Alternating optimization of design and stress for stress-constrained topology optimization

Structural and Multidisciplinary Optimization https://doi.org/10.1007/s00158-021-02985-1 RESEARCH PAPER Alternating optimization of design and stress for stress-constrained topology optimization Xiaoya Zhai1,2 · Falai Chen1 · Jun Wu2 Received: 1 April 2020 / Revised: 9 June 2021 / Accepted: 13 June 2021 © The Author(s) 2021 Abstract Handling stress constraints is an important topic in topology optimization. In this paper, we introduce an interpretation of stresses as optimization variables, leading to an augmented Lagrangian formulation. This formulation takes two sets of optimization variables, i.e., an auxiliary stress variable per element, in addition to a density variable as in conventional density-based approaches. The auxiliary stress is related to the actual stress (i.e., computed by its definition) by an equality constraint. When the equality constraint is strictly satisfied, an upper bound imposed on the auxiliary stress design variable equivalently applies to the actual stress. The equality constraint is incorporated into the objective function as linear and quadratic terms using an augmented Lagrangian form. We further show that this formulation is separable regarding its two sets of variables. This gives rise to an efficient augmented Lagrangian solver known as the alternating direction method of multipliers (ADMM). In each iteration, the density variables, auxiliary stress variables, and Lagrange multipliers are alternatingly updated. The introduction of auxiliary stress variables enlarges the search space. We demonstrate the effectiveness and efficiency of the proposed formulation and solution strategy using simple truss examples and a dozen of continuum structure optimization settings. Keywords Topology optimization · Stress constraints · Augmented Lagrangian · Alternating direction method of multipliers 1 Introduction Design of structures with local stresses upper-bounded by a critical stress value is of paramount importance in engineering. To this end, the incorporation of stress constraints has been an important field of study in topology optimization of continuum structures (Duysinx and Sigmund 1998). Over the past two decades, three computational challenges have been recognized (Le et al. 2010; Holmberg et al. 2013), and solutions for some of them have been proposed: – – The “singularity” problem — the feasible design space contains degenerate sub-spaces of a lower dimension (Kirsch 1990; Rozvany 2001). The globally optimal Responsible Editor: Gregoire Allaire  Jun Wu 1 School of Mathematical Sciences, University of Science and Technology of China, Hefei, China 2 Department of Sustainable Design Engineering, Delft University of Technology, Delft, The Netherlands – solution, which often locates in the degenerate subspaces, is not accessible to nonlinear programming algorithms. It has been shown that the problem of degenerate sub-spaces can be alleviated by relaxing the stress constraints, i.e., the -relaxation originally developed for trusses (Cheng and Guo 1997) and its variants for continuum structures (Duysinx and Bendsøe 1998; Bruggi 2008; Le et al. 2010). The local nature of stress constraints — the stress limit applies to every material point in the domain. This results in a large number of constraints. An often used solution strategy is to approximate these local constraints by a global one which can then be more efficiently addressed, e.g., the p-norm (Duysinx and Sigmund 1998) and Kreisselmeier-Steinhauser (KS) function (Yang and Chen 1996). The highly nonlinear dependence of stress on design variables. Especially at stress concentration regions, stresses are sensitive to density changes in neighborhoods. This leads to convergence problems: a large number of iterations, fluctuations in the objective and constraints, and a suboptimal objective value which potentially could be further reduced. X. Zhai et al. This last challenge is coupled with solutions of the first two. For instance, the aggregation for reducing the number of constraints may further exacerbate the nonlinearity. It is found in the literature that research has been mostly focusing on the first two challenges by reformulating the optimization problem, and effective alternative approaches have been proposed, e.g., Verbart et al. (2016, 2017), Wang and Qian (2018). Stress-constrained topology optimization problems (and more general structural optimization) are typically solved by using sequential convex programming. Notable algorithms include the Convex Linearization method (CONLIN) (Fleury 1989), the Method of Moving Asymptotes (MMA) (Svanberg 1987) and its globally convergent version GCMMA (Svanberg 2002). These algorithms make use of (first order) approximations of objective and constraint functions. An assessment of these optimization algorithms, as well as general methods such as the primal-dual interior point methods (Forsgren and Gill 1998) and Sequential Quadratic Programming (SQP) (Boggs and Tolle 1995), is presented by Rojas-Labanda and Stolpe (2015). The benchmark problems in the comparative study include compliance/volume minimization and mechanism design. Unfortunately stress constraints are not included. An alternative solution strategy to stress-constrained topology optimization is to incorporate local stress constraints in the objective function using an augmented Lagrangian formulation (Pereira et al. 2004). It has been used for topology optimization based on density (Fancello 2006; da Silva and Cardoso 2017; da Silva et al. 2019) and level sets (James et al. 2012; Emmendoerfer and Fancello 2014). Very recently (da Silva et al. 2020) demonstrated that this formulation allows handling very large problems in 3D manufacturing-tolerant topology optimization, with hundreds of millions of stress constraints. Also aiming for 3D large scale optimization, Senhora et al. (2020) proposed to modify both the penalty and objective function terms of the augmented Lagrangian function, leading to consistent solutions under mesh refinement and driving the mass minimization towards black and white solutions. GiraldoLondoño and Paulino (2020) applied this to handle multiple classical failure criteria with a unified yielding function. In this paper we introduce an interpretation of local stresses as optimization variables, using an augmented Lagrangian formulation. We consider auxiliary stresses as optimization variables, in addition to the design variables (i.e., densities) representing the material distribution. The stress limit is then imposed upon the auxiliary stresses as an upper bound. Given a material distribution and boundary conditions, the actual stress is computed by a finite element analysis. The auxiliary stress is related to the actual stress by an equality constraint. This equality constraint is then incorporated into the objective function as linear and quadratic terms using an augmented Lagrangian form. This reformulation offers some conceptual (...truncated)