Numerical study of a relaxed variational problem from optimal design

Jonathan Goodman, Robert V. Kohn, Luis Reyna

Research output: Contribution to journalArticlepeer-review


We revisit a well-known problem of optimal design, the placement of two elastic materials in the cross-section of a rod for maximum torsional rigidity. Another interpretation is the arrangement of two viscous fluids in a pipe for maximum flux under Poiseuille flow. The existence theory allows mixing on a microscopic scale, producing composite materials, and solving a relaxed version of the original design problem. This paper demonstrates that relaxation is as important for calculation as it is for existence. We minimize a discretized version of the relaxed problem using Newton's method; each quadratic approximation is solved by a multigrid method. This allows for greater resolution than previously published calculations, which were based on gradient flow.

Original languageEnglish (US)
Pages (from-to)107-127
Number of pages21
JournalComputer Methods in Applied Mechanics and Engineering
Issue number1
StatePublished - Aug 1986

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • General Physics and Astronomy
  • Computer Science Applications


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