On the topology optimization of variable-thickness fiber-reinforced plates

Abstract

Two-field topology optimization algorithm is suggested for the optimal design of rib-stiffened fiber-reinforced plates. Additional nodal variables related to the local thickness of plate and local angle of fiber orientation are incorporated in a standard topology optimization approach. Minimal compliance design problems are solved for several examples with rectangular Reissner–Mindlin plates subjected to the transverse concentrated and distributed loads. Possibility for simultaneous assessments on the optimal geometry of curvilinear ribs and fibers trajectories is shown. Efficiency of algorithm is evaluated and compared to the conventional variable-thickness topology optimization results for isotropic plates.

Keywords

Topology optimization variable-thickness plates transverse loading fiber-reinforced composites curvilinear reinforcement minimal compliance design

1. Introduction

Modern automated layup and three-dimensional (3D)-printing technologies provide possibilities for the manufacturing of the fiber-reinforced structures with complex geometry and curvilinear trajectories of fibers [1 –3]. Corresponding numerical optimization methods and design approaches have been developed and widely studied during last decades [4, 5]. The geometry optimization of composite structures maintains on the general methods of structural mechanics and takes into account the complex anisotropic and layered nature of fiber-reinforced composites [6 –8]. Optimization for the curvilinear trajectories of fibers allows one to reduce the stress concentration and improve the load-bearing capacity and durability of composite structures [4, 5]. Conventionally, the optimization of the geometry of the composite structures and the choice of their reinforcement schemes are performed sequentially (iterative) [9, 10] or within the shape optimization approaches [5, 11].

In the present study, we evaluate the possibility for the simultaneous optimization of the geometry and fibers trajectories within the topology optimization (TO) algorithm. Namely, we consider the minimal compliance design problems for the rectangular rib-stiffened fiber-reinforced plates. We generalize the conventional variable-thickness approach [12, 13] for the anisotropic structures and introduce two additional field variables. The first additional variable (“density”) is related to the local thickness of the plate, while the second one (“orientation”) is responsible for the local direction of fibers. From the mechanical viewpoint, we consider the Reissner–Mindlin plates with continuous variation of thickness and orientation of principal axes of the orthotropy.

Previously, application of TO methods for the variable-thickness plates has been used to evaluate the optimal shape of the ribs in isotropic thin-walled structures [14 –16]. Such methods allow one to reduce the dimension of the problem (in comparison to the 3D TO) and directly evaluate the trajectories of curvilinear stiffeners [16]. The drawback of this method for the plates under transverse loads is the mesh-dependent solutions that cannot be avoided without additional relaxation or regularization concepts [14]. Relaxation concepts have been suggested involving the generalized plates models and Young-measure relaxation theory [12, 14, 17]. Additional regularization constraints pertaining to the thickness gradient have been used in the research study [16, 18]. A regularization approach for the 3D shape optimization process in thinning domains has been established in Bouchitté et al. [19]. Furthermore, a proposed additional condition for enforcing a minimum distance between ribs is detailed in Lam and Santhikumar [20].

Validation of the considered approach for the isotropic rib-stiffened plates was presented in our recent works considering convergence theoretical analysis and experimental results [16] and by using comparison with available optimal analytical solutions for benchmark problem [21]. In particular, it has been shown that solutions for the displacements in Reissner–Mindlin plates with even high thickness gradients are accurate enough for solving the compliance optimization problems [16].

Generally, the results of optimal design for the variable-thickness plates can be accepted without additional regularization methods if one defines the smallest mesh size related to the thinnest ribs that can be manufactured by using considered technological methods. By fixing the minimum mesh size, one can obtain a design variant that can be sufficiently optimal for the considered task (although, it is not a global optimal solution). Examples of solving such optimization problems have been presented [10, 15]. Furthermore, often, when reducing the mesh size, the main stiffeners of the plate do not significantly change their shape, but there only arise more detailed fine and thin ribs [16]. Therefore, it is precisely this approach that we consider in the present study with anisotropic plates. We investigate how the mesh size affects the optimization results, but do not introduce additional regularization conditions. This is done in order not to complicate the optimization algorithm, which is already overloaded by the presence of two design variables (“density” and “orientation”).

