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On Methods for Discrete Topology Optimization of Continuum Structures
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Optimization and Systems Theory.
2008 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

This thesis consists of an introduction and seven appended papers. The purpose of the introduction is to give an overview of the field of topology optimization of discretized load carrying continuum structures. It is assumed that the design domain has been discretized by the finite element method and that the design variable vector is a binary vector indicating presence or absence of material in the various finite elements. Common to all papers is the incorporation of von Mises stresses in the problem formulations.

In the first paper the design variables are binary but it is assumed that the void structure can actually take some load. This is equivalent to adding a small positive value, epsilon, to all design variables, both those that are void and those that are filled with material. With this small positive lower bound the stiffness matrix becomes positive definite for all designs. If only one element is changed (from material to void or from void to material) the new global stiffness matrix is just a low rank modification of the old one and thus the Sherman-Morrison-Woodbury formula can be used to compute the displacements in the neighbouring designs efficiently. These efficient sensitivity calculations can then be applied in the context of a neighbourhood search method. Since the computed displacements are exact in the 1-neighbourhood (when one design variable is changed) the neighbourhood search method will find a local optimum with respect to the 1-neighbourhood.

The second paper presents globally optimal zero-one solutions to some small scale topology optimization problems defined on discretized continuum design domains. The idea is that these solutions can be used as benchmarks when testing new algorithms for finding pure zero-one solutions to topology optimization problems.

In the third paper the results from the first paper are extended to include also the case where there is no epsilon>0. In this case the stiffness matrix will no longer be positive definite which means that the Sherman-Morrison-Woodbury formula can no longer be applied. The changing of one or two binary design variables to their opposite binary values will still result in a low rank change, but the size of the reduced stiffness matrix will change with the design. It turns out, however, that it is possible to compute the effect of these low rank changes efficiently also without the positive lower bound. These efficient sensitivity calculations can then be used in the framework of a neighbourhood search method. In this case the complete 1-neighbourhood and a subset of the 2-neighbourhood is investigated in the search for a locally optimal solution.

In the fourth paper the sensitivity calculations developed in the third paper are used to generate first and partial second order approximations of the nonlinear functions usually present in topology optimization problems. These approximations are then used to generate subproblems in two different sequential integer programming methods (SLIP and SQIP, respectively). Both these methods generate a sequence of iteration points that can be proven to converge to a local optimum with respect to the 1-neighbourhood. The methods are tested on some different topology optimization problems.

The fifth paper demonstrates that the SLIP method developed in the previous paper can be applied also to the mechanism design problem with stress constraints. In order to generate the subproblems in a fast way small displacements are assumed, which implies that the efficient sensitivity calculations derived in the third paper can be used. The numerical results indicate that the method can be used to lower the stresses and still get a functional mechanism.

In the sixth paper the SLIP method developed in the fourth paper is used as a post processor to obtain locally optimal zero-one solutions starting from a rounded solution to the corresponding continuous problem. The numerical results indicate that the method can perform well as a post processor.

The seventh paper is a theoretical paper that investigates the validity of the commonly used positive lower bound epsilon on the design variables when stating and solving topology optimization problems defined on discretized load carrying continuum structures. The main result presented here is that an optimal "epsilon-1" solution to an "epsilon-perturbed" discrete minimum weight problem with constraints on compliance, von Mises stresses and strain energy densities, is optimal, after rounding to zero-one, to the corresponding "unperturbed" discrete problem. This holds if the constraints in the perturbed problem are carefully defined and epsilon>0 is sufficiently small.

