Topology optimization of adaptive fluid-actuated cellular structures with arbitrary polygonal motor cells

In the microscale computation step of the PMsFEM, the multiscale base functions for both the solid matrix and fluid phase in the closed inclusions are defined and constructed on sub-grid meshes of the polygonal motor cells. For a polygonal fluidic motor cell with N_S sides, the multiscale base functions can be expressed as

$$\begin{eqnarray}&&\left\{\begin{array}{c}u=\displaystyle \sum _{i=1}^{{N}_{S}}({N}_{ixx}{\tilde{U}}_{i}+{N}_{iyx}{\tilde{V}}_{i})\\ v=\displaystyle \sum _{i=1}^{{N}_{S}}({N}_{ixy}{\tilde{U}}_{i}+{N}_{iyy}{\tilde{V}}_{i})\\ {p}_{{\rm{fluid}}}=\displaystyle \sum _{i=1}^{{N}_{S}}({N}_{pix}{\tilde{U}}_{i}+{N}_{piy}{\tilde{V}}_{i})\end{array}\right.\end{eqnarray} \tag{ 1 }$$

where ${N}_{ixx},$ ${N}_{iyx},$ ${N}_{ixy}$ and ${N}_{iyy}$ denote the multiscale displacement base functions of the solid matrix; ${N}_{pix}$ and ${N}_{piy}$ represent multiscale pressure base functions of the fluid phase; $u$ and $v$ denote the displacements of sub-grid nodes in different directions; ${p}_{{\rm{fluid}}}$ denotes the fluid pressure inside the inclusion, and ${\tilde{U}}_{i}$ and ${\tilde{V}}_{i}$ denote the displacements of the node i at the polygonal coarse-grid element.

The multiscale base functions defined in equation (1) are numerical values which are constructed on the sub-grid meshes of the unit cells through a FE analysis. The construction method of the multiscale base functions for the polygonal coarse-grid element can be found in detail in the works by Lv et al [15, 16]. In this construction, linear boundary conditions are applied to the sub-grid meshes of the unit cells, and then boundary value problems are performed on the sub-grids via a standard FE analysis. The calculated multiscale base functions are some numerical values which contain small-scale information about the heterogeneous and materials of the polygonal motor cells.

Once the microscale computations are finished for each motor cell, the element stiffness of the polygonal coarse-grid elements can be calculated with the multiscale base functions of the displacement field. Take the polygonal motor cells with N_s sides as an example—the motor cells are meshed with m sub-grid elements. Therefore, the equivalent stiffness matrix ${{\boldsymbol{K}}}_{{\rm{E}}}$ of the polygonal coarse-grid element E can be calculated by

$$\begin{eqnarray}&&{{\boldsymbol{K}}}_{{\rm{E}}}=\displaystyle \sum _{e=1}^{m}{{\boldsymbol{G}}}_{e}^{{\rm{T}}}{{\boldsymbol{K}}}_{{e}}{{\boldsymbol{G}}}_{e}\end{eqnarray} \tag{ 2 }$$

where, ${{\boldsymbol{K}}}_{e}$ denotes the element stiffness matrix of the sub-grid element e; ${{\boldsymbol{G}}}_{e}$ is the transition matrix and it can be written as:

$$\begin{eqnarray}{{\boldsymbol{G}}}_{e}={\left[\begin{array}{cccc}{{\boldsymbol{R}}}_{1}^{{\rm{T}}} & {{\boldsymbol{R}}}_{2}^{{\rm{T}}} & \ldots & {{\boldsymbol{R}}}_{{\rm{n}}}^{{\rm{T}}}\end{array}\right]}^{{\rm{T}}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}{{\boldsymbol{R}}}_{s}=\left[\begin{array}{cccc}{{\boldsymbol{Q}}}_{1}(s) & {{\boldsymbol{Q}}}_{2}(s) & \mathrm{...} & {{\boldsymbol{Q}}}_{{N}_{s}}(s)\end{array}\right],{{\boldsymbol{Q}}}_{k}(s)\\ &&\quad \;\;=[{N}_{ijk}(s)]\\ &&(i,j=x,y;k=\mathrm{1,\; 2,...},{N}_{s};s=\mathrm{1,\; 2},\ldots ,n)\end{eqnarray} \tag{ 4 }$$

where n is the number of the nodes on the fine-grid e within the unit cell. Once the element stiffness matrices of all the polygonal coarse-grid elements are computed, the mechanical response of the fluidic cellular structure can be easily simulated on the polygonal coarse-grid elements.