2. Formulation of the model

Considered optimization problem for the variable-thickness fiber-reinforced plate can be stated as follows:

\begin{matrix} minimize : J (ρ, α) = \int_{Ω} f (x) w (x) d Ω \\ subject to : equilibrium equations, \\ boundary conditions, \\ h = h_{skin} + ρ h_{rib}, \\ ϕ = ϕ_{0} + α π, \end{matrix}

(1)

\begin{matrix} \bar{ρ} = \frac{\int_{Ω} hd Ω}{\int_{Ω} d Ω} = {\bar{ρ}}_{\max}, \\ 0 \leq ρ \leq 1, - 1 \leq α \leq 1, \end{matrix}

where $x \in Ω$ is a position vector; $Ω \in R^{2}$ is a bounded domain related to the reference plane of the plate that has boundary $\partial Ω$ ; $J (ρ, α)$ is the compliance functional that is the objective for the optimization problem under consideration; $f (x)$ is the prescribed transverse load; $w (x)$ is the deflections function that obeys equilibrium equations and boundary conditions of Reissner–Mindlin plate theory; $ρ (x) \in [0, 1]$ is the first additional field variable of optimization problem (“density”), which is introduced to define the local values of the plate thickness $h (x)$ in the prescribed range between minimum $h_{skin}$ and maximum ( $h_{skin} + h_{rib}$ ) values; linear dependence of thickness on “density” is used according to the preliminary performed convergence analysis with isotropic plates [16]; $\bar{ρ}$ is an averaged density (or averaged thickness) that is constraint to be equal to some prescribed value ${\bar{ρ}}_{\max}$ ; $α (x) \in [- 1, 1]$ is the second additional field variable of optimization problem (“orientation”) that defines the in-plane direction of continuous reinforcing fibers; angle of orientation has initial value $ϕ_{0}$ and it varies in a range $ϕ_{0} \pm π$ so that both rotation directions (clockwise and counterclockwise) are allowable for the optimization algorithm.

Given statement equation (1) implies the finding of the optimal functions $ρ (x)$ and $α (x)$ so that they minimize the compliance functional $J (ρ, α)$ under prescribed constraints for the total volume of plate ( ${\bar{ρ}}_{\max}$ defines the remaining fraction of solid material), for the minimum thickness of its skin ( $h_{skin}$ ) and maximum height of its ribs ( $h_{rib}$ ). The compliance of plate $J (ρ, α)$ is defined as a work done by the external forces $f (x)$ . Found solution should provide the highest rigidity of plate for given loading and supporting conditions.

Equilibrium equations to be satisfied by the solution of the problem have standard form of Reissner–Mindlin plate theory:

N_{α β, β} = 0, M_{α β, β} - V_{α} = 0, V_{α, α} + f = 0,

(2)

where Greek indices $α$ and $β$ take values 1 and 2, derivatives are denoted by commas, standard summation notation of repeated indices is employed and the definitions of membrane forces $N_{α β}$ , bending moments $M_{α β}$ , and shear forces $V_{α}$ are given taking into account a non-symmetric structure (with one-side ribs) and anisotropic material of plate [6]:

\begin{matrix} \begin{matrix} {\begin{matrix} N_{11} \\ N_{22} \\ N_{12} \\ M_{11} \\ M_{22} \\ M_{12} \end{matrix}} = (\begin{matrix} A_{11} & A_{12} & A_{16} & B_{11} & B_{12} & B_{16} \\ A_{12} & A_{22} & A_{26} & B_{12} & B_{22} & B_{26} \\ A_{61} & A_{62} & A_{66} & B_{61} & B_{62} & B_{66} \\ B_{11} & B_{12} & B_{16} & D_{11} & D_{12} & D_{16} \\ B_{21} & B_{22} & B_{26} & D_{21} & D_{22} & D_{26} \\ B_{61} & B_{62} & B_{66} & D_{61} & D_{62} & D_{66} \end{matrix}) {\begin{matrix} ε_{11} \\ ε_{22} \\ 2 ε_{12} \\ κ_{11} \\ κ_{22} \\ 2 κ_{12} \end{matrix}} \end{matrix} \\ {\begin{matrix} V_{1} \\ V_{2} \end{matrix}} = k (\begin{matrix} S_{44} & S_{45} \\ S_{54} & S_{55} \end{matrix}) {\begin{matrix} 2 ε_{13} \\ 2 ε_{23} \end{matrix}}, \end{matrix}

