Three-dimensional adaptive mesh refinement in stress-constrained topology optimization


Structural optimization software that can produce high-resolution designs optimized for arbitrary cost and constraint functions is essential to solve real-world engineering problems. Such requirements are not easily met due to the large-scale simulations and software engineering they entail. In this paper, we present a large-scale topology optimization framework with adaptive mesh refinement (AMR) applied to stress-constrained problems. AMR allows us to save computational resources by refining regions of the domain to increase the design resolution and simulation accuracy, leaving void regions coarse. We discuss the challenges necessary to resolve such large-scale problems with AMR, namely, the need for a regularization method that works across different mesh resolutions in a parallel environment and efficient iterative solvers. Furthermore, the optimization algorithm needs to be implemented with the same discretization that is used to represent the design field. To show the efficacy and versatility of our framework, we minimize the mass of a three-dimensional L-bracket subject to a maximum stress constraint and maximize the efficiency of a three-dimensional compliant mechanism subject to a maximum stress constraint.

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    Such data structures are also required in parallel high-order finite difference simulation codes (Tegeler et al. 2017).

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    The normal gradient ∇n is defined such that ∇na = ∇an


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The author thanks the Livermore Graduate Scholar Program for its support.


This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.

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Correspondence to Miguel A. Salazar de Troya.

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Replication of results

The software used to generate the results shown in Section 5 is property of the US Department of Energy and has not yet been approved for public release, and therefore is not currently openly distributed. The computational meshes used to generate the results can be obtained by contacting the corresponding author. All the details necessary to reproduce the results in Section 5 (loads, boundary conditions, constraints, objectives, optimization parameters, etc.) have been defined in the paper. We summarize the most important parameters in Table 1.

Table 1 Relevant parameters

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Salazar de Troya, M.A., Tortorelli, D.A. Three-dimensional adaptive mesh refinement in stress-constrained topology optimization. Struct Multidisc Optim (2020).

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  • Topology optimization
  • Stress constrained
  • Adaptive mesh refinement
  • Large-scale design