Place, publisher, year, edition, pages
Stockholm: KTH , 2008. , xi, 26 p.
Series
Trita-MAT. OS, ISSN 1401-2294 ; 08:01
Keyword [en]
Topology optimization, Stress constraints, Sensitivity calculations, Neighbourhood search methods, Sequential integer programming
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-4644ISBN: 978-91-7178-872-6 (print)OAI: oai:DiVA.org:kth-4644DiVA: diva2:13221
Public defence
2008-03-14, D3, Lindstedtsv. 5, Stockholm, 10:00
Opponent
Supervisors
Note
QC 20100917Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2010-09-17Bibliographically approved
List of papers
1. A hierarchical neighbourhood search method for topology optimization
Open this publication in new window or tab >>A hierarchical neighbourhood search method for topology optimization
2005 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 29, no 5, 325-340 p.Article in journal (Refereed) Published
Abstract [en]

This paper presents a hierarchical neighbourhood search method for solving topology optimization problems defined on discretized linearly elastic continuum structures. The design of the structure is represented by binary design variables indicating material or void in the various finite elements.

Two different designs are called neighbours if they differ in only one single element, in which one of them has material while the other has void. The proposed neighbourhood search method repeatedly jumps to the "best" neighbour of the current design until a local optimum has been found, where no further improvement can be made. The "engine" of the method is an efficient exploitation of the fact that if only one element is changed (from material to void or from void to material) then the new global stiffness matrix is just a low-rank modification of the old one. To further speed up the process, the method is implemented in a hierarchical way. Starting from a coarse finite element mesh, the neighbourhood search is repeatedly applied on finer and finer meshes.

Numerical results are presented for minimum-weight problems with constraints on respectively compliance, strain energy densities in all non-void elements, and von Mises stresses in all non-void elements.

Keyword
neighbourhood search, stress constraints, topology optimization
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8008 (URN)10.1007/s00158-004-0493-x (DOI)000229106000001 ()2-s2.0-18744364715 (Scopus ID)
Note
QC 20100917Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2017-12-14Bibliographically approved
2. Globally optimal benchmark solutions to some small-scale discretized continuum topology optimization problems
Open this publication in new window or tab >>Globally optimal benchmark solutions to some small-scale discretized continuum topology optimization problems
2006 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 32, no 3, 259-262 p.Article in journal (Refereed) Published
Abstract [en]

In this note, globally optimal solutions to three sets of small-scale discretized continuum topology optimization problems are presented. All the problems were discretized by the use of nine-node isoparametric finite elements. The idea is that these solutions can be used as benchmark problems when testing new algorithms for finding pure 0-1 solutions to topology optimization problems defined on discretized ground structures.

Keyword
benchmark problems, global optimum, topology optimization
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8009 (URN)10.1007/s00158-006-0015-0 (DOI)000248944600009 ()2-s2.0-33746656874 (Scopus ID)
Note
QC 20100917Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2017-12-14Bibliographically approved
3. Topology optimization by a neighbourhood search method based on efficient sensitivity calculations
Open this publication in new window or tab >>Topology optimization by a neighbourhood search method based on efficient sensitivity calculations
2006 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 67, no 12, 1670-1699 p.Article in journal (Refereed) Published
Abstract [en]

This paper deals with topology optimization of discretized load-carrying continuum structures, where the design of the structure is represented by binary design variables indicating material or void in the various finite elements. Efficient exact methods for discrete sensitivity calculations are developed. They utilize the fact that if just one or two binary variables are changed to their opposite binary values then the new stiffness matrix is essentially just a low-rank modification of the old stiffness matrix, even if some nodes in the structure may disappear or re-enter. As an application of these efficient sensitivity calculations, a new neighbourhood search method is presented, implemented, and applied on some test problems, one of them with 6912 nine-node finite elements where the von Mises stress in each non-void element is considered.

