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Research Papers

# Identifying Key Parameters for Design Improvement in High-Dimensional Systems With Uncertainty

[+] Author and Article Information
Johannes Fender

BMW Group Research and Innovation Center,
Knorrstr. 147,
Munich 80937, Germany
e-mail: Johannes.Fender@bmw.de

L. Graff

BMW Group Research and Innovation Center,
Knorrstr. 147,
Munich 80937, Germany
e-mail: Lavinia.Graff@bmw.de

H. Harbrecht

Department of Mathematics
and Computer Science,
University of Basel,
Rheinsprung 21,
Basel 4051, Switzerland
e-mail: Helmut.Harbrecht@unibas.ch

Markus Zimmermann

BMW Group Research and Innovation Center,
Knorrstr. 147,
Munich 80937, Germany
e-mail: markusz@alum.mit.edu

On a Linux workstation, 2x Intel Xeon X5550, 2.67 GHz, 12GB RAM

The forces measured in components 5, 6, 8 and 9 cross the corridor lines only when unloading, that is, when $u·<0$.

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 19, 2013; final manuscript received January 24, 2014; published online February 28, 2014. Assoc. Editor: Rikard Söderberg.

J. Mech. Des 136(4), 041007 (Feb 28, 2014) (10 pages) Paper No: MD-13-1175; doi: 10.1115/1.4026647 History: Received April 19, 2013; Revised January 24, 2014

## Abstract

Key parameters may be used to turn a bad design into a good design with comparatively little effort. The proposed method identifies key parameters in high-dimensional nonlinear systems that are subject to uncertainty. A numerical optimization algorithm seeks a solution space on which all designs are good, that is, they satisfy a specified design criterion. The solution space is box-shaped and provides target intervals for each parameter. A bad design may be turned into a good design by moving its key parameters into their target intervals. The solution space is computed so as to minimize the effort for design work: its shape is controlled by particular constraints such that it can be reached by changing only a small number of key parameters. Wide target intervals provide tolerance against uncertainty, which is naturally present in a design process, when design parameters are unknown or cannot be controlled exactly. In a simple two-dimensional example problem, the accuracy of the algorithm is demonstrated. In a high-dimensional vehicle crash design problem, an underperforming vehicle front structure is improved by identifying and appropriately changing a relevant key parameter.

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## Figures

Fig. 1

USNCAP front crash

Fig. 2

(a) Vehicle structure of example problem consisting of two deformable components and (b) force-deformation characteristics of the structural components 1 and 2

Fig. 3

Changes necessary to meet the design goal: (a) F1 and F2, (b) only F1, (c) only F2

Fig. 4

Algorithm to compute solution boxes

Fig. 5

Evolution of the candidate box in the exploration phase (top row) (a) and consolidation phase (bottom row) for constraints of scenario (b) ensuring that only F1 needs to be changed, that is, F2low≤Fc,2low = 425 kN and F2up≥Fc,2up = 475 kN

Fig. 6

Computed solution boxes (a) without constraints, (b) constraints ensuring that only F1 needs to be changed (F2low≤Fc,2low = 425 kN,F2up≥Fc,2up = 475 kN), and (c) constraints ensuring that only F2 needs to be changed (F1low≤Fc,1low = 250 kN and F1up≥Fc,1up = 300 kN)

Fig. 7

(a) Component of a vehicle front structure. (b) Detailed and coarse force-deformation characteristics F∧(u). (c) Detail model for a full vehicle finite element simulation. (d) Simplified model with longitudinal deformation elements. Black boxes in (c) and (d) indicate the location of component shown in (a).

Fig. 8

Force-deformation characteristics and their nonlinear influence on the maximum deceleration. (a) Original design. (b) and (c) two modifications yielding bad designs each. (d) Combined modification yielding a good design.

Fig. 9

(a) Measured force-deformation characteristics of design 1 with a > ac and corridors Ω5 and (b) the front rail undeformed and deformed

Fig. 10

(a) Measured force-deformation characteristics of design 2 with a < ac and corridors Ω5 and (b) the reinforced front rail undeformed and deformed

Fig. 11

Deceleration of the good and bad design

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