(3)

where we use the Voigt notation; $k = 5 / 6$ is the shear correction factor; $ε_{α β} = (u_{α, β} + u_{β, α}) / 2$ and $κ_{α β} = (θ_{α, β} + θ_{β, α}) / 2$ are the components of strain and curvature tensors, respectively; $A_{ij} (x)$ , $B_{ij} (x)$ , $D_{ij} (x)$ , $S_{ij} (x)$ are the membrane, coupling, bending and shear stiffness coefficients of plate, respectively, that are defined taking into account the variable thickness of plate and anisotropy of fiber-reinforced material:

\begin{matrix} A_{ij} (x) = C_{ij}^{*} (x) h (x), B_{ij} (x) = C_{ij}^{*} (x) \frac{h^{2} (x)}{2}, (i, j = 1, 2, 6), \\ D_{ij} (x) = C_{ij}^{*} (x) \frac{h^{3} (x)}{3}, (i, j = 1, 2, 6), \\ S_{ij} (x) = C_{ij}^{*} (x) h (x), (i, j = 4, 5), \end{matrix}

(4)

where the Latin indices take the values that are explicitly listed in the formulas, and it is assumed that the reference plane is located at the outer surface of a plate skin, and $C_{ij}^{*} (x)$ are the components of the anisotropic material stiffness matrix. This matrix can be found by using standard relations for unidirectional composites taking into account the local properties of a ply $C_{ij}$ and the fiber orientation angle $ϕ = ϕ (x)$ . Using matrix notation, these relations for the in-plane properties can be given in the following compact form [22]:

\begin{matrix} C^{*} (ϕ) = T^{- 1} Q T^{- T}, \\ T = [\begin{matrix} \cos^{2} ϕ & \sin^{2} ϕ & \sin 2 ϕ \\ \sin^{2} ϕ & \cos^{2} ϕ & - \sin 2 ϕ \\ - \frac{1}{2} \sin 2 ϕ & \frac{1}{2} \sin 2 ϕ & \cos 2 ϕ \end{matrix}], \\ Q = [\begin{matrix} Q_{11} & Q_{12} & 0 \\ Q_{12} & Q_{22} & 0 \\ 0 & 0 & Q_{66} \end{matrix}], \end{matrix}

(5)

where $C^{*}$ is the stiffness matrix in global coordinates that has the components $C_{ij}^{*}$ ; $T$ is the rotation matrix in Voigt notation; $Q$ is the stiffness matrices in principal material coordinates that contains the stiffness constants $Q_{ij}$ that are defined according to plain stress assumption via the engineering constants as follows:

Q_{11} = \frac{E_{1}}{1 - ν_{12} ν_{21}}, Q_{22} = \frac{E_{2}}{1 - ν_{12} ν_{21}}, Q_{12} = \frac{ν_{12} E_{1}}{1 - ν_{12} ν_{21}}, Q_{66} = G_{12},

where $E_{1}$ , $E_{2}$ are the elastic moduli along and across the fibers in the unidirectional ply of composite material, $ν_{12}$ , $ν_{21}$ are the Poisson ratios, and $G_{12}$ is the in-plane shear modulus. Stiffness constants for the transverse shear along and across the fibers in equation (4) are assumed to be equal to the in-plane shear stiffness, i.e., $C_{44} = C_{55} = G_{12}$ , $C_{45} = C_{54} = 0$ .

