Structural and Multidisciplinary Optimization manuscript No. (will be inserted by the editor) Direct Lagrange Multiplier Updates in Topology Optimization Revisited Tej Kumar · Krishnan Suresh Received: date / Accepted: date Abstract In topology optimization, the bisection method is typically used for computing the Lagrange multiplier associ- ated with a constraint. While this method is simple to imple- ment, it leads to oscillations in the objective and could pos- sibly result in constraint failure if proper scaling is not ap- plied. 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 70s and 80s 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 bench- mark problems, we demonstrate several advantages of the direct method over the bisection method including: (1) fewer and faster update iterations, (2) smoother and robust conver- gence, and (3) insensitivity to material and force parame- ters. 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 Tej Kumar Department of Mechanical Engineering University of Wisconsin-Madison E-mail: tkumar3@wisc.edu ORCiD: 0000-0001-8762-8121 Krishnan Suresh Department of Mechanical Engineering University of Wisconsin-Madison E-mail: ksuresh@wisc.edu 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 in- clude density methods [6, 29], level-set [27, 37], topologi- cal derivative [31], evolutionary methods [40], etc. Density methods, in particular, “Solid Isotropic Material with Penal- ization” (SIMP) are the most popular today. In SIMP, finite element method is used as the analysis engine, and each fi- nite element e is associated with a design variable x e . Then the topology optimization problem is posed as: minimize x J (x, u) (1a) subject to K(x)u = f (1b) g(x, u) ≤ g * (1c) x ≤ x e ≤ x ∀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 Eqn. (1b) where K(x) is the stiffness matrix and f is the force vector. Note that the design constraint is defined via Eqn. (1c), while Eqn. (1d) are the box constraints that sets lower (x ) and upper bounds ( x) on the design variables. A typical in- stance of the above problem is compliance minimization where: J (x, u)= u T K(x)u (2) subject to a volume constraint: g(x)= ∑ e x e v e (3) where v e is the volume of element-e. There are several open- source SIMP-based codes for solving such problems; the 99- line code [28] being the first. This was later improved for