TY - JOUR
AU - Choi, Young
AB - Abstract Additive manufacturing has enabled the fabrication of complex structures such as cellular structures. Although numerous design frameworks have been proposed for cellular structures, their effectiveness was limited owing to the use of B-rep-based representation. To address the limitations in previous research, this study proposes an implicit-based computer-aided design framework customized for additively manufactured functionally graded cellular structures (AM-FGCSs). The proposed design framework effectively aids in both single- and multiscale structural optimization for designing FGCSs. Moreover, implicit-based modeling afforded a reliable geometric representation that could efficiently assist computation tasks such as visualization, validation, and process planning for fabrication. In addition, two case studies were conducted to demonstrate the effectiveness of the proposed framework for designing FGCSs. The first case study on a three-point bending beam design problem proved the practicality of implicit-based representation in multiscale structural design. Meanwhile, the second case study validated the proficiency of the proposed framework in process planning for fabrication and engineering analysis, the two most vital computation tasks in designing cellular structures. Graphical Abstract Open in new tabDownload slide Graphical Abstract Open in new tabDownload slide implicit representation, design for additive manufacturing, functionally graded cellular structures, optimum structural design Highlights Proposed an implicit-based computer-aided design framework customized for functionally graded cellular structures. Multiscale implicit-based modeling is used for geometric representation at design stage. Either single-scale or multiscale optimization could be chosen for cellular structure design. Voxel-based reconstructed geometry is used for design validation and fabrication. 1. Introduction 1.1. Design for additive manufacturing of cellular structures Additive manufacturing (AM) is a unique fabrication process whereby components are manufactured by adding the material layer by layer. In practice, complicated structures in terms of shape, material compositions, and hierarchy can be fabricated using AM (Sossou et al., 2018). Furthermore, AM could also aid the maintenance and remanufacturing processes, increasing products’ life cycles (Ceruti et al., 2019). This innovative manufacturing technique has prompted a new research direction termed as “design for additive manufacturing” (DFAM). Among the several topics in DFAM, additively manufactured cellular structures (AM-CSs) have garnered significant attention owing to their lightweight, multifunctionality, and wide applicability. Cellular structures comprise connected trusses or plates (Gibson & Ashby, 1997) and can be categorized into two main groups: stochastic and nonstochastic structures (Tamburrino et al., 2018). Nonstochastic structures have gathered considerable attention owing to the emergence of AM and is the focus of this study. Functionally graded cellular structure (FGCS) is a type of cellular structure with locally customized geometric configurations, featuring suitable structural density and shapes for specific applications (Fig. 1). According to literature, FGCSs are more advantageous than uniform structures as they afford enhanced performances under the same working conditions. Recently, researchers have attempted to derive adequate design frameworks for FGCSs (Nguyen et al., 2013, 2021; Tang et al., 2015; Cheng et al., 2017). Figure 1: Open in new tabDownload slide (a) Uniform cellular structure. (b) FGCS. Figure 1: Open in new tabDownload slide (a) Uniform cellular structure. (b) FGCS. Structural optimization constitutes a practical approach for designing FGCSs. The design problem for AM with cellular structures could be formulated as presented in Table 1. In this design problem, the multiscale performance–property–structure–process relationship (Rosen, 2007) is employed and the design variables could either pertain to unit cell relative density (macroscale variables) or shape (mesoscale variables). The goal of the design is to maximize structure performance under process-based constraints. Several other constraints, such as those related to performance and design variable boundaries, should also be satisfied. Figure 7: Open in new tabDownload slide Data flow in proposed design framework. Figure 7: Open in new tabDownload slide Data flow in proposed design framework. Table 1: Design problem formulation for DFAM of FGCSs. Given • Base material and AM process • Loading condition • Design domain Find • Geometric configuration of FGCSs • Density distribution (macroscale variables) • Unit cell topology (mesoscale variables) Satisfy • Design constraints (size and volume) • AM constraints  ○ Geometric (overhang, bridging, etc.)  ○ Mechanical (anisotropy, etc.) Maximize • Structural performance (Compliance, maximum stress, etc.) Given • Base material and AM process • Loading condition • Design domain Find • Geometric configuration of FGCSs • Density distribution (macroscale variables) • Unit cell topology (mesoscale variables) Satisfy • Design constraints (size and volume) • AM constraints  ○ Geometric (overhang, bridging, etc.)  ○ Mechanical (anisotropy, etc.) Maximize • Structural performance (Compliance, maximum stress, etc.) Open in new tab Table 1: Design problem formulation for DFAM of FGCSs. Given • Base material and AM process • Loading condition • Design domain Find • Geometric configuration of FGCSs • Density distribution (macroscale variables) • Unit cell topology (mesoscale variables) Satisfy • Design constraints (size and volume) • AM constraints  ○ Geometric (overhang, bridging, etc.)  ○ Mechanical (anisotropy, etc.) Maximize • Structural performance (Compliance, maximum stress, etc.) Given • Base material and AM process • Loading condition • Design domain Find • Geometric configuration of FGCSs • Density distribution (macroscale variables) • Unit cell topology (mesoscale variables) Satisfy • Design constraints (size and volume) • AM constraints  ○ Geometric (overhang, bridging, etc.)  ○ Mechanical (anisotropy, etc.) Maximize • Structural performance (Compliance, maximum stress, etc.) Open in new tab The stated design problem is complex and multidisciplinary with the appearance of plenty of computation issues including geometric representation, structural optimization, engineering analysis, and process planning. Therefore, an appropriate framework is required to accomplish such product development tasks for FGCSs, with the following objectives: R1: The framework should appropriately aid multiscale and multidisciplinary structure optimization. R2: The influence of AM processes should be reflected in the design with geometric and physical constraints. R3: A suitable geometric representation method is required for cellular structures in computation tasks such as structure characterization, optimum design, visualization, engineering analysis, and process planning for fabrication. 1.2. Contribution of proposed framework 1.2.1. Limitations of previous works The proposed design framework is a continuation of prior studies for designing AM-FGCSs. Previously, a design framework that successfully captured the build orientation-dependent anisotropic stiffness of AM-FGCSs was proposed (Nguyen et al., 2021). In addition, the developed anisotropic stiffness modeled was utilized for a concurrent structure–process design with conformal AM-FGCSs (Nguyen & Choi, 2020). However, the previously proposed framework did not satisfy all the requirements cited in Section 1.1. The limitations were primarily due to the B-rep-based geometric representation, which weakened the ability of the framework in multiscale design and postprocessing optimum designed structures. In this study, a computer-aided design (CAD) framework relying on implicit-based representation was proposed to overcome the abovementioned drawbacks of previous research. 1.2.2. Novelty of the proposed design framework In the proposed framework, first, anisotropic mechanical properties were considered in the design of AM-FGCSs, similar to previous studies. However, unlike previous research, implicit-based modeling and the representative volume elements (RVEs) were applied for efficient geometric and physical modeling. The implicit-based modeling technique allowed for convenient transformations into voxel models for visualization, validation, and process planning for fabrication. Furthermore, the proposed framework could either serve as a CAD/CAE plugin or a stand-alone software package with efficient data transformation between the computation platforms. This characteristic increases the flexibility of the proposed framework in practice. 2. Literature Review 2.1. Current status of academic research The performance-driven design of AM-FGCSs via structure optimization involves two major approaches: truss element-based and volume element-based methods. Truss element-based models have been developed in parallel with truss-based cellular structures (Chang & Rosen, 2013; Nguyen et al., 2013; Alzahrani et al., 2015; Tang et al., 2015, 2018). However, volume element-based models are more desirable owing to the limitations of the truss-based approach in terms of the number of solvable problems. Volume element-based methods can solve various types of problems, such as those pertaining to structural static (Cheng et al., 2017, 2019), heat transfer (Cheng et al., 2018a, b; Kim & Yoo, 2020), multiscale (Wang et al., 2017, 2018), and multifunctional structures (Tang & Zhao, 2018). The design procedure for cellular structures using the volume element-based approach has been summarized by Tamburrino et al. (2018). From literature, the reconstruction of optimally designed structures for visualization and fabrication primarily relies on B-rep; however, this approach also involves several difficulties. As the geometry of cellular structures is usually complicated, being constructed by numerous Boolean operations of geometric primitives, the B-rep approach is not viable. In contrast, implicit-based representation provides a more robust solution, regardless of the size of the problem. In addition, as implicit-based representation presents shape relying on mathematical operations of scalar fields, the problems arising in B-rep, such as singularity and nonmanifold, are avoided. As a result, implicit-based representation is exceptionally suitable for cellular structure modeling. Fryazinov et al. (2013) demonstrated the advantages of implicit-based representation with a function representation for designing infinite-periodic cellular structures. Moreover, level-set-based optimization is another implicit-based approach that has proved its capabilities in multidisciplined structure and material designs (Li et al., 2016; Nanthakumar et al., 2016; Sivapuram et al., 2016; Ghasemi et