Strains and curvatures in constitutive equations (3) are defined through the displacements $u_{i}$ and rotations $θ_{i}$ according to Reissner–Mindlin theory: $u_{α} = u_{α}^{0} (x) + z θ_{α} (x)$ ( $α = 1, 2$ ) and $u_{3} = w (x)$ , where $u_{α}^{0}$ are the in-plane displacements at the reference plane and $w$ is the deflection of plate.

In the test problems, we consider the loading cases for the square plates under central and offset concentrated forces, and under constant pressure (Figure 1). Considered examples of optimization will be given for the simply supported plates with restricted in-plane and transverse displacements over the whole boundary (in total, five types of boundary conditions at the plate edges):

\begin{matrix} u_{α} = 0, & M_{α β} n_{α} n_{β} = 0, θ_{α} s_{α} = 0, w = 0 x \in \partial Ω, \end{matrix}

(6)

where $n_{α}$ and $s_{α}$ are the outer normal and tangent unit vectors to the plate boundary, respectively.

Implementation of considered optimization problem (1) and plate model (2)–(6) was performed within the “Shell” interface in Comsol Multiphisics. Additional nodal variable $ρ (x)$ (“density”) was introduced by using “Density model” interface and linear Lagrangian shape functions. Additional nodal variable $α (x)$ (“orientation”) was introduced by using optimization interface and linear Lagrangian shape functions. Evaluation of plate stiffnesses (4) and (5) was programmed by using “Variables” interface together with material model for anisotropic elastic plates. Global constraint (for the total volume) and local constraints (for the thickness and fibers orientation ranges) were introduced by using corresponding nodes of optimization interface. Compliance function was defined in a form of total elastic energy of the plate. Triangular mesh elements with Lagrangian quadratic polynomial shape functions were used. Solution of the plate theory problem was found by using MUMS direct solver. Optimization problem was solved by using method of moving asymptotes. Maximum allowable number of iterations was 100. Optimality tolerance factor was 0.001. Initial conditions (initial thickness of plate) were defined by using constant value of “density” variable $ρ (x) \equiv \bar{ρ}$ and constant value of fiber orientation angle $ϕ_{0} = 0$ , $π / 2$ or $π / 4$ (with $α (x) \equiv 0$ ) in the whole domain. Initial direction of fibers is evaluated from the central line shown in Figure 1(b) so that $ϕ_{0} = 0$ and $ϕ_{0} = π / 2$ will be denoted as horizontal and vertical orientations, respectively. Normalization of the objective function was done by using solution of the first iteration with initial uniform thickness and straight fibers.

3. Results and discussion

In the calculations, we consider the plates with the edge size $L = 300$ mm and with thickness of skin $h_{skin} = 1$ mm. Maximum height of ribs was set to be $h_{rib} = 29$ mm so that maximum total thickness was $h = 30$ mm. Constraint on the remaining volume in TO was ${\bar{ρ}}_{\max} = 5 %$ . Typical elastic properties of polymer matrix composites reinforced with unidirectional carbon fibers were used: $E_{1} = 120$ GPa, $E_{2} = 10$ GPa, $ν_{21} = 0.3$ , $ν_{12} = ν_{21} E_{2} / E_{1} = 0.025$ , $G_{12} = 7$ GPa. Magnitude of the loads was $P = 10$ N. Density is $γ_{cfrp} = 1700$ kg/m ³ (for mass evaluation). Magnitude of pressure was $p = 10$ kPa. All calculations were performed within the linear elastic formulation.

3.1. Influence of mesh size

Analysis on the influence of mesh elements size is performed for the relative seed size ( $h$ ) varied in the range $h / L = 1 / 30 . . . 1 / 100$ . Initial orientation of fibers was horizontal. Optimal solution was compared from the viewpoint of resulting geometry, deflections ( $w$ ), and total mass ( $m$ ). In Figure 2, we show the optimal solutions for the thickness and orientation of fibers in the solutions with different mesh size.

