Shape optimization for the generalized Graetz problem

We apply shape optimization tools to the generalized Graetz problem which is a convection-diffusion equation. The problem boils down to the optimization of generalized eigenvalues on a two phases domain. Shape sensitivity analysis is performed with respect to the evolution of the interface between the fluid and solid phase. In particular physical settings, counterexamples where there is no optimal domains are exhibited. Numerical examples of optimal domains with different physical parameters and constraints are presented. Two different numerical methods (level-set and mesh-morphing) are show-cased and compared.

[1]  Dorin Bucur,et al.  OPTIMISATION DE FORME SOUS CONTRAINTE CAPACITAIRE , 1994 .

[2]  Pekka Neittaanmäki,et al.  Optimization of Elliptic Systems: Theory and Applications , 2005 .

[3]  R. Viskanta,et al.  Three-dimensional conjugate heat transfer in the microchannel heat sink for electronic packaging , 2000 .

[4]  P. Neittaanmäki,et al.  Fixed domain approaches in shape optimization problems , 2012 .

[5]  R. Gorenflo,et al.  Multi-index Mittag-Leffler Functions , 2014 .

[6]  Grégoire Allaire,et al.  A level set method for the numerical simulation of damage evolution , 2009 .

[7]  A. H. Reis,et al.  Optimization of fluid flow and internal geometric structure of volumes cooled by forced convection in an array of parallel tubes , 2011 .

[8]  T. E. Bruns,et al.  Topology optimization of convection-dominated, steady-state heat transfer problems , 2007 .

[9]  Doraiswami Ramkrishna,et al.  The extended Graetz problem with Dirichlet wall boundary conditions , 1980 .

[10]  F. Murat,et al.  Sur le controle par un domaine géométrique , 1976 .

[11]  G. Allaire,et al.  Structural optimization using sensitivity analysis and a level-set method , 2004 .

[12]  Johanna Weiss,et al.  Optimal Shape Design For Elliptic Systems , 2016 .

[13]  V. Sverák,et al.  On optimal shape design , 1992 .

[14]  R. Monneau,et al.  DERIVATION AND STUDY OF DYNAMICAL MODELS OF DISLOCATION DENSITIES , 2009, 0901.2330.

[15]  N. Olhoff,et al.  Multiple eigenvalues in structural optimization problems , 1994 .

[16]  Frédéric de Gournay,et al.  Velocity Extension for the Level-set Method and Multiple Eigenvalues in Shape Optimization , 2006, SIAM J. Control. Optim..

[17]  Antoine Henrot,et al.  Variation et optimisation de formes , 2005 .

[18]  L. Gary Leal,et al.  Laminar Flow and Convective Transport Processes , 1992 .

[19]  C. DeWitt-Morette,et al.  Mathematical Analysis and Numerical Methods for Science and Technology , 1990 .

[20]  S. Osher,et al.  Level Set Methods for Optimization Problems Involving Geometry and Constraints I. Frequencies of a T , 2001 .

[21]  Giampietro Fabbri,et al.  Heat transfer optimization in internally finned tubes under laminar flow conditions , 1998 .

[22]  William Rundell,et al.  The determination of a discontinuity in a conductivity from a single boundary measurement , 1998 .

[23]  J OsherStanley,et al.  Level set methods for optimization problems involving geometry and constraints I. Frequencies of a two-density inhomogeneous drum , 2001 .

[24]  Edward J. Haug,et al.  Design Sensitivity Analysis in Structural Mechanics.II. Eigenvalue Variations , 1980 .

[25]  Carlos Conca,et al.  SHAPE DERIVATIVE FOR A TWO-PHASE EIGENVALUE PROBLEM AND OPTIMAL CONFIGURATIONS IN A BALL ; , 2009 .

[26]  S. Nishiwaki,et al.  Topology optimization for thermal conductors considering design-dependent effects, including heat conduction and convection , 2009 .

[27]  G. Allaire,et al.  Shape optimization by the homogenization method , 1997 .

[28]  J. Cea.  Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût , 1986 .

[29]  Olivier Pironneau,et al.  SENSITIVITY OF DARCY'S LAW TO DISCONTINUITIES , 2003 .

[30]  Søren Tørholm Christensen.  Design Sensitivity Analysis in Structural Acoustics , 1998 .

[31]  Franck Plouraboué,et al.  The Generalized Graetz Problem in Finite Domains , 2012, SIAM J. Appl. Math..

[32]  V. Komkov.  Optimal shape design for elliptic systems , 1986 .

[33]  Yaochu Jin,et al.  Optimization of micro heat exchanger: CFD, analytical approach and multi-objective evolutionary algorithms , 2006 .

[34]  L. Graetz,et al.  Ueber die Wärmeleitungsfähigkeit von Flüssigkeiten , 1882 .

[35]  박일한,et al.  전자장시스템에서 Level Set Method , 2008 .

[36]  Olivier Pantz,et al.  Sensibilité de l'équation de la chaleur aux sauts de conductivité , 2005 .

[37]  Franck Plouraboué,et al.  Numerical Analysis of a New Mixed Formulation for Eigenvalue Convection-Diffusion Problems , 2009, SIAM J. Appl. Math..

[38]  Robert V. Kohn,et al.  Topics in the Mathematical Modelling of Composite Materials , 1997 .

[39]  Dan Tiba,et al.  Optimization of Elliptic Systems , 2006 .

[40]  Denise Chenais,et al.  On the existence of a solution in a domain identification problem , 1975 .

[41]  Haim H. Bau,et al.  Optimization of conduits' shape in micro heat exchangers , 1998 .