al., 2018; Ye et al., 2019). In terms of postprocessing design results such as the process planning for fabrication, implicit-based modeling also posed potential for convenient tool path generation (Sharma & Gurumoorthy, 2020). As AM is a unique fabrication process, the properties of as-fabricated components should be considered in the design, including the geometric and physical issues. Nevertheless, AM constraints such as allowable unit cell size, truss diameter, or relative density ranges are primarily geometric (Nguyen et al., 2021). In general, the mechanical properties of as-fabricated cellular structures are not considered in a majority of the design approaches. One unique characteristic of additively manufactured structures is the anisotropic mechanical properties (Ahn et al., 2002). Although this issue has been reflected in several design frameworks (Stanković et al., 2016, 2017; Nguyen & Choi, 2020; Nguyen et al., 2021), the scope for further explorations still remains. 2.2. Current status in industry In general, design toolkits for cellular structures are becoming popular in the CAD software industry, where makers have started developing CAD applications for cellular structures with different implementation levels, varying from plugins to stand-alone packages. Several CAD tools for cellular structures, including their degree of fulfillment for the requirements stated in Section 1.1, in comparison with the proposed framework are summarized in Table 2. Table 2: Summary of available industrial CAD solutions for cellular structure design. Software/software module . Type (*) . Integration with conventional CAD . Supported cellular structure (**) . Cellular database . Autodesk Netfabb 1 △ 1 △ nTopology 2 △ 2 ○ Altair OptiStructure 3 × 2 ○ Materialise 3-Matic 3 × 2 ○ Intralattice 2 △ 1 × Proposed system 2 ○ 2 ○ Software/software module . Type (*) . Integration with conventional CAD . Supported cellular structure (**) . Cellular database . Autodesk Netfabb 1 △ 1 △ nTopology 2 △ 2 ○ Altair OptiStructure 3 × 2 ○ Materialise 3-Matic 3 × 2 ○ Intralattice 2 △ 1 × Proposed system 2 ○ 2 ○ ○: Fully supported; △: Partly supported; ×: Not supported. (*) 1. 3D Printing slicing software; 2. CAD software; 3.CAE software. (**) 1. Homogeneous lattice structure; 2. Heterogeneous lattice structure. Open in new tab Table 2: Summary of available industrial CAD solutions for cellular structure design. Software/software module . Type (*) . Integration with conventional CAD . Supported cellular structure (**) . Cellular database . Autodesk Netfabb 1 △ 1 △ nTopology 2 △ 2 ○ Altair OptiStructure 3 × 2 ○ Materialise 3-Matic 3 × 2 ○ Intralattice 2 △ 1 × Proposed system 2 ○ 2 ○ Software/software module . Type (*) . Integration with conventional CAD . Supported cellular structure (**) . Cellular database . Autodesk Netfabb 1 △ 1 △ nTopology 2 △ 2 ○ Altair OptiStructure 3 × 2 ○ Materialise 3-Matic 3 × 2 ○ Intralattice 2 △ 1 × Proposed system 2 ○ 2 ○ ○: Fully supported; △: Partly supported; ×: Not supported. (*) 1. 3D Printing slicing software; 2. CAD software; 3.CAE software. (**) 1. Homogeneous lattice structure; 2. Heterogeneous lattice structure. Open in new tab 3. Methodology 3.1. Overview The proposed design framework for the optimum design of AM-CSs is illustrated in Fig. 2. The concept of implicit-based representation and the RVE were adopted for cellular structure modeling and are the backbone of the framework (Section 3.2). The design process for AM-FGCSs involved three steps. Initially, the process-induced anisotropic mechanical properties of AM-CSs were characterized and modeled (Section 3.3). Subsequently, optimization was conducted for the performance-driven design of AM-FGCSs (Section 3.4). Finally, a voxel-based approach that allows the exploitation of the benefits of implicit-based representation was applied for both process planning and design validation via the finite element analysis (FEA; Section 3.5). The implementation and data exchange issues within the proposed design framework are elaborated in Section 3.6. Figure 2: Open in new tabDownload slide Overall design framework. Figure 2: Open in new tabDownload slide Overall design framework. 3.2. Cellular structure modeling 3.2.1. Implicit-based representation for geometric modeling Implicit representation is a volume-based modeling technique in which the boundary of an object is derived by extracting the “zero set” of an implicit signed distance function, such as $$\begin{eqnarray} \left\{ \begin{array}{@{}*{1}{c}@{}} {{\rm{\Phi }}\left( {{\bm{x}},{\bm{s}}} \right) \ge 0\ {\bm{x}} \in {\rm{\Omega }}}\\ {{\rm{\ \Phi }}\left( {{\bm{x}},{\bm{s}}}\right)=\ 0\ {\bm{x}} \in {\rm{\Gamma \ }}}\\ {{\rm{\Phi }}\left( {{\bm{x}},{\bm{s}}} \right) \textless \ 0\ {\bm{x}} \notin {\rm{\Omega }}} \end{array}, \right. \end{eqnarray}$$(1) where |$\Phi $| denotes the level-set function, |${\bm{x}}( {{x_1},\ {x_2},\ \ldots {x_n}} )$| is the evaluation point in an n-dimensional space, |${\bm{s}}( {{s_1},\ {s_2},\ \ldots ,\ {s_n}} )$| is the control parameter vector, and |${\rm{\Phi }}( {{\bm{x}},\ {\bm{s}}} )$| represents the level-set descriptor at evaluation point |${\bm{x}}$|⁠. |${{\bf \Omega }}$| denotes the domain inside the structure and Γ denotes the structural boundary. A 2D wall-based honeycomb unit cell in implicit-based representation is portrayed as in Fig. 3a. In addition, the same approach can be applied to three-dimensional (3D) structures without losing generality. Unlike