Keyword
topology optimization, von Mises stresses, neighbourhood search, 0-1 problems, sensitivities
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8010 (URN)10.1002/nme.1677 (DOI)000240611600002 ()2-s2.0-33748636231 (Scopus ID)
Note
QC 20100917Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2017-12-14Bibliographically approved
4. Sequential integer programming methods for stress constrained topology optimization
Open this publication in new window or tab >>Sequential integer programming methods for stress constrained topology optimization
2007 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 34, no 4, 277-299 p.Article in journal (Refereed) Published
Abstract [en]

This paper deals with topology optimization of load carrying structures defined on a discretized design domain where binary design variables are used to indicate material or void in the various finite elements. The main contribution is the development of two iterative methods which are guaranteed to find a local optimum with respect to a 1-neighbourhood. Each new iteration point is obtained as the optimal solution to an integer linear programming problem which is an approximation of the original problem at the previous iteration point. The proposed methods are quite general and can be applied to a variety of topology optimization problems defined by 0-1 design variables. Most of the presented numerical examples are devoted to problems involving stresses which can be handled in a natural way since the design variables are kept binary in the subproblems.

Keyword
topology optimization, stress constraints, sequential integer programming
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8011 (URN)10.1007/s00158-007-0118-2 (DOI)000255419500001 ()2-s2.0-34548203501 (Scopus ID)
Note
QC 20100917Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2017-12-14Bibliographically approved
5. Designing compliant mechanisms with stress constraints using sequential linear integer programming
Open this publication in new window or tab >>Designing compliant mechanisms with stress constraints using sequential linear integer programming
(English)Manuscript (Other academic)
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8012 (URN)
Note
QC 20100917Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2010-09-17Bibliographically approved
6. Using the sequential linear integer programming method as a post-processor for stress-constrained topology optimization problems
Open this publication in new window or tab >>Using the sequential linear integer programming method as a post-processor for stress-constrained topology optimization problems
2008 (English)In: International Journal for Numerical Methods in Engineering, ISSN 0029-5981, E-ISSN 1097-0207, Vol. 76, no 10, 1544-1567 p.Article in journal (Refereed) Published
Abstract [en]

This paper deals with topology optimization of load-carrying structures defined on discretized continuum design domains. In particular, the mininium compliance problem with stress constraints is considered. The finite element method is used to discretize the design domain into n finite elements and the design of a certain structure is represented by an n-dimensional binary design variable vector. In order to solve the problems, the binary constraints on the design variables are initially relaxed and the problems are solved with both the method of moving asymptotes and the sparse non-linear optimizer solvers for continuous optimization in order to compare the two solvers. By solving a sequence of problems with a sequentially lower limit on the amount of grey allowed, designs that are close to 'black-and-white' are obtained. In order to get locally optimal solutions that are purely {0, 1}(n). a sequential linear integer programming method is applied as a post-processor. Numerical results are presented for some different test problems.

Keyword
topology optimization, stress constraints, 0-1 problems, post-processing
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8013 (URN)10.1002/nme.2378 (DOI)000261277900005 ()2-s2.0-60949087151 (Scopus ID)
Note
QC 20100917. Uppdaterad från submitted till published (20100917).Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2017-12-14Bibliographically approved
7. On the validity of using small positive lower bounds on design variables in discrete topology optimization
Open this publication in new window or tab >>On the validity of using small positive lower bounds on design variables in discrete topology optimization
2009 (English)In: Structural and multidisciplinary optimization (Print), ISSN 1615-147X, E-ISSN 1615-1488, Vol. 37, no 4, 325-334 p.Article in journal (Refereed) Published
Abstract [en]

It is proved that an optimal {epsilon, 1}(n) solution to a "epsilon-perturbed" discrete minimum weight problem with constraints on compliance, von Mises stresses and strain energy densities, is optimal, after rounding to {0, 1}(n), to the corresponding "unperturbed" discrete problem, provided that the constraints in the perturbed problem are carefully defined and epsilon > 0 is sufficiently small.

Keyword
Topology optimization, Discrete problems, Perturbation
National Category
Computational Mathematics
Identifiers
urn:nbn:se:kth:diva-8014 (URN)10.1007/s00158-008-0248-1 (DOI)000261037600001 ()2-s2.0-56749163912 (Scopus ID)
Note
QC 20100917. Uppdaterad från accepted till published (20100917).Available from: 2008-02-21 Created: 2008-02-21 Last updated: 2017-12-14Bibliographically approved

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