TOPOLOGY OPTIMIZATION OF 3D AUXETIC METAMATERIALS USING RECONCILED LEVEL-SET METHOD

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1 Proceedings of the ASME 6 International Design Engineering echnical Conferences and Computers and Information in Engineering Conference IDEC/CIE 6 August -4, 6, Charlotte, North Carolina DEC OPOLOGY OPIMIZAION OF 3D AUXEIC MEAMAERIALS USING RECONCILED LEVEL-SE MEOD Shikui Chen Computational Modeling, Analysis and Design Optimization Research Laboratory Department of Mechanical Engineering State University of New York at Stony Brook Stony Brook, NY, Shikui.Chen@stonybrook.edu Panagiotis Vogiatzis Computational Modeling, Analysis and Design Optimization Research Laboratory Department of Mechanical Engineering State University of New York at Stony Brook Stony Brook, NY, Panagiotis.Vogiatzis@stonybrook.edu ABSRAC Metamaterials with extraordinary material properties, not easily found in nature, are of great interest. In this paper, the authors employ topology optimization with a reconciled levelset method in order to design 3D negative Poisson s ratio designs. he strain energy method has been used to predict the effective properties of the periodically assembled metamaterial, which is based on the resulted unit cell. he sensitivity analysis is derived utilizing the adjoint variable method with the general linear elastic equation combined with a weak imposition of the Dirichlet boundary condition. 3D single-material and multimaterial designs have been generated showing the effectiveness of the proposed method. he advantage of the method can be located on the fact that the design can be manufactured by 3D printing, without any post-processing work on the result of the topology optimization, due to the clear boundaries between the different materials. optimization of trusses, created 3D ground structures showing different Poisson s ratio behavior in different planes. For 3D continuum mechanics, there are conceptual designs [, ], extruded designs based on a D topology optimization [5] and 3D designs [, 3] using SIMP. he target of this paper is to design 3D metamaterials that will have negative Poisson s ratio. he final design will be a periodically assembled structure based on the resulted unit cell, which consists of one or more materials plus void. In the following sections, the prediction of the elastic properties, the problem setting and numerical examples are presented. PREDICION OF ELASIC PROPERIES Given a specific design, one can predict the macroscopic effective properties of the metamaterial using the strain energy method [4, 5]. In the 3D case, 6 loading scenarios are needed in order to calculate the homogenized elastic tensor (ab. ). able. LOADING SCENARIOS FOR 3D CASE. INRODUCION Materials with negative Poisson s ratio, also known as auxetic materials, when compressed in one direction, contract in the other two perpendicular directions. Lakes [] firstly developed a negative Poisson s ratio foam back in 987. Since then, D auxetic metamaterials have been also achieved in literature by using mathematical algorithms based on inverse homogenization [, 3], Solid Isotropic Material with Penalization (SIMP) method [4, 5], Bi-directional Evolutionary Structural Optimization (BESO) [6] and Level-Set Method (LSM) [7, 8]. In 994, Sigmund [9], using topology Address ε () ε() ε (33) ε (3) ε (3) ε () For each loading scenario, all the remaining domains (without an applied loading) are simply supported. all correspondence to this author. Copyright 6 by ASME

2 he strain energy for the first scenario can be calculated by: C C C C C C C C C U V C () C 33 C V C. At the same time, it can also be calculated as a domain integration of the strain energy density: U ε D ε d. () ijkl For unit volume, combining Eqn. ()-() and since For the last scenario: C U. (3) U U : C C C C C C C C C U V C (4) C 33 C V C C C C, where C can be derived as: C U U U. (5) aving calculated the strain energy of each loading scenario, the elastic tensors for the 3D case can be found concentrated in ab.. he Poisson s ratio values, finally, are obtained considering the compliance matrix for orthotropic materials. able OMOGENIZED ELASIC PROPERIES. C U C U C U U U U C U U U C U U U C OPOLOGY OPIMIZAION For the single material optimization, the well-known classical level-set method [6-] has been employed. In LSM, a one-higher dimensional level-set function Φ represents the design. he properties of each point of the design domain are defined by the sign of Φ, and the boundaries are implicitly determined by the zero level-set. hus, for a 3D design, a 4D Φ is needed. On the other hand, for the multi-material optimization, several schemes have been proposed [-3]. For level-set based topology optimization, Wang and Wang [4] developed the Color level-set method (CLSM) [4-6] separating the different materials by the sign combination of the level-set functions. Moreover, the Piecewise constant level-set method [7-9] divides the design domain into different areas by using different values of the level-set functions. In this work, the reconciled level-set method [3-3] (RLSM) has been selected due to its straightforward connection of each material with one single Φ i. he potential problem of an overlap between the different level-set functions (Φ i and Φ j) is overcome by employing the Meriam-Bence-Osher [3] (MBO) operator as shown in the following equation: temp temp max. i i i j j An example, including the corresponding D design of a 3D level-set function, has been selected to show the contribution of the MBO operator (Fig. ). he advantage of the RLSM combined with the MBO operator can be appreciated during the manufacturing process, since each level-set function is directly used to generate the stl file of each material. Figure MBO OPERAOR IN ACION: (a) Φ BEFORE EVOLUION. (b) EMPORARY Φ AFER EVOLUION. (c) Φ AFER MBO OPERAOR. (6) he calculation of the effective properties along with the Poisson s ratios is a necessary step of the optimization process, which will be described in the next section. Copyright 6 by ASME