explicit-based representations, implicit functions are required to be evaluated over the entire sampling domain to extract the “zero set” that defines the structure boundary. Although that computation process is expensive, the application of parallel computing on the Graphics Processing Unit (GPU) could drastically reduce computational costs (Fig. 3b). Figure 3: Open in new tabDownload slide Implicit-based representation: (a) representation of 2D wall-based honeycomb structure and (b) GPU- and CPU-based function evaluation of 3D cubic unit cell. Figure 3: Open in new tabDownload slide Implicit-based representation: (a) representation of 2D wall-based honeycomb structure and (b) GPU- and CPU-based function evaluation of 3D cubic unit cell. 3.2.2. RVE for physical modeling The RVE concept is widely adopted for simplifying computational models of composite structures (Andreassen & Andreasen, 2014). The modeling with RVE is also suitable for cellular structures owing to their similarity with composite structures (void domains are considered as materials with extremely weak properties). In the design approach with RVE, a unit cell with a complex geometry is replaced by a solid element assigned with the effective material properties (Fig. 4), which are defined as a function of the relative density, an essential property of cellular structures (Gibson & Ashby, 1997). These relationships are also known as cellular “scaling laws.” Thus, the determination of the relative density of a unit cell could indicate its physical properties. In addition, scaling laws could be derived either by empirical or simulation-based methods. Figure 4: Open in new tabDownload slide Octet cellular unit cell and its RVE. Figure 4: Open in new tabDownload slide Octet cellular unit cell and its RVE. 3.2.3. Multiscale structure modeling Upon combining implicit-based representation and the modeling with RVE, a component filled with cellular structures could be mathematically represented as the union of infill region I and boundary region B. Typically, the level-set description of a sampling point |${{\rm{\Phi }}_S}$| can be determined at an evaluation point as $$\begin{eqnarray} {{\rm{\Phi }}_{\rm{S}}} = {\rm{max}}({{\rm{\Phi }}_B}\ ,{{\rm{\Phi }}_I}). \end{eqnarray}$$(2) The infill region comprises the RVEs and is trimmed by the infill region boundary |${R_I}$|⁠, such that $$\begin{eqnarray} {{\rm{\Phi }}_I} = \min \left( {\max \left( {{\Phi _{\mathrm{ RV}{\mathrm{ E}_i}}}} \right),{R_I}} \right)\ . \end{eqnarray}$$(3) The geometry of an RVE was determined using a predefined implicit function based on the selected cellular structure. Furthermore, this approach can be repeated without compromising generality at lower scales for providing a compact multiscale representation toward efficient modeling: $$\begin{eqnarray} {{\rm{\Phi }}_{{S_{\mathrm{ upper}}}}} = \max \left( {{\Phi _{{B_{\mathrm{ upper}}}}},{\Phi _{{I_{\mathrm{ upper}}}}}} \right)\ , \end{eqnarray}$$(4) $$\begin{eqnarray} {\Phi _{{I_{\mathrm{ upper}}}}} = \min \left( {\max \left( {{\Phi _{\mathrm{ RV}{\mathrm{ E}_{\mathrm{ lower}}}_i}}} \right),{R_{{I_{\mathrm{ upper}}}}}} \right)\ . \end{eqnarray}$$(5) Multiscale structure modeling for a piston rod with cellular structures by using implicit representation and RVEs is presented in Fig. 5. Figure 5: Open in new tabDownload slide Multiscale modeling using RVE and implicit representation. Figure 5: Open in new tabDownload slide Multiscale modeling using RVE and implicit representation. 3.3. Characterization of anisotropic mechanical properties of AM-CSs The anisotropy in the properties of as-fabricated AM-CSs is induced by their geometric configurations and the AM processes. As mentioned in previous studies, the orthotropic material model can be employed for such a scenario. In general, nine independent scaling laws are required for the orthotropic material model, which are derived by regression from the multiscale homogenization analysis results. As the simulation-based approach was discussed and validated in previous studies, interested readers can refer to Nguyen and Choi (2020) and Nguyen et al. (2021) for further details. 3.4. Optimum design of components with cellular structures A typical structure optimization for designing parts with FGCSs could be formulated as $$\begin{eqnarray} \begin{array}{@{}*{1}{l}@{}} {\mathop {\mathrm{ min}}\limits_x f\left( {\bm{x}} \right)|\ {\bm{x\ }} = \left[ {{\bm{\rho \ }} = {\bm{\ }}\left( {{\rho _1},{\rho _2}, \ldots } \right);{\bm{s\ }} = {\bm{\ }}\left( {{{\bm{s}}_1},{{\bm{s}}_2}, \ldots } \right)} \right]}\\ {s.t:}\\ {{{\bm{g}}_{\mathrm{ eq}}}\ \left( {\bm{x}} \right) = {\bm{g}}_{\mathrm{ eq}}^*\ ,}\\ {{{\bm{g}}_{\mathrm{ ineq}}}({\bm{x}}) \le {\bm{g}}_{\mathrm{ ineq}}^*,}\\ {{x_{\mathrm{ low}}} \le {x_i} \le {x_{\mathrm{ high}}}.