Here and in the following, thickness is shown by the background color scheme in gray scale, while fibers trajectories are given by the red lines (Figure 2). Parts of plates with highest thickness (black color) correspond to the found positions of the ribs. The white background in the presented solution corresponds to the thinnest regions of plate (skin). It can be seen that the optimal solutions for concentrated forces have single horizontal ribs. These ribs have variable width along the length with maximal width at the position of concentrated force. It is important to note that the mesh size does not strongly affect the optimal geometry of the ribs. Starting from maximal considered mesh size ( $h / L = 0.331$ ), we have approximate shape of the ribs that become only more smooth in the solutions with the finest mesh ( $h / L = 0.01$ ).

In the problem with pressure, the main ribs become somewhat thinner with the mesh size reduction. Also, additional small stiffeners arise under distributed loading and form the tree-type structure in the solutions with smaller mesh elements. These effects are well known and correspond to the mesh-dependent TO solution for variable-thickness plates [14, 17]. It can be also seen, in Figure 3, where we show the dependence of maximal deflections and total mass in TO solutions on the mesh size. In Figure 3(a), it is shown that the reduced size of the mesh provides better solution from the viewpoint of plate stiffness, i.e., the deflections are reduced. Mesh dependency of stiffness is more pronounced for the case of distributed loading (yellow line in Figure 3(a)). At the same time, the total mass of optimized plates is almost independent on the mesh size and corresponds to the prescribed average thickness of the plates (Figure 3(b)).

Thus, for possible applications, we can just avoid to reduce the mesh size, when the desired small width of the ribs is obtained in TO solutions. In other case, one should involve some variants of regularization or relaxation methods, although this will require additional computational time [16]. For the following analysis, we will use the mesh size $h / L = 0.01$ . This mesh size allows us to find the optimal thin-walled structure of the stiffeners (that is of interest for application) preserving the computing efficiency of the method. Note that, in comparison to isotropic structures, where the convergence of TO algorithm was reached in 20–30 iterations [16], here for the two-field TO for anisotropic plates it took up to 80–100 iterations to reach the prescribed optimality tolerance (0.001).

Now, let us evaluate the results for the fiber orientation. The most important result is that TO algorithm always finds the longitudinal orientation of fibers inside the ribs. It can be seen that even in the small and thin ribs (Figure 2, results for “Pressure”), the orientation of fibers corresponds to the orientation of the ribs. It is well known that such reinforcement is the most optimal for the composite ribs. Such structures, formed by the beams with unidirectional reinforcement, are known as isogrid and anisogrid lattices [23, 24]. Thus, the presented approach can be possibly involved in the design of the structures reinforced by such lattices [25, 26].

Around the thick ribs, TO algorithm found the optimal trajectories of fibers in the skin. These trajectories correlate with the directions of principal stresses in the corresponding isotropic plate with constant thickness (see Figure 4). However, the presence of ribs can change the stress trajectories in the variable-thickness plates, especially in the case of distributed loading, that also affect the resulting orientation of fibers. The approximately optimal solutions for orientation of fibers can be found even on the coarse meshes if one neglects some refinement of the solution in the area of thinner ribs (see Figure 2, $h / L = 0.033$ ).

It is worth noting that in the case of pressure loading, the found orientation of the ribs can be also attributed to the stress trajectories in the corresponding isotropic plate (cf. Figures 2 and 4(c), results for “Pressure”). In this TO result, we obtain the single central rib oriented in horizontal direction that is due to used initial condition for the horizontal orientation of fibers. Generally, the resulting geometry will be strongly affected by the initial conditions and additional symmetry conditions that can be imposed on the considered problems. These effects are detailed in the next subsection.

3.2. Influence of symmetry conditions and initial orientation of fibers

Initial conditions significantly affect the TO results of isotropic and anisotropic bodies. In this section, we consider solution variants for panels with initial fiber orientations in the horizontal (0) and vertical (90) directions. For the case of distributed loading, we also investigate options for initial inclined reinforcement with ±45 angles, which can be effective considering the optimization results depicted in Figure 2. In addition, we examine the influence of symmetry conditions that can be prescribed in the problems under consideration. Specifically, in the problem with central force and pressure, we can take into account the presence of two symmetry planes and consider 1/4 of the panels. In the case of eccentric force, we can consider a symmetry plane and find the solution for 1/2 of the panel. By symmetry conditions, we mean the standard requirements for zero displacements and rotations in the normal direction to the plane of symmetry.

