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Topology optimization considering stress, fatigue and load uncertainties
Linköping University, Department of Management and Engineering, Solid Mechanics. Linköping University, Faculty of Science & Engineering.
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: urn:nbn:se:liu:diva-123008DOI: 10.3384/diss.diva-123008ISBN: 978-91-7685-883-7 (print)OAI: oai:DiVA.org:liu-123008DiVA: diva2:875698
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
List of papers
1. Stress constrained topology optimization
Open this publication in new window or tab >>Stress constrained topology optimization
2013 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 48, no 1, 33-47 p.Article in journal (Refereed) Published
Abstract [en]

This paper develops and evaluates a method for handling stress constraints in topology optimization. The stress constraints are used together with an objective function that minimizes mass or maximizes stiffness, and in addition, the traditional stiffness based formulation is discussed for comparison. We use a clustering technique, where stresses for several stress evaluation points are clustered into groups using a modified P-norm to decrease the number of stress constraints and thus the computational cost. We give a detailed description of the formulations and the sensitivity analysis. This is done in a general manner, so that different element types and 2D as well as 3D structures can be treated. However, we restrict the numerical examples to 2D structures with bilinear quadrilateral elements. The three formulations and different approaches to stress constraints are compared using two well known test examples in topology optimization: the L-shaped beam and the MBB-beam. In contrast to some other papers on stress constrained topology optimization, we find that our formulation gives topologies that are significantly different from traditionally optimized designs, in that it actually manage to avoid stress concentrations. It can therefore be used to generate conceptual designs for industrial applications.

Keyword
Topology optimization, Stress constraints, Clusters, SIMP, MMA
National Category
Engineering and Technology
Identifiers
urn:nbn:se:liu:diva-88092 (URN)10.1007/s00158-012-0880-7 (DOI)000320865900003 ()
Available from: 2013-01-30 Created: 2013-01-30 Last updated: 2017-12-06Bibliographically approved
2. Fatigue constrained topology optimization
Open this publication in new window or tab >>Fatigue constrained topology optimization
2014 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 50, no 2, 207-219 p.Article in journal (Refereed) Published
Abstract [en]

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 minimal mass that can withstand prescribed 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 makes 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.

Place, publisher, year, edition, pages
Springer-Verlag New York, 2014
National Category
Mechanical Engineering
Identifiers
urn:nbn:se:liu:diva-88093 (URN)10.1007/s00158-014-1054-6 (DOI)000339944100003 ()
Available from: 2013-01-30 Created: 2013-01-30 Last updated: 2017-12-06Bibliographically approved
3. Worst-case topology optimization of self-weight loaded structures using semi-definite programming
Open this publication in new window or tab >>Worst-case topology optimization of self-weight loaded structures using semi-definite programming
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.

Keyword
Topology optimization; Semi-definite programming; Worst-case compliance; Self-weight; Robust optimization
National Category
Applied Mechanics Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-123002 (URN)10.1007/s00158-015-1285-1 (DOI)000366590800006 ()
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-12-01Bibliographically approved
4. Large-scale robust topology optimization under load-uncertainty
Open this publication in new window or tab >>Large-scale robust topology optimization under load-uncertainty
2015 (English)In: Advances in Structural and Multidisciplinary Optimization - Proceedings of the 11th World Congress of Structural and Multidisciplinary Optimization(WCSMO-11) / [ed] Qing Li, Grant P Steven and Zhongpu (Leo) Zhang, 2015, 255-260 p.Conference paper, Published paper (Refereed)
Abstract [en]

Structures designed by topology optimization (TO) are frequently sensitive to loads different from the ones accounted for in the optimization. In extreme cases this means that loads differing ever so slightly from the ones it was designed to carry may cause a structure to collapse. It is therefore clear that handling uncertainty regarding the actual loadings is important. To address this issue in a systematic  manner is one of the main goals in the field of robust TO. In this work we present a deterministic robust formulation of TO for maximum stiffness design which accounts for uncertain variations around a set of nominal loads. The idea is to find a design which minimizes the maximum compliance obtained as the loads vary in infinite, so-called uncertainty sets. This naturally gives rise to a semi-infinite optimization problem, which we here reformulate into a non-linear, semi-definite program. With appropriate numerical algorithms this optimization problem can be solved at a cost similar to that of solving a standard multiple load-case TO problem with the number of loads equal to the number of spatial dimensions plus one, times the number of nominal loads. In contrast to most previously suggested methods, which can only be applied to small-scale problems, the presented method is – as illustrated by a numerical example – well-suited for large-scale TO problems.

Keyword
Robust optimization, Topology optimization, Large-scale optimization, Non-linear semi-definite programming
National Category
Applied Mechanics
Identifiers
urn:nbn:se:liu:diva-123004 (URN)978-0-646-94394-7 (ISBN)
Conference
11th World Congress on Structural and Multidisciplinary Optimization (WCSMO-11), Sydney Australia, 7–12 June
Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-05-15Bibliographically approved
5. Game theory approach to robust topology optimization with uncertain loading
Open this publication in new window or tab >>Game theory approach to robust topology optimization with uncertain loading
2017 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 55, no 4, 1383-1397 p.Article in journal (Refereed) Published
Abstract [en]

The paper concerns robustness with respect to uncertain loading in topology optimization problems with essentially arbitrary objective functions and constraints. Using a game theoretic framework we formulate problems, or games, defining Nash equilibria. In each game a set of topology design variables aim to find an optimal topology, while a set of load variables aim to find the worst possible load. Several numerical examples with uncertain loading are solved in 2D and 3D. The games are formulated using global stress, mass and compliance as objective functions or constraints.

Place, publisher, year, edition, pages
Springer, 2017
Keyword
Topology optimization, Robust optimization, Game theory, Nash equilibrium, Stress constraints
National Category
Applied Mechanics
Identifiers
urn:nbn:se:liu:diva-123006 (URN)10.1007/s00158-016-1548-5 (DOI)000398951100015 ()
Note

Funding agencies: NFFP [2013-01221]; Swedish Armed Forces; Swedish Defence Materiel Administration; Swedish Governmental Agency for Innovation Systems; Swedish Foundation for Strategic Research [AM13-0029]

Available from: 2015-12-01 Created: 2015-12-01 Last updated: 2017-05-18Bibliographically approved

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