Article
Full entry |
PDF
(1.0 MB)
Feedback
PDF
(1.0 MB)
Feedback
Keywords:
building design; heat transfer; inverse and optimization problems; Nelder-Mead algorithm
building design; heat transfer; inverse and optimization problems; Nelder-Mead algorithm
Summary:
Advanced building design is a rather new interdisciplinary research branch, combining knowledge from physics, engineering, art and social science; its support from both theoretical and computational mathematics is needed. This paper shows an example of such collaboration, introducing a model problem of optimal heating in a low-energy house. Since all particular function values, needed for optimization are obtained as numerical solutions of an initial and boundary value problem for a sparse system of parabolic partial differential equations of evolution with at least two types of physically motivated nonlinearities, the usual gradient-based methods must be replaced by the downhill simplex Nelder-Mead approach or its quasi-gradient modifications. One example of the real low-energy house in Moravian Karst is demonstrated with references to other practical applications.
References:
[1] Abdelhalim, A., Nakata, K., El-Alem, M., Eltawil, A.: Guided particle swarm optimization method to solve general nonlinear optimization problems. Eng. Optim. 50 (2018), 568-583. DOI 10.1080/0305215X.2017.1340945 | MR 3766656
[2] Castro, A. Bermúdez de: Continuum Thermomechanics. Progress in Mathematical Physics 43, Birkhäuser, Basel (2005). DOI 10.1007/3-7643-7383-0 | MR 2145925 | Zbl 1070.74001
[3] Bomberg, M.: Building science or building physics. Frontiers of Architectural Research 1 (2012), 421-423. DOI 10.1016/j.foar.2012.10.003
[4] Brandts, J., Korotov, S., Křížek, M., Šolc, J.: On nonobtuse simplicial partitions. SIAM Rev. 51 (2009), 317-335. DOI 10.1137/060669073 | MR 2505583 | Zbl 1172.51012
[5] Chen, S., Wang, X.: A derivative-free optimization algorithm using sparse grid integration. American Journal of Computational Mathematics 3 (2013), 16-26. DOI 10.4236/ajcm.2013.31003
[6] Colaço, M. J., Orlande, H. R. B., Dulikravich, G. S.: Inverse and optimization problems in heat transfer. J. Braz. Soc. Mech. Sci. & Eng. 28 (2006), 1-24. DOI 10.1590/S1678-58782006000100001
[7] Coope, I. D., Price, C. J.: Frame based methods for unconstrained optimization. J. Optim. Theory Appl. 107 (2000), 261-274. DOI 10.1023/A:1026429319405 | MR 1800945 | Zbl 0983.90074
[8] Directive 2010/31/EU of the European Parliament and of the Council on the Energy Performance of Buildings. Official Journal of the European Union L 153/13 (2010). DOI 10.3000/17252555.L_2010.153.eng
[9] Feist, W.: Gestaltungsgrundlagen Passivhäuser. Das Beispiel, Darmstadt (2000), German.
[10] Fiedler, M.: Special Matrices and Their Applications in Numerical Mathematics. Dover Publications, Mineola (1981). MR 2517585 | Zbl 1170.65018
[11] Gao, F., Han, L.: Implementing the Nelder-Mead simplex algorithm with adaptive parameters. Comput. Optim. Appl. 51 (2012), 259-277. DOI 10.1007/s10589-010-9329-3 | MR 2872499 | Zbl 1245.90121
[12] Gavin, H. P.: The Nelder-Mead algorithm in two dimensions. CEE 201L Uncertainty, Design, and Optimization Lecture Notes, Duke University, Durham (2016).
[13] Hicken, J. E., Alonso, J. J.: Multidisciplinary Design Optimization. Lecture Notes, Stanford University, Stanford (2012).
[14] Hudec, M., Johanisová, B., Mansbart, T.: Pasivní domy z přírodních materiálů. Grada, Praha (2012), Czech.
[15] Jarošová, P.: Optimalizace vybraných parametrů pro návrh objektů s nízkou energetickou náročností. Dissertation thesis, Brno University of Technology, Brno (2017), Avai-lable at https://www.vutbr.cz/www$_-$base/zav$_-$prace$_-$soubor$_-$verejne.php?file$_-$id=156981\kern0pt Czech.
[16] Jarošová, P., Vala, J.: New approaches to the thermal design of energy saving buildings. Advanced Materials Research 1126 (2015), 59-66. DOI 10.4028/www.scientific.net/AMR.1126.59
[17] Kelley, C. T.: Detection and remediation of stagnation in the Nelder-Mead algorithm using a sufficient decrease condition. SIAM J. Optim. 10 (1999), 43-55. DOI 10.1137/S1052623497315203 | MR 1722147 | Zbl 0962.65048
[18] Klaus, V.: Blue Planet in Green Shackles: What Is Endangered: Climate or Freedom?. Competitive Enterprise Institute, Washington (2008).
