Design of an aircraft engine bracket using stress-constrained bi-directional evolutionary structural optimization method

Abstract

This work improves our previous stress-constrained topology optimization method (Fan et al., in Struct Multidisc Optim 59:647–658, 2019) and provides an application of the improved method to a typical aircraft engine bracket design problem. The original method was built upon the bi-directional evolutionary structural optimization (BESO) method with an extension to account for stress constraints. In this work, we first improve the method by means of a more efficient and versatile self-adaptive scheme for the determination of the Lagrange multiplier. The improved method is then applied for the design of a typical aircraft engine bracket considering multiple practical load conditions. The resulting bracket topology from stress-constrained design is further smoothed and detailed using basic CAD (Computer-Aided Design) primitives. Numerical results show that the reconstructed bracket design evidently outperforms than the original bracket design in terms of weight, stiffness, and strength.

Appendices

Appendix 1: Sensitivity analysis

By the BESO method, sensitivities are evaluated only for solid elements while are set directly to zero for void elements. The derivative of the modified design objective ${f}_{2}$ w.r.t. ${x}_{i}$ equals

$$\frac{\partial {f}_{2}}{\partial {x}_{i}}=\frac{\partial c}{\partial {x}_{i}}+\lambda \frac{\partial {\sigma }_{PN}}{\partial {x}_{i}}$$

(A1)

in which the first term can be easily derived as

$$\frac{\partial c}{\partial {x}_{i}}=-{\mathbf{u}}_{i}^{T}{\mathbf{k}}_{i}^{\left(0\right)}{\mathbf{u}}_{i}$$

(A2)

and ${\mathbf{k}}_{i}^{\left(0\right)}$ is the stiffness matrix of the i-th element with solid material.

According to the definition in (5), the second term in (A1) is derived as

$$\frac{\partial {\sigma }_{PN}}{\partial {x}_{i}}={\sigma }_{PN}^{1-p}\left(\sum_{j=1}^{N}{\sigma }_{vm,j}^{p-1}\frac{\partial {\sigma }_{vm,j}}{\partial {x}_{i}}\right)$$

(A3)

Recalling the definition of von Mises stress,

$${\sigma }_{vm}={\left({{\varvec{\sigma}}}^{{\varvec{T}}}\mathbf{V}{\varvec{\sigma}}\right)}^{1/2}$$

(A4)

with ${\varvec{\sigma}}$ the stress in the Voigt notation and $\mathbf{V}$ the stress coefficient matrix, the derivative of j-th element von Mises stress w.r.t. the stress vector equals

$$\frac{\partial {\sigma }_{vm,j}}{\partial {{\varvec{\sigma}}}_{j}}=\frac{1}{2}{\left({{\varvec{\sigma}}}_{j}^{T}\mathbf{V}{{\varvec{\sigma}}}_{j}\right)}^{-1/2}2{{\varvec{\sigma}}}_{j}^{T}\mathbf{V}={\sigma }_{vm,j}^{-1}{{\varvec{\sigma}}}_{j}^{T}\mathbf{V}$$

(A5)

and by the chain rule, the derivative of ${\sigma }_{vm,j}$ w.r.t. ${x}_{i}$ equals

$$\frac{\partial {\sigma }_{vm,j}}{\partial {x}_{i}}={\sigma }_{vm,j}^{-1}{{\varvec{\sigma}}}_{j}^{T}\mathbf{V}\frac{\partial {{\varvec{\sigma}}}_{j}}{\partial {x}_{i}} .$$

(A6)

It is defined and assumed in (1) that ${\mathbf{D}}_{0}$ and ${\mathbf{B}}_{i}$ are independent with the associated design variable and thus (A6) can be further written as

$$\frac{\partial {\sigma }_{vm,j}}{\partial {x}_{i}}={\sigma }_{vm,j}^{-1}{{\varvec{\sigma}}}_{j}^{T}\mathbf{V}{\mathbf{D}}_{0}{\mathbf{B}}_{j}{\mathbf{L}}_{j}\frac{\partial \mathbf{U}}{\partial {x}_{i}}$$