3 where the Courant-Friedrichs-Lewy (CFL) condition must be satisfied. At this step, the material time derivative approach is followed to construct the design velocity field, which will enable the evolution of the boundaries. he objective function is coupled with the governing equation through the Lagrangian multiplier method and the steepest-decent method is applied [35-37]: L J g. () ere, the design velocity field for the first 3 scenarios are being derived: Figure OPIMIZAION FLOWCAR. ' ' ij u C ijklkl vd ' u vds D ij ucijklkl v Vi ds wijkl Cijkl Cijkl ij u Cijkl kl u d DL Dt wijkl Cijkl Cijkl ij u Cijkl kl u Vi ds. Setting the first bracket of Eqn. () equal to zero: () he design procedure includes initially set targets for all the elastic tensors and, therefore, the least square optimization problem fits to the needs of this process [9]. hat is to minimize the difference between the homogenized elastic tensor and the specified targets, subject to the governing elasticity equation and a volume constraint: 3 Minimize: J wijkl Cijkl Cijkl (7) i, j,k,l ij udijkl kl vd u u vds subject to: D V V, where test function, (8) w ijkl is the weighting factor for each elastic tensor, v the the eaviside function and * denotes the targets. he first constraint is the weak form of the governing equation with a weak imposition of Dirichlet boundary condition [33], where u is the prescribed displacement on the boundary. As shown in the flowchart of the optimization (Fig. ), the procedure is iterative. At the end of each iteration, the design is getting updated by finding the steady-state solution of the amilton-jacobi equation with respect to the pseudo time t [6, 34]: ' ' ij ijklkl ' u vds wijkl Cijkl Cijkl ij u Cijkl kl u d u C v d () D wijkl Cijkl Cijkl u, in v,, on D (3) v is obtained and replaced in the second bracket of Eqn. (), leading to the design velocity field: ij ucijklkl uvds i ij ucijklkl u Vds i ijkl ijkl ijkl ij ijklkl i w C C u C u Vds u C v Vds ijkl ijkl ijkl ij ijkl kl i ij ijkl kl i w C C ijkl ijkl ijkl w C C u C u Vds V w C C u C u. (5) i ijkl ijkl ijkl ij ijkl kl (4) Similarly, the velocities for the last 3 cases can be obtained and the velocity will be the sum of all 6 cases (ab. 3): 6 f V V i. i (6) V n (x), t (9) 3 Copyright 6 by ASME

4 able 3 DESIGN VELOCIY FIELD FOR EAC LOADING. V w C C uc u V w C C uc u 3 V w C C uc u V u u u u u u u C w C C u C u 33 uc uc33 3 w 33 C 33 C 33 u C u 33 uc uc w C C u C u uc V 5 V 6 able 4 COMPARISON BEWEEN D AND 3D CASE. D case 3D case Loading Scenarios 3 6 Number of elements 6 64 Degrees of freedom 7,5,594,33 RAM needed.gb 6GB ime per iteration a 9s h3m a For given computer specifications (for comparison). Finally, the curvature flow κ and the volume constraint are being added to obtain the final normal velocity field: f Vn V V V. (7) RESULS For the optimization procedure, the commercial software COMSOL was coupled with the optimization code developed in MALAB. In contrast to the D case, the optimization in 3D has higher computational cost. Apart from the double number of scenarios that have to be examined, the number of the elements and subsequently the number of the degrees of freedom are increasing dramatically (ab. 4). For demonstration of the proposed method, a 3D singlematerial (Fig. 3), and both D and 3D multi-material examples are presented (Fig. 5 and 7). he design domain of the unit cell has been discretized with 4x4x4 elements. he initial design has spheres of.3 radius without material distributed on the whole design. he design consists of one material (E=.3MPa and ν=.3) and void, which is represented by a dummy material (E=.MPa and ν=.3). he targets of the elastic properties are set in a manner that the design will have a desired ν=-.5 in all 3 planes. he final result can be seen in Fig. 3 and a verification through simulation with a finer mesh (8x8x8 elements) in Fig. 4. he resulted effective properties are: C C C.668MPa 3333 C C C.536MPa (8) Figure 3 INIIAL AND FINAL DESIGN FOR 3D SINGLE- MAERIAL WI RESULED NEGAIVE POISSON S RAIO ν=-.49 IN ALL PLANES. Figure 4 DEFORMAION OF A 3D SINGLE-MAERIAL DESIGN AFER APPLYING A ORIZONAL DISPLACEMEN IN X DIRECION. 4 Copyright 6 by ASME

