Modelling and process optimization for functionally graded materials

Ravi S. Bellur-Ramaswamy, Robert Haber, Nahil A. Sobh, Daniel A Tortorelli

Research output: Contribution to journalArticlepeer-review


We optimize continuous quench process parameters to produce functionally graded aluminium alloy extrudates. To perform this task, an optimization problem is defined and solved using a standard non-linear programming algorithm. Ingredients of this algorithm include (1) the process parameters to be optimized, (2) a cost function: the weighted average of the precipitate number density distribution, (3) constraint functions to limit the temperature gradient (and hence distortion and residual stress) and exit temperature, and (4) their sensitivities with respect to the process parameters. The cost and constraint functions are dependent on the temperature and precipitate size which are obtained by balancing energy to determine the temperature distribution and by using a reaction-rate theory to determine the precipitate particle sizes and their distributions. Both the temperature and the precipitate models are solved via the discontinuous Galerkin finite element method. The energy balance incorporates non-linear boundary conditions and material properties. The temperature field is then used in the reaction rate model which has as many as 105 degrees-of-freedom per finite element node. After computing the temperature and precipitate size distributions we must compute their sensitivities. This seemingly intractable computational task is resolved thanks to the discontinuous Galerkin finite element formulation and the direct differentiation sensitivity method. A three-dimension example is provided to demonstrate the algorithm.

Original languageEnglish (US)
Pages (from-to)186-204
Number of pages19
JournalInternational Journal for Numerical Methods in Engineering
Issue number2
StatePublished - Jan 14 2005


  • Aluminum extrusion
  • Discontinuous Galerkin
  • Functionally graded materials
  • Non-linear thermal
  • Process optimization
  • Quenching
  • Sensitivity

ASJC Scopus subject areas

  • Numerical Analysis
  • General Engineering
  • Applied Mathematics


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