Fatigue constrained topology optimization

Abstract

We present a contribution to a relatively unexplored application of topology optimization: structural topology optimization with fatigue constraints. A probability based high-cycle fatigue analysis is combined with principal stress calculations in order to find the topology with minimum mass that can withstand prescribed variable-amplitude loading conditions for a specific life time. This allows us to generate optimal conceptual designs of structural components where fatigue life is the dimensioning factor. We describe the fatigue analysis and present ideas that make it possible to separate the fatigue analysis from the topology optimization. The number of constraints is kept low as they are applied to stress clusters, which are created such that they give adequate representations of the local stresses. Optimized designs constrained by fatigue and static stresses are shown and a comparison is also made between stress constraints based on the von Mises criterion and the highest tensile principal stresses. The paper is written with focus on structural parts in the avionic industry, but the method applies to any load carrying structure, made of linear elastic isotropic material, subjected to repeated loading conditions.

Appendix

Differentiating (13) with respect to design variable x _b gives

$$\frac{\partial S_{ij}}{\partial x_{b}}\phi_{j}^{\alpha}+S_{ij}\frac{\partial\phi_{j}^{\alpha}}{\partial x_{b}}=\frac{\partial\lambda^{\alpha}}{\partial x_{b}}I_{ij}\phi_{j}^{\alpha}+\lambda^{\alpha}\frac{\partial I_{ij}}{\partial x_{b}}\phi_{j}^{\alpha}+\lambda^{\alpha}I_{ij}\frac{\partial\phi_{j}^{\alpha}}{\partial x_{b}}. $$

Using $\frac {\partial I_{ij}}{\partial x_{b}}=0$ and rearranging gives

$$ \frac{\partial S_{ij}}{\partial x_{b}}\phi_{j}^{\alpha}+(S_{ij}-\lambda^{\alpha}I_{ij})\frac{\partial\phi_{j}^{\alpha}}{\partial x_{b}}=\frac{\partial\lambda^{\alpha}}{\partial x_{b}}I_{ij}\phi_{j}^{\alpha}. $$

(24)

The eigenvector $\phi _{j}^{\alpha }$ is a unit vector, i.e.,

$$\phi_{i}^{\alpha}I_{ij}\phi_{j}^{\alpha}=\phi_{j}^{\alpha}\phi_{j}^{\alpha}=1, $$

therefore, if we premultiply (25) by $\phi _{i}^{\alpha }$, we get

$$ \phi_{i}^{\alpha}\frac{\partial S_{ij}}{\partial x_{b}}\phi_{j}^{\alpha}+\phi_{i}^{\alpha}(S_{ij}-\lambda^{\alpha}I_{ij})\frac{\partial\phi_{j}^{\alpha}}{\partial x_{b}}=\phi_{i}^{\alpha}\frac{\partial\lambda^{\alpha}}{\partial x_{b}}I_{ij}\phi_{j}^{\alpha}, $$

(25)

and rearranging the right hand side of (26) gives

$$ \phi_{i}^{\alpha}\frac{\partial\lambda^{\alpha}}{\partial x_{b}}I_{ij}\phi_{j}^{\alpha}=\frac{\partial\lambda^{\alpha}}{\partial x_{b}}\phi_{i}^{\alpha}I_{ij}\phi_{j}^{\alpha}=\frac{\partial\lambda^{\alpha}}{\partial x_{b}}. $$

(26)

As both S _{i j} and I _{i j} are symmetric, we can rewrite as

$$ \phi_{i}^{\alpha}(S_{ij}-\lambda^{\alpha}I_{ij})=(S_{ji}-\lambda^{\alpha}I_{ji})\phi_{i}^{\alpha}=0, $$

(27)

where the zero comes from (13). Combining (26), (27) and (28), we now find that the derivative of the eigenvalues with respect to the design variable reads

$$ \frac{\partial\lambda^{\alpha}}{\partial x_{b}}=\phi_{i}^{\alpha}\frac{\partial S_{ij}}{\partial x_{b}}\phi_{j}^{\alpha}. $$

(28)