The motions of the fluidic cellular structures are caused by the expansion and shrinkage deformation of each motor cell. As illustrated in figure 3, the cell deformation is caused by the volume change of water in the fluid inclusion, which can be regarded as the driving forces. As indicated by various researchers [3, 7, 14], in higher plants the fluid lost or diffused into the cells can be controlled by the biological transport systems in a membrane of the plant cells. The controlled mass transport of charge and fluid across a selectively permeable membrane employing biological processes can achieve bulk deformation of the cells. With this inspiration, some bioinspired driving units, such as the electroosmotic transporting mechanism [22, 23], have been developed in real applications. Recently, some special pressurization devices, such as inflatable rubber tubes [9] and a fluid transfer system with sealing caps [13] have been developed and tested in experiments. Although pressurization systems with a different efficiency can be used to generate the pressure, the motor cells will work in a similar way, i.e. expansion or shrinkage under the pressure generated.

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In this work, in order to model the pressured motor cells, it is assumed that the pressure is generated by the enclosed pressured fluid in the inclusion; further that each inclusion in the motor cells is fully filled with incompressible fluid; moreover, that the inclusion will have no cavitation when the cells are under a tension state. Thus, while the fluid with a volume of ${\rm{\Delta }}{V}_{{\rm{fluid}}}$ is pumped into the inclusion, the total volume of the fluid inclusion is satisfied as follows:

$$\begin{eqnarray}&&V-{V}_{{\rm{0}}}={\rm{\Delta }}{V}_{{\rm{fluid}}}\end{eqnarray} \tag{ 5 }$$

where V and V₀ denote the current and initial volume of the fluid inclusion. With the incompressibility condition, the actuation behaviors of the motor cells can be directly simulated using the standard FEM on the micromechanical models.

An equivalent method needs to be proposed to convert the driving force of the fluidic motor cells into the nodal forces acting on the polygonal coarse-grid elements. It can be found in figure 3 that the driving forces act within the fluid inclusions at the small-scale level, which cannot be directly utilized in the PMsFEM. Zhang et al [24] proposed an equivalent method to transform the volume changes of water inside the motor cells to the equivalent nodal forces on structural coarse-grid elements. In this article, the equivalent method is extended to calculate the equivalent nodal forces of the fluid on the irregular polygonal coarse-grid elements. As indicated in figure 4, the main idea of the equivalent method is that the response of the motor cells under the volume changes of the fluid inside (figure 4(a)) is equal to the response of the polygonal coarse-grid element under equivalent nodal forces $F$ (figure 4(b)). Moreover, the local response of the volume expansion within the unit cell, which is important for the downscaling computation, should not be neglected. More details about the computation can be found in the work by Zhang et al [24].

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Once the equivalent nodal forces are calculated, the actuation capability of the fluidic motor cells can be easily evaluated. It can be observed from the equivalent method introduced above, that the actuation capability of the fluidic motor cells depends on various factors, such as the shape and size of the fluid inclusions, as well as the geometries and physical parameters of the solid matrix in the motor cells. Here we focus on the effect on the physical property parameters, especially the elastic modulus of the solid matrix on the actuation capability of the motor cells. For simplicity, here it is assumed that the solid matrix of the motor cells is homogeneous. Take the motor cell shown in figure 4(a) as an example—the elastic modulus of its solid matrix varies from $40\;{\rm{MPa}}$ to $200\;{\rm{MPa}}.$ The nodal forces acting on the polygonal coarse-grid element (figure 4(b)) calculated by the equivalent method are plotted in figure 5. It can be observed that the equivalent nodal forces of the motor cells vary linearly with respect to the elastic modulus of the solid matrix. The nodal forces can be expressed as

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{{\rm{\Delta }}V}(x)=f(x){{\boldsymbol{F}}}_{{\rm{\Delta }}V=1}{\rm{\Delta }}V\end{eqnarray} \tag{ 6 }$$

where ${{\boldsymbol{F}}}_{{\rm{\Delta }}V=1}$ denotes the equivalent nodal force of which a unit volume of incompressible fluid is pumped into the cell, and $f(x)$ is a linear function about the elastic modulus of the solid matrix in the motor cells.