[19] Korjenic, A., Steuer, R., Šťastník, S., Vala, J., Bednar, T.: Beitrag zur Lösung des Problems der Algenbildung auf Aussenwänden mit Wärmedämmverbundsystemen (WDVS). Bauphysik 31 (2009), 343-353 German. DOI 10.1002/bapi.200910045
[20] Lagarias, J. C., Poonen, B., Wright, M. H.: Convergence of the restricted Nelder-Mead algorithm in two dimensions. SIAM J. Optim. 22 (2012), 501-532. DOI 10.1137/110830150 | MR 2968864 | Zbl 1246.49029
[21] Lagarias, J. C., Reeds, J. A., Wright, M. H., Wright, P. E.: Convergence properties of the Nelder-Mead simplex method in low dimensions. SIAM J. Optim. 9 (1998), 112-147. DOI 10.1137/S1052623496303470 | MR 1662563 | Zbl 1005.90056
[22] Luersen, M. A., Riche, R. Le, Guyon, F.: A constrained, globalized, and bounded Nelder-Mead method for engineering optimization. Struct. Multidiscip. Optim. 27 (2004), 43-54. DOI 10.1007/s00158-003-0320-9
[23] Lukšan, L.: Numerické Optimalizační metody. Technical report 930, Institute of Informatics AS CR, Praha (2005), Czech.
[24] McKinnon, K. I. M.: Convergence of the Nelder-Mead simplex method to a nonstationary point. SIAM J. Optim. 9 (1998), 148-158. DOI 10.1137/S1052623496303482 | MR 1662567 | Zbl 0962.65049
[25] Meir, I. A., Pearlmutter, D.: Building for climate change: planning and design considerations in time of climatic uncertainty. Corrosion Engineering Science and Technology 45 (2010), 70-75. DOI 10.1179/147842209X12476568584548
[26] Nawaz, A., Malik, T. H., Saleem, N., Mustafa, E.: Globalized Nelder Mead trained artificial neural networks for short term load forecasting. J. of Basic and Applied Scientific Research 5 (2015), 1-13.
[27] Nelder, J. A., Mead, R.: A simplex method for function minimization. Computer J. 7 (1965), 308-313. DOI 10.1093/comjnl/7.4.308 | MR 3363409 | Zbl 0229.65053
[28] Pacheco-Vega, A.: Soft computing applications in thermal energy systems. Soft Computing in Green and Renewable Energy Systems K. Gopalakrishnan, S. K. Khaitan, S. Kalogirou Studies in Fuzziness and Soft Computing 269, Springer, Berlin (2011), 1-35. DOI 10.1007/978-3-642-22176-7_1
[29] Pham, N., Wilamowski, B. M.: Improved Nelder-Mead's simplex method and applications. J. Computing 3 (2011), 55-63.
[30] Price, C. J., Coope, I. D., Byatt, D.: A convergent variant of the Nelder-Mead algorithm. J. Optim. Theory Appl. 113 (2002), 5-19. DOI 10.1023/A:1014849028575 | MR 1896704 | Zbl 1172.90508
[31] Rode, C.: Global building physics. J. Building Phys. 36 (2013), 337-352. DOI 10.1177/1744259112462107
[32] Roubíček, T.: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics 153, Birkhäuser, Basel (2005). DOI 10.1007/978-3-0348-0513-1 | MR 2176645 | Zbl 1087.35002
[33] Šťastník, S., Vala, J.: On the thermal stability in dwelling structures. Building Research J. 52 (2004), 31-55.
[34] Vala, J.: Computational approaches to some inverse problems from engineering practice. Programs and Algorithms of Numerical Mathematics. Proc. of the 17th Seminar J. Chleboun et al. Academy of Sciences of the Czech Republic, Institute of Mathematics, Praha (2015), 215-230. Zbl 1374.80001
[35] Vala, J.: Computational design optimization of low-energy buildings. Proc. 14th Equadiff in Bratislava Spektrum STU, Bratislava (2017), 265-274.
[36] Vala, J.: On some non-gradient approaches to inverse and optimization problems of thermal transfer. Proc. 23rd Thermophysics in Smolenice American Institute of Physics, Melville (2018), 5. Submitted.
[37] Vala, J., Jarošová, P.: Computational optimization of energy consumption in the design of buildings. Proc. 15th International Conference on Numerical Analysis and Applied Mathematics (ICNAAM) American Institute of Physics, Melville (2018), 4. In print.
[38] Wright, M. H.: Nelder, Mead, and the other simplex method. Doc. Math. (Bielefeld) Extra Vol., Optimization Stories (2012), 271-276. MR 2991490 | Zbl 1262.49003
Search
Partner of
