Shape optimization of conductive-media interfaces using an IGA-BEM solver

Abstract In this paper, we present a method that combines the Boundary Element Method (BEM) with IsoGeometric Analysis (IGA) for numerically solving the system of Boundary Integral Equations (BIE) arising in the context of a 2-D steady-state heat conduction problem across a periodic interface separating two conducting and conforming media. Our approach leads to a fast solver with high convergence rate when compared with low-order BEM. Additionally, an optimization framework comprising a parametric model for the interface’s shape, our IGA-BEM solver, and evolutionary and gradient-based optimization algorithms is developed and tested. The optimization examples demonstrate the efficiency of the framework in generating optimum interfaces for maximizing heat transfer under various geometric constraints.

[1]  M. Kotsonis,et al.  Optimum isothermal surfaces that maximize heat transfer , 2013 .

[2]  Ricardo H. Nochetto,et al.  Adaptive finite element method for shape optimization , 2012 .

[3]  W. Ang.  A Beginner's Course in Boundary Element Methods , 2007 .

[4]  M. Sellier,et al.  Optimal Shape Design in Heat Transfer Based on Body-Fitted Grid Generation , 2013 .

[5]  Chao Wang,et al.  Implementation of isogeometric boundary element method for 2-D steady heat transfer analysis , 2018, Adv. Eng. Softw..

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

[7]  Franck Plouraboué,et al.  Shape optimization for the generalized Graetz problem , 2014 .

[8]  Jeng-Tzong Chen,et al.  A Practical Guide to Boundary Element Methods with the Software Library BEMLIB , 2002 .

[9]  Panagiotis D. Kaklis,et al.  A BEM-isogeometric method for the ship wave-resistance problem , 2013 .

[10]  Thomas J. R. Hughes,et al.  Isogeometric boundary-element analysis for the wave-resistance problem using T-splines , 2014 .

[11]  T. Rabczuk,et al.  A two-dimensional Isogeometric Boundary Element Method for elastostatic analysis , 2012 .

[12]  M. Fyrillas.  Critical Depth of Buried Isothermal Circular Pipes , 2018 .

[13]  Panagiotis D. Kaklis,et al.  Ship-hull shape optimization with a T-spline based BEM-isogeometric solver , 2015 .

[14]  M. Sellier,et al.  Three-Dimensional Optimal Shape Design in Heat Transfer Based on Body-fitted Grid Generation , 2013 .

[15]  Chin-Hsiang Cheng, Chun-Yin Wu.  AN APPROACH COMBINING BODY-FITTED GRID GENERATION AND CONJUGATE GRADIENT METHODS FOR SHAPE DESIGN IN HEAT CONDUCTION PROBLEMS , 2000 .

[16]  E. Kontoghiorghes,et al.  Numerical Calculation of Mass Transfer from Elliptical Pools in Uniform Flow Using the Boundary Element Method , 2004 .

[17]  J. Trevelyan,et al.  Extended isogeometric boundary element method (XIBEM) for two-dimensional Helmholtz problems , 2013 .

[18]  Marios M. Fyrillas,et al.  Conductive heat transport across rough surfaces and interfaces between two conforming media , 2001 .

[19]  L. Shampine.  Vectorized adaptive quadrature in MATLAB , 2008 .

[20]  M. Fyrillas.  Shape optimization for 2D diffusive scalar transport , 2009 .

[21]  Kang Li,et al.  Isogeometric analysis and shape optimization via boundary integral , 2011, Comput. Aided Des..

[22]  Panagiotis D. Kaklis,et al.  Shape-optimization of 2D hydrofoils using an Isogeometric BEM solver , 2017, Comput. Aided Des..

[23]  M. Fyrillas.  Advection–dispersion mass transport associated with a non-aqueous-phase liquid pool , 2000, Journal of Fluid Mechanics.

[24]  Stéphane Bordas,et al.  Implementation of regularized isogeometric boundary element methods for gradient‐based shape optimization in two‐dimensional linear elasticity , 2016 .

[25]  T. Leontiou,et al.  Critical Biot Number of a Periodic Array of Rectangular Fins , 2016 .

[26]  Chin-Hsiang Cheng,et al.  Shape design for a cylinder with uniform temperature distribution on the outer surface by inverse heat transfer method , 2003 .

[27]  C. Pozrikidis.  Conductive mass transport from a semi-infinite lattice of particles , 2000 .

[28]  M. Fyrillas,et al.  Critical Biot Numbers of Periodic Arrays of Fins , 2017 .

[29]  C. Doumanidis,et al.  Approximate solution to Fredholm integral equations using linear regression and applications to heat and mass transfer , 2012 .

[30]  T. Leontiou,et al.  Heat transfer enhancement of a periodic array of isothermal pipes , 2016 .

[31]  Les A. Piegl,et al.  The NURBS Book , 1995, Monographs in Visual Communication.

[32]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[33]  Helmut Harbrecht,et al.  On the Numerical Solution of a Shape Optimization Problem for the Heat Equation , 2013, SIAM J. Sci. Comput..

[34]  Frank P. Incropera,et al.  Fundamentals of Heat and Mass Transfer , 1981 .

[35]  T. Leontiou,et al.  Shape optimization with isoperimetric constraints for isothermal pipes embedded in an insulated slab , 2014 .

[36]  Helcio R. B. Orlande,et al.  Inverse and Optimization Problems in Heat Transfer , 2006 .

[37]  Tinh Quoc Bui,et al.  Structural shape optimization by IGABEM and particle swarm optimization algorithm , 2018 .

[38]  M. Klein,et al.  An Ewald summation method for planar surfaces and interfaces , 1992 .

[39]  Marios M. Fyrillas,et al.  Heat conduction in a solid slab embedded with a pipe of general cross-section: Shape Factor and Shape Optimization , 2008 .

[40]  M. Diligenti,et al.  New efficient assembly in Isogeometric Analysis for Symmetric Galerkin Boundary Element Method , 2017 .

[41]  T. Leontiou,et al.  Optimum interfaces that maximize the heat transfer rate between two conforming conductive media , 2017 .

[42]  Seonho Cho,et al.  Isogeometric shape design sensitivity analysis of elasticity problems using boundary integral equations , 2016 .

[43]  M. Scott,et al.  Acoustic isogeometric boundary element analysis , 2014 .

[44]  M. Fyrillas.  Shape factor and shape optimization for a periodic array of isothermal pipes , 2010 .

[45]  H. Stone,et al.  Critical insulation thickness of a rectangular slab embedded with a periodic array of isothermal strips , 2011 .

[46]  K. Atkinson.  The Numerical Solution of Integral Equations of the Second Kind , 1997 .

[47]  John A. Evans,et al.  Isogeometric boundary element analysis using unstructured T-splines , 2013 .