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The polygon-based topology optimization method firstly focuses on the optimization problems about minimizing the system compliance of the fluidic cellular structures. In this design problem, the fluidic cellular structures are served as load-bearing structures. It is assumed that the motor cells in the structures do not export driving forces. The optimization problem here is to optimize the arrangement of the polygonal motor cells to minimize the system compliance of structures under external loading and boundary conditions. With the upscaling procedure of the PMsFEM, the optimization problem can be converted into finding the macroscopic stiffest configuration of the polygonal coarse-grid elements. Thus, the optimization formulation defined on the polygonal coarse-grid elements can be expressed as follows:

$$\begin{eqnarray}\underset{X}{{\rm{Minimize}}} & : & C({\boldsymbol{X}})={{\boldsymbol{U}}}^{{\rm{T}}}{\boldsymbol{KU}}=\displaystyle \sum _{E\in N}{{\boldsymbol{U}}}_{E}^{{\rm{{\rm T}}}}{{\boldsymbol{K}}}_{E}^{P}{{\boldsymbol{U}}}_{E}\\ {\rm{Subject}}\;{\rm{to}} & : & \left\{\begin{array}{c}{\boldsymbol{KU}}={{\boldsymbol{F}}}_{{\rm{ext}}}\\ \displaystyle \sum _{E\in {\rm{N}}}{X}_{E}{\upsilon }_{E}\leqslant {V}_{0}f\\ 0\lt {X}_{{\rm{min}}}\leqslant {X}_{E}\leqslant 1,\;E=\mathrm{1,}\;\mathrm{2,}\mathrm{...}N\end{array}\right.\end{eqnarray} \tag{ 8 }$$

where C is the structural compliance of the fluidic cellular structure, which can be directly calculated on the coarse-grid elements at the macroscopic level; ${\boldsymbol{X}}$ denotes the design variable; U denotes the displacements matrix of the polygonal coarse-grid elements; K is the global stiffness matrix of the structures; ${{\boldsymbol{F}}}_{{\rm{ext}}}$ is the external load applied on the polygonal coarse-grid mesh; the constant ${{\rm{V}}}_{0}$ is the volume of the design domain; ${\upsilon }_{E}$ is the volume of the coarse-grid element E; f is the prescribed volume fraction and N is the number of the polygonal coarse-grid elements of the cellular structures.

To effectively solve the optimization formulation defined in equation (8), the sensitivity of the object functions with respect to the pseudo densities of polygonal coarse-grid elements should be firstly derived. Similar to those standard topology optimization problems defined on the structural elements, the sensitivity of the objection function $C$ to the design variables of the irregular polygonal coarse-grid element can be easily expressed as:

$$\begin{eqnarray}&&\displaystyle \frac{\partial C}{\partial {X}_{E}}=-p{X}_{E}^{p-1}({X}_{0}-{X}_{\mathrm{min}}){{\boldsymbol{U}}}_{E}^{{\rm{T}}}{{\boldsymbol{K}}}_{E}^{0}{{\boldsymbol{U}}}_{E}\end{eqnarray} \tag{ 9 }$$

where ${X}_{E}$ is the pseudo density of the polygonal coarse-grid element E.

