##
**A quadratically convergent unstructured remeshing strategy for shape optimization.**
*(English)*
Zbl 1122.74478

Summary: A novel unstructured remeshing environment for gradient-based shape optimization using triangular finite elements is presented. The remeshing algorithm is based on a truss structure analogy; in solving for the equilibrium position of the truss system, the quadratically convergent Newton’s method is used. Exact analytical sensitivity information is therefore made available to the shape optimization algorithm. The overall computational efficiency in gradient-based shape optimization is very high.In solving the truss structure analogy, we compare our quadratically convergent Newton solver with a previously proposed forward Euler solver; this includes notes regarding mesh uniformity, element quality, convergence rates and efficiency.We present three numerical examples; it is then shown that remeshing may introduce discontinuities and local minima. We demonstrate that the effects of these on gradient-based algorithms are alleviated to some extent through mesh refinement, and may largely be overcome with a simple multi-start strategy.

### MSC:

74P10 | Optimization of other properties in solid mechanics |

74S05 | Finite element methods applied to problems in solid mechanics |

### Keywords:

shape optimization; unstructured remeshing; truss analogy; analytical sensitivity analysis; consistent tangent; local minima### Software:

DistMesh
PDF
BibTeX
XML
Cite

\textit{D. N. Wilke} et al., Int. J. Numer. Methods Eng. 65, No. 1, 1--17 (2006; Zbl 1122.74478)

Full Text:
DOI

### References:

[1] | . Numerical methods in sensitivity analysis and shape optimization. Modeling and Simulation in Science, Engineering and Technology. BirkhĂ¤user: Basel, 2003. |

[2] | Garcia, Structural and Multidisciplinary Optimization 26 pp 92– (2004) |

[3] | Li, Mechanics Research Communications 26 pp 657– (1999) |

[4] | Xie, Computers and Structures 49 pp 885– (1993) |

[5] | Imam, International Journal for Numerical Methods in Engineering 18 pp 661– (1982) |

[6] | Belegundu, Computer Methods in Applied Mechanics and Engineering 66 pp 87– (1988) |

[7] | Sienz, Computers and Structures 64 pp 31– (1997) |

[8] | Allaire, Journal of Computational Physics 194 pp 363– (2004) |

[9] | Mattheck, International Journal of Fatigue 12 pp 185– (1990) |

[10] | Persson, SIAM Review 46 pp 329– (2004) |

[11] | Snyman, Computers and Mathematics with Applications 44 pp 1589– (2002) |

[12] | , . Nonlinear Programming Theory and Algorithms. Wiley: New York, 1993. |

[13] | Geometry and Topology for Mesh Generation. Cambridge University Press: Cambridge, MA, 2001. |

[14] | Olhoff, Mechanics of Structures and Machines 21 pp 1– (1993) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.