} \end{array} \end{eqnarray}$$(6) First, a predefined design domain was tessellated and populated with RVEs. The design variables |${\bm{x}}$| contained relative densities, |${\rho _e}$|⁠, and shape parameter vectors, |${{\bm{s}}_e}$|⁠, of each RVE. In addition, the optimization process aimed at maximizing the structural performance |$f( {\bm{x}} )$|⁠, i.e. minimizing maximum stress, structure total volume, or compliance. Moreover, the given boundary conditions and material properties, e.g. anisotropic scaling law, could be formulated as equality constraints, |${{\bm{g}}_{\mathrm{ eq}}}( {\bm{x}} )$|⁠. Similarly, performance-related constraints could be formulated either as inequality, |${{\bm{g}}_{\mathrm{ ineq}}}( {\bm{x}} )$|⁠, or equality constraints, depending on the design problem. Finally, a boundary, |${x_{\mathrm{ low}}} < {x_i} < {x_{\mathrm{ high}}}$|⁠, must be set. Subsequently, the gradient-based method was applied to solve the optimization problem, and FEA with RVEs was used to estimate the structural performance after each iteration. The structure optimization process is a multiscale optimization problem with macroscale (length scale > 10 mm) and mesoscale (length scale varies from 0.1 to 10 mm) structures. Also, it could be transformed into a single-scale design problem depending on the level of design. In the single-scale design problem, either the density distribution (macroscale structures) or the shape of the RVEs (mesoscale structures) is designed, and the other is treated as a design constraint. In contrast, they are optimized simultaneously in multiscale design. 3.5. Structure reconstruction, visualization, validation, and process planning for fabrication A reconstruction process is required for the visualization, validation, and process planning for fabrication of the optimum designed result. In implicit-based modeling, the reconstructed model derived by evaluating the structure implicit function and containing the level-set descriptions of all evaluation points is called the “level-set model.” This model is different from the “implicit model” that denotes the implicit function of the structure. In addition, the reconstruction process can be performed based on the smallest element, owing to the hierarchical modeling of cellular structures (Fig. 6). This hierarchy also allows for parallel computation, which reduces the computation costs. In the first step, the level-set model of each unit cell was derived and assembled up to the global level-set model. In addition, a Boolean operation was performed with a level-set model representing the boundary of the infill region, to derive the final infill level-set model. Furthermore, the derived infill level-set model was combined with the boundary level-set model using a union operation to derive the final level-set model of components filled with FGCSs. Figure 6: Open in new tabDownload slide Level-set model reconstruction of optimally designed cellular structures for visualization, process planning, and validation. Figure 6: Open in new tabDownload slide Level-set model reconstruction of optimally designed cellular structures for visualization, process planning, and validation. Thereafter, the reconstructed level-set model was directly utilized to exploit all the benefits of the implicit-based representation. The level-set models of components filled with FGCSs were visualized using the ray-casting visualization technique, which is commonly employed to visualize volume data in medical applications. The voxel-based FE approach, which directly converts the level-set models of AM-FGCSs to finite element models, was utilized for the validation of the designed structures. Lastly, process planning for fabrication was performed with voxel-based AM. 3.6. Implementation and data flow within the framework A prototype test bed was implemented with self-developed components, and an efficient data flow was established within the proposed framework. The vital components with information on third-party libraries and software are summarized in Table 3, and the data flow among the processes is depicted in Fig. 7. In particular, a majority of the components were implemented with MATLAB, including the structure characterization and design optimization. Moreover, the “.mat” format of MATLAB was used to store data for visualization, engineering analysis, and process planning. In addition, ray-casting visualization was implemented with Visualization Toolkit Library, and OpenVDB was used to support the voxel-based process planning. An open-source software for voxel printing, Voxelizer 2, was used for process planning, and ANSYS APDL was used for FEA with the voxel-based FE models. Finally, the components were fabricated using a 3D WOX fused deposition modeling (FDM) printer. Table 3: Implemented components of proposed design framework. Sl. No. . Process . Status . Note . 1 Model preparation Self-developed 2 Structure characterization Self-developed CALFEM for internal FEA 3 Design optimization Self-developed 4 Structure reconstruction Self-developed MATLAB–GPU computing 5 Visualization Self-developed VTK ray-casting 6 Process planning Voxelizer 2 OpenVDB as input file 7 Engineering analysis ANSYS APDL CSV input Sl. No. . Process . Status . Note . 1 Model preparation Self-developed 2 Structure characterization Self-developed CALFEM for internal FEA 3 Design optimization Self-developed 4 Structure reconstruction Self-developed MATLAB–GPU computing 5 Visualization Self-developed VTK ray-casting 6 Process planning Voxelizer 2 OpenVDB as input file 7 Engineering analysis ANSYS APDL CSV input Open in new tab Table 3: Implemented components of proposed design framework. Sl. No. . Process . Status . Note . 1 Model preparation Self-developed 2 Structure characterization Self-developed CALFEM for internal FEA 3 Design optimization Self-developed 4 Structure reconstruction Self-developed MATLAB–GPU computing 5 Visualization Self-developed VTK ray-casting 6 Process planning Voxelizer 2 OpenVDB as input file 7 Engineering analysis ANSYS APDL CSV input Sl. No. . Process . Status . Note . 