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Direct lagrange multiplier updates in topology optimization revisited

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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.

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Acknowledgments

The authors would like to thank the support of National Science Foundation through grant CMMI 1561899. Prof. Suresh is a consulting Chief Scientific Officer of SciArt, Corp

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Authors and Affiliations

  1. Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, USA

    Tej Kumar & Krishnan Suresh

Authors
  1. Tej Kumar
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  2. Krishnan Suresh
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Correspondence to Krishnan Suresh.

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Replication of results

Modifications in various open-source Matlab codes have been provided in Appendix to help readers reproduce the results. The complete set of Matlab code is available at ersl.wisc.edu/software/DirectLagrangeMultiplier.zip.

Appendix:

Appendix:

Here, we summarize the required changes to popular topology optimization codes, in order to replace the bisection method with the direct method. These changes can be easily adapted to other codes such as fast topology optimization based on reanalysis and conjugate gradient solvers (Amir et al. 2014; Amir 2015).

1.1 Modifications to 99-line code

For the classic 99-line code (top.m) (Sigmund 2001), all that is needed is to replace the OC function with the following:

figure d

1.2 Modifications to 88-line code

In the 88-line code (top88.m) (Andreassen et al. 2011), one must replace lines 70-80 with the following:

figure e

To incorporate non-design regions, further modifications to the 88-line code is given below. A logical array named passive with true entry for every non-design element must be created. Then lines 79–81 in the modified 88-line code becomes,

figure f

In addition, line 91 changes to

figure g

Lines 93–94 must be modified identically to the modifications in lines 80–81. Finally, xnew(passive) = 1; should be appended to line 83 and 96 so that they read:

figure h

1.3 Modifications to 3D code

In the 3D code (top3d.m) (Liu and Tovar 2014), one must replace lines 82–88 with the following

figure i

Further, to update the display in every iteration, line 6 (i.e. the displayflag) must be removed, and line 94 in the original code must be replaced with volshow(xPhys).

1.4 Modifications for nonlinear constraint

The following code can replace the bisection method in any volume-minimization code (Amir et al. 2014; Amir 2015) available at github.com/odedamir/topopt-mgcg-matlab. For instance in the code minV.m, replace lines 83-96 with the lines below:

figure j

Then, depending on the approximation used, the following changes are sufficient:

Linear approximation

:

figure k

Reciprocal approximation

:

figure l

Exponential approximation

:

figure m

1.5 Modifications to PolyMat

Finally, for the PolyMat code (Sanders et al. 2018b), the function UpdateScheme must be replaced with the following.

figure n

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Kumar, T., Suresh, K. Direct lagrange multiplier updates in topology optimization revisited. Struct Multidisc Optim 63, 1563–1578 (2021). https://doi.org/10.1007/s00158-020-02740-y

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