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Worst-case topology optimization of self-weight loaded structures using semi-definite programming
Linköping University, Department of Management and Engineering, Solid Mechanics. Linköping University, Faculty of Science & Engineering. Saab AB, SE 581 88, Linköping, Sweden .
Linköping University, Department of Management and Engineering, Solid Mechanics. Linköping University, Faculty of Science & Engineering.
Linköping University, Department of Management and Engineering, Solid Mechanics. Linköping University, Faculty of Science & Engineering.ORCID iD: 0000-0001-8460-0131
2015 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 52, no 5, 915-928 p.Article in journal (Refereed) Published
Abstract [en]

The paper concerns worst-case compliance optimization by finding the structural topology with minimum compliance for the loading due to the worst possible acceleration of the structure and attached non-structural masses. A main novelty of the paper is that it is shown how this min-max problem can be formulated as a non-linear semi-definite programming (SDP) problem involving a small-size constraint matrix and how this problem is solved numerically. Our SDP formulation is an extension of an eigenvalue problem seen previously in the literature; however, multiple eigenvalues naturally arise which makes the eigenvalue problem non-smooth, whereas the SDP problem presented in this paper provides a computationally tractable problem. Optimized designs, where the uncertain loading is due to acceleration of applied masses and the weight of the structure itself, are shown in two and three dimensions and we show that these designs satisfy optimality conditions that are also presented.

Place, publisher, year, edition, pages
2015. Vol. 52, no 5, 915-928 p.
Keyword [en]
Topology optimization; Semi-definite programming; Worst-case compliance; Self-weight; Robust optimization
National Category
Applied Mechanics Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-123002DOI: 10.1007/s00158-015-1285-1ISI: 000366590800006OAI: oai:DiVA.org:liu-123002DiVA: diva2:875660
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved
In thesis
1. Topology optimization considering stress, fatigue and load uncertainties
Open this publication in new window or tab >>Topology optimization considering stress, fatigue and load uncertainties
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This dissertation concerns structural topology optimization in conceptual design stages. The objective of the project has been to identify and solve problems that prevent structural topology optimization from being used in a broader sense in the avionic industry; therefore the main focus has been on stress and fatigue constraints and robustness with respect to load uncertainties.

The thesis consists of two parts. The first part gives an introduction to topology optimization, describes the new contributions developed within this project and motivates why these are important. The second part includes five papers.

The first paper deals with stress constraints and a clustered approach is presented where stress constraints are applied to stress clusters, instead of being defined for each point of the structure. Different approaches for how to create and update the clusters, such that sufficiently accurate representations of the local stresses are obtained at a reasonable computational cost, are developed and evaluated.

High-cycle fatigue constraints are developed in the second paper, where loads described by a variable-amplitude load spectrum and material data from fatigue tests are used to determine a limit stress, for which below fatigue failure is not expected. A clustered approach is then used to constrain the tensile principal stresses below this limit.

The third paper introduces load uncertainties and stiffness optimization considering the worst possible loading is then formulated as a semi-definite programming problem, which is solved very efficiently. The load is due to acceleration of point masses attached to the structure and the mass of the structure itself, and the uncertainty concerns the direction of the acceleration. The fourth paper introduces an extension to the formulated semi-definite programming problem such that both fixed and uncertain loads can be optimized for simultaneously.

Game theory is used in the fifth paper to formulate a general framework, allowing essentially any differentiable objective and constraint functions, for topology optimization under load uncertainty. Two players, one controlling the structure and one the loads, are in conflict such that a solution to the game, a Nash equilibrium, is a design optimized for the worst possible load.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2016. 63 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1730
National Category
Applied Mechanics
Identifiers
urn:nbn:se:liu:diva-123008 (URN)10.3384/diss.diva-123008 (DOI)978-91-7685-883-7 (ISBN)
Public defence
2016-01-15, C3, C-huset, Campus Valla, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-05-15Bibliographically approved

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Holmberg, ErikThore, Carl-JohanKlarbring, Anders

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