1 Model preparation Self-developed 2 Structure characterization Self-developed CALFEM for internal FEA 3 Design optimization Self-developed 4 Structure reconstruction Self-developed MATLAB–GPU computing 5 Visualization Self-developed VTK ray-casting 6 Process planning Voxelizer 2 OpenVDB as input file 7 Engineering analysis ANSYS APDL CSV input Open in new tab 4. Case Studies The effectiveness of the proposed design framework in designing AM-FGCSs is discussed in this section, specifically for the multiscale structural optimization (case study 1) and postprocessing of optimum designed results including visualization, validation, and process planning (case study 2). 4.1. Case study 1: multiscale design of three-point bending beam In this case study, the advantages of applying the proposed framework for the multiscale optimum design were verified via a design problem of a 2D three-point bending beam design with a honeycomb structure. 4.1.1. Design problem formulation A three-point bending beam with dimensions of 120 × 43.3 |$( {\mathrm{ mm}} )$| was tessellated into a 24 × 5 RVE model, under the loading conditions depicted in Fig. 8a. Honeycomb unit cells, with dimensions of |$5$| × |$8.66$||$( {\mathrm{ mm}} )$|⁠, were selected for the infill cellular structure (Fig. 8b). The multiscale design problem, in the context of decoupled design, is mathematically expressed in (7) and (8) for macroscale and mesoscale design problems, respectively. In addition, as the fabrication process, FDM with polylactic acid material was selected. A volume constraint with a maximum allowable volume fraction of |$0.45$| was set as the performance-related constraint. Figure 8: Open in new tabDownload slide Design input information: (a) design domain and boundary conditions and (b) honeycomb cellular structures. Figure 8: Open in new tabDownload slide Design input information: (a) design domain and boundary conditions and (b) honeycomb cellular structures. Moreover, the flexibility of the proposed framework was evaluated through the use of both single- and multiscale design approaches. The objective function of the macroscale structure optimization was to minimize the total compliance of the structure under external loads, whereas that of the mesoscale structure optimization was maximizing the stiffness of the structure under a prescribed strain. In macroscale structure design problem, unit cell shapes are fixed and predetermined scaling laws are given. Meanwhile, the mesoscale structure design problem was conducted with the volume constraints derived from the macroscale design. $$\begin{eqnarray} \begin{array}{@{}*{1}{l}@{}} {\mathop {\mathrm{ min}\ }\limits_{\bm{\rho }} c = {{\bm{F}}^{\bm{T}}}{\bm{ u}};\ \left\{ {{\bm{\rho }}\left( {{\rho _1}, \ldots ,{\rho _N}} \right)\ } \right\}}\\ {s.t:}\\ {{\bm{Ku}} = {\bm{F}};}\\ {{\bm{K}} = \mathop \bigwedge \limits_{e = 1}^N {{\bm{K}}_e};\ {{\bm{K}}_e} = {{\bm{K}}_e}\left( {{\bm{C}}_e^*\left( {{\rho _e}} \right)} \right);}\\ {\ \displaystyle\frac{{C_{11}^*}}{{{C_{11}}}} = \ 0.78\rho _e^{2.38} + 0.03;\ \ \displaystyle\frac{{C_{22}^*}}{{{C_{22}}}} = \ 0.77\rho _e^{2.32} + 0.03;}\\ {\ \displaystyle\frac{{C_{12}^*}}{{{C_{12}}}} = \ 0.48\rho _r^{2.34} + 0.09;\ \ \displaystyle\frac{{C_{33}^*}}{{{C_{33}}}} = \ 1.07\rho _r^{2.39} - 0.02;}\\ {{\bm{C}} = \left[ {\begin{array}{@{}*{3}{c}@{}} {4962.8} & \quad {2260.2} & \quad 0\\ {2260.2} & \quad {4118.5} & \quad 0\\ 0 & \quad 0& \quad{1053.2} \end{array}} \right];}\\ {\sum\limits_{e\ = \ 1}^n {{V_e}{\rho _e} \le 0.45} }\\ {0 \le {\rho _{\mathrm{ low}}} \le {\rho _e} \le {\rho _{\mathrm{ high}}} \le 1;} \end{array} \end{eqnarray}$$(7) $$\begin{eqnarray} \begin{array}{@{}*{1}{l}@{}} {\mathop {\mathrm{ max }}\limits_{\bm{s}} \,\,J\left( {\bm{s}} \right)\ = {{\bm{v}}^{\bm{T}}}\ {{\bm{K}}_{\bm{m}}}{\bm{v}};}\\ {s.t:}\\ {\bar{\varepsilon } = {\varepsilon _0}}\\ {{\bm{K}} = \mathop \bigwedge \limits_{e = 1}^M {{\bm{K}}_e}\ ;\ \ {{\bm{K}}_e} = {{\bm{K}}_e}\ \left( {{\bm{C}}_e^*\left( {{E^*}\left( {{\eta _e}} \right)} \right)} \right);}\\ {{E^*} = \ E{\eta _e};}\\ {V\left( s \right)\ = {V_{\mathrm{ target}}}\ }\\ {{s_{\mathrm{ min}}} \le {s_i} \le {s_{\mathrm{ max}}};i = 1 \ldots K} \end{array} \end{eqnarray}$$(8) In addition, a joint-based modeling approach (Fig. 8b) was utilized to support the design for the effective modeling of cellular structures. In particular, the honeycomb unit cell can be mathematically expressed as $$\begin{eqnarray} {C_B} = KS\left( {{J_i}\left( x \right),\beta } \right), \end{eqnarray}$$(9) $$\begin{eqnarray} {J_i} \left( x \right) = KS\left( {{W_j}\left( x \right),{\beta _i}} \right), \end{eqnarray}$$(10) $$\begin{eqnarray} {W_j} \left( x \right) = - \left[ {d\left( {x,W{C_j}} \right) - \frac{{{t_j}}}{2}} \right]. \end{eqnarray}$$(11) Generally, the cellular unit cell |${C_B}$| is the union of cellular joints |${J_i}$| as in (9), which are formed by the union of cellular walls |${W_j}$| as in (10). The level-set description of cellular walls at a sampling point is derived by subtracting the wall thickness |${t_j}$| to the distance from the sampling point to the wall’s centerline |$d( {x,\ W{C_i}} )$|⁠. Moreover, the Kreisselmeier–Steinhauser (KS) blending function (12) was employed in this design example instead of the standard union operator, as follows: $$\begin{eqnarray} \mathrm{ KS }\left( {{\phi _i},\beta } \right) = \frac{1}{\beta }\ln \left( {\mathop \sum \limits_{i = 1}^N {\mathrm{ e}^{ - \beta {\phi _i}}}} \right). \end{eqnarray}$$(12) 4.1.2. Optimum