. Auxetic materials with negative Poisson’s ratio have potential applications across a broad range of engineering fields. Several design techniques have been developed to obtain auxetics with targeted mechanical properties. However, many of these finite element based techniques are difficult to use directly fo r auxetics, particularly during the design optimization stage which involves evolving boundary parts with large curvatures. This paper focusses on a series of smoothed petal auxetics , with lower stress concentrations at connecti ng parts, compared to the reference star shaped structures. An isogeometric shape optimization framework to achieve target Poisson’s ratios at large deformation is presented. Several smoothed petal auxetic desig ns with target constant Poisson’s ratios up to an effective tensile strain of 30% are shown to demonstrate the capability of the optimization framework. Keywords : Auxetic ; Negative Poisson’s Ratio; Isogeometric analysis; Shape optimization; Sensitivity analysis ; 1. INTRODUCTION In contrast to conventional materials, a uxetics are a class of structural and/or functional materials that expand (contract) transversely when stretched (compressed) in the axial direction, exhibiting a negative Poisson’s ratio (NPR) phenomenon (Alderson 1999). This counter - intuitive behavior has been shown to enhance the mechanical properties of engineering structures. Auxetics have thus been a subject of increased interest in recent years. The deformation mechanisms (Lakes 1991) and mechanical properties of auxetics such as indentation , shear resistance ( Alderson et al. 2000, Dirrenberger et al. 2014 ), energy absorption (Zhu et al. 2018, Zhu et al. 2019 ) and wave propagation characteristics ( Spadoni et al. 2009 ) have been studied for various engineering applications. Design techniques based on modern numerical methods started with the works of Sigmund (Sigmund 199 4, 1995 ). Subsequently, several design frameworks adopting topology optimization such as Solid Isotropic Material with Penalization ( SIMP ) (Schwerdtfeger et al. 2011 ), evolutionary hybrid algorithms (Kaminakis et al. 2015 ), and Bi-directional Evolutionary Structural Optimization ( BESO ) (Da et al. 2018, Xia et al. 2018 ) have been developed for design of auxetic structures. Recently, a level -set based topology optimization was successfully employed to obtain designs with prescribed Poisson’s ratios in the elastic regime (Vogiatzis et al. 2017, Wang et al. 2017a ). Although auxetics are subjected to large deformation in many potential applications, the above -mentioned design approaches are limited to the linear regime. Experiment al results have shown that auxetics exhibit significant nonlinear behavior at finite stra ins (Alderson et al. 2000 , Quek et al. 2 Smith et al. 1999 ). Consequently, the optimized effective properties based on linear theory may degrade rapidly with deformation (Clausen et al. 2015, Wang et al. 2014a ). To overcome this, topology optimization is adopted for the design of auxetics with prescribed non -linear properties at finite strains (Clausen et al. 2015, Wang et al. 2014a ). However, m ost published numerical optimization frameworks for auxetics are based on the finite