5 Figure 5 INIIAL AND FINAL DESIGN FOR MULI- MAERIAL WI RESULED NEGAIVE POISSON S RAIO ν=-.48 IN ONE PLANE. he D multi-material example includes an optimization targeting to negative Poisson s ratio (ν=-.5) in only one plane. In this case: one hard material (E=.8GPa and ν=.3), one soft material (E=.3MPa and ν=.3) and a dummy material for the void (E=.MPa and ν=.3). he initial design consists of a domain with hard material (green color), holes of.5 radius (void) and rings of soft material, in between, with radius zone.5~. (red color). he obtained geometry is shown in Fig. 5 and a verification in Fig. 6. he effective properties of the multi-material design are: Figure 6 VERICAL DISPLACEMEN IN MM OF A 3X3 MULI-MAERIAL SRUCURE (3X3mm) AFER APPLYING A ORIZONAL 3mm DISPLACEMEN ON E RIG BOUNDARY GPa.4 C C / C.487 (9) Figure 7 INIIAL AND FINAL /8 OF E DESIGN, AND FINAL FULL UNI CELL FOR 3D MULI-MAERIAL WI RESULED NEGAIVE POISSON S RAIO.95, 3.3 AND 3.5. From the resulted design, it can be observed that the soft material has been concentrated at the location of the hinges, permitting more flexibility to the final result. Designs made only by the soft material would achieve high auxetic behavior with low stiffness. On the other hand, designs made by the hard material would show high stiffness, but it would need a significant force to show the negative Poisson s ratio effect. he multi-material design balances these two characteristics by achieving high overall stiffness and auxetic behavior with less force applied. Similarly, a 3D multi-material design has been generated with.95, 3.3 and 3.5 (Fig. 7 and 8). Figure 8 DEFORMAION OF A 3D MULI-MAERIAL DESIGN AFER APPLYING A ORIZONAL DISPLACEMEN IN X DIRECION. 5 Copyright 6 by ASME

6 Figure 9 FABRICAED 3x3 SRUCURE BASED ON E 3D UNI CELL WI NEGAIVE POISSON S RAIO ν=-.49 IN ALL PLANES. 3D printing In Figure 9, a structure of 3x3 periodically assembled unit cells has been fabricated using 3D printer (Objet 6 Connex, Stratasys Ltd) with angoplus material. he current method gives the flexibility of having a design with more than one material. he multi-material design obtained by the optimization procedure can also be 3D printed. he printed structure consists of 3x3 periodically assembled unit cells using two different materials: VeroWhite for the hard material and angoplus for the soft material, white and transparent in Fig respectively. Similarly, a printed 3D multimaterial unit cell is shown in Fig.. Figure FABRICAED 3D MULI-MAERIAL UNI CELL WI NEGAIVE POISSON S RAIO.95, 3.3 AND 3.5. CONCLUSION In conclusion, in the reconciled level-set method, used in this paper, the boundaries between the multiple materials are clear and the design is free of areas with intermediate material properties. hat allows the designer to 3D print prototypes of the design and by extension to directly connect the design procedure to the manufacturing process. he proposed method was used to generate 3D single-material and multi-material designs with negative Poisson s ratio, and they were 3D-printed to provide a complete design procedure: form the problem setting to the realization of the design through rapid prototyping. ACKNOWLEDGMENS he authors acknowledge the support from the National Science Foundation (CMMI467), Ford Motor Company, the Region University ransportation Research Center (URC) and the start-up funds from State University of New York at Stony Brook. Figure FABRICAED 3x3 SRUCURE BASED ON E MULI-MAERIAL UNI CELL WI NEGAIVE POISSON S RAIO ν=-.48 IN ONE PLANE. REFERENCES [] Lakes, R., 987, "Foam structures with a negative Poisson's ratio," Science, 35(479), pp [] Sigmund, O., 994, "Design of Material Structuresusing opology Optimization." [3] Sigmund, O., 994, "Materials with prescribed constitutive parameters: An inverse homogenization problem," International Journal of Solids and Structures, 3(7), pp [4] Sigmund, O., and orquato, S., 999, "Design of smart composite materials using topology optimization," Smart Materials and Structures, 8(3), p Copyright 6 by ASME

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