Research Papers

On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems

[+] Author and Article Information
Stefan Erschen

Vehicle Dynamics, Preliminary Design,
BMW Group Research and Innovation Center,
Knorrstrasse 147,
Munich 80788, Germany
e-mail: stefan.erschen@mytum.de

Fabian Duddeck

Computational Mechanics,
Technical University of Munich,
Arcisstr. 21,
Munich 80333, Germany
e-mail: duddeck@tum.de

Matthias Gerdts

Institut für Mathematik and Rechneranwendung,
Universität der Bundeswehr München,
Werner-Heisenberg-Weg 39, Neubiberg,
Munich 85577, Germany
e-mail: matthias.gerdts@unibw.de

Markus Zimmermann

Vehicle Dynamics, Preliminary Design,
BMW Group Research and Innovation Center,
Knorrstrasse 147,
Munich 80788, Germany
e-mail: markusz@alum.mit.edu

1Corresponding author.

Manuscript received February 8, 2017; final manuscript received July 9, 2017; published online October 5, 2017. Assoc. Editor: Alba Sofi.

ASME J. Risk Uncertainty Part B 4(2), 021008 (Oct 05, 2017) (15 pages) Paper No: RISK-17-1011; doi: 10.1115/1.4037485 History: Received February 08, 2017; Revised July 09, 2017

In the early development phase of complex technical systems, uncertainties caused by unknown design restrictions must be considered. In order to avoid premature design decisions, sets of good designs, i.e., designs which satisfy all design goals, are sought rather than one optimal design that may later turn out to be infeasible. A set of good designs is called a solution space and serves as target region for design variables, including those that quantify properties of components or subsystems. Often, the solution space is approximated, e.g., to enable independent development work. Algorithms that approximate the solution space as high-dimensional boxes are available, in which edges represent permissible intervals for single design variables. The box size is maximized to provide large target regions and facilitate design work. As a result of geometrical mismatch, however, boxes typically capture only a small portion of the complete solution space. To reduce this loss of solution space while still enabling independent development work, this paper presents a new approach that optimizes a set of permissible two-dimensional (2D) regions for pairs of design variables, so-called 2D-spaces. Each 2D-space is confined by polygons. The Cartesian product of all 2D-spaces forms a solution space for all design variables. An optimization problem is formulated that maximizes the size of the solution space, and is solved using an interior-point algorithm. The approach is applicable to arbitrary systems with performance measures that can be expressed or approximated as linear functions of their design variables. Its effectiveness is demonstrated in a chassis design problem.

Copyright © 2018 by ASME
Topics: Space , Design , Optimization
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Boyd, S. , and Vandenberghe, L. , 2004, Convex Optimization, Cambridge University Press, Cambridge, UK. [CrossRef]
Beyer, H.-G. , and Sendhoff, B. , 2007, “ Robust Optimization—A Comprehensive Survey,” Comput. Methods Appl. Mech. Eng., 196(33–34), pp. 3190–3218. [CrossRef]
Al-Ashaab, A. , Howell, S. , Usowicz, K. , Hernando Anta, P. , and Gorka, A. , 2009, “ Set-Based Concurrent Engineering Model for Automotive Electronic/Software Systems Development,” 19th CIRP Design Conference–Competitive Design, Cranfield, UK, Mar. 30–31. https://core.ac.uk/download/pdf/139040.pdf?repositoryId=23
Sobek, D. K., II , Ward, A. C. , and Liker, J. K. , 1999, “ Toyota's Principles of Set-Based Concurrent Engineering,” MIT Sloan Manage. Rev., 40(2), pp. 67–83. https://www.researchgate.net/publication/248139929_Toyota%27s_Principles_of_Set-Based_Concurrent_Engineering
Zimmermann, M. , and Hoessle, J. E. , 2013, “ Computing Solution Spaces for Robust Design,” Int. J. Numer. Methods Eng., 94(3), pp. 290–307. [CrossRef]
Graff, L. , 2013, “ A Stochastic Algorithm for the Identification of Solution Spaces in High-Dimensional Design Spaces,” Ph.D. thesis, University of Basel, Basel, Switzerland. http://edoc.unibas.ch/30278/
Fender, J. , Graff, L. , Harbrecht, H. , and Zimmermann, M. , 2014, “ Identifying Key Parameters for Design Improvement in High-Dimensional Systems With Uncertainty,” ASME J. Mech. Des., 136(4), p. 041007. [CrossRef]
Lehar, M. , and Zimmermann, M. , 2012, “ An Inexpensive Estimate of Failure Probability for High-Dimensional Systems With Uncertainty,” Struct. Saf., 36–37, pp. 32–38. [CrossRef]
Eichstetter, M. , Redeker, C. , Kvasnicka, P. , Müller, S. , and Zimmermann, M. , 2014, “ Solution Spaces for Damper Design in Vehicle Dynamics,” Fifth International Munich Chassis Symposium, Munich, Germany, June 24–25, pp. 107–132.
Münster, M. , Lehner, M. , Rixen, D. , and Zimmermann, M. , 2014, “ Vehicle Steering Design Using Solution Spaces for Decoupled Dynamical Subsystems,” 26th Conference on Noise and Vibration Engineering (ISMA), Leuven, Belgium, Sept. 15–17, pp. 279–288.
Graff, L. , Harbrecht, H. , and Zimmermann, M. , 2012, “ On the Computation of Solution Spaces in High Dimensions,” Struct. Multidiscip. Optim., 54(4), pp. 811–829. [CrossRef]
Eichstetter, M. , Müller, S. , and Zimmermann, M. , 2015, “ Product Family Design With Solution Spaces,” ASME J. Mech. Des., 137(12), p. 121401. [CrossRef]
Fender, J. H. W. , 2014, “ Solution Spaces for Vehicle Crash Design,” Ph.D. thesis, Technical University of Munich, Munich, Germany. http://www.shaker.nl/Online-Gesamtkatalog-Download/2017.08.11-11.19.53-
Erschen, S. , Duddeck, F. , and Zimmermann, M. , 2015, “ Robust Design Using Classical Optimization,” PAMM, 15(1), pp. 565–566. [CrossRef]
Beer, M. , and Liebscher, M. , 2008, “ Designing Robust Structures—A Nonlinear Simulation Based Approach,” Comput. Struct., 86(10), pp. 1102–1122. [CrossRef]
Götz, M. , Liebscher, M. , and Graf, W. , 2012, “ Efficient Detection of Permissible Design Spaces in an Early Design Stage,” 11th LS-Dyna Forum, Ulm, Germany, Oct. 9–10, pp. 9–10. https://www.dynamore.de/de/download/papers/ls-dyna-forum-2012/documents/optimization-1-1
Rocco, C. M. , Moreno, J. A. , and Carrasquero, N. , 2003, “ Robust Design Using a Hybrid-Cellular-Evolutionary and Interval-Arithmetic Approach: A Reliability Application,” Reliab. Eng. Syst. Saf., 79(2), pp. 149–159. [CrossRef]
Fender, J. , Duddeck, F. , and Zimmermann, M. , 2017, “ Direct Computation of Solution Spaces,” Struct. Multidiscip. Optim., 55(5), pp. 1787–1796. [CrossRef]
ISO, 2012, “ Passengers Cars—Steady-State Circular Driving Behavior—Open Loop Test Methods,” International Organization for Standardization, Geneva, Switzerland, Standard No. ISO 4138-2012. https://www.iso.org/standard/54143.html
Fung, G. , Sandilya, S. , and Rao, R. B. , 2005, “ Rule Extraction From Linear Support Vector Machines,” 11th ACM SIGKDD International Conference on Knowledge Discovery in Data Mining (KDD), Chicago, IL, Aug. 21–24, pp. 32–40.
Allgower, E. L. , and Schmidt, P. H. , 1986, “ Computing Volumes of Polyhedra,” Math. Comput., 46(173), pp. 171–174. [CrossRef]
Gerdts, M. , and Lempio, F. , 2011, Mathematische Optimierungsverfahren des Operations Research, Walter de Gruyter, Berlin. [CrossRef]
Byrd, R. H. , Gilbert, J. C. , and Nocedal, J. , 2000, “ A Trust Region Method Based on Interior Point Techniques for Nonlinear Programming,” Math. Program., 89(1), pp. 149–185. [CrossRef]
Byrd, R. H. , Hribar, M. E. , and Nocedal, J. , 1999, “ An Interior Point Algorithm for Large-Scale Nonlinear Programming,” SIAM J. Optim., 9(4), pp. 877–900. [CrossRef]
Waltz, R. A. , Morales, J. L. , Nocedal, J. , and Orban, D. , 2006, “ An Interior Algorithm for Nonlinear Optimization That Combines Line Search and Trust Region Steps,” Math. Program., 107(3), pp. 391–408. [CrossRef]


