Structural and Multidisciplinary Optimization

, Volume 55, Issue 5, pp 1787–1796 | Cite as

Direct computation of solution spaces

  • Johannes Fender
  • Fabian Duddeck
  • Markus Zimmermann


In engineering, it is often desirable to find a subset of the set of feasible designs, a solution space, rather than a single solution. A feasible design is defined as a design which does not violate any constraints and has a performance value below a desired threshold. Performance measure, threshold value and constraints depend on the specific problem. For evaluation of a design with respect to feasibility, a model is required which maps the design parameters from the input space onto the performance measures in the output space. In state-of-the-art methodology, iterative sampling is used to generate an estimate of the frontier between feasible and infeasible regions in the input space. By evaluating each sample point with respect to feasibility, areas which contain a large fraction of feasible designs are identified and subsequently resampled. The largest hypercube containing only feasible designs is sought, because this results in independent intervals for each design parameter. Estimating this hypercube with sufficient precision may require a large number of model evaluations, depending on the dimensionality of the input space. In this paper, a novel approach is proposed for modeling the inequality constraints and an objective function in a way for which a linear formulation can be used, independently of the dimensionality of the problem. Thereby the exact solution for the largest feasible hypercube can be calculated at much lower cost than with stochastic sampling as described above, as the problem is reduced to solving a linear system of equations. The method is applied to structural design with respect to the US-NCAP frontal impact. The obtained solution is compared to numerical solutions of an identical system, which are computed using reduced order models and stochastic methods. By this example, the high potential of the new direct method for solution space computation is shown.


Solution Space Crashworthiness Optimization Surrogate Model Systems Engineering 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Johannes Fender
    • 1
  • Fabian Duddeck
    • 2
  • Markus Zimmermann
    • 1
  1. 1.BMW Group, Research and Innovation CenterMunichGermany
  2. 2.Technische Universität MünchenMünchenGermany

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