(A7)

where ${\mathbf{L}}_{j}$is the gathering matrix (${\mathbf{u}}_{j}={\mathbf{L}}_{j}\mathbf{U}$). As the external force is assumed to be independent with topology change, (A7) can thus be further written as

$$\frac{\partial {\sigma }_{vm,j}}{\partial {x}_{i}}=-{\sigma }_{vm,j}^{-1}{{\varvec{\sigma}}}_{j}^{T}\mathbf{V}{\mathbf{D}}_{0}{\mathbf{B}}_{\rm {j}}{\mathbf{L}}_{\rm {j}}{\mathbf{K}}^{-1}\frac{\partial \mathbf{K}}{\partial {x}_{i}}\mathbf{U}$$

(A8)

using the differential property of

$$\frac{\partial \mathbf{K}}{\partial {x}_{i}}\mathbf{U}+\mathbf{K}\frac{\partial \mathbf{U}}{\partial {x}_{i}}=0.$$

(A9)

Substituting (A8) into (A3) yields

$$\frac{\partial {\sigma }_{PN}}{\partial {x}_{i}}=-{\sigma }_{PN}^{1-p}\left(\sum_{j=1}^{N}{\sigma }_{vm,j}^{p-2}{{\varvec{\sigma}}}_{j}^{T}\mathbf{V}{\mathbf{D}}_{0}{\mathbf{B}}_{j}{\mathbf{L}}_{j}\right){\mathbf{K}}^{-1}\frac{\partial \mathbf{K}}{\partial {x}_{i}}\mathbf{U}$$

(A10)

and with the adjoint solution of following

$$\mathbf{K}{\varvec{\Phi}}=\sum_{j=1}^{N}{\sigma }_{vm,j}^{p-2}{\left({\mathbf{D}}_{0}{\mathbf{B}}_{j}{\mathbf{L}}_{j}\right)}^{T}\mathbf{V}{{\varvec{\sigma}}}_{j}$$

(A11)

(A10) can be simplified to

$$\frac{\partial {\sigma }_{PN}}{\partial {x}_{i}}=-{\sigma }_{PN}^{1-p}{\mathbf{\varphi }}_{i}^{T}{\mathbf{k}}_{i}^{\left(0\right)}{\mathbf{u}}_{i}$$

(A12)

where ${\mathbf{\varphi }}_{i}$is the vector of the adjoint nodal values of the i-th element (${\mathbf{\varphi }}_{j}={\mathbf{L}}_{j}{\varvec{\Phi}}$).

Finally, substituting (A2) and (A12) into (A1), the sensitivity of the modified objective function w.r.t. ${x}_{i}$ euqals

$$\frac{\partial {f}_{2}}{\partial {x}_{i}}=-{\mathbf{u}}_{i}^{T}{\mathbf{k}}_{i}^{\left(0\right)}{\mathbf{u}}_{i}-\lambda \cdot {\sigma }_{PN}^{1-p}{\mathbf{\varphi }}_{i}^{T}{\mathbf{k}}_{i}^{\left(0\right)}{\mathbf{u}}_{i} .$$

(A13)

Appendix 2: BESO variables update scheme

Starting from full solid material design, the allowed material usage at the current design iteration is determined as

$${V}^{(l)}=\rm {max}\left\{{V}^{*}, (1-{c}_{\rm {er}}){V}^{(l-1)}\right\}$$

(A14)

in which the evolutionary ratio ${c}_{\rm {er}}$ determines the material removal rate, the superscript $l$ denotes the design iteration.