As the fluidic cellular structures can be self-actuated with the motor cells imbedded, it is important to optimize the bioinspired actuator to export displacement in a predefined direction. In this section, the polygon-based topology optimization method is proposed to design the fluid-actuated compliant mechanism. Differing from the system compliance optimization problem, there are no external loads acting on the fluidic cellular structures in this problem. In contrast, the motions of the fluidic cellular structure are induced by internal pressure/volume changes of the fluid inside the motor cells. As introduced in section 2.3, these driving forces can be easily equivalent to the nodal forces at the nodes of the polygonal coarse-grid elements. Therefore, according to standard topology optimization of the compliant mechanism problems, the design of a fluid-actuated compliant mechanism can be naturally performed on polygonal coarse-grid meshes of the structures.

The design problems of the fluid-actuated compliant mechanism can be interpreted as maximizing the displacement in a predefined direction ${U}_{{\rm{out}}},$ in response to the equivalent nodal forces of each polygonal coarse-grid element. The optimization formulation of the fluid-actuated complaint mechanism based on the polygonal coarse-grid meshes can be written as follows:

$$\begin{eqnarray}{\rm{Minimize}} & : & -{U}_{{\rm{out}}}\\ {\rm{Subject}}\;{\rm{to}} & : & \left\{\begin{array}{c}{\boldsymbol{K}}({\boldsymbol{X}}){\boldsymbol{U}}={\boldsymbol{F}}({\boldsymbol{X}},{\rm{\Delta }}{\boldsymbol{V}})\\ \displaystyle \sum _{e\in N}{X}_{E}{\upsilon }_{E}\leqslant {{\rm{V}}}_{0}f\\ 0\lt {X}_{{\rm{min}}}\leqslant {X}_{E}\leqslant 1,E=\mathrm{1,\; 2,...}{\rm{N}}\end{array}\right.\end{eqnarray} \tag{ 10 }$$

where ${\boldsymbol{F}}({\boldsymbol{X}},{\rm{\Delta }}{\boldsymbol{V}})$ is the macroscopic equivalent forces applied on the polygonal coarse-grid element, which is dependent on the pseudo densities of the motor cells. A spring of stiffness ${K}_{{\rm{out}}}$ is applied on the reference point, so the optimal compliant mechanism solved will be dependent on the stiffness of the spring. The irregular polygonal motor cells must be distributed in a structurally efficient way to maximize the work on the output displacement.

The sensitivity of the output displacement with respect to the pseudo densities of the polygonal coarse-grid elements can be derived as:

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}{U}_{{\rm{out}}}}{{\rm{d}}{X}_{E}}={{\boldsymbol{\lambda }}}^{{\rm{T}}}p{{X}_{E}}^{p-1}({X}_{0}-{X}_{{\rm{min}}})({{\boldsymbol{K}}}_{E}^{{\rm{0}}}{\boldsymbol{U}}-{{\boldsymbol{F}}}_{{\rm{\Delta }}V}^{0})\end{eqnarray} \tag{ 11 }$$

where λ is the adjoint vector. More details of the derivation of the sensitivity analysis can be found in appendix A below. 

For the last problems, the polygon-based topology optimization method is used for the lightweight design of fluidic cellular structures. The aim of the method here is to minimize the number of motor cells distributed in the structures under the constraints for the compliance of the whole system. Based on the PMsFEM, the lightweight design problem can be transformed into the minimum volume problem based on the polygonal coarse-grid elements at the macroscale level. The considered settings may be framed within the following discrete formulation

$$\begin{eqnarray}{\rm{Minimise}} & : & W=\displaystyle \sum _{E\in N}{X}_{E}{\upsilon }_{E}\\ {\rm{Subject}}\;{\rm{to}} & : & \left\{\begin{array}{c}{\boldsymbol{K}}({\boldsymbol{X}}){\boldsymbol{U}}={\boldsymbol{F}}({\boldsymbol{X}},{\rm{\Delta }}{\boldsymbol{V}})\\ {U}_{out}\leqslant {U}_{{\rm{const}}}\\ 0\lt {X}_{{\rm{min}}}\leqslant {X}_{E}\leqslant 1,E=\mathrm{1,\; 2,...}{\rm{N}}\mathrm{.}\end{array}\right.\end{eqnarray} \tag{ 12 }$$

