# Analysis of optimality conditions for some topology optimization problems in elasticity

International Journal of Applied Mathematics and Computer Science (2002)

- Volume: 12, Issue: 1, page 127-137
- ISSN: 1641-876X

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topTrabucho, Luis. "Analysis of optimality conditions for some topology optimization problems in elasticity." International Journal of Applied Mathematics and Computer Science 12.1 (2002): 127-137. <http://eudml.org/doc/207564>.

@article{Trabucho2002,

abstract = {The subject of topology optimization has undergone an enormous practical development since the appearance of the paper by Bendso e and Kikuchi (1988), where some ideas from homogenization theory were put into practice. Since then, several engineering applications as well as different approaches have been developed successfully. However, it is difficult to find in the literature some analytical examples that might be used as a test in order to assess the validity of the solutions obtained with different algorithms. As a matter of fact, one is often faced with numerical instabilities requiring a fine tuning of the algorithm for each specific case. In this work, we develop a family of analytical solutions for very simple topology optimization problems, in the framework of elasticity theory, including bending and extension of rods, torsion problems as well as plane stress and plane strain elasticity problems. All of these problems are formulated in a simplified theoretical framework. A key issue in this type of problems is to be able to evaluate the sensitivity of the homogenized elastic coefficients with respect to the microstructure parameter(s). Since we are looking for analytical solutions, we use laminates for which an explicit dependence of the homogenized coefficients on the microstructure is known.},

author = {Trabucho, Luis},

journal = {International Journal of Applied Mathematics and Computer Science},

keywords = {elasticity; optimality conditions; topology optimization},

language = {eng},

number = {1},

pages = {127-137},

title = {Analysis of optimality conditions for some topology optimization problems in elasticity},

url = {http://eudml.org/doc/207564},

volume = {12},

year = {2002},

}

TY - JOUR

AU - Trabucho, Luis

TI - Analysis of optimality conditions for some topology optimization problems in elasticity

JO - International Journal of Applied Mathematics and Computer Science

PY - 2002

VL - 12

IS - 1

SP - 127

EP - 137

AB - The subject of topology optimization has undergone an enormous practical development since the appearance of the paper by Bendso e and Kikuchi (1988), where some ideas from homogenization theory were put into practice. Since then, several engineering applications as well as different approaches have been developed successfully. However, it is difficult to find in the literature some analytical examples that might be used as a test in order to assess the validity of the solutions obtained with different algorithms. As a matter of fact, one is often faced with numerical instabilities requiring a fine tuning of the algorithm for each specific case. In this work, we develop a family of analytical solutions for very simple topology optimization problems, in the framework of elasticity theory, including bending and extension of rods, torsion problems as well as plane stress and plane strain elasticity problems. All of these problems are formulated in a simplified theoretical framework. A key issue in this type of problems is to be able to evaluate the sensitivity of the homogenized elastic coefficients with respect to the microstructure parameter(s). Since we are looking for analytical solutions, we use laminates for which an explicit dependence of the homogenized coefficients on the microstructure is known.

LA - eng

KW - elasticity; optimality conditions; topology optimization

UR - http://eudml.org/doc/207564

ER -

## References

top- Bendsøe M.P. and Kikuchi N. (1988): Generating optimal topologies in structural design using a homogenization method. -Comp. Meth. Appl. Mech. Eng., No. 71, pp. 192-224. Zbl0671.73065
- Bendsøe M.P. (1995): Optimization of Structural Topology, Shape, and Material. - Berlin: Springer. Zbl0822.73001
- Chenais D. (1987): Optimal design of midsurface of shells: differentiability proof and sensitivity computation. - Appl. Math.Optim., No. 16, pp. 93-133. Zbl0626.73097
- Trabucho L. and Viaño J.M. (1996): Mathematical Modeling of Rods, In: Handbook of Numerical Analysis IV (P.G. Ciarlet and J.L. Lions, Eds.). - Amsterdam: Elsevier. Zbl0873.73041

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