Sensitivity numbers denoting the relative ranking of elemental contribution of the design objective are used to guide material removal and addition. When uniform meshes are used, the sensitivity number for is defined using the sensitivity computed from (A13)

$${\alpha }_{i}=-{x}_{i}\frac{\partial {f}_{2}}{\partial {x}_{i}}={x}_{i}\left({\mathbf{u}}_{i}^{T}{\mathbf{k}}_{i}^{\left(0\right)}{\mathbf{u}}_{i}+\lambda \cdot {\sigma }_{PN}^{1-p}{\mathbf{\varphi }}_{i}^{T}{\mathbf{k}}_{i}^{\left(0\right)}{\mathbf{u}}_{i}\right)$$

(A15)

where ${x}_{i}$ serves as an indicator to simply ensure that sensitivity numbers are evaluated only for solid elements $({x}_{i}=1)$, while are enforced to zero for void elements $({x}_{i}=0)$. In view of the highly nonlinear stress behavior, both sensitivity numbers and topology design variables are filtered to stabilize the optimization procedure. To maintain the material recovery mechanism, sensitivity filter radius is set ${r}_{\rm {sen}}=2{h}_{e}$ with ${h}_{e}$ the typical element length. The filtered sensitivity numbers are stabilized with their historical information to improve the convergence.

$${\alpha }_{i}^{(l)}\leftarrow \left({\alpha }_{i}^{\left(l\right)}+{\alpha }_{i}^{\left(l-1\right)}+{\alpha }_{i}^{\left(l-2\right)}\right)/3 for l>2. $$

(A16)

All elements are sorted into a vector according to their sensitivity number values. Then, the update of the topology design variables is realized by means of a threshold parameters ${\alpha }_{\rm {th}}$

$${x}_{i}^{(l+1)}=\left\{\begin{array}{cc}0& \rm {if} {\alpha }_{i}<{\alpha }_{\rm {th}} \rm {and} {x}_{i}^{(l)}=1\\ 1& \rm {if} {\alpha }_{i}>{\alpha }_{\rm {th}} \rm {and} {x}_{i}^{(l)}=0\\ {x}_{i}^{(l)}& \rm {otherwise}.\end{array}\right.$$

(A17)

The present scheme indicates that solid elements are removed when their sensitivity numbers are less than ${\alpha}_{\rm {del}}^{\rm {th}}$ and void elements are recovered when their sensitivity numbers are greater than ${\alpha }_{\rm {add}}^{\rm {th}}$. The parameters ${\alpha }_{\rm {del}}^{\rm {th}}$ and ${\alpha }_{\rm {add}}^{\rm {th}}$ are obtained from the following iterative algorithm:

• 1. Let ${\alpha }_{\rm {del}}^{\rm {th}}={\alpha }_{\rm {add}}^{\rm {th}}={\alpha }_{\rm {th}}$, where the value of ${\alpha }_{\rm {th}}$ is determined iteratively such that the current allowed material volume is met.

• 2. Compute the admission ratio ${c}_{\rm {ar}}$ as the volume of recovered elements divided by the total volume. If ${c}_{\rm {ar}}\le {c}_{\rm {ar}}^{\rm {max}}$, the maximum admission ratio, skip the next steps; otherwise, ${\alpha }_{\rm {del}}^{\rm {th}}$ and ${\alpha }_{\rm {add}}^{\rm {th}}$ are re-determined in the next steps.

• 3. Determine ${\alpha }_{\rm {add}}^{\rm {th}}$ iteratively from the sensitivity numbers of void elements until the maximum admission ratio is met, i.e., ${c}_{\rm {ar}}\approx {c}_{\rm {ar}}^{\rm {max}}$.

• 4. Determine ${\alpha }_{\rm {del}}^{\rm {th}}$ iteratively from the sensitivity numbers of the solid elements until the current allowed material volume is met.

Finally, the obtained binary topology variables are filtered with a radius ${r}_{\rm{den}}$ to ensure the convergence and mesh independency of the design. The resulting intermediate densities are then converted into the discrete design through a post-processing procedure according to a determined density threshold satisfying the current material volume usage constraint.