In the above equation, the structural weight $W$ is computed by multiplying the element density ${X}_{E}$ for the relevant volume ${\upsilon }_{E}$ over all the N polygonal coarse-grid elements. ${\boldsymbol{F}}({\boldsymbol{X}},{\rm{\Delta }}{\boldsymbol{V}})$ denotes the force vector about the polygonal coarse-grid elements, which are composed of the equivalent loads of the fluid actuation and the direct external forces. ${U}_{{\rm{out}}}$ is the output displacement at a prescribed direction where a spring with stiffness ${K}_{{\rm{out}}}$ is applied; ${U}_{{\rm{const}}}$ is the upper limit of the output displacement. The spring here is used to maintain the structural compliance. The sensitivities for the object function are the same as those for the constraints used in the compliant mechanism problem, while the sensitivities for the constraints are equivalent to those for the object function.

The three topology optimization problems defined above can be effectively carried out based on the sensitivity analyses (see equations (9) and (11)) of many theoretically well-founded optimization algorithms, such as the optimality criteria (OC), aequential linear programming (SLP), sequential quadratic programming (SQP), and the method of moving asymptotes (MMA) [26]. In this research, due to its simple form and high efficiency, the OC is selected to solve the design problems about the system compliance minimization. Furthermore, the compliant mechanism design of the fluidic cellular structures described above has the same form as the system compliance minimization. So the OC is also applied to solve compliant mechanism design problems. However, since the sensitivity of the objective function calculated by equation (11) may take both positive and negative signs, a heuristic modification in the update process should be performed in the OC method for the compliant mechanism design problems [25]. On the other hand, the OC method seems to have some difficulties in solving the lightweight design problems of the fluidic cellular structure proposed here. Instead, the MMA which solves a sequence of linearized sub-problems and uses information from prior iterations to improve the approximation of the design space, is used for the lightweight design problems of fluidic cellular structures. The corresponding computation procedures for all the three optimization problems can be described in a unified way as illustrated in figure 6.

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As demonstrated in existing works about topology optimization methods based on polygonal elements [19, 20], the polygonal coarse-grid elements with different edges can naturally avoid checkerboard problems. Furthermore, in order to ensure the existence of solutions to the topology optimization problem and avoid mesh-dependency problems, a filtering matrix for the standard polygonal grid elements is introduced for polygonal coarse-grid elements in fluidic cellular structures. For better readability of this article, the filtering matrix of the linear kernel proposed by Talischi et al [19] is briefly introduced here. The filtered field can be written as:

$$\begin{eqnarray}&&{y}_{l}=\displaystyle \frac{\displaystyle {\sum }_{k\in S(l)}{z}_{k}|{{\rm{\Omega }}}_{k}|(1-|{{\boldsymbol{x}}}_{l}^{*}-{{\boldsymbol{x}}}_{k}^{*}|)/R}{\displaystyle {\sum }_{k\in S(l)}|{{\rm{\Omega }}}_{k}|(1-|{{\boldsymbol{x}}}_{l}^{*}-{{\boldsymbol{x}}}_{k}^{*}|)/R}\end{eqnarray} \tag{ 13 }$$

where ${z}_{k}$ is the constant value of the design variable over element ${{\rm{\Omega }}}_{k};$ R is the radius of the linear filter; $S(l)$ denotes the set of indices of the element ${{\rm{\Omega }}}_{k}$ of which the centroid falls within radius $R$ of the centroid of element ${{\rm{\Omega }}}_{l};$ $|{{\rm{\Omega }}}_{k}|$ denotes the area of the polygonal element k; ${{\boldsymbol{x}}}_{l}^{*}$ and ${{\boldsymbol{x}}}_{k}^{*}$ are the coordinates of the centroids of elements l and k, respectively. The expression can be written in the vector representation as [19]:

$$\begin{eqnarray}&&{({\boldsymbol{P}})}_{lk}=\displaystyle \frac{{\rm{max}}(0,|{{\rm{\Omega }}}_{k}|(1-|{{\boldsymbol{x}}}_{l}^{*}-{{\boldsymbol{x}}}_{k}^{*}|)/R)}{\displaystyle {\sum }_{k\in S(l)}|{{\rm{\Omega }}}_{k}|(1-|{{\boldsymbol{x}}}_{l}^{*}-{{\boldsymbol{x}}}_{k}^{*}|)/R}\end{eqnarray} \tag{ 14 }$$

where ${({\boldsymbol{P}})}_{lk}$ is the filter matrix and can be stored as a sparse matrix, which is more appropriate for implementation in the computations.