element (FE) framework. In general, a FE based structural optimization framework is time consuming because of the need to ensure a proper integration between the geometrical description and the corresponding FE analysis (Lian et al. 2016 ). Moreover, approximating the design doma in with FE meshes reduces the accuracy of the geometrical description ( Xie et al. 2018 ). To this end , an isogeometric shape optimization framework for smoothed petal auxetics was presented in Wang et al. (2017b ). Subsequently, isogeometric shape optimization was employed to design planar isotropic auxetics with prescribed effective properties in the linear regime ( Wang and Poh 2018 ) and prescribed Poisson’s ratios in the nonlinear regime ( Choi and Cho 2018, Weeger et al. 2019 ). In this paper, the isogeometric shape optimization for petal auxetics is extended for achieving targeted effective properties in the nonlinear deformation regime. The versatility of the presented approach is demonstrated by designing petal auxetic structures with constant NPRs within -0.5, considering both plane strain and plane stress conditions , in the strain interval [0.00, 0.30]. 2. ISOGEOMETRIC ANALYSIS FOR SMOOTHED PETAL AUXETICS The auxetic behavior of star shaped re -entrant structures h as been studied by several researchers . The auxeticity of such structures is found to depend largely on hinge -functional connections at the vertex connections (Carneiro et al. 2016 ). This poses a great challenge in manufacturi ng and may lead to significant stress concentrations. Consequently, a series of petal auxetics with smoothe d connections , was proposed in Wang et al. (2017 b). This study focusses on such smoothed petal auxetics that are amenable to the manufacturing proces s and exhibit lower stress concentrations, compared to the reference star shaped structures. As highlighted in Wang et al. (2017b), a key challenge for the shape optimization of petal auxetics is the proper integration between the curved geometrical descr iption and the corresponding FE analysis. The iso geometric analysis (IGA) ( Hughes et al. 2005 ) integrates the analysis and design processes together . IGA is thus well suited for shape optimization in terms of the exact geometrical description and enhanced sensitivity analysis (Cho and Ha 2009, Wall et al. 2008). In IGA, the control points representing the design boundary of the geometry can directly serve as optimization variables, thus eliminat ing the need for re-meshing ( Sun et al. 2018, Weeger et al. 2019 ). In the following, the IGA framework for smoothed petal auxetics in the nonlinear deformation regime is presented. 2.1. Initial Shape Setting and Updating Scheme The shape updating scheme for smoothed petal auxetics presented in Kumar et al. (201 9), is summarized as follows. Commencing from reference petal -shaped structures shown in Fig. 1 , this study seeks an optimized geometry which achieves a targeted Poisson’s ratio in the nonlinear deformation regime. The entire geometry can be obtained through the patterning of one petal, henceforth termed as parent petal and addition of connecting bars. To simplify the design problem, only the exterior boundary of the design problem is considered. The