Topology optimization with finite-life fatigue constraints

Abstract

This work investigates efficient topology optimization for finite-life high-cycle fatigue damage using a density approach and analytical gradients. To restrict the minimum mass problem to withstand a prescribed finite accumulated damage, constraints are formulated using Palmgren-Miner’s linear damage hypothesis, S-N curves, and the Sines fatigue criterion. Utilizing aggregation functions and the accumulative nature of Palmgren-Miner’s rule, an adjoint formulation is applied where the amount of adjoint problems that must be solved is independent of the amount of cycles in the load spectrum. Consequently, large load histories can be included directly in the optimization with minimal additional computational costs. The method is currently limited to proportional loading conditions and linear elastic material behavior and a quasi-static structural analysis, but can be applied to various equivalent stress-based fatigue criteria. Optimized designs are presented for benchmark examples and compared to stress optimized designs for static loads.

Appendix

In the following the partial derivatives in the sensitivity analysis of the fatigue constraint are written using the chain rule of differentiation. The independence or linear dependence on the load scaling factors is indicated.

The partial derivative of the fatigue constraint with respect to a physical design variable ${\tilde{x}}_j$ can be written as:

$$ \begin{array}{l}\frac{\partial {g}_D}{\partial {\tilde{x}}_j}=\frac{\partial {g}_D}{\partial {D}_j}\ \sum_{i=1}^{n_{RF}}\left(\frac{\partial {D}_{j_i}}{\partial {N}_{j_i}}\frac{\partial {N}_{j_i}}{\partial {\tilde{\sigma}}_{j_i}}\left({c}_{a_i}\left(\frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\sigma}}_{j_x}}\frac{\partial {\hat{\sigma}}_{j_x\ }}{\partial {\tilde{x}}_j}\right.\right.\right.\\ {}\left.\kern3.5em +\frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\sigma}}_{j_y}}\frac{\partial {\hat{\sigma}}_{j_y}}{\partial {\tilde{x}}_j}+\frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\tau}}_j}\frac{\partial {\hat{\tau}}_j}{\partial {\tilde{x}}_j}\right)\\ {}\kern3.5em \left.+{c}_{m_i}\left.\left(\frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\sigma}}_{j_{m x}}}\frac{\partial {\hat{\sigma}}_{j_x}}{\partial {\tilde{x}}_j}+\frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\sigma}}_{j_{m y}}}\frac{\partial {\hat{\sigma}}_{j_y}}{\partial {\tilde{x}}_j}\right)\right)\right),\kern1em \forall j\end{array} $$

(31)

In this equation, the Sines stress differentiated with respect to the stress components and the stress components differentiated with respect to the physical design variables do not need to be calculated directly for each stress cycle i. Consequently, only the computational inexpensive partial derivatives $\partial {D}_{j_i}/\partial {N}_{j_i}$ and $\partial {N}_{j_i}/\partial {\tilde{\sigma}}_{j_i}$ need to be calculated for every load cycle.

The partial derivative of the aggregated constraint function with respect to the accumulated damage is given by:

$$ \frac{\partial {g}_D}{\partial {D}_j}={\left(\sum_{e=1}^{n_e}\kern0.2em {D}_e^P\right)}^{\frac{1}{P}-1}\cdot {D}_j^{P-1} $$

(32)

The partial derivative of the damage with respect to the estimated amount of cycles to failure is:

$$ \frac{\partial {D}_{j_i}}{\partial {N}_{j_i}}=-{c}_D\frac{n_{j_i}}{{N_j}_i^2} $$

(33)

The partial derivative of the estimated cycles to failure with respect to the Sines equivalent stress is:

$$ \frac{\partial {N}_{j_i}}{\partial {\tilde{\sigma}}_{j_i}}=\frac{1}{2}\ {\left(\frac{\left(\frac{{\tilde{\sigma}}_{j_i}}{\sigma_f^{\hbox{'}}}\right)}{{\tilde{\sigma}}_{j_i}}\right)}^{\frac{1}{b}} $$

(34)

The partial derivatives of the Sines equivalent stress are found by:

$$ \frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\sigma}}_{j_x}}=\frac{1}{4}\ \frac{\left(4{\hat{\sigma}}_{j_x}-2{\hat{\sigma}}_{j_y}\right)\sqrt{2}}{\sqrt{{\left({\hat{\sigma}}_{j_x}-{\hat{\sigma}}_{j_y}\right)}^2+{\hat{\sigma}}_{j_x}^2+{\hat{\sigma}}_{j_y}^2+6{\hat{\tau}}_j^2}} $$

(35)

$$ \frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\sigma}}_{j_y}}=\frac{1}{4}\frac{\left(4{\hat{\sigma}}_{j_y}-2{\hat{\sigma}}_{j_x}\right)\sqrt{2}}{\sqrt{{\left({\hat{\sigma}}_{j_x}-{\hat{\sigma}}_{j_y}\right)}^2+{\hat{\sigma}}_{j_x}^2+{\hat{\sigma}}_{j_y}^2+6{\hat{\tau}}_j^2}} $$

