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Dynamical systems, SIMP, bone remodeling and time dependent loads

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Abstract

The dynamical systems approach to sizing and SIMP topology optimization, introduced in a previous paper, is extended to the case of time-varying loads. A general dynamical system, satisfying a Lyaponov-type descent condition, is derived and specialized to a goal function combining stiffness and mass. For a cyclic time-dependent load it is indicated how, in the limit of short cycles compared to the overall time scale, this can be handled by multiple load cases. Numerical examples, both for a convex and a non-convex case, illustrates the theory.

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Notes

  1. The acronym SIMP stands for Solid Isotropic Material with Penalization and is essentially an approach for penalizing intermediate design variable values when optimizing stiffness under a volume constraint (or the reversed). For a detail account we refer to Bendsøe and Sigmund (2003) and Christensen and Klarbring (2009).

  2. In the generalized version of bone remodeling theory, presented by Harrigan and Hamilton (1994a), q = N/m, where N is the exponent of the stiffness relation, m is a positive constant and \(\rho^{1/m}_i\) is the apparent density.

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Authors and Affiliations

  1. Mechanics, Department of Management and Engineering, The Institute of Technology, Linköping University, 581 83, Linköping, Sweden

    Anders Klarbring & Bo Torstenfelt

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  1. Anders Klarbring
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  2. Bo Torstenfelt
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Correspondence to Anders Klarbring.

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Klarbring, A., Torstenfelt, B. Dynamical systems, SIMP, bone remodeling and time dependent loads. Struct Multidisc Optim 45, 359–366 (2012). https://doi.org/10.1007/s00158-011-0724-x

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  • DOI: https://doi.org/10.1007/s00158-011-0724-x

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