control points defining the exterior boundary of the parent petal are thus chosen as design variables. Quek et al. 3 Figure 1. Smoothed tri - and hexa - petals auxetic structures. The initial parent petal characterization starts with the des cription of its interior boundary by considering practical engineering and manufacturing prerequisites. Next, the control points of the interior boundary are simply offset to obtain the initial exterior boundary of the parent petal. A closed boundary for p arent petal is formed by connecting the interior and exterior boundaries via straight lines at the planes of symmetry. Henceforth, the space where the parent petal is characterized is called design space . The external boundary of the petal will be updated during the iterative process of the optimization framework. 2.2. Characterization of Nonlinear Deformation In this study, a biocompatible silicone -based rubber (commercial name: Elite Double 32, Zhermack) is considered as the base material . The Yeoh hyperelastic model (Yeoh 1993 ) is adopted to characterize the nonlinear response . Following Yeoh (1993) , the strain energy density is given as 𝜓(𝐹)=𝐶1(𝐼1̅−3)+𝐶2(𝐼1̅−3)2+𝐶3(𝐼1̅−3)3+1 𝐷1(𝐽−1)2+1 𝐷2(𝐽−1)4 +1 𝐷3(𝐽−1)6, (1) where 𝐶𝑖 and 𝐷𝑖 are material parameters (𝑖=1,2,3). Here, 𝐼1̅=𝐽−2 3𝑡𝑟[𝑭𝑭𝑇] is the first invariant for incompressible material, 𝐽=det(𝑭) and 𝑭 is the deformation gradient. For the silicone -based rubber base material, 𝐶1=131 𝐾𝑃𝑎, 𝐶2=0 𝐾𝑃𝑎 , 𝐶3=3.5 𝐾𝑃𝑎 , 𝐷1=𝐷2= 𝐷3=154 𝐺𝑃𝑎−1 (Babaee et al. 2013 ). The first Piola -Kirchioff stress (𝑷) can be obtained as 𝑃𝑖𝑗=𝜕𝜓 𝐹𝑖𝑗=2 𝐶1 𝐽−2 3 [𝐹𝑖𝑗−1 3𝐼1𝐹𝑖𝑗−𝑇]+ 4 𝐶2 𝐽−2 3 (𝐼1̅−3)[𝐹𝑖𝑗−1 3𝐼1𝐹𝑖𝑗−𝑇]+6 𝐶3 𝐽−2 3 (𝐼1̅− 3)2[𝐹𝑖𝑗−1 3𝐼1𝐹𝑖𝑗−𝑇]+ 2 𝐷1 (𝐽−1) 𝐽 𝐹𝑖𝑗−𝑇+ 4 𝐷2 (𝐽−1)3 𝐽 𝐹𝑖𝑗−𝑇+ 6 𝐷3 (𝐽−1)3 𝐽 𝐹𝑖𝑗−𝑇 . (2) The paper focuses on achieving prescribed NPRs in the large deformation regime. The deformation of representative volume element (RVE) shown in Fig. 2 is characterized via the average Biot strain ( Biot 1939) defined as 𝑬=√𝑭𝑇𝑭−𝟏 (3) where 1 is the second -order identity tensor. In the following , the fine scale analysis within a RVE is termed as “micro”, and the effe ctive response of a RVE as “macro”. Within a RVE, the bulk material forming the structure occupies a sub -domain Ω⊂𝑉 with boundary Γ. This structure characterized by region Ω is the design object, while the rest of the RVE is empty. The updated desig ns of region Ω in an iterative design process, can be considered as a sequence that follows a time -like parameter 𝜏 (Choi and Kim 2016 ). Quek et al. 4 Figure 2. RVE of smoothed (a) tri -petals structure and (b) hexa -petals structure. (c) Traction and displacement boundaries, Γ=Γ𝑡+Γ𝑢. Following the standard homogenization procedure, a boundary value problem (BVP) for the microstructure in material configuration with a Biot (macro) strain 𝑬 can be obtained by imposing periodic boundary conditions and is given by, {𝒄[𝒖]≔div 𝑷=𝟎 in Ωτ 𝒕̂=𝟎 on Γ𝑡𝜏 𝒙+−𝒙−=𝑭⋅(𝑿+−𝑿−) on Γ𝑢𝜏 (4) where 𝑷 denotes the first Piola -Kirchoff stress tensor, with symbols ‘+’ and ‘-’ denoting the opposing sides of a RVE . 