. The four parameters of the lattice cell pattern are set as design variables, and the minimization of relative global mass of the lattice deck structure with the decrease of the maximum value of relative Von Mises stress are set as two objectives (rM = f1(x) and rVM = f2(x)) under the constraints of material yield stress and variable value boundaries. 8 > > > > > > <
8 W > > < f 1 ðxÞ ¼ 1 W0 min s max > > : f 2 ðxÞ ¼
example, the average gain of mass is more than 75%. The following action is to select a non-dominated solution from the Pareto front according to design preferences. If required, new alternative lattice pattern can be selected to launch the second iteration of the proposed method to search for better result. The whole lattice modelling and optimization process is conducted in an ordinary PC with a configuration as: CPU E31535M [email protected] GHz, RAM, 32Go. However, due to the use of beam and shell element, the average calculation time for each FEM analysis is only about 11 min. Therefore, the total computation cost is about 660 min since the main contribution is the FEM analysis for the random sample solutions to build the surrogate model. This could not be imaged by other existing methods where FEM analysis is embedded into the optimization iteration. Another strong point of this proposed method is that the lattice pattern knowledge base can provide efficient pre-configuration. A good pre-configuration can greatly reduce the work of fine tuning on parameter optimization.
5. Conclusion
sy > > > > Such that : s max a:s y > > : With : xl x xu where W1 denotes the global structure weight; W0 is the original solid structure weight; s max is maximum VM (Von Mises) stress and s y is the yield stress. x is the design variable of the four selected lattice parameters. More details about GA coding can be found in author’s work [15]. The optimization result, a set of nondominated solutions on the Pareto front, is presented in Fig. 7.
Via the use of benchmarking knowledge and surrogate model based optimization, lattice configuration optimization can be realized for median or large size structure design with acceptable computation cost. Reliable design solution can be obtained by using current design and simulation tools, which implies that lattice structure would be possibly adopted widely by ordinary designers for light weight design. It is the main contribution of this paper. However, this work is limited to uniformed lattice configuration. More work should be done in the future to deal with conformal or gradient lattice configuration and optimization.
References
Fig. 7. GA based optimization result.
Step 6: Since the obtained optimal solutions come from the optimization on the surrogate model, there is a need to validate via the FEM analysis. Three solutions (marked by blue, orange and red colour respectively as shown in Fig. 7) are selected for calculation validation. Table 2 shows the validation result. Table 2 Optimization result analysis (demi-deck). Design solutions
Hc /mm
Lb /mm
Ab /mm
D /mm
Mass /kg
VM /MPa
Surrogate-B FEM-B Surrogate-O FEM-O Surrogate-R FEM-R Original solid
7.45 7.45 8.93 8.93 8.99 8.99 –
6.99 6.99 6.99 6.99 6.99 6.99 –
43 43 37.74 37.74 44.42 44.42 –
1.89 1.89 1.75 1.75 1.81 1.81 –
0.155 0.154 0.116 0.113 0.115 0.110 0.687
82.2 90 139.6 159.5 149 166 160
Observation: Average error of surrogate model is 5.23% for mass & VM. (Note: Bblue, O-orange, R-red).
The result shows that the average error between surrogate model-based optimization result and the FEM simulation result is about 5%, which is acceptable for engineering design, especially when a relative large security coefficient value is set. For this
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