Analytical and Numerical Shape Optimization of a Class of Structures under Mass Constraints and Self-Weight

Bingbing San*, Haim Waisman, Isaac Harari

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


This paper first extends a classical solution concerning the shape optimization of a hanging bar. The well-known solution determines the optimal cross section of a homogeneous bar that minimizes elongation under its own weight and a given applied force, subject to a total volume constraint. Herein, the analytical solution is generalized to materials with a variable density and elastic modulus along the bar, subject to a total mass constraint. A gradient-based numerical optimization algorithm is developed and then used to solve the inverse problem to validate the analytical results. The approach is then extended to two-dimensional structures through the parameterization of the external boundary using nonuniform rational B-splines (NURBS) functions and the solution of repeated forward problems with updated meshes. Three different cases are studied: (1) homogeneous elastic, (2) homogeneous hyperelastic, and (3) inhomogeneous elastic materials. The results show the differences between the optimal shape of one- and two-dimensional models and the effect of material models on the optimal solutions.

Original languageEnglish
Article number04019109
JournalJournal of Engineering Mechanics - ASCE
Issue number1
StatePublished - 1 Jan 2020


  • Analytical solutions
  • Euler-Lagrange equation
  • Nonuniform rational B-splines (NURBS)
  • Numerical optimization
  • Shape optimization


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