2.3. Isogeometric Discretization Non-uniform Rational B -Splines ( NURBS ) offer great flexibility in modelling smoothed petal auxetics with the combination of control points located in the physical space and the basis function defined in the index space (Nguyen et al. 2019). The isogeometric shape optimization adopts the locations and weights of the control points as design variables . Hence, it allows the smooth shape variation of petal auxetics during an iterative design process. A general expression for a NURBS surface with parameters 𝜉 and 𝜂 is expressed mathematically as 𝑿[𝜉,𝜂]=∑∑𝑅𝑖𝑗𝑚 𝑗=1𝑛 𝑖=1 [𝜉,𝜂]𝑿𝑖𝑗, (5) where 𝑛×𝑚 control points denoted by 𝑿𝑖𝑗 form a control net and 𝑅𝑖𝑗 represents the NURBS basis function. Based on the non -decreasing knot vectors 𝝃={𝜉1,𝜉2,𝜉3,…𝜉𝑛+𝑝+1} and 𝜼= {𝜂1,𝜂2,𝜂3,…𝜂𝑛+𝑝+1}, the NURBS basis function 𝑅𝑖𝑗 can be defined as 𝑅𝑖𝑗[𝜉,𝜂]=𝑁𝑖,𝑝[𝜉]𝑀𝑗,𝑞[𝜂]𝑤𝑖,𝑗 ∑∑𝑁𝑖′,𝑝[𝜉]𝑀𝑗′,𝑞[𝜂]𝑤𝑖′,𝑗′ 𝑚 𝑗′=1𝑛 𝑖′=1 (6) where 𝑁𝑖,𝑝[𝜉] and 𝑀𝑗,𝑞[𝜂] are the B -spline basis functions, 𝑤𝑖,𝑗 is the weight of control point 𝑿𝑖𝑗, 𝑝 and 𝑞 denote the degrees in the parametric axes 𝜉 and 𝜂, respectively. In IGA, the displacement fields are discretized using NURBS parametrization as follows: 𝒖[𝜉,𝜂]=∑∑𝑅𝑖𝑗𝑚 𝑗=1𝑛 𝑖=1 [𝜉,𝜂]𝒖𝑖𝑗, (7) where 𝒖𝑖𝑗 is the displacement corresponding to the control point 𝑿𝑖𝑗. The static equilibrium of a RVE is governed by the fol lowing equation: 𝒓=𝒇𝑒𝑥𝑡−𝒇𝑖𝑛𝑡(𝒖)=0, (8) Quek et al. 5 where 𝒓 is the residual force vector, 𝒇𝑒𝑥𝑡 is the external force vector, and 𝒇𝑖𝑛𝑡(𝒖) is the internal force vector. The static equilibrium given by Eq. (8), is solved iteratively using the Newton -Raphson method to give 𝐾𝑡 𝑑𝑢= 𝑟, (9) where 𝑑𝑢 is the correction displacement vector and 𝐾𝑡 is the tangent stiffness matrix. The NURBS model is discretized differently for efficient analysis and design in the context of isog eometric shape optimization. In the design space, a coarse discretization is used to minimize the number of design variables whereas a refined discretization is employed in the analysis space to accurately compute the state fields ( Lian et al. 2017 ). 3. OPTIMIZATION FORMULATION FOR 2D AUXETIC DESIGN In this section, an isogeometric shape optimization framework for achieving the prescribed Poisson’s ratios at large deformation is presented. The optimization is performed using “fmincon” function in MATLA B, which necessitates computation of design objective, constraints and their gradients with respect to design variables. 3.1. Design Objective and Sensitivity Analysis The aim is to find the optimized shape of the design boundary, such that the prescribed Poisson’s ratio s can be achieved . For the petal auxetics, it is also desirable to preserve the geometrical symmetry. Correspondingly , the objective function can be defined as the minimiz ation of least square difference between the actual and targeted Poisson’s ratios , given as Φ≔∑(𝜈−𝜈̅)𝛼2= ∑1 (𝐸11)𝛼2(𝐸22−𝐸22̅̅̅̅̅)𝛼2 𝛽 𝛼=1 𝛽 𝛼=1 (10) where 𝐸22 and 𝐸̅22 are the actual and targeted tr ansverse strains at a given applied longitudinal strain (𝐸11), and 𝛽 is the number of target strains to achieve. To simplify the sensitivity analysis, t he objective function can be reformulated as Φ≔∑ {∑1 𝐸11(𝑢2(𝑘)−𝑢̅2(𝑘) 𝑋2(𝑘)) 𝑘 } 𝛼2 =∑{∑1 (𝐸11𝑊)2(𝑢2−𝑢̅2)𝑘2 𝑘 } 𝛼,𝛽 𝛼=1 𝛽 𝛼=1 (11) where 𝑋1(𝑘) and 𝑋2(𝑘) are the 𝑋1- and 𝑋2-coordinates of the master control point 𝑘, respectively and 𝑊=𝑋2(𝑘) (refer Fig. 2). The actual and targeted