Abstract
This note addresses the use of the finite volume method (FVM) for topology optimization of a heat conduction problem. Issues pertaining to the proper choice of cost functions, sensitivity analysis, and example test problems are used to illustrate the effect of applying the FVM as an analysis tool for design optimization. This involves an application of the FVM to problems with nonhomogeneous material distributions, and the arithmetic and harmonic averages have here been used to provide a unique value for the conductivity at element boundaries. It is observed that when using the harmonic average, checkerboards do not form during the topology optimization process.
Similar content being viewed by others
References
Barth T, Ohlberger M (2004) Finite volume methods: foundation and analysis. In: Stein E, Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 1. Wiley, West Sussex, England
Bartholdi JJ III, Gue KR (2004) The best shape for a crossdock. Transp Sci 38(2):235–244. DOI:10.1287/trsc.1030.0077
Bejan A (2000) Shape and structure, from engineering to nature. Cambridge University Press, Cambridge, UK
Bejan A, Ledezma GA (1998) Streets tree networks and urban growth: optimal geometry for quickest access between a finite-size volume and one point. Physica A 255(1–2):211–217. DOI:10.1016/S0378-4371(98)00085-5
Bendsøe MP, Sigmund O (1999) Material interpolation schemes in topology optimization. Arch Appl Mech 69(9–10):635–654. DOI:10.1007/s004190050248
Bendsøe MP, Sigmund O (2004) Topology optimization—theory, methods, and applications. Springer, Berlin Heidelberg New York
Borrvall T, Petersson J (2003) Topology optimization of fluids in Stokes flow. Int J Numer Methods Fluids 41:77–107. DOI:10.1002/fld.426
Craig KJ, de Kock DJ, Snyman JA (2001) Minimizing the effect of automotive pollution in urban geometry using mathematical optimization. Atmos Environ 35:579–587. DOI:10.1016/S1352-2310(00)00307-1
Chalot FL (2004) Industrial aerodynamics. In: Stein E, Borst R, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 3. Wiley, West Sussex, England
Choi KK, Kim N-H (2005) Structural sensitivity analysis and optimization, 1 and 2. Springer, Berlin Heidelberg New York
Cockburn B, Karniadakis GE, Shu C-W (1999) Discontinuous Galerkin methods: theory, computation and applications. Springer, Berlin Heidelberg New York
Diaz AR, Benard A (2003) Topology optimization of heat-resistant structures. Proc ASME Des Eng Tech Conf 2A:633–639
Donoso A, Sigmund O (2004) Topology optimization of multiple physics probelms modelled by Poisson’s equation. Latin Am J Solid Struct 1:169–184
Evgrafov A (2004) Topology optimization of slightly incompressible fluids. Ph.D. thesis: Approximation of topology optimization problems using sizing optimization problems, Department of Mathematics, Chalmers University of Technology, Göteborg, Sweden, pp 55–81. ISBN 91-7291-466-1
Gersborg-Hansen A, Sigmund O, Haber RB (2005a) Topology optimization of channel flow problems. Struct Multidiscipl Optim 30(3):181–192. DOI:10.1007/s00158-004-0508-7
Gersborg-Hansen A, Bendsøe MP, Sigmund O (2005b) Topology optimization using the finite volume method. In: Proceedings of the 6th world congress of structural and multidisciplinary optimization, Rio de Janeiro, 2005
Guillaume P, Idris KS (2005) Topological sensitivity and shape optimization for the stokes equations. SIAM J Control Optim 43(1):1–31. DOI:10.1137/S0363012902411210
Habbal A, Petersson J, Thellner M (2004) Multidisciplinary topology optimization solved as a Nash game. Int J Numer Methods Eng 61(7):949–963. DOI:10.1002/nme.1093
Hashin Z (1983) Analysis of composite-materials—a survey. J Appl Mech 50(3):481–505
Hashin Z, Shtrikman S (1962) A variational approach to the theory of the effective magnetic permeability of multiphase materials. J Appl Phys 33(10):3125–3131
Hovedstadens Udviklingsråd (2003) Trafikplan 2003 (in Danish). Available at www.hur.dk. ISBN 87-7971-110-3
Jenny P, Lee SH, Tchelepi HA (2004) Adaptive multiscale finite-volume method for multiphase flow and transport in porous media. Multiscale Model Simul 3(1):50–64. DOI:10.1137/030600795
Jha MK, Schonfeld P (2004) A highway alignment optimization model using geographic information systems. Transp Res Part A 38:455–481. DOI:10.1016/j.tra.2004.04.001