The total mass is out of consideration in this section, as it has been shown that with the prescribed identical target average “density” ( ${\bar{ρ}}_{\max}$ ), the masses of plates turn out to be practically the same (Figure 3(a)). In the present case, it is approximately 0.375 kg.

Comparison of the obtained optimization results for the geometry and stiffness parameters of the panels is shown in Figures 5 –8. Figure 5 presents the panel geometries for all the examined loading cases, in which initial conditions for horizontal/vertical fiber orientation were utilized, and symmetry conditions were either considered or not considered. The main result here is that the primary load-bearing ribs in anisotropic panels predominantly appear along the initially specified reinforcement direction. It is also noteworthy that accounting for symmetry leads to the halving of central massive stiffness ribs (so that we have two parallel ribs), which occurs in solutions with concentrated forces as well as in the case of distributed loading. The geometry found for panels subjected to pressure with an initial fiber direction of ± 45 degrees is presented in Figure 6. Here, it can be seen that these solutions also fundamentally differ from those presented in Figure 5.

Comparison of the stiffness parameters of the optimized panels subjected to concentrated forces is presented in Figure 7. Here, it can be observed that the deflections of the panels do not change significantly in the examined test cases when considering (not considering) symmetry conditions. So, the appearance of two parallel thin ribs instead of one thick rib turns out to be a solution artifact (Figure 5). It is likely that this artifact can be resolved by formulating additional symmetry conditions on the “density” and “orientation” functions. In contrast, the influence of the initial fiber direction significantly affects the stiffness of the optimized panel. In particular, specifying the initial horizontal fiber direction proves to be more effective (yellow bars in Figure 7). This is presumably related to the fact that in problems with eccentric forces, the plane of symmetry also lies horizontally.

In the case of distributed loading, on the contrary, symmetry conditions turn out to be significant in terms of the stiffness of the resulting structures. This is demonstrated in Figure 8. Here, it can be seen that considering symmetry, for any fiber orientation, allows reducing the maximum deflections of the panel by 2 to 2.5 times. In this case, the most effective option is the initial conditions with inclined initial orientation at +45 degrees, which leads to the formation of diagonal stiffening ribs connecting the corners of the panel (see Figure 6).

Thus, it is evident that initial and symmetry conditions strongly influence the results for variable-thickness plates. Therefore, a comprehensive approach to designing such structures involves considering various initial conditions and calculation schemes, with the results then compared to identify the most optimal design.

3.3. Comparison with optimal design for isotropic plates

Next, we compare the efficiency of TO solutions for anisotropic and isotropic panels. To do this, we additionally consider panels made of aluminum alloy with standard properties: Young’s modulus of 70 GPa and a Poisson’s ratio of 0.3, and density of $γ_{al} = 2700$ kg/m ³. The validation of TO solutions for such isotropic panels was presented in our recent works [16, 21]. In the following, we compare the geometry of the resulting rib-stiffened isotropic and fiber-reinforced structures and their stiffness. We consider two variants of solutions for anisotropic panels: (1) with the same average thickness and (2) with the same mass. In the first case, we use as previously ${\bar{ρ}}_{\max} = 0.05$ and directly evaluate the optimality of TO solutions for typical properties of metal and composite materials. It is obvious that a panel made of carbon-fiber reinforced plastic (CFRP) will always be lighter than metal panel but not always stiffer. In the second case, we consider a CFRP panel and allow the algorithm to use a larger volume of material so that the mass of the composite panel equals the mass of the aluminum panel. To achieve this, we determine the maximum average “density” as ${\bar{ρ}}_{\max} = 0.05 γ_{al} / γ_{cfrp}$ . For the optimization of anisotropic panels, we will use the best settings identified in the previous subsection. Specifically, for cases of concentrated forces, we will use the solution with initial conditions with horizontal fiber orientation and without considering symmetry. For distributed loading, we will consider symmetry conditions and set initial fiber orientation conditions at 45 degrees.

