Generative design of stiffened plates based on homogenization method

Abstract

Stiffened plates are widely used in aerospace structures as load-bearing components. In order to obtain a novel design of stiffened structures with excellent performance, a generative design method of stiffened plates (GDMSP) based on the homogenization method is proposed in this paper, which optimizes the stiffener layout based on an equivalent model. Then, the detailed model can then be obtained by extracting the stiffener path from the discrete distribution of stiffener angles. Moreover, the optimized design can be obtained by size optimization based on the detailed model. Two examples are used to illustrate the proposed framework, including the stiffness maximization of a rectangle stiffener plate and the buckling load maximization of a square stiffener plate. The optimized stiffener configurations are characterized by streamlines and uniform lines, respectively. For the first example, the stiffness of the stiffener design has an improvement of 17%. For the second example, the optimized design improves the buckling load by 35%. Results indicate that the proposed method can effectively provide a novel generative design for stiffened plates. Moreover, the obtained results have a clear stiffener path and have a noticeable improvement in performance, which can be directly used to establish a detailed model.

Appendix: The sensitivity analysis of the linear buckling load

In this paper, the linear buckling load is calculated with the finite element method by solving the eigenvalue equation as follows.

$${{\{ }}{\mathbf{K}} + \lambda_{j} {\mathbf{K}}_{{\varvec{\sigma}}} {{\} }}{{\varvec{\Phi}}}_{j} = {\mathbf{0}} \,$$

(22)

For the DMO, the sensitivities are necessary. The sensitivities are calculated with the adjoint method, which is briefly derived as follows. The direct approach to obtain the eigenvalue sensitivity in case of a distinct, i.e., the simple eigenvalue $\lambda_{j}$ is to differentiate Eq. (22) to the design variable $x_{ij}$ premultiply by ${{\varvec{\Phi}}}_{j}^{T}$ and make use of Eq. (22). Then the following expression can be obtained as

$$\frac{{d\lambda_{j} }}{{dx_{ij} }} = - {{\varvec{\Phi}}}_{j}^{T} \left( {\frac{{d{\mathbf{K}}}}{{dx_{ij} }} + \lambda_{j} \frac{{d{\mathbf{K}}_{{{\varvec{\upsigma}}}} }}{{dx_{ij} }}} \right){{\varvec{\Phi}}}_{j}$$

(23)

where it has been assumed that the eigenvectors have been ${\mathbf{K}}_{{{\varvec{\upsigma}}}}$ -orthonormalized, such that ${{\varvec{\Phi}}}_{i}^{T} {\mathbf{K}}_{{{\varvec{\upsigma}}}} {{\varvec{\Phi}}}_{i} = 1$. The geometric matrix is an implicit function of the displacement field, i.e., ${\mathbf{K}}_{{{\varvec{\upsigma}}}} = {\mathbf{K}}_{{{\varvec{\upsigma}}}} \left( {{\mathbf{u}}({\mathbf{x),x}}} \right)$, which has to be taken into account as

$$\frac{{d{\mathbf{K}}_{{{\varvec{\upsigma}}}} }}{{dx_{ij} }} = \frac{{\partial {\mathbf{K}}_{{{\varvec{\upsigma}}}} }}{{\partial x_{ij} }} + \frac{{\partial {\mathbf{K}}_{{{\varvec{\upsigma}}}} }}{{\partial {\mathbf{u}}}}\frac{{d{\mathbf{u}}}}{{dx_{ij} }}$$

(24)

Then the static equilibrium equation ${\mathbf{K}}{\mathbf{u}}{\mathbf{ = F}}$ is differentiated to the design variable $x_{ij}$ as

$${\mathbf{K}}\frac{{d{\mathbf{u}}}}{{dx_{ij} }} = \frac{{\partial {\mathbf{F}}}}{{\partial x_{ij} }} - \frac{{\partial {\mathbf{K}}}}{{\partial x_{ij} }}{\mathbf{u}}$$

(25)

where the load sensitivity $\partial {\mathbf{F}}/\partial x_{ij}$ is zeros. Then, substituting Eq. (25) into Eq. (24) yields

$$\frac{{d{\mathbf{K}}_{{{\varvec{\upsigma}}}} }}{{dx_{ij} }} = \frac{{\partial {\mathbf{K}}_{{{\varvec{\upsigma}}}} }}{{\partial x_{ij} }} - \frac{{\partial {\mathbf{K}}_{{{\varvec{\upsigma}}}} }}{{\partial {\mathbf{u}}}}{\mathbf{K}}^{{{\mathbf{ - 1}}}} \frac{{d{\mathbf{K}}}}{{dx_{ij} }}{\mathbf{u}}$$

(26)

Thus, the adjoint vector ${{\varvec{\Lambda}}}$ that satisfies Eq. (27) can be introduced.

$${\mathbf{K\Lambda }} = \left( {{{\varvec{\Phi}}}_{j}^{T} \frac{{\partial {\mathbf{K}}_{{{\varvec{\upsigma}}}} }}{{\partial {\mathbf{u}}}}{{\varvec{\Phi}}}_{j} } \right)^{T}$$

(27)

Finally, substituting the adjoint vector ${{\varvec{\Lambda}}}$ into Eq. (23), the full analytically design sensitivities can be obtained as follows:

$$\frac{{d\lambda_{j} }}{{dx_{ij} }} = - \left( {{{\varvec{\Phi}}}_{j}^{T} \frac{{d{\mathbf{K}}}}{{dx_{ij} }}{{\varvec{\Phi}}}_{j} + \lambda_{j} {{\varvec{\Phi}}}_{j}^{T} \frac{{\partial {\mathbf{K}}_{{{\varvec{\upsigma}}}} }}{{\partial x_{ij} }}{{\varvec{\Phi}}}_{j} - \lambda_{j} {{\varvec{\Lambda}}}^{T} \frac{{d{\mathbf{K}}}}{{dx_{ij} }}{\mathbf{u}}} \right)$$

(28)