Direct lagrange multiplier updates in topology optimization revisited
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EDUCATIONAL PAPER
Direct lagrange multiplier updates in topology optimization revisited Tej Kumar1
· Krishnan Suresh1
Received: 20 February 2020 / Revised: 7 August 2020 / Accepted: 9 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In topology optimization, the bisection method is typically used for computing the Lagrange multiplier associated with a constraint. While this method is simple to implement, it leads to oscillations in the objective and could possibly result in constraint failure if proper scaling is not applied. In this paper, we revisit an alternate and direct method to overcome these limitations. The direct method of Lagrange multiplier computation was popular in the 1970s and 1980s but was later replaced by the simpler bisection method. In this paper, we show that the direct method can be generalized to a variety of linear and nonlinear constraints. Then, through a series of benchmark problems, we demonstrate several advantages of the direct method over the bisection method including (1) fewer and faster update iterations, (2) smoother and robust convergence, and (3) insensitivity to material and force parameters. Finally, to illustrate the implementation of the direct method, drop-in replacements to the bisection method are provided for popular Matlab-based topology optimization codes. Keywords Topology optimization · Optimality criteria · Bisection · Design constraints · Lagrange multiplier · Design update
1 Introduction Topology optimization is now a well-established method for computing optimal material distribution within a design domain that extremizes an objective while meeting a set of constraints. Popular topology optimization methods include density methods (Bendsøe 1989; Stolpe and Svanberg 2001), level-set (Sethian and Wiegmann 2000; Wang et al. 2003), topological derivative (Suresh 2010), and evolutionary methods (Xie and Steven 1993). Density methods, in particular, “Solid Isotropic Material with Penalization” (SIMP) are the most popular today. In SIMP, the finite element method is used as the analysis engine, and each finite element e is associated with a design variable xe . Then, the topology optimization problem is posed as
Responsible Editor: Ole Sigmund Krishnan Suresh
[email protected] 1
Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, USA
minimize J (x, u)
(1a)
subject to K(x)u = f
(1b)
x
g(x, u) ≤ g ∗ x ≤ xe ≤ x
(1c) ∀e
(1d)
where the objective function J is dependent on the design variables x and state variables u. The latter is computed via the governing Eq. (1b) where K(x) is the stiffness matrix and f is the force vector. Note that the design constraint is defined via (1c), while Eq. (1d) are the box constraints that set lower (x) and upper bounds (x) on the design variables. A typical instance of the above problem is compliance minimization where J (x, u) = uT K(x)u
(2)
subject to a volume constraint: g(x) = xe ve
(3)
e
where ve is the volume of element-e. There are several open-s
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