displacements of master control point 𝑘 are denoted by 𝑢2(𝑘) and 𝑢̅2(𝑘). Following the Lagrangian multiplier approach, t he shape gradient with respect to the design parameter 𝜏 can be derived using the adjoint method (Wang and Turteltaub 2015 , Wang and Kumar 2017 ). Given the objective function in Eq. (11), in order to satisfy the equality constraints from the equilibrium condition in Eq. (4) for any point 𝑿∈Ω𝜏 at each target strain 𝛼, an augmented functional is introduced as Φ̃=Φ+∑{〈𝒄[𝒖],𝒖∗〉Ω𝜏}𝛼𝛽 𝛼=1 , (12) where 〈𝒄[𝒖],𝒖∗〉Ω𝜏=∫𝒖∗⋅(div 𝑷) dΩ=−∫𝑷:∇𝒖∗ dΩ+ ∫𝒕⋅𝒖∗ dΓ.Γ𝜏 Ω𝜏 Ω𝜏 The Lagrange multiplier 𝒖∗ is the so called adjoint displacement field and 𝒕 is the traction on boundary Γ𝜏. Recalling that 𝒕̂=𝟎 on Γ𝑡𝜏, gives 〈𝒄[𝒖],𝒖∗〉Ω𝜏=−∫𝑷:∇𝒖∗ dΩ+ ∫𝒕⋅𝒖∗ dΓ.Γ𝑢𝜏Ω𝜏 (13) The next step is to derive the total derivative of Eq. (12) with respect to design parameter 𝜏 Φ̃̇=DΦ̃ D𝜏=Φ̇+∑{D Dτ〈𝒄[𝒖],𝒖∗〉Ω𝜏} 𝛼𝛽 𝛼=1 , (14) Quek et al. 6 where (̇)=D() Dτ represents the total design derivative. Given a continuous differentiable function ℎ defined in Ω𝜏 and Γ𝜏, the volume and surface transport relations are giv en, respectively, as D Dτ∫ℎ dΩΩ𝜏= ∫ℎ′ dΩ+ ∫ℎ𝑣𝑛 dΓΓ𝜏 Ω𝜏 , and (15) D Dτ∫ℎ dΓΓ𝜏= ∫(ℎ′+∇ℎ⋅𝒏𝑣𝑛−𝜅ℎ𝑣𝑛) dΓΓ𝜏 , (16) where ()′ denotes the partial design derivative, 𝜅 is the total curvature, 𝒗 is the so -called design velocity at 𝑿 defined as 𝒗= 𝑿̇[𝜏]=D𝑿[𝜏] D𝜏, (17) and 𝑣𝑛=𝒗⋅𝒏, in which 𝒏 is the unit outward normal vector. Using the transport relation in Eq. (16), the first term on the right side of Eq. (14) can be written as Φ̇=∑{∑1 (𝐸11𝑊)2 𝑘(2⋅(𝑢2−𝑢2̅̅̅)⋅𝑢2′)𝑘} 𝛼𝛽 𝛼=1 +∑{∑1 (𝐸11𝑊)2 𝑘(∇((𝑢2−𝑢2̅̅̅)2)⋅𝛽 𝛼=1 𝑛𝑦𝑣𝑛−𝑘((𝑢2−𝑢2̅̅̅)2)𝑣𝑛)𝑘} 𝛼. (18) Similarly, using the transport relations in Eqs. (15) and (16), the derivative of the second term on the right side of Eq. (12) becomes ∑{𝐷 𝐷𝜏〈𝒄[𝒖],𝒖∗〉Ω𝜏} 𝛼𝛽 𝛼=1 =∑{−∫𝑷′:∇𝒖∗ dΩΩ𝜏 −∫(𝑷:∇𝒖∗)𝑣𝑛 dΓ+ ∫𝒕′⋅Γ𝑢𝜏Γ𝜏𝛽 𝛼=1 𝒖∗ dΓ} 𝛼+∑{∫(∇(𝒕⋅𝒖∗)⋅𝒏𝑣𝑛−𝑘(𝒕⋅𝒖∗)𝑣𝑛)Γ𝑢𝜏 dΓ} 𝛼𝛽 𝛼=1 , (19) where the relation 〈𝒄[𝒖],𝒖∗′〉Ω𝜏=0 is utilized to give −∫𝑷:∇𝒖∗′ 𝑑Ω+∫𝒕⋅𝒖∗′ dΓ=0Γ𝑢𝜏Ω𝜏 . (20) From Eqs. (18) and (19), Eq. (14) can be rewritten as Φ̃̇=Ψ1+Ψ2, (21) where Ψ1=∑{−∫𝑷′:∇𝒖∗ dΩ+ ∫𝒕′⋅𝒖∗ dΓ+ ∑1 (𝐸11𝑊)2(2⋅(𝑢2−𝑢2̅̅̅)⋅𝑢2′)𝑘 𝑘 Γ𝑢𝜏Ω𝜏 } 𝛼,𝛽 𝛼=1 and Ψ2=∑{−∫ (𝑷:∇𝒖∗)𝑣𝑛 dΩ+ ∫(∇(𝒕⋅𝒖∗)⋅𝒏𝑣𝑛−𝑘(𝒕⋅𝒖∗)𝑣𝑛) dΓ+Γ𝑢𝜏Γ𝜏𝛽 𝛼=1 ∑1 (𝐸11𝑊)2(∇((𝑢2−𝑢2̅̅̅)2)⋅𝑛𝑦𝑣𝑛−𝑘((𝑢2−𝑢2̅̅̅)2)𝑣𝑛)𝑘𝑘 } 𝛼. Referring to the incremental constitutive relation 𝑃′=ℂ:∇𝒖′, Ψ1 can be written as Ψ1=∑{∫div(ℂ:∇𝒖∗)⋅𝒖′ dΩ−∫(ℂ:∇𝒖∗)𝒏⋅𝒖′ dΓΓ𝜏 Ω𝜏 +∫𝒕′⋅𝒖∗ dΓΓ𝑢𝜏 } 𝛼+𝛽 𝛼=1 ∑{∑1 (𝐸11𝑊)2(2⋅(𝑢2−𝑢2̅̅̅)⋅𝑢2′)𝑘 𝑘 } 𝛼𝛽 𝛼=1 , (22) where ℂ is the material tangent modulus. In order to eliminate the implicitly dependent terms 𝑢2′ and 𝒕′ in Eq. (22), displacement based adjoint BVPs, each corresponding to a target strain 𝛼, are introduced such that { 𝒄∗[𝒖∗]≔div (ℂ:∇𝒖∗)=𝟎 in Ωτ (ℂ:∇𝒖∗)𝒏− 𝒕∗̂=0; 𝒕∗̂=[0,1 (𝐸11𝑊)2⋅2⋅(𝑢2−𝑢2̅̅̅)]𝑇 on Γ𝑡∗𝜏 𝒕∗̂= (ℂ:∇𝒖∗)𝒏=0 on Γ𝑡𝜏 𝒖∗+= 𝒖∗− on Γ𝑢𝜏 (23) in which 𝒕∗̂ is the prescribed adjoint traction along the boundary Γ𝑡∗𝜏 comprising of the 𝑘(= 2,3,5,6) master control points (refer to Fig. 2). The adjoint BVP formulated for a given target strain 𝛼 is solved using the tangent stiffness operator of the primary BVP at equilibrium, as described in (Santos and Choi 1962). Quek et al. 7 Utilizing the adjoint model, Ψ1 reduces to Ψ1=∑{∫𝒕∗⋅𝒖′ dΓ+∫𝒕′⋅𝒖∗ dΓΓ𝑢𝜏Γ𝑢𝜏−Γ𝑡∗𝜏 } 𝛼,𝛽 𝛼=1 (24) Considering