Discrete material selection and structural topology optimization of composite frames for maximum fundamental frequency with manufacturing constraints

Abstract

This paper proposes a methodology for simultaneous optimization of composite frame topology and its material design considering specific manufacturing constraints for the maximum fundamental frequency with a bound formulation. The discrete material optimization (DMO) approach is employed to couple two geometrical scales: frame structural topology scale and microscopic composite material parameter scale. The simultaneous optimization of macroscopic size or topology of the frame and microscopic composite material design can be implemented within the DMO framework. Six types of manufacturing constraints are explicitly included in the optimization model as a series of linear inequality or equality constraints. Sensitivity analysis with respect to variables of the two geometrical scales is performed using the semi-analytical sensitivity analysis method. Corresponding optimization formulation and solution procedures are also developed and validated through numerical examples. Numerical study shows that the proposed simultaneous optimization model can effectively enhance the frame fundamental frequency while including specific manufacturing constraints that reduce the risk of local failure of the laminated composite. The proposed multi-scale optimization model for the maximum fundamental frequency is expected to provide a new choice for the design of composite frames in engineering applications.

Appendices

Appendix 1. Evaluation of convergence

The convergence measure given in Stegmann and Lund (2005) is adopted to describe whether the optimization has converged to a satisfactory result, i.e., a single candidate ply angle has been chosen in a specified element and all other materials have been discarded. For each layer, the following inequality is evaluated as

$$ {\omega}_{i,j,c}\ge \upvarepsilon \sqrt{\omega_{i,j,1}^2+{\omega}_{i,j,2}^2+\cdots +{\omega}_{i,j,{\mathrm{N}}^{\mathrm{cand}}}^2} $$

(20)

where ε is a tolerance, typically, ε ∈ [0.95~0.99]. If the inequality in (20) is satisfied for any ω_{i, j, c} in the jth layer, then the layer is flagged as converged. The convergence assessment criterion H_ε is defined as the ratio between the number of converged layers ${N}_c^{l, tot}$ and the total number of layers N^{l, tot}. N^lay is the number of layers in each tube, and it is assumed that each tube has the same number of layers in the paper. N^tub is the number of tubes in the frame structure. Thus, N^{l, tot} can be expressed as the number of tubes multiplied by the number of layers in a tube, that is, N^{l, tot} = N^tub ∙ N^lay. Then, the convergence assessment H_ε can be expressed as

$$ {H}_{\varepsilon }=\frac{N_c^{l, tot}}{N^{l, tot}} $$

(21)

If the tolerance is 0.95 and the optimization is fully converged, i.e., H_{ε = 0.95} = 1, all layers have a single weight factor that contributes more than 95% to the Euclidian norm of the weight factors. More discussion about convergence criteria can be found in the references (Xu et al. 2019; Duan et al. 2015).

Appendix 2. Determination of cross-sectional stiffness matrix

The determination of K_s entails a two-dimensional problem solution associated with the determination of three-dimensional deformation of the cross-section. The solution is obtained from the cross-section equilibrium equations given by

$$ \mathbf{K}\mathbf{\mathcal{R}}=\mathbf{F} $$

(22)

where the components of K are associated with the stiffness of the cross-section. Furthermore, the solution matrix $\mathbf{\mathcal{R}}$ contains the cross-section rigid body motions φ and the three-dimensional warping displacements u. Finally, the load array F is associated with a series of unit load vectors θ. The solution $\mathbf{\mathcal{R}}$ from (22) is subsequently used in the determination of the cross-sectional compliance matrix ${\mathbf{\mathcal{C}}}_s$ as

$$ {\mathbf{\mathcal{C}}}_s={\mathbf{\mathcal{R}}}^T\mathbf{G}\mathbf{\mathcal{R}} $$

(23)

where the coefficient matrix G is defined in Blasques and Stolpe (2012). For most practical applications, and in all cases considered in this paper, ${\mathbf{\mathcal{C}}}_s$ is symmetric and positive definite. Hence, the cross-sectional stiffness matrix is obtained from ${\mathbf{K}}_s={{\mathbf{\mathcal{C}}}_s}^{-1.}$

6 × 6 cross-sectional mass matrix M_s relates the linear and angular velocities in ϕ to the generalized inertial linear and angular momentum in γ through ϕ = M_sγ. According to Hodge (2006), M_s is expressed as

$$ \left[{\mathbf{M}}_s\right]=\left[\begin{array}{cccccc}m& 0& 0& 0& 0& -m{y}_m\\ {}0& m& 0& 0& 0& m{x}_m\\ {}0& 0& m& m{y}_m& -m{x}_m& 0\\ {}0& 0& m{y}_m& {I}_{xx}& -{I}_{xy}& 0\\ {}0& 0& -m{x}_m& -{I}_{xy}& {I}_{yy}& 0\\ {}-m{y}_m& m{x}_m& 0& 0& 0& {I}_{xx}+{I}_{yy}\end{array}\right] $$

(24)

where $m$ is the mass per unit length, and I_xx and I_yy are the moment of inertia with respect to x and y, respectively. I_xy is the product of inertia. The off-diagonal terms are due to the offset between the position of the cross-section reference center and the mass center m_c = (x_m, y_m). Here, the reference center is defined as the point through which the reference line goes through and is coincident with the beam finite element discretization. All of the terms in M_s are determined through integration of the mass properties in the finite element mesh of cross-sections.