(36)

$$ \frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\tau}}_j}=\frac{3{\hat{\tau}}_j\sqrt{2}}{\sqrt{{\left({\hat{\sigma}}_{j_x}-{\hat{\sigma}}_{j_y}\right)}^2+{\hat{\sigma}}_{j_x}^2+{\hat{\sigma}}_{j_y}^2+6{\hat{\tau}}_j^2}} $$

(37)

$$ \frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\sigma}}_{j_{mx}}}=\frac{\partial {\tilde{\sigma}}_j}{\partial {\hat{\sigma}}_{j_{my}}}=\frac{1}{2}\beta\ \sqrt{2} $$

(38)

Note that these values are unscaled and it is therefore not necessary to calculate these derivatives for each cycle i. Similarly, the stress components differentiated with respect to the physical design variables can be calculated independently of the cycles, i.e. without the scaling factors:

$$ \frac{\partial {\hat{\boldsymbol{\sigma}}}_j}{\partial {\tilde{x}}_j}=\left[\begin{array}{c}\hfill \frac{\partial {\hat{\sigma}}_{j_x}}{\partial {\tilde{x}}_j}\hfill \\ {}\hfill \frac{\partial {\hat{\sigma}}_{j_y}}{\partial {\tilde{x}}_j}\hfill \\ {}\hfill \frac{\partial {\hat{\tau}}_j}{\partial {\tilde{x}}_j}\hfill \end{array}\right]= q{\tilde{x}}_j{\left(\boldsymbol{x}\right)}^{q-1}{\boldsymbol{E}}_j{\boldsymbol{B}}_j{\hat{\boldsymbol{u}}}_j $$

(39)

Likewise, the partial derivative of the constraint function with respect to the amplitude and mean displacement, that are required to solve the adjoint problem can be found by:

$$ \frac{\partial {g}_D}{\partial {\boldsymbol{u}}_{j_a}}=\frac{\partial {g}_D}{\partial {D}_j}\sum_{i=1}^{n_{RF}}\left({c}_{a_i}\frac{\partial {D}_{j_i}}{\partial {\tilde{\sigma}}_{j_i}}\left(\frac{\partial {\tilde{\sigma}}_j}{\partial {\widehat{\sigma}}_{j_x}}\frac{\partial {\widehat{\sigma}}_{j_x}}{\partial {\widehat{\boldsymbol{u}}}_j}\right.\right.\left.\left.+\frac{\partial {\tilde{\sigma}}_j}{\partial {\widehat{\sigma}}_{j_y}}\frac{\partial {\widehat{\sigma}}_{j_y}}{\partial {\widehat{\boldsymbol{u}}}_j}+\frac{\partial {\tilde{\sigma}}_j}{\partial {\widehat{\tau}}_j}\frac{\partial {\widehat{\tau}}_j}{\partial {\widehat{\boldsymbol{u}}}_j}\right)\right) $$

(40)

$$ \frac{\partial {g}_D}{\partial {\boldsymbol{u}}_{j_m}}=\frac{\partial {g}_D}{\partial {D}_j}\sum_{i=1}^{n_{RF}}\left({c}_{m_i}\frac{\partial {D}_{j_i}}{\partial {\tilde{\sigma}}_{j_i}}\left(\frac{\partial {\tilde{\sigma}}_j}{\partial {\widehat{\sigma}}_{j_{m x}}}\frac{\partial {\widehat{\sigma}}_{j_x}}{\partial {\widehat{\boldsymbol{u}}}_j}\right.\right.\left.\left.+\frac{\partial {\tilde{\sigma}}_j}{\partial {\widehat{\sigma}}_{j_{m y}}}\frac{\partial {\widehat{\sigma}}_{j_y}}{\partial {\widehat{\boldsymbol{u}}}_j}\right)\right) $$

(41)

Here the partial derivatives of the amplitude and mean stress components with respect to the reference displacements are constant in each iteration and calculated by:

$$ \frac{\partial {\hat{\boldsymbol{\sigma}}}_j}{\partial {\hat{\boldsymbol{u}}}_j}=\left[\begin{array}{c}\hfill \frac{\partial {\hat{\sigma}}_{j_x}}{\partial {\hat{\boldsymbol{u}}}_j}\hfill \\ {}\hfill \frac{\partial {\hat{\sigma}}_{j_y}}{\partial {\hat{\boldsymbol{u}}}_j}\hfill \\ {}\hfill \frac{\partial {\hat{\tau}}_j}{\partial {\hat{\boldsymbol{u}}}_j}\hfill \end{array}\right]={\tilde{x}}_j{\left(\boldsymbol{x}\right)}^q{\boldsymbol{E}}_j{\boldsymbol{B}}_j,\kern1em \forall j $$

(42)

Note that the constitutive matrix and strain-displacement matrix are constant and equal for all elements in this work.

As can be seen from the above equations, the additional computational costs as compared with stress constraints are the analysis and derivatives of Basquin’s equation and Palmgren-Miner’s equation, assuming that the computational cost of the Sines equivalent stress criterion is similar to a stress criterion. Furthermore, the equations that must be evaluated for each cycle i are either computational inexpensive or can be found by linear scaling. Consequently, the sensitivity analysis for fatigue constraints can be done very efficiently.