design result The results for both the single- and multiscale design problems are illustrated in Fig. 9. The multiscale design approach exhibited the greatest enhancement in structural performance (42.94%), as compared with the application of the uniform cellular structure. The single-scale design for the macrostructure also displayed a remarkable improvement in structure performance, with a total compliance reduction of 35.84%. Although single-scale design for mesostructures is not as effective as the other two options, it can still decrease the total compliance of the structure by 3%, compared with the uniform cellular structure. This result proves the validity of the approach in performing multiscale structure optimization with implicit-based representation. Figure 9: Open in new tabDownload slide Optimum design result: (a) uniform density map, (b) functionally graded density map, (c) uniform cellular structure (⁠|${\bm{c\ }} = {\bm{\ }}0.1691$|⁠), (d) single-scale density-graded cellular structure (c = 0.1085), (e) single-scale unit cell shape-graded cellular structure (c = 0.1655), and (d) multiscale FGCS (c = 0.096). Figure 9: Open in new tabDownload slide Optimum design result: (a) uniform density map, (b) functionally graded density map, (c) uniform cellular structure (⁠|${\bm{c\ }} = {\bm{\ }}0.1691$|⁠), (d) single-scale density-graded cellular structure (c = 0.1085), (e) single-scale unit cell shape-graded cellular structure (c = 0.1655), and (d) multiscale FGCS (c = 0.096). 4.1.3. Benefits of implicit-based representation in structural optimization The adoption of implicit-based representation for multiscale structural optimization provides several benefits, especially when solving mesoscale structural design problems, i.e. the RVE problem. First, the RVE problem requires the application of periodic boundary conditions (PBCs) (13), which, in turn, requires identical meshes at two opposite sides of the RVE model. The B-rep-based approach barely achieves this condition as the meshing process involves numerous manual operations. In contrast, the uniform grid in implicit-based representation effortlessly affords the identical meshes for the PBCs (Fig. 10a). $$\begin{eqnarray} {{\bm{u}}^{k + }} - {\rm{\ }}{{\bm{u}}^{k - }} = \bar{\varepsilon }\ \left( {\bm{x}} \right) \cdot ({{\bm{y}}^{k + }} - {{\bm{y}}^{k - }}) \end{eqnarray}$$(13) Figure 10: Open in new tabDownload slide Advantages of applying implicit-based modeling: (a) identical mesh for PBCs and (b) conversion from geometric to mechanical models of the honeycomb unit cell. Figure 10: Open in new tabDownload slide Advantages of applying implicit-based modeling: (a) identical mesh for PBCs and (b) conversion from geometric to mechanical models of the honeycomb unit cell. In addition, iterative evaluation of the implicit function within the optimization is no longer a bottleneck with GPU computing. Furthermore, the conversion from geometric to mechanical models for structural performance estimation was convenient with the voxel-based FE approach. Figure 10b depicts the geometric and mechanical models of a honeycomb unit cell. It is noted that voxel-FEs at boundary were trimmed, and its effective material properties were derived using the density approach (Bendsøe & Sigmund, 2004). Finally, the mesoscale design problem was solved using gradient-based methods, and it is necessary to evaluate the shape derivatives. In this context, another advantage of applying implicit-based representation was revealed as remeshing was not required. This benefit is owing to the uniform meshes and helps reducing the computation complexity. 4.2. Case study 2: validation and process planning of optimally designed 3D FGCSs The effectiveness of the proposed framework in the process planning for fabrication and the FEA for validation is discussed in this section. Three engineering design examples that are commonly used in the automotive and aerospace industries were conducted: a pillow bracket, a control arm, and an unmanned aerial vehicle (UAV) wing (Fig. 11a). In addition, three different types of cellular structures were selected: the 3D cubic (3D Lattice group), Gyroid (Triply Periodic Minimal Surface group), and Voronoi (customized group) structures (Fig. 11d). As this case study focused on the postprocessing stage, density distributions derived from the optimization were provided in advance (Fig. 11b). Furthermore, visualizations via ray-casting of the three examples are illustrated in Fig. 11d. Figure 11: Open in new tabDownload slide Examples of the proposed framework for 3D application (from top to bottom): pillow bracket, control arm, and UAV wing. (a) Design infill and boundary domain. (b) Optimally designed density map. (c) Ray-casting visualization. (d) Applied unit cells: (from top to bottom) 3D Cubic, Gyroid, and Voronoi-based structures. Figure 11: Open in new tabDownload slide Examples of the proposed framework for 3D application (from top to bottom): pillow bracket, control arm, and UAV wing. (a) Design infill and boundary domain. (b) Optimally designed density map. (c) Ray-casting visualization. (d) Applied unit cells: (from top to bottom) 3D Cubic, Gyroid, and Voronoi-based structures. 