WIAS Preprint No. 2362, (2016)

Stochastic topology optimisation with hierarchical tensor reconstruction


  • Eigel, Martin
    ORCID: 0000-0003-2687-4497
  • Neumann, Johannes
  • Schneider, Reinhold
  • Wolf, Sebastian

2010 Mathematics Subject Classification

  • 35R60 47B80 60H35 65C20 65N12 65N22 65J10


  • partial differential equations with random coefficients, tensor representation, tensor train, uncertainty quantification, topology optimization, phase field, adaptive methods, low-rank, tensor reconstruction, risk measures




A novel approach for risk-averse structural topology optimization under uncertainties is presented which takes into account random material properties and random forces. For the distribution of material, a phase field approach is employed which allows for arbitrary topological changes during optimization. The state equation is assumed to be a high-dimensional PDE parametrized in a (finite) set of random variables. For the examined case, linearized elasticity with a parametric elasticity tensor is used. Instead of an optimization with respect to the expectation of the involved random fields, for practical purposes it is important to design structures which are also robust in case of events that are not the most frequent. As a common risk-aware measure, the Conditional Value at Risk (CVaR) is used in the cost functional during the minimization procedure. Since the treatment of such high-dimensional problems is a numerically challenging task, a representation in the modern hierarchical tensor train format is proposed. In order to obtain this highly efficient representation of the solution of the random state equation, a tensor completion algorithm is employed which only required the pointwise evaluation of solution realizations. The new method is illustrated with numerical examples and compared with a classical Monte Carlo sampling approach.

Appeared in

  • Computer Methods Appl. Math. Engrg., 334 (2018), pp. 470--482, DOI 10.1016/j.cma.2018.02.003; changed title: Risk averse stochastic structural topology optimization.

Download Documents