On the co-rotational method for geometrically nonlinear topology optimization

Peter D Dunning*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)
12 Downloads (Pure)


This paper investigates the application of the co-rotational method to solve geometrically nonlinear topology optimization problems. The main benefit of this approach is that the tangent stiffness matrix is naturally positive definite, which avoids some numerical issues encountered when using other approaches. Three different methods for constructing the tangent stiffness matrix are investigated: a simplified method, where the linear elastic stiffness matrix is simply rotated, the consistent method, where the tangent stiffness is derived by differentiating residual forces by displacements, and a symmeterized method, where the consistent tangent stiffness is approximated by a symmetric matrix. The co-rotational method is implemented for 2D plane quadrilateral elements and 3-node shell elements. Matlab code is given in the appendix to modify an existing, freely available, density-based topology optimization code so it can solve 2D problems with geometric nonlinear analysis using the co-rotational method. The approach is used to solve four benchmark problems from the literature, including optimizing for stiffness, compliant mechanism design and a plate problem. The solutions are comparable to those obtained with other methods, demonstrating the potential of the co-rotational method as an alternative approach for geometrically nonlinear topology optimization. However, there are differences between the methods in terms of implementation effort, computational cost, final design and objective value. In summary, schemes involving the symmetrized tangent stiffness did not outperform the other schemes. For problems where the optimal design has relatively small displacments, then the simplified method is suitable. Otherwise, it is recommended to use the consistent method, as it is the most accurate.
Original languageEnglish
Pages (from-to)2357-2374
Number of pages18
JournalStructural and multidisciplinary optimization
Issue number5
Early online date9 May 2020
Publication statusPublished - Nov 2020

Bibliographical note

Acknowledgements The author would like to thank the Numerical Analysis Group at the Rutherford Appleton Laboratory for their FORTRAN HSL packages (HSL, a collection of Fortran codes for large-scale scientific computation.
See http://www.hsl.rl.ac.uk/).


  • Nonlinear geometry
  • Topology optimization
  • Co-rotational method
  • Compliant mechanism


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