Topology optimization with local stress constraints: a stress aggregation-free approach
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Topology optimization with local stress constraints: a stress aggregation-free approach ˜ 1 · Ivan F. M. Menezes2 · Glaucio H. Paulino1 Fernando V. Senhora1,2 · Oliver Giraldo-Londono Received: 23 July 2018 / Revised: 27 February 2020 / Accepted: 9 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This paper presents a consistent topology optimization formulation for mass minimization with local stress constraints by means of the augmented Lagrangian method. To solve problems with a large number of constraints in an effective way, we modify both the penalty and objective function terms of the augmented Lagrangian function. The modification of the penalty term leads to consistent solutions under mesh refinement and that of the objective function term drives the mass minimization towards black and white solutions. In addition, we introduce a piecewise vanishing constraint, which leads to results that outperform those obtained using relaxed stress constraints. Although maintaining the local nature of stress requires a large number of stress constraints, the formulation presented here requires only one adjoint vector, which results in an efficient sensitivity evaluation. Several 2D and 3D topology optimization problems, each with a large number of local stress constraints, are provided. Keywords Consistent topology optimization · Augmented Lagrangian · Stress constraints · Stress relaxation · von Mises stress · Aggregation-free
1 Introduction Cauchy was a visionary mathematician, physicist, and engineer who made pioneering contributions to several fields of knowledge, including continuum mechanics and elasticity (Bell 1986). Inspired by his work on continuum mechanics, we introduce a consistent stress-constrained topology optimization formulation that treats stress as a local quantity both in the solution of the boundary value problem and in the optimization phase. By treating stresses locally, we follow its definition as a fundamental quantity obtained by means of a limiting process known as Cauchy’s tetrahedron Responsible Editor: Kurt Maute Dedicated to the memory of Augustin-Louis Cauchy (August 21, 1789 – May 23, 1857) Glaucio H. Paulino
[email protected] 1
School of Civil and Environmental Engineering, Georgia Institute of Technology, 790 Atlantic Drive, Atlanta, GA 30332, USA
2
Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rua Marques de Sao Vicente, 225, Rio de Janeiro, R.J. 22453, Brazil
argument (Cauchy 1827; Love 1892; Timoshenko and Goodier 1951; Malvern 1969; Gurtin 1981). The argument states that the stress vector acting on a small area da oriented perpendicular to its normal vector n and located at a point in a continuous medium depends on the infinitesimal internal force vector df(n) acting on that surface, and it is defined as (Cauchy 1827): f(n) , a→0 a
σ · n = lim
(1)
where σ is the stress tensor. In this paper, we treat stress consistently, i.e., as a local quantity. The solution of a consistent topology optimization proble
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