that 𝑢′and 𝑢∗are periodic whi le 𝑡′and 𝑡∗are anti -periodic for the primary and adjoint BVPs, gives Ψ1=0. Since the design velocity 𝒗 is non -vanishing on the design boundary Γ𝑑𝜏, the shape gradient in Eq. (1 4) reduces to DΦ̃ D𝜏=∑{−∫(𝑷:∇𝒖∗)𝑣𝑛 dΓ Ω𝜏 } 𝛼𝛽 𝛼=1 . (25) Following Eq. (5) , the design velocity can be discretized as 𝒗=𝑅𝑖𝑗[𝑿]D𝑿𝑖𝑗[𝜏] D𝜏. (26 ) Substituting Eq. (26) into Eq. (25), the discretized isogeometric shape gradie nt with respect to the design control point 𝑿𝑖𝑗 can be obtained as Φ,𝑿𝑖𝑗=𝜕Φ 𝜕𝑿𝑖𝑗=∑{∫𝑅𝑖𝑗(−𝑷:∇𝒖∗)𝒏 dΓ Γ𝑑𝜏 } 𝛼𝛽 𝛼=1 . (27) By s olving for the unknown displacement fields 𝒖 and 𝒖∗ in the primary and adjoint problems, respectively, the discretized shape derivatives in Eq. (27) can be determined. The shape gradient is normalized to ensure the balance movement of the boundary design control points ( Kiendl et al. 2014 ) and is given as Φ̂,𝑿𝑖𝑗=Φ,𝑿𝑖𝑗 ∫𝑅𝑖𝑗dΓ Γ𝑑𝜏 (28) 3.2. Geometr ical Constraints The shape gradient obtained from Eq. ( 28) will result to a non -symmetric petals design and thus exhibits distinct behavior when loaded along two orthogonal directions. Hence, a symmetry constraint is necessary to preserve the geometrical symmetry of petal auxetics during shape optimization. The approach presented in Wang et al. (2017b) is adopted in this study. The above -mentioned symmetr ic constraint , by itself, may result in a design with thin and weak connecting parts , which are difficult to manufacture. Consequently, it is necessary to impose a thickness constraint for the petal s. Following polynomial -based thickness constraint presented in Wang et al. (2017b), an improved spline -spline based thickness constraint is developed in this study. The spline -based thickness constraint is accurate in terms of curve fitting the points gen erated to impose thickness constraint for a smooth petal shape . 4. RESULTS In this section, the above -mentioned optimization formulation is utilized to design petal auxetics with different prescribed Poisson’s ratios in the nonlinear regime. The first part focuses on designing tri -petal auxetics to achieve different values of constant Poisson’s ratio up to 30% strain in plane strain condition. In the second part, hexa-petal auxetics are designed for prescribed constant Poisson’s ratios within 30% strain in plane stress condition. For the optimization, the geometrical specifications of the petal auxetics are as follows: the area of the RVE is chosen as 800 and 400 for tri -and hexa -petal structures. The connecting bars between the petals have fi xed thickness of 1. The min imum thickness of petal arms is set to 0.20 to ensure manufacturability of the optimized designs. 4.1. Smoothed Tri -petals Structures with Constant Poisson’s Ratio in Plane Strain This subsection describes the design of tri-petals structures with different targeted constant Poisson’s ratios up to 30% strain in plane strain condition. The design optimization of the smoothed tri -petals structure begins with an initial design having uniform petal thickness of 1 and a fixed interior boundary as depicted in Fig. 3 (a). Quek et al. 8 Figure 3. Smoothed tri -petals structure: (a) Initial design, (b) Performance of optimized structures with various prescribed constant Poisson’s