4.2.1. Validation with voxel-FEA The validation results of the voxel-based approach are presented in Fig. 12a. As illustrated, the voxel-based approach delivered the expected results, whereas the conventional FEA failed to process due to heavy input B-rep-based model. The voxel-FEA enables reliable analyses for engineering validations with relatively less manual effort as the volume nature of the implicit representation allows for fast and robust conversions from geometric to mechanical models. Despite producing quite heavy FE models, the robustness and automatability of the proposed approach are considerable advantages over conventional FEA. Figure 12: Open in new tabDownload slide Validation and process planning results with the voxel-based approach: (a) pillow bracket, (b) control arm, and (c) UAV wing. Figure 12: Open in new tabDownload slide Validation and process planning results with the voxel-based approach: (a) pillow bracket, (b) control arm, and (c) UAV wing. 4.2.2. Process planning for fabrication with voxel-based models Figure 12b illustrates the voxel-based process planning for fabrication. Furthermore, the computational efficiencies and storage requirements of the conventional approach with the stereolithography (.stl) format and the proposed approach with the OpenVDB (.vdb) format are summarized in Table 4. The experiment was conducted on a laboratory computer with an AMD® Ryzen Threadripper CPU @3.50 GHz (12 cores and 20 threads) and 64.0 GB RAM. GPU computing was performed using a NVIDA GeForce GTX 2080 Ti with 68-Multiprocessor and 11 GB memory. As reported, the proposed approach delivered superior performance in terms of both computational efficiency and storage requirements, as compared with the B-rep-based approach with the STL format. Moreover, the advantage of the proposed approach was more significant as the complexity of the design problem increased, such as in the case of the UAV wing design with the Voronoi cellular structure. In addition, the storage of level-set models in the VDB format was not mandatory, as they can be conveniently derived from the optimally designed shape. Therefore, the optimally designed shape parameters can be saved, which is considerably less expensive than saving the level-set models in the VDB form. Table 4: Comparison of computational efficiencies and storage requirements of the B-rep-based approach with STL and the proposed approach with OpenVDB. . . Pillow bracket . Control arm . UAV wing . . . .stl . .vdb . .stl . .vdb . .stl . .vdb . Time consumption (s) Exporting to physical file 26.7 1.5 78.3 3.1 729.4 4.6 Inputting data for slicers 5.3 5.9 10.2 8.9 25.9 11.5 G-code generation 33.6 7.4 45.1 18.1 76.6 39.9 Total 65.6 14.8 133.6 30.1 831.9 56.0 Physical file size (MB) 90.7 24.6 250.3 54.3 682.4 101.2 . . Pillow bracket . Control arm . UAV wing . . . .stl . .vdb . .stl . .vdb . .stl . .vdb . Time consumption (s) Exporting to physical file 26.7 1.5 78.3 3.1 729.4 4.6 Inputting data for slicers 5.3 5.9 10.2 8.9 25.9 11.5 G-code generation 33.6 7.4 45.1 18.1 76.6 39.9 Total 65.6 14.8 133.6 30.1 831.9 56.0 Physical file size (MB) 90.7 24.6 250.3 54.3 682.4 101.2 Open in new tab Table 4: Comparison of computational efficiencies and storage requirements of the B-rep-based approach with STL and the proposed approach with OpenVDB. . . Pillow bracket . Control arm . UAV wing . . . .stl . .vdb . .stl . .vdb . .stl . .vdb . Time consumption (s) Exporting to physical file 26.7 1.5 78.3 3.1 729.4 4.6 Inputting data for slicers 5.3 5.9 10.2 8.9 25.9 11.5 G-code generation 33.6 7.4 45.1 18.1 76.6 39.9 Total 65.6 14.8 133.6 30.1 831.9 56.0 Physical file size (MB) 90.7 24.6 250.3 54.3 682.4 101.2 . . Pillow bracket . Control arm . UAV wing . . . .stl . .vdb . .stl . .vdb . .stl . .vdb . Time consumption (s) Exporting to physical file 26.7 1.5 78.3 3.1 729.4 4.6 Inputting data for slicers 5.3 5.9 10.2 8.9 25.9 11.5 G-code generation 33.6 7.4 45.1 18.1 76.6 39.9 Total 65.6 14.8 133.6 30.1 831.9 56.0 Physical file size (MB) 90.7 24.6 250.3 54.3 682.4 101.2 Open in new tab Notably, technologies for facilitating visualization, validation, and process planning for fabrication with voxel models have been available for a long time. In engineering design, the application of such volume-based representation has been limited owing to hardware capacities. Moreover, the employment of subtractive manufacturing, involving the requirement for geometric boundary information, is an additional factor responsible for the lack of volume-based representation utilization. Nevertheless, the adoption of volume-based approaches has become viable with the developments in GPUs and AM. Furthermore, the complex and hierarchical characteristics of AM-CSs necessitate the application of volume-based representation. In this context, the proposed implicit-based design framework offers appropriate representation and effortless conversion to available voxel-based solutions for postprocessing the designed results. Therefore, a robust CAD framework can be established given the minimization of development efforts. 5. Conclusion This paper proposed a complete CAD framework with implicit-based representation for designing AM-FGCSs. The proposed framework satisfied the requirements of a performance-driven design framework for AM-FGCSs. 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TI - Implicit-based computer-aided design for additively manufactured functionally graded cellular structures
JO - Journal of Computational Design and Engineering
DO - 10.1093/jcde/qwab016
DA - 2021-05-13
UR - https://www.deepdyve.com/lp/oxford-university-press/implicit-based-computer-aided-design-for-additively-manufactured-9zeTdiZ0P3
SP - 813
EP - 823
VL - 8
IS - 3
DP - DeepDyve
ER -