Non-intrusive polynomial chaos expansion for topology optimization using polygonal meshes

Nilton Cuellar (1) , Anderson Pereira (1) , Ivan Fabio Mota De Menezes (1) , Americo Cunha Jr (2)

1 PUC-Rio - Pontifícia Universidade Católica do Rio de Janeiro [Brasil] = Pontifical Catholic University of Rio de Janeiro [Brazil] = Université catholique pontificale de Rio de Janeiro [Brésil]
2 UERJ - Universidade do Estado do Rio de Janeiro [Brasil] = Rio de Janeiro State University [Brazil] = Université d'État de Rio de Janeiro [Brésil]

This paper deals with the applications of stochastic spectral methods for structural topology optimization in the presence of uncertainties. A non-intrusive polynomial chaos expansion is integrated into a topology optimization algorithm to calculate low-order statistical moments of the mechanical-mathematical model response. This procedure, known as robust topology optimization, can optimize the mean of the compliance while simultaneously minimizing its standard deviation. In order to address possible variabilities in the loads applied to the mechanical system of interest, magnitude and direction of the external forces are assumed to be uncertain. In this probabilistic framework, forces are described as a random field or a set of random variables. Representation of the random objects and propagation of load uncertainties through the model are efficiently done through Karhunen-Lòeve and polynomial chaos expansions. We take advantage of using polygonal elements, which have been shown to be effective in suppressing checkerboard patterns and reducing mesh dependency in the solution of topology optimization problems. Accuracy and applicability of the proposed methodology are demonstrated by means of several topology optimization examples. The obtained results, which are in excellent agreement with reference solutions computed via Monte Carlo method, show that load uncertainties play an important role in optimal design of structural systems, so that they must be taken into account to ensure a reliable optimization process.

Mots clés: topology optimization; stochastic spectral approach; polynomial chaos; Karhunen–Loève expansion; robust optimization; polygonal finite element

Domaines: Génie mécanique [physics.class-ph]; Mécanique des structures [physics.class-ph]; Mécanique des solides [physics.class-ph]; Structures; Optimisation et contrôle [math.OC]; Probabilités [math.PR]

Nilton Cuellar, Anderson Pereira, Ivan Fabio Mota De Menezes, Americo Cunha Jr. Non-intrusive polynomial chaos expansion for topology optimization using polygonal meshes. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2018, 40 (12), pp.561. ⟨10.1007/s40430-018-1464-2⟩. ⟨hal-01927228⟩