Jong JC, Schonfeld P (2003) An evolutionary model for simultaneously optimizing three-dimensional highway alignments. Transp Res Part B 37:107–128. DOI:10.1016/S0191-2615(01)00047-9
Klarbring A, Petersson J, Torstenfelt B, Karlsson M (2003) Topology optimization of flow networks. Comput Methods Appl Mech Eng 192(35–36):3909–3932. DOI:10.1016/S0045-7825(03)00393-1
Lesaint P, Raviart P-A (1974) On a finite element method for solving the neutron transport equation. In: de Boor C (ed) Mathematical aspects of finite elements in partial differential equations. Academic, New York
Li Q, Steven GP, Xie YM, Querin OM (2004) Evolutionary topology optimization for temperature reduction of heat conducting fields. Int J Heat Mass Transfer 47:5071–5083. DOI:10.1016/ j.ijheatmasstransfer.2004.06.010
Mohammadi B, Pironneau O (2004) Shape optimization in fluid mechanics. Annu Rev Fluid Mech 36:255–279. DOI:10.1146/annurev.fluid.36.050802.121926
Olesen LO, Okkels F, Bruus H (2005) A high-level programming—language implementation of topology optimization applied to steady-state Navier–Stokes flow. Int J Numer Methods Eng. DOI:10.1002/nme.1468
Patankar SV (1980) Numerical heat transfer and fluid flow. Hemisphere, New York
Pedersen NL (2004) On optimization of bio-probes. Int J Numer Methods Eng 61:791–806. DOI:10.1002/nme.1026
Sigmund O (2001a) A 99 line topology optimization code written in MATLAB. Struct Multidiscipl Optim 21:120–127 (MATLAB code available online at: www.topopt.dtu.dk). DOI:10.1007/ s001580050176
Sigmund O (2001b) Design of multiphysics actuators using topology optimization—Part I: one-material structures. Comput Methods Appl Mech Eng 190(49–50):6577–6604. DOI:10.1016/ S0045-7825(01)00251-1
Sigmund O (2001c) Design of multiphysics actuators using topology optimization—Part II: two-material structures. Comput Mehtods Appl Mech Eng 190(49–50):6605–6627. DOI:10.1016/S0045-7825(01)00252-3
Sigmund O, Gersborg-Hansen A, Haber RB (2003) Topology optimization for multiphysics problems: a future FEMLAB application? In: Gregersen L (ed) Nordic Matlab conference (held in Copenhagen). Comsol, Søborg, Denmark, pp 237–242
Squire W, Trapp G (1998) Using complex variables to estimate derivatives of real functions. SIAM Rev 40(1):110–112. DOI:10.1137/S003614459631241X
Stolpe M, Svanberg K (2001) An alternative interpolation scheme for minimum compliance topology optimization. Struct Multidiscipl Optim 22(2):116–124. DOI:10.1007/s001580100129
Svanberg K (1987) The method of moving asymptotes—a new method for structural optimization. Int J Numer Methods Eng 24:359–373
Svanberg K (2002) A class of globally convergent optimization methods based on conservative convex separable approximations. SIAM J Optim 12(2):555–573. DOI:10.1137/S1052623499362822
Thellner M (2005) Topology optimization of convection–diffusion problems. Ph.D. thesis: Multi-parameter topology optimization in continuum mechanics. Linköping Studies in Science and Technology, Dissertations no. 934, pp 71–87. ISBN 91-85297-71-2
Torquato S, Gibiansky LV et al (1998) Effective mechanical and transport properties of cellular solids. Int J Mech Sci 40(1):71–82. DOI:10.1016/S0020-7403(97)00031-3
van Keulen F, Haftka RT, Kim NH (2005) Review of options for structural design sensitivity analysis. Part 1: linear systems. Comput Methods Appl Mech Eng 194(30–33):3213–3243. DOI:10.1016/j.cma.2005.02.002
Versteeg HK, Malalasekera W (1995) An introduction to computational fluid dynamics: the finite volume method. Longman Scientific & Technica, London
Vos JB et al (2002) Navier–Stokes solvers in European aircraft design. Prog Aerosp Sci 38:601–697. DOI:10.1016/S0376-0421(02)00050-7
Author information
Authors and Affiliations
Corresponding author
Additional information
Preliminary results of the work reported here were presented at the WCSMO 6 in Rio de Janeiro 2005, see Gersborg-Hansen et al. (2005b).
Rights and permissions
About this article
Cite this article
Gersborg-Hansen, A., Bendsøe, M.P. & Sigmund, O. Topology optimization of heat conduction problems using the finite volume method. Struct Multidisc Optim 31, 251–259 (2006). https://doi.org/10.1007/s00158-005-0584-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-005-0584-3