Structural and Multidisciplinary Optimization

, Volume 44, Issue 4, pp 443–453 | Cite as

Eliminating beta-continuation from Heaviside projection and density filter algorithms

  • James K. GuestEmail author
  • Alireza Asadpoure
  • Seung-Hyun Ha
Research Paper


Projection methods and density filters based on the Heaviside step function are an effective means for producing discrete (0–1) solutions in continuum topology optimization. They naturally impose a minimum length scale on designed features and thereby prevent numerical instabilities of checkerboards and mesh dependence, as well as provide the designer a tool to influence solution manufacturability. A drawback of the Heaviside approach is that a continuation method must be applied to the continuous approximation so as not to approach the step function too quickly. This is achieved by gradually increasing a curvature parameter known as β as the optimization progresses. This is not only inefficient, but also causes slight, artificial perturbations to the topology. This note offers simple modifications to optimizer algorithms and/or Heaviside formulation that allow this continuation method to be eliminated. The modifications are tested on minimum compliance and compliant inverter benchmark problems and are shown to be effective and efficient.


Topology optimization Projection methods Filters SIMP Volume preservation Continuation methods 



This work was supported by National Science Foundation under Grant No. CMMI-0928613 with Dr. Christina Bloebaum serving as program officer. This support is gratefully acknowledged. The authors thank Krister Svanberg for providing the MMA code.


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Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • James K. Guest
    • 1
    Email author
  • Alireza Asadpoure
    • 1
  • Seung-Hyun Ha
    • 1
  1. 1.Department of Civil EngineeringJohns Hopkins UniversityBaltimoreUSA

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