ratios. Note that only half of tri -petals RVE is shown in (a). Figure 4. Optimized smoothed tri -petals structures with different constant Poisson’s ratios up to 30% strain in plane strain. Figure 5. Resultant displacement (U) contour plot of deformed optimized tri -petals RVE with constant Poisson’s ratio of -0.4 up to 30% in plane strain. The smoothed tri -petals str ucture is optimized for different targeted Poisson’s ratio of 𝜈⊂ {0,−0.1,−0.2,−0.3,−0.4}.The performance of the optimized tri -petals structures is summarized in Fig. 3(b) , where the Poisson’s ratios are close to the target Poisson’s ratios in the desired strain interval. The optimized tr i-petals structures are shown in Fig. 4. The deformed configuration s of the optimized structure with a Poisson’s ratio of -0.4, at 15% and 30% strains are shown in Fig. 5. It can be observed that the auxetic behavio r of smoothed tri -petals is induced by the opening of the petals, together with the movement of connecting bars. Quek et al. 9 4.2. Smoothed Hexa -petals Structures with Constant Poisson’s Ratio in Plane Stress The presentation here focuses a t designing hexa -petals structures with different prescribed constant Poisson’s ratios up to 30% strain in plane str ess condition. The design optimization of the smoothed hexa -petals structure begins with an initial design having uniform petal thickness of 0.8 and a fixed interior boundary as depicted in Fig. 6(a). Figure 6. Smoothed hexa -petals structure: (a) Initial design, (b) Performance of optimized structures with various prescribed constant Poisson’s ratios. Figure 7. Optimized smoothed hexa -petals structures with different constant Poisson’s ratios up to 30% strai n in plane stress. The smoothed hexa -petals structure is optimized for different targeted Poisson’s ratio of 𝜈⊂ {0,−0.1,−0.2,−0.3,−0.4,−0.5}. The performance of the optimized hexa -petals structures is summarized in Fig. 6(b), where the Poisson’s ratios a re close to the target Poisson’s ratios in the desired strain interval. The optimized hexa -petals structures are shown in Fig. 7. The deformed configurations of the optimized structure with a Poisson’s ratio of -0.5, at 15% and 30% strains are shown in Fig . 8. It can be seen from Fig. 8 that the initial auxetic behavior is due to the opening of two petals along the vertical axis of symmetry. Subsequently, the opening of other four petals triggers the auxetic behavior at very large strains. Quek et al. 10 Figure 8. Resultant displacement (U) contour plot of deformed optimized hexa -petals RVE with constant Poisson’s ratio of -0.5 up to 30% in plane stress. 5. CONCLUSIONS In this paper, an isogeometric shape optimization framework to design smoothed petals auxetics with prescribed nonlinear deformation is presented. Geometry constrains are imposed to ensure symmetry and manufacturability of the optimized designs. The exterior boundary of smoothed tri - and hexa -petals structures is optimized to obtain a series of smoothed petals structures with different constant Poisson’s ratios within -0.5. Numerical results demonstrate that the optimized designs can achieve nearly constant Poisson’s ratios up to 30% st rain in both plane strain and plane stress conditions.