Grahic Jump Location
Fig. 1

Front axle of a passenger vehicle with chassis components damper (1), bump stop (2), rebound stop (3), and anti-roll bar (4)

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Fig. 2

Complete solution space (white area) and box-shaped solution space (box, right upper corner) for the design variables xca,f and xca,r (all other variables constant)

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Fig. 3

Decomposing the d-dimensional inequality of the hyperplane problem into d/2 two-dimensional inequalities. For each two-dimensional inequality, the threshold is chosen to be 1/(d/2) of the threshold of the overall problem (=d/2) to guarantee that the obtained design, with each pair of design variables satisfying the associated two-dimensional inequality, satisfies the high-dimensional inequality.

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Fig. 4

A solution box versus solution spaces expressed as product of a 2D-space and an interval for the hyperplane problem in three dimensions (white area, top row) and product of two 2D-spaces for the hyperplane problem in four dimensions (white area, bottom row)

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Fig. 5

Decomposing the complete solution space into two different 2D-spaces: Solution space expressed as a product of two 2D-spaces (white areas) for the hyperplane problem in four dimensions for two different sets of values for the thresholds of the two-dimensional inequalities

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Fig. 6

Solution space decomposed into 2D-spaces Ω1,Ω2,...,Ωn. Each two-dimensional solution space (2D-space) is enclosed by a polygon. A corner of the polygon is an intersection point of two active boundaries, an intersection point of an active boundary with a boundary of the design space, or an intersection point of two design-space boundaries. The design space of the kth 2D-space is denoted by Ωdsk.

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Fig. 7

Derivative of the objective function for the case that condition (9) holds (left) and for the case that condition (9) does not hold (right). The boundary of the first constraint moves with increasing values of gc,jk toward the boundary of the second constraint.

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Fig. 8

Size of an optimal solution space based on 2D-spaces divided by the size of an optimal box-shaped solution space versus the number of dimensions

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Fig. 9

Results of the 2D-space approach (polygons) and interval approach (boxes) for the chassis design problem with twelve design variables and nine constraints on six vehicle performance measures. Here, the 2D-spaces represent a different subset of the complete solution space compared to the boxes and hence, the boxes are not entirely included in the polygons. The 2D-spaces are larger than the solution box by a factor of 7625.

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Fig. 10

Top left: Relationship between a change in μ(Ωk) with a change in t; Top right: Relationship between a change in t with a change in gc,jk; Bottom left: Relationship between a change in the width wjk with a change in gc,rs. Bottom right: Relationship between a change in w with a change in t.




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