In the first example, three square fluidic cellular structures which are actuated with the imbedded motor cells are considered to validate the proposed optimization method. As illustrated in figure 7, the square structures with a side length of 300 mm consist of 924 triangular, 900 quadrilateral and 900 polygonal motor cells, respectively. Each motor cell contains a circular fluid inclusion with a radius (r = 2.26 mm), and the volume expansion of the fluid in each inclusion is set to ${\rm{\Delta }}V=10\;{{\rm{mm}}}^{3}.$ For all the structures, three sides are constrained in all directions, and a spring with a stiffness of ${K}_{{\rm{out}}}=5\times {{\rm{10}}}^{{\rm{3}}}\;{\rm{N}}\;{{\rm{m}}}^{-1}$ is attached at the center of the right side of the structure. The Young's modulus and Poisson's ratio of the solid matrix of the motor cells are set to 100 MPa and $v=0.3,$ respectively.

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In order to perform the polygon-based optimization method on these structures, the fluidic cellular structures are meshed with different coarse-grid elements according to the size and shape of the motor cell distributed inside. As illustrated in figure 7, the cellular structures composed of triangular and quadrilateral motor cells are directly meshed with standard triangle and quadrilateral coarse-grid elements, respectively. At the same time, the cellular structures with irregular polygonal motor cells are meshed with 900 unstructured polygonal coarse-grid elements. Each coarse-grid element contains a single fluid inclusion. It should be noted that although different types of coarse-grid elements are utilized in the three structures, the codes developed in this article based on the theories proposed can solve these problems in a consistent format.

The final results about the three structures shown in figure 8 were obtained using a linear filter of radius (R = 14 mm) and the SIMP model with continuation performed on the penalty parameter p [19] is used in this example. In order to get the ideal clear topology of the optimal distributions of the motor cells in the structures, the value of p was increased from 4 to 6 using increments of 0.5 in size and a maximum of 150 iterations was allowed for each value of p. It can be observed from figure 8 that the optimization method applied on the three structures with different mesh grids can achieve compliant mechanisms with similar geometries. Among them, the optimization result calculated on the cellular structure meshed with triangular coarse-grid elements (see figure 8(a)) indicates that checkboard problems occur in some local parts of the resulting compliant mechanism. Furthermore, the optimization method can achieve the clearest results on the polygonal coarse-grids (see figure 8(c)), while the compliant mechanism obtained has a smoother boundary than those on structures with standard quadrilateral motor cells (see figure 8(b)).

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As introduced above, an important application of the bioinspired fluidic cellular structure is that it can be used to design smart wings for morphing aircraft structures. In this section, the polygon-based optimization method is utilized to design the smart wings structure of fluidic cellular materials. As illustrated in figure 9(a), the Quabeck HQ 3.0/14 R/C sailplane airfoil is considered. The central part of the wing structure is set to the design domain, which is composed of 1800 irregular polygonal motor cells, as shown in figure 9(b). Each motor cell contains a closed inclusion fully filled with incompressible fluid. The fluid inclusions are distributed in the centers of the motor cells. The material properties of the solid matrix in the smart wing are the same as those used in the first example.

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In this example, the smart wing structure serves as a load-bearing structure. Further, there is no volume change of fluid inside the motor cells. The purpose here is to illustrate how the distribution of the motor cells affects the mechanical behaviors of smart wing structures. The design domain of the fluidic cellular part in the structure is meshed with 1800 polygonal motor cells according to the shapes of the motor cells. As illustrated in figure 9(b), the left side of the design domain is constrained in both of the two directions, while a uniform distribution load ($F=-2.5\times {{\rm{10}}}^{{\rm{4}}}\;{\rm{N}}$ in total) is subjected at the tail of the wing structure.

