Abstract
This article introduces a new inverse method for thermal model parameter identification that stands out from standard inverse methods by its formulation. While these latter methods aim at identifying all the model parameters in order to fit the experimental data at best, the proposed goal-oriented inverse method focuses on the prediction of a specific quantity of interest, automatically identifying and updating the model parameters involved in its computation alone. To further reduce the computational time, the goal-oriented inverse method is associated with a model order reduction method referred to as Proper Generalized Decomposition (PGD). The objective of this original approach is to robustly predict the sought quantity of interest in a reduced computational time while using a limited measurement data set. The goal-oriented inverse method is developed and illustrated on transient heat transfer models encountered in building thermal problems. The first application deals with a simplified 1D heat transfer problem through a building wall with synthetic data, and the second one is dedicated to a real building with measured data. The performance of the approach is numerically assessed by comparing the results with those obtained using the classical least squares method (with Tikhonov's regularization). It is shown that the goal-oriented inverse method allows to robustly predict the sought quantities of interest, with an error of less than 5% by updating only the model parameters that affect it the most and thus leads to save computation time compared to standard inversion methods.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alekseev AK, Navon IM (2010). Criteria of optimality for sensors' location based on adjoint transformation of observation data interpolation error. International Journal for Numerical Methods in Fluids, 62: 74–89.
Allix O, Ladevèze P, Gilletta D, Ohayon R (1989). A damage prediction method for composite structures. International Journal for Numerical Methods in Engineering, 27: 271–283.
Allix O, Feissel P, Nguyen HM (2005). Identification strategy in the presence of corrupted measurements. Engineering Computations, 22: 487–504.
Alzetto F, Pandraud G, Fitton R, Heusler I, Sinnesbichler H (2018). QUB: A fast dynamic method for in situ measurement of the whole building heat loss. Energy and Buildings, 174: 124–133.
Artiges N (2016). De l'instrumentation au contrôle optimal prédictif pour la performance énergétique du bâtiment. PhD Thesis, Université Grenoble Alpes, France, (in French)
Bartaud du Chazaud E, Baleynaud J (2016). Contribution à la mise au point d'un modèle “en boite grise” pour le contrôle prédictif de la consommation énergétique des bâtiments. In: Congrès de la Société Française de thermique.
Becker R, Vexler B (2004). A posteriori error estimation for finite element discretization of parameter identification problems. Numerische Mathematik, 96: 435–459.
Berger J, Gasparin S, Chhay M, Mendes N (2016a). Estimation of temperature-dependent thermal conductivity using proper generalised decomposition for building energy management. Journal of Building Physics, 40: 235–262.
Berger J, Orlande HRB, Mendes N, Guernouti S (2016b). Bayesian inference for estimating thermal properties of a historic building wall. Building and Environment, 106: 327–339.
Berger J, Mendes N, Guernouti S, Woloszyn M, Chinesta F (2017a). Review of reduced order models for heat and moisture transfer in building physics with emphasis in PGD approaches. Archives of Computational Methods in Engineering, 24: 655–667.
Berger J, Orlande HRB, Mendes N (2017b). Proper Generalized Decomposition model reduction in the Bayesian framework for solving inverse heat transfer problems. Inverse Problems in Science and Engineering, 25: 260–278.
Beringhier M, Gigliotti M (2015). A novel methodology for the rapid identification of the water diffusion coefficients of composite materials. Composites Part A: Applied Science and Manufacturing, 68:212–218.
Berthou T, Stabat P, Salvazet R, Marchio D (2013). Optimal control for building heating: An elementary school case study. In: Proceedings of the 13th International IBPSA Building Simulation Conference, Chambéry, France.
Binev P, Cohen A, Mula O, Nichols I (2018). Greedy algorithms for optimal measurements selection in state estimation using reduced models. ASA Journal on Uncertainty Quantification, 6: 1101–1126.
Boisson P, Bouchié R (2014). ISABELE method: In-Situ Assesslent of the Building EnveLope pErformances. In: Proceedings of the 9th International Conference on System Simulation in Buildings, Liege, Belgium.
Bonnet M, Aquino W (2014). Three-dimensional transient elastodynamic inversion using the modified error in constitutive relation. Journal of Physics: Conference Series, 542: 012003.
