Risk averse stochastic structural topology optimization. (English) Zbl 1440.74291
Summary: A novel approach for risk-averse structural topology optimization under uncertainties is presented, which takes into account stochastic data of the state equation, specifically random material properties and random forces. For the distribution of material, a phase field approach is employed, which allows for arbitrary topological changes during the iterative optimization. The state equation is assumed to be a high-dimensional PDE parametrized in a (truncated finite) set of random variables. The examined case employs linearized elasticity with a parametric elasticity tensor. For practical purposes, instead of an optimization with respect to the expectation of the involved random fields, the designed structures should in particular be robust with respect to rather unlikely and possibly critical events. For this, as a common risk measure, the Conditional Value at Risk (CVaR), is introduced to the cost functional of the minimization procedure. The proposed method is illustrated with numerical examples based on Monte Carlo sampling for different risk values and compared with the result of the deterministic formulation. It is observed that the resulting shapes dependent on the risk parameter of the functional and can deviate significantly from the deterministic case.
MSC:
| 74P15 | Topological methods for optimization problems in solid mechanics |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 47B80 | Random linear operators |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N75 | Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs |
Keywords:
partial differential equations with random coefficients; risk averse optimization; phase field; topology optimization; conditional value at risk; uncertainty quantificationReferences:
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