Integral Criterion of the Non-uniformity of Stress Distribution for the Topology Optimization of 2D-Models

Journal Journal of Mechanical Engineering – Problemy Mashynobuduvannia
Publisher A. Pidhornyi Institute for Mechanical Engineering Problems
National Academy of Science of Ukraine
ISSN  2709-2984 (Print), 2709-2992 (Online)
Issue Vol. 24, no. 1, 2021 (March)
Pages 65-74
Cited by J. of Mech. Eng., 2021, vol. 24, no. 1, pp. 65-74



Ihor V. Yanchevskyi, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” (37, Peremohy Ave., Kyiv, 03056, Ukraine), e-mail:, ORCID: 0000-0002-7113-2276

Volodymyr F. Kryshtal, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” (37, Peremohy Ave., Kyiv, 03056, Ukraine), e-mail:, ORCID: 0000-0002-5597-2435



The emergence of new technologies for the production of structural elements gives impetus to the development of new technologies for their design, in particular with the involvement of a topology optimization method. The most common algorithm for designing topologically optimal structures is focused on reducing their elastic flexibility at a given volume of material. However, a closer to the engineering design approach is the minimization of the volume of a structural element while limiting the resulting mechanical stresses. In contrast to the classical algorithms of this approach, which limit the values of stresses at certain points, this paper develops an alternative criterion: the formation of the image of a structural element is based on minimizing the integral parameter of stress distribution non-uniformity. The developed algorithm is based on the method of proportional topology optimization, and when mechanical stresses are calculated, the classical relations of the finite element method are used. The above parameter can be interpreted as the ratio of the deviation of the values, ordered in ascending order, of equivalent von Mises stresses in the finite elements of a calculation model from their linear approximation to the corresponding mean value. The search for the optimal result is carried out for the full range of possible values of the averaged “density” of the calculation area, which is associated with a decrease in the amount of input data. The proposed integrated strength criterion provides better uniformity of the optimized topology, allows us to smooth the effect of the local peak values of mechanical stresses, determining a single optimization result that is resistant to calculation errors. The algorithm is implemented in the MatLab software environment for two-dimensional models. The efficiency of the approach is tested on the optimization of a classical beam (mbb-beam), a cantilever beam, and an L-shaped beam. A comparative analysis of the obtained results with those available in the literature is given. It is shown that in the absence of constraint on the average value of the density of a finite element model, the proposed criterion gives a ″less dense″ optimization result compared to the classical one (approximately 40%), while the values of “contrast index” are quite close.

Keywords: topology optimization; two-dimensional problem; strength condition; integral criterion; algorithm; finite element method; equivalent von Mises stresses.


Full text: Download in PDF



  1. Bendsøe, M. P. & Kikuchi, N. (1988). Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, vol. 71, iss. 2, pp. 197–224.
  2. Borovikov, A. A. & Tenenbaum, S. M. (2016). Topologicheskaya optimizatsiya perekhodnogo otseka kosmicheskogo apparata [Topology optimization of spacecraft transfer compartment]. Aerokosmicheskiy nauchnyy zhurnalAerospace Scientific Journal, vol. 2, no. 5, pp. 16–30. (in Russian).
  3. Bendsoe, M. P. & Sigmund, O. (2003). Topology optimization: theory, methods and applications. Berlin: Springer-Verlag, 390 p.
  4. Sigmund, O. (2001). A 99 line topology optimization code written in Matlab. Structural and Multidisciplinary Optimization, vol. 21, iss. 2, pp. 120–127.
  5. Andreassen, E., Clausen, A., Schevenels, M., Lazarov, B. S., & Sigmund, O. (2011). Efficient topology optimization in MATLAB using 88 lines of code. Structural and Multidisciplinary Optimization, vol. 43, iss. 1, pp. 1–16.
  6. Liu, K. & Tovar, A. (2014). An efficient 3D topology optimization code written in MatLab. Structural and Multidisciplinary Optimization, vol. 50, iss. 6, pp. 1175–1196.
  7. Xie, Y. M. & Steven, G. P. (1993). A simple evolutionary procedure for structural optimization. Computers & Structures, vol. 49, iss. 5, pp. 885–896.
  8. Huang, X. & Xie, Y. M. (2010). Evolutionary topology optimization of continuum structures: methods and applications. UK: Wiley, 223 p.
  9. Xia, L., Xia, Q., Huang, X., & Xie, Y. M. (2018). Bi-directional evolutionary structural optimization on advanced structures and materials: A comprehensive review. Archives of Computational Methods in Engineering, vol. 25, iss. 2, pp. 437–478.
  10. Cysoyeva, V. V. & Chedrik, V. V. (2011). Algoritmy optimizatsii topologii silovykh konstruktsiy [Optimization algorithms for the topology of power structures]. Uchenyye zapiski TSAGICentral Aerohydrodynamic Institute scientific notes, Vol. XLII, no. 2, pp. 91–101 (in Russian).
  11. Kirsch, U. (1990). On singular topologies in optimum structural design. Structural and Multidisciplinary Optimization, vol. 2, iss. 3, pp. 133–142.
  12. Duysinx, P. & Bendsøe, M. P. (1998). Topology optimization of continuum structures with local stress constraints. International Journal for Numerical Methods in Engineering, vol. 43, pp. 1453–1478.<1453::AID-NME480>3.0.CO;2-2.
  13. Le, C., Norato, J., Bruns, T., Ha, C., & Tortorelli, D. (2010). Stress based topology optimization for continua. Structural and Multidisciplinary Optimization, vol. 41, iss. 4, pp. 605–620.
  14. Lee, E., James, K. A., & Martins, J. R. (2012). Stress-constrained topology optimization with design-dependent loading. Structural and Multidisciplinary Optimization, vol. 46, iss. 5, pp. 647–661.
  15. Biyikli, E. & To, A. C. (2014). Proportional Topology Optimization: A new non-gradient method for solving stress constrained and minimum compliance problems and its implementation in MATLAB. PLoS ONE, no. 10, pp. 1–18.


Received 12 February 2021

Published 30 March 2021