Bouchié R, Alzetto F, Brun A, Boisson P, Thebaut S (2014). Short methodologies for in-situ assessment of the intrinsic thermal performance of the building envelope. In: Proceedings of Sustainable Places, Nice, France.
Bouclier R, Louf F, Chamoin L (2013). Real-time validation of mechanical models coupling PGD and constitutive relation error. Computational Mechanics, 52: 861–883.
Braack M, Ern A (2003). A posteriori control of modeling errors and discretization errors. Multiscale Modeling & Simulation, 1:221–238.
Brouns J, Nassiopoulos A, Bourquin F, Limam K (2016). Dynamic building performance assessment using calibrated simulation. Energy and Buildings, 122: 160–174.
Bui HD, Constanctinescu A (2000). Spatial localization of the error of constitutive law for the identification of defects in elastic bodies. Archives of Mechanics, 52(4–5): 511–522.
Chamoin L, Ladevèze P, Waeytens J (2014). Goal-oriented updating of mechanical models using the adjoint framework. Computational Mechanics, 54: 1415–1430.
Chatterjee A (2000). An introduction to the proper orthogonal decomposition. Current Science, 78(7): 10.
Chavent G, Kunisch K, Roberts J (1996). Primal-dual formulations for parameter identification problems. Research report, INRIA Research Report RR-2891.
Chinesta F, Ladeveze P, Cueto E (2011). A short review on model order reduction based on proper generalized decomposition. Archives of Computational Methods in Engineering, 18: 395–404.
Chinesta F, Keunings R, Leygue A (2014). The Proper Generalized Decomposition For Advanced Numerical Simulations. Cham, Switzerland: Springer.
Chinesta F, Huerta A, Rozza G, Willcox K (2017). Model order reduction. In: Stein E, Borst R, Hughes JTR (eds), Encyclopedia of Computational Mechanics, 2nd edn. Chichester, UK: John Wiley & Sons.
Chouaki A, Ladevèze P, Proslier L (1996). An updating of structural dynamic model with damping. In: Proceedings of the 2nd International Conference on Inverse Problems in Engineering: Theory and Practice, Le Croisic, France, pp. 335–342.
Daescu DN, Carmichael GR (2003). An adjoint sensitivity method for the adaptive location of the observations in air quality modeling. Journal of the Atmospheric Sciences, 60: 434–450.
De Simon L, Iglesias M, Jones B, Wood C (2018). Quantifying uncertainty in thermophysical properties of walls by means of Bayesian inversion. Energy and Buildings, 177: 220–245.
Deng K, Barooah P, Mehta PG, Meyn SP (2010). Building thermal model reduction via aggregation of states. In: Proceedings of American Control Conference, Baltimore, MD, USA.
Deraemaeker A, Ladevèze P, Leconte P (2002). Reduced bases for model updating in structural dynamics based on constitutive relation error. Computer Methods in Applied Mechanics and Engineering, 191:2427–2444.
Derkx F, Lebental B, Bourouina T, Bourquin F, Cojocaru C-S, Robine E, Van Damme H (2012). The Sense-City project. In: Proceedings of XVIIIth Symposium on Vibrations, Shocks and Noise.
Faggianelli GA, Brun A, Wurtz E, Muselli M (2015). Grey-box modelling for naturally ventilated buildings. In: Proceedings of the 14th International IBPSA Building Simulation Conference, Hyderabad, India.
Feissel P, Allix O (2007). Modified constitutive relation error identification strategy for transient dynamics with corrupted data: The elastic case. Computer Methods in Applied Mechanics and Engineering, 196: 1968–1983.
Gagnon R, Gosselin L, Decker S (2018). Sensitivity analysis of energy performance and thermal comfort throughout building design process. Energy and Buildings, 164: 278–294.
Golub GH, Reinsch C (1970). Singular value decomposition and least squares solutions. Numerische Mathematik, 14: 403–420.
Gonzalez D, Masson F, Poulhaon F, Leygue A, Cueto E, Chinesta F (2012). Proper Generalized Decomposition based dynamic data driven inverse identification. Mathematics and Computers in Simulation, 82: 1677–1695.
Goyal S, Barooah P (2011). A method for model-reduction of nonlinear building thermal dynamics. In: Proceedings of American Control Conference, San Francisco, CA, USA.
