Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: Front axle of a passenger vehicle with chassis components damper (1), bump stop (2), rebound stop (3), and anti-roll bar (4)

Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: 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)

Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: 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.

Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: 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)

Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: 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

Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: 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.

Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: 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.

Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: 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

Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: 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.

Date of download: 10/26/2017 Copyright © ASME. All rights reserved. From: On the Optimal Decomposition of High-Dimensional Solution Spaces of Complex Systems ASME J. Risk Uncertainty Part B. 2017;4(2):021008-021008-15. doi:10.1115/1.4037485 Figure Legend: 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.