The found geometry of the panels is presented in Figures 9 and 10. It can be seen that isotropic and anisotropic panels are significantly different. The similarity lies only in the fact that for concentrated forces, the algorithm finds a small number of thick stiffeners, while for distributed loading, it selects numerous thin ribs. However, their orientations are fundamentally different in anisotropic and isotropic panels. Solutions in the form of a cross for a central force (Figure 9) and a three-bar cross for eccentric forces (Figure 10) can be validated for isotropic structures based on analytical solutions. In particular, it can be shown that a cross beam is always more efficient than a single longitudinal beam. Moreover, the angle of opening of the three-bar cross corresponds to the exact solution in TO results [21]. Validating these effects for anisotropic structures is more challenging, as the additional load-bearing capacity of the reinforced skin can play a significant role in the resulting deflections.

For anisotropic panels, TO suggests the selection of massive longitudinal edges with additional thickening directly under the loading point. For the problem with increased averaged “density” ( ${\bar{ρ}}_{\max} = 0.05 γ_{al} / γ_{cfrp}$ ), i.e., with a mass close to that of an isotropic metal construction, the geometry of the solution remains unchanged, and only the thickness of the stiffness ribs is increased. A similar effect is realized in the case of pressure (see Figure 9). This is interesting because the TO solution proves to be stable with respect to changes (albeit small) in the target value of the average “density” ( ${\bar{ρ}}_{\max}$ ).

The comparison of stiffness and mass of panels subjected to central force and pressure is presented in Figure 11. It is visible that in the case of a central force, the anisotropic panel exhibits approximately 1.5 times lower stiffness compared to the metallic one, while its mass is 40% lower. By increasing the allowable amount of composite material, we can achieve equivalent mass and stiffness for the structures. This is apparently because the presence of concentrated forces is not well accommodated by anisotropic panels and requires the use of multilayer composite structures with different reinforcement angles.

In the case of pressure, on the contrary, the composite panel exceeds the isotropic one in stiffness by almost three times. Moreover, additional increase in the volume of composite material allows to enhance this superiority up to 5 times (see Figure 11, results for “Pressure”). Thus, in the case of distributed loading, the anisotropic structure proves to be significantly more effective than the isotropic one, which is not obvious in general. Notice that the mass of composite panels with increased allowable material content does not exactly match the mass of metal panels (Figure 11(b)). This is due to algorithm inaccuracies in satisfying global constraints. However, all results confirm the made conclusions: for distributed loading, composite panels unequivocally have the advantage (when typical elastic properties are implemented), even when using a locally unidirectional reinforcement scheme.

For the purpose of simplifying the comparisons, we introduce a figure of merit as the product of the panel’s mass and its relative deflection under a given load (FM $= mw / L$ ). Its values should be minimal for the best panel designs. In particular, for an isotropic panel under pressure, this indicator equals 169.2, whereas for a composite panel, it is 20.8 (for the same average “density”) and 28 (for increased average “density”). The values of figure of merit, obtained in solutions for panels under concentrated loads (Figure 10), are shown in Figure 12. Here, we show the dependence of figure of merit obtained in the problems with different shift of the loading point from the panel’s center ( $ℓ / L$ ). It can be seen, that in all cases, except the central loading, the composite panels have more optimal design in comparison to the isotropic plates (red line in Figure 12). In the problem with central force, the metal panel are stiffer in comparison to the composite panels of similar volume as stated above.

4. Conclusion

Methods of TO for variable-thickness plates receive much less attention from researchers compared to standard TO methods for 3D solid models. In the present contribution, we suggest an approach for simultaneous optimization of fiber trajectories and curvilinear ribs in anisotropic plates. We show the efficiency of the method and its peculiarities for the problems with concentrated and distributed forces. Namely, we show that the initial conditions for fiber orientation and symmetries of the problem are essential for the design of optimal structures. The designed rib-stiffened anisotropic plates can be manufactured by using 3D-printing technologies. Corresponding experimental studies and validation of supposed method are planned for the future research.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Ministry of Science and Higher Education of the Russian Federation (Agreement No. 075-15-2024-535).

ORCID iD

Yury Solyaev

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