Hadamard J (1923). Lecture on Cauchy's Problem in Linear Partial Differential Equations. New Haven, CT, USA: Yale University Press.
Heiselberg P, Brohus H, Hesselholt A, Rasmussen HES, Seinre E, Thomas S (2007). Application of sensitivity analysis in design of sustainable buildings. In: Proceedings of Sustainable Development of Building and Environment (SDBE), Chongqing, China.
Hong T, Jiang Y (1997). A new multizone model for the simulation of building thermal performance. Building and Environment, 32: 123–128.
IEA (2019). Energy efficiency: Buildings. International Energy Agency. Available at https://www.iea.org/topics/energyefficiency/buildings/. Accessed 18 Sept 2019.
Johansson H, Runesson K, Larsson F (2007). Parameter identification with sensitivity assessment and error computation. GAMM- Mitteilungen, 30: 430–457.
Johansson H, Larsson F, Runesson K (2011). Application-specific error control for parameter identification problems. International Journal for Numerical Methods in Biomedical Engineering, 27: 608–618.
Kleiber M, Antúnez P, Hien TD, Kowalczyk H (1997). Parameter Sensitivity in Nonlinear Mechanics: Theory and Finite Element Computations. New York: John Wiley & Sons.
Klema VC, Laub AJ (1980). The singular value decomposition: Its computation and some applications. IEEE Transactions on Automatic Control, 25: 164–176.
Kristensen MH, Petersen S (2016). Choosing the appropriate sensitivity analysis method for building energy model-based investigations. Energy and Buildings, 130: 166–176.
Ladevèze P (1977). Nouvelle procédure d'estimation d'erreur relative à la méthode des éléments finis et applications. Publications Mathématiques et informatiques de Rennes, pp. 1–19. (in French)
Ladevèze P, Nedjar D, Reynier M (1994). Updating of finite element models using vibration tests. AIAA Journal, 32: 1485–1491.
Ladevèze P, Chouaki A (1999). Application of a posteriori error estimation for structural model updating. Inverse Problems, 15: 49–58.
Ladevèze P, Moës N, Douchin B (1999). Constitutive relation error estimators for (visco)plastic finite element analysis with softening. Computer Methods in Applied Mechanics and Engineering, 176: 247–264.
Ladevèze P, Pelle JP (2005). The constitutive relation error method for linear problems. In: Ling FF, Gloyna EF, Hart WH (eds), Mastering Calculations in Linear and Nonlinear Mechanics. New York: Springer, pp. 29–50.
Ladevèze P, Puel G, Deraemaeker A, Romeuf T (2006). Validation of structural dynamics models containing uncertainties. Computer Methods in Applied Mechanics and Engineering, 195: 373–393.
Ladevèze P, Chamoin L (2015). The constitutive relation error method: A general verification tool. In: Chamoin L, Diez P (eds), Verifying Calculations-Forty Years On. An Overview Of Classical Verification Techniques for FEM Simulations. Cham, Switzerland: Springer, pp. 59–89.
Lawson CL, Hanson RJ (1974). Solving Least Squares Problem. Englewood Cliffs, NJ, USA: Prentice-Hall
Li X, Nassiopoulos A, Waeytens J, Chakir R (2015). A posteriori estimation of modeling error for a building thermal model. In: Proceedings of VII International Conference on Adaptive Modelling and Simulation (Admos 2015).
Li W, Tian Z, Lu Y, Fu F (2018). Stepwise calibration for residential building thermal performance model using hourly heat consumption data. Energy and Buildings, 181: 10–25.
Maday Y, Mula O (2013). A generalized empirical interpolation method: Application of reduced basis techniques to data assimilation. In: Brezzi F, Colli Franzone P, Gianazza U, Gilardi G (eds), Analysis and Numerics of Partial Differential Equations. Milano: Springer, pp. 221–235.
Maday Y, Mula O, Patera AT, Yano M (2015a). The generalized empirical interpolation method: stability theory on Hilbert spaces with an application to the stokes equation. Computer Methods in Applied Mechanics and Engineering, 287: 310–334.
Maday Y, Patera AT, Penn JD, Yano M (2015b). A parameterized-background data-weak approach to variational data assimilation: formulation, analysis, and application to acoustics. International Journal for Numerical Methods in Engineering, 102: 933–965.
