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# Data-fitted constraint aggregation schemes

Publikationstyp

Conference Presentation

Publikationsdatum

2022-06

Sprache

English

Citation

8th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2022)

Contribution to Conference

Peer Reviewed

true

Constraint aggregation is a way to reduce the size of the dual problem of a constrained optimization problem by replacing nc similar constraints with a single, aggregated con-straint. In this contribution, especially the local volume constraints by Wu et al. [1] are

studied, since employing them in a topology optimization (TO) problem usually leads to one constraint per finite element of the model.

Commonly used constraint aggregation functions, like e.g. the p-norm, p-mean, or Kreisselmeier-Steinhauser (KS) [2] function, include a tunable aggregation parameter p. Increasing p also increases the accuracy of the approximation of the maximum (worst) value

of all nc original constraints by the aggregated constraint. On the downside, an increase of p also increases the nonlinearity of the aggregated constraint, possibly leading to convergence issues. Therefore, it is desirable to keep p as low as possible without compromising the quality of the approximation by the aggregated constraint.

However, the choice of a speci c aggregation parameter p is often not justi ed in the literature. Local volume constraints are used in the context of TO e.g. to enforce redundant designs [1], for which the robustness to partial failure is implicitly increased, or to enforce

a maximum feature size [3]. Although the same constraint equations can be used for both cases, a value of p = 16 is sufficient in [1], and, depending on the studied geometry, values up to p = 300 are used in [3]. In this contribution, very high values of p are seen

as an indication, that the chosen aggregation function is not well suited for the studied problem. Therefore, alternative aggregation functions based on the p-norm are introduced, that offer an additional parameter to adapt the aggregation characteristics to the expected

distribution of the data being aggregated. This results in higher accuracy also for very low p values when compared to known aggregation functions.

REFERENCES

[1] Wu, J., Aage, N.,Westermann, R., and Sigmund, O., Infi ll Optimization for Additive Manu-

facturing -- Approaching Bone-like Porous Structures. IEEE Transactions on Visualization

and Computer Graphics, Vol. 24(2), pp. 1127--1140, 2018.

[2] Kreisselmeier, G. and Steinhauser, R., Systematic Control Design by Optimizing a Vector

Performance Index. IFAC Proceedings Volumes, Vol. 12(7), pp. 113--117, 1979.

[3] Fernández, E., Collet, M., Alarcón, P., Bauduin, S. and Duysinx, P., An aggregation strat-

egy of maximum size constraints in density-based topology optimization. Structural and

Multidisciplinary Optimization, Vol. 60(5), pp. 2113--2130, 2019.

studied, since employing them in a topology optimization (TO) problem usually leads to one constraint per finite element of the model.

Commonly used constraint aggregation functions, like e.g. the p-norm, p-mean, or Kreisselmeier-Steinhauser (KS) [2] function, include a tunable aggregation parameter p. Increasing p also increases the accuracy of the approximation of the maximum (worst) value

of all nc original constraints by the aggregated constraint. On the downside, an increase of p also increases the nonlinearity of the aggregated constraint, possibly leading to convergence issues. Therefore, it is desirable to keep p as low as possible without compromising the quality of the approximation by the aggregated constraint.

However, the choice of a speci c aggregation parameter p is often not justi ed in the literature. Local volume constraints are used in the context of TO e.g. to enforce redundant designs [1], for which the robustness to partial failure is implicitly increased, or to enforce

a maximum feature size [3]. Although the same constraint equations can be used for both cases, a value of p = 16 is sufficient in [1], and, depending on the studied geometry, values up to p = 300 are used in [3]. In this contribution, very high values of p are seen

as an indication, that the chosen aggregation function is not well suited for the studied problem. Therefore, alternative aggregation functions based on the p-norm are introduced, that offer an additional parameter to adapt the aggregation characteristics to the expected

distribution of the data being aggregated. This results in higher accuracy also for very low p values when compared to known aggregation functions.

REFERENCES

[1] Wu, J., Aage, N.,Westermann, R., and Sigmund, O., Infi ll Optimization for Additive Manu-

facturing -- Approaching Bone-like Porous Structures. IEEE Transactions on Visualization

and Computer Graphics, Vol. 24(2), pp. 1127--1140, 2018.

[2] Kreisselmeier, G. and Steinhauser, R., Systematic Control Design by Optimizing a Vector

Performance Index. IFAC Proceedings Volumes, Vol. 12(7), pp. 113--117, 1979.

[3] Fernández, E., Collet, M., Alarcón, P., Bauduin, S. and Duysinx, P., An aggregation strat-

egy of maximum size constraints in density-based topology optimization. Structural and

Multidisciplinary Optimization, Vol. 60(5), pp. 2113--2130, 2019.