The optimization object here is to minimize the system compliance of the whole structure with a given number of motor cells. The volume function f of the motor cells for this problem is set to 0.4. The linear filter of radius (R = 8 mm) is used in the optimization codes. The penalty parameter p used in the SIMP-like model is increased from 4 to 6 using increments 0.5 in size. The maximum iterations for each value of p are set to 150. The results of the cellular structure with three different sizes of fluid inclusion (r = 0.97 mm, 1.62 mm and 2.11 mm) are illustrated in figure 10. It can be observed that the polygon-based optimization method can successfully solve such a compliance minimization problem. Additionally, clear optimization topologies of the geometries can be found in all smart wing structures composed of motor cells with different shapes. Moreover, it can be found that the sizes of the fluid inclusion in the motor cells can affect the final compliance of the cellular structure, but have little influence on resulting distributions of motor cells in the fluidic cellular structures. The main reason for this is that the stiffness of polygonal motor cells will be weakened simultaneously in all directions by increasing the radius of the fluid inclusion.

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The fluid-actuated cellular structures exhibit multifunctional features, such as self-actuation, shape variability and being lightweight, which makes them useful in the design of morphing flexible aircraft wings. In the last example, we focus on the lightweight design of the smart wing structure to verify the versatility of the polygon-based optimization method proposed in this article. The wing structure of the Quabeck HQ 3.0/14 R/C sailplane airfoil is still considered in this example. The design domain is composed of 1800 polygonal motor cells which separately contain a fluid inclusion. At the same time, different volume increments of the fluid are pumped into inclusions, and their values varied linearly with respect to the vertical positions of the inclusions in the cellular structure. As indicated in figure 11, the left side of the design domain is constrained in both directions. An external uniform force ($F=75\times {{\rm{10}}}^{{\rm{3}}}\;{\rm{N}}$ in total) is subjected to the top side of the structure, while a spring with a stiffness of ${K}_{{\rm{out}}}=5\times {{\rm{10}}}^{{\rm{3}}}\;{\rm{N}}\;{{\rm{m}}}^{-1}$ is attached at the bottom of the tails. The material properties of the solid matrix are the same as those used in the last example detailed above.

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The polygon-based topology optimization method is utilized to minimize the number of motor cells distributed in the smart wing. As introduced above, the lightweight design problem can be converted into a volume minimization problem based on the polygonal coarse-grid mesh, since the PMsFEM is used to bring the small-scale information of the motor cells to the macroscale coarse-grid elements. Here the system compliance constraint is induced to the problems by the upper limit of the output displacement in a prescribed direction ${U}_{{\rm{const}}}=30\;{\rm{mm}}$ in equation (12). The linear filter of radius in the optimization codes is set to R = 10 mm. The penalty parameter p used in the SIMP-like and the maximum number of iterations are the same as those in the proposed two examples. The final results of the optimization with a different shape of fluid inclusions in the motor cells are given in figure 12. It should be noted that the area of these fluid inclusions is the same in all the motor cells. The results indicated that the polygon-based optimization method can successfully perform the lightweight designs of the fluid-actuated cellular structure. At the same time, the influence of the shape of the motor cell on the final topology of the structure can be also easily captured. It can be found that the weight of the fluidic cellular structure which contains vertical elliptical fluid inclusions (figure 12(d)) can be heavily reduced, while the fluidic cellular structure containing transverse elliptical fluid inclusions (figure 12(b)) cannot lose any weight under these constraints. The main reason for this is that fluidic cells with vertical elliptical inclusions (figure 12(d)) can provide larger driving forces in the x direction than those with transverse elliptical inclusions (figure 12(b)). Therefore, the fluidic cellular structure which contains vertical elliptical fluid inclusions (figure 12(b)) can provide the maximum bending deformation, while the minimum bending deformation is generated by the fluidic cellular structure containing transverse elliptical fluid inclusions (figure 12(d)). These conclusions are consistent with those provided in the works by Lv et al [16].

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