Mangematin E, Pandraud G, Roux D (2012). Quick measurements of energy efficiency of buildings. Comptes Rendus Physique, 13: 383–390.
Marchand B, Chamoin L, Rey C (2016). Real-time updating of structural mechanics models using Kalman filtering, modified constitutive relation error, and proper generalized decomposition. International Journal for Numerical Methods in Engineering, 107: 786–810.
Marchand B, Chamoin L, Rey C (2019). Parameter identification and model updating in the context of nonlinear mechanical behaviors using a unified formulation of the modified Constitutive Relation Error concept. Computer Methods in Applied Mechanics and Engineering, 345: 1094–1113.
Martinez S, Erkoreka A, Eguia P, Granada E, Febrero L (2019). Energy characterization of a PASLINK test cell with a gravel covered roof using a novel methodology: Sensitivity analysis and Bayesian calibration. Journal of Building Engineering, 22: 1–11.
Morozov VA (1966). On the solution of functional equations by the method of regularization. Doklady Akademii Nauk SSSR, 167(3): 510–512.
Nassiopoulos A, Kuate R, Bourquin F (2014). Calibration of building thermal models using an optimal control approach. Energy and Buildings, 76: 81–91.
Nguyen H-M, Allix O, Feissel P (2008). A robust identification strategy for rate-dependent models in dynamics. Inverse Problems, 24: 065006.
Nouy A (2010). A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations. Computer Methods in Applied Mechanics and Engineering, 199: 1603–1626.
Oden J, Prudhomme S (2002). Estimation of modeling error in computational mechanics. Journal of Computational Physics, 182: 496–515.
Ohlberger M, Rave S (2016). Reduced basis methods: Success, limitations and future challenges. In: Proceedings of ALGORITMY.
Papadimitriou C, Lombaert G (2012). The effect of prediction error correlation on optimal sensor placement in structural dynamics. Mechanical Systems and Signal Processing, 28: 105–127.
Quarteroni A, Manzoni A, Negri F (2016). Reduced Basis Methods for Partial Differential Equations. Cham, Switzerland: Springer.
Raillon L, Ghiaus C (2018). An efficient Bayesian experimental calibration of dynamic thermal models. Energy, 152: 818–833.
Rouchier S (2018). Solving inverse problems in building physics: An overview of guidelines for a careful and optimal use of data. Energy and Buildings, 166: 178–195.
Rozza G (2009). An introduction to reduced basis method for parametrized PDEs. In: Proceedings of WSPC.
Rozza G, Malik H, Demo N, Tezzele M, Girfoglio M, Stabile G, Mola A (2018). Advances in Reduced Order Methods for parametric industrial problems in computational fluid dynamics. arXiv: 1811.08319.
Rubio P-B, Louf F, Chamoin L (2018). Fast model updating coupling Bayesian inference and PGD model reduction. Computational Mechanics, 62: 1485–1509.
Saltelli A, Tarantola S, Campolongo F (2000). Sensitivity analysis as an ingredient of modeling. Statistical Science, 15: 377–395.
Signorini M, Zlotnik S, Diez P (2017). Proper generalized decomposition solution of the parameterized Helmholtz problem: application to inverse geophysical problems. International Journal for Numerical Methods in Engineering, 109: 1085–1102.
Thébault S, Bouchié R (2018). Refinement of the ISABELE method regarding uncertainty quantification and thermal dynamics modelling. Energy and Buildings, 178: 182–205.
Tikhonov TN, Arsenin Y (1977). Solutions to ill-posed problems. New York: Winston-Widley.
Waeytens J, Mahfoudhi I, Chabchoub MA, Chatellier P (2017). Adjoint-based numerical method using standard engineering software for the optimal placement of chlorine sensors in drinking water networks. Environmental Modelling and Software, 92: 229–238.
Acknowledgments
The authors acknowledge support from the project Sense-City funded by ANR (France) within the Investment for the Future Program under reference number ANR-10-EQPX-48. We would like to thank the Sense-City team, and particularly Erick Merliot for his technical advice and support during the sensors deployment.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Djatouti, Z., Waeytens, J., Chamoin, L. et al. Coupling a goal-oriented inverse method and proper generalized decomposition for fast and robust prediction of quantities of interest in building thermal problems. Build. Simul. 13, 709–727 (2020). https://doi.org/10.1007/s12273-020-0603-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12273-020-0603-8