Towards adaptation of the NURBS weights in shape optimization

Bézier based parametrisations in shape optimization have the drawback of using high degree polynomials to draw more complex shapes.  To overcome this drawback, Non-Uniform Rational B-Splines (NURBS) are usually used.  But, by considering the NURBS weights, in addition to the locations of the control points, as optimization variables, the dimension of the problem greatly increases and this would make the optimization process stiffer.  In this work we propose, then, an algorithm to adapt the weights of NURBS in the parametrization of shape optimization problems.  Unlike the coordinates of the control points, the weights are not considered, in this case, as variables of the optimization process.  From the knowledge of an approximate optimal shape, we consider the set of all NURBS parametrizations of the same degree that approximate the shape in the sense of least squares.  Then, we elect the parametrization associated with the most regular control polygon (least length of the  control polygon).  Numerical results show that the adaptive parametrization  improves  the performance of the optimization process.

  1. Rogers D. F.  An introduction to NURBS: with historical perspective.  Morgan Kaufman Publishers, San Fransisco (2001).
  2. Piegl L., Tiller W.  The NURBS book.  Springer, New York (1997).
  3. Yashchuk Yu. O., Tajs-Zielinska K.  Solving topology optimization problems using cellular automata and mortar finite element method.  Mathematical Modeling and Computing.  7 (2), 239–247 (2020).
  4. Hassouna S., Timesli A.  Optimal variable support size for mesh-free approaches using genetic algorithm.  Mathematical Modeling and Computing.  8 (4), 678–690 (2021).
  5. Saffah Z., Hassouna S., Timesli A., Azouani A., Lahmam H.  RBF collocation path-following approach: optimal choice for shape parameter based on genetic algorithm.  Mathematical Modeling and Computing.  8 (4), 770–782 (2021).
  6. Bélahcène F., Désidéri J.-A.  Paramétrisation de Bézier adaptative pour l'optimisation de forme en aérodynamique.  Technical Report 4943, INRIA, September 2003.
  7. Hollig K., Horner J.  Approximation and modeling with B-splines.  SIAM (2013).
  8. Fußeder D., Simeon B., Vuong A.-V.  Fundamental aspects of shape optimization in the context of isogeometric analysis.  Computer Methods in Applied Mechanics and Engineering.  286, 313–331 (2015).
  9. Wang Y., Wang Z., Xia Z., Hien P. L.  Structural design optimization using isogeometric analysis: A comprehensive review.  Computer Modeling in Engineering and Sciences.  117 (3), 455–507 (2018).
  10. Polishchuk O.  Finite element approximations in projection methods for solution of some fredholm integral equation of the first kind.  Mathematical Modeling and Computing.  5 (1), 74–87 (2018).
  11. Wall W. A., Frenzel M. A., Cyron C.  Isogeometric structural shape optimization.  Computer Methods in Applied Mechanics and Engineering.  197 (33–40), 2976–2988 (2008).
  12. Qin X. C., Dong C. Y.  NURBS-based isogeomtric shape and material optimization of curvilinearly stiffened plates with FGMs.  Thin-Walled Structures.  162, 107601 (2021).
  13. Song Y.-U., Hur J. Y., Youn S.-K.  Study of the shape optimization in spline FEM considering both nurbs control point positions and weights as design variables.  Transactions of the Korean Society of Mechanical Engineers A.  The Korean Society of Mechanical Engineers.  38 (4), 363–370 (2014).
  14. Taheri A. H., Abolghasemi S., Suresh K.  Generalizations of Non-Uniform Rational B-Splines via decoupling of the weights: theory, software and applications.  Engineering with Computers.  36, 1831–1848 (2020).
  15. Clarich A., Désidéri J.-A.  Self-adaptive parameterization for aerodynamic optimum-shape design.  Technical Report 4428, INRIA, March 2002.
  16. Tang Z., Désidéri J.-A.  Towards self-adaptive parameterization for {B}\'ezier curves for airfoil aerodynamic design.  Technical Report 4572, INRIA, September 2002.
  17. Cox M. G.  The numerical evaluation of B-splines.  IMA Journal of Applied Mathematics.  10 (2), 134–149 (1972).
  18. De Boor C.  On calculating with B-splines.  Journal of Approximation Theory.  6 (1), 50–62 (1972).
  19. De Boor C.  A pactical guide to splines.  New York, Springer–Verlag (1978).
  20. Ziani M., Duvigneau R.  On the role played by NURBS weights in isogeometric structural shape optimization.  In Proceedings of PICOF 10 (2009).
Mathematical Modeling and Computing, Vol. 9, No. 4, pp. 959–967 (2022)