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It is of great concern to produce numerically efficient methods for moisture diffusion through porous media, capable of accurately calculate moisture distribution with a reduced computational effort. In this way, model reduction methods are promising approaches to bring a solution to this issue since they do not degrade the physical model and provide a significant reduction of computational cost. Therefore, this article explores in details the capabilities of two model-reduction techniques - the Spectral Reduced-Order Model (Spectral-ROM) and the Proper Generalised Decomposition (PGD) - to numerically solve moisture diffusive transfer through porous materials. Both approaches are applied to three different problems to provide clear examples of the construction and use of these reduced-order models. The methodology of both approaches is explained extensively so that the article can be used as a numerical benchmark by anyone interested in building a reduced-order model for diffusion problems in porous materials. Linear and non-linear unsteady behaviors of unidimensional moisture diffusion are investigated. The last case focuses on solving a parametric problem in which the solution depends on space, time and the diffusivity properties. Results have highlighted that both methods provide accurate solutions and enable to reduce significantly the order of the model around ten times lower than the large original model. It also allows an efficient computation of the physical phenomena with an error lower than 10^{-2} when compared to a reference solution.
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Transport in Porous Media (2018) 124:965–994
https://doi.org/10.1007/s11242-018-1106-2
Advanced Reduced-Order Models for Moisture Diffusion in
Porous Media
Suelen Gasparin1,3 ·Julien Berger2·Denys Dutykh3·Nathan Mendes1
Received: 10 October 2017 / Accepted: 6 June 2018 / Published online: 27 June 2018
© Springer Nature B.V. 2018
Abstract
It is of great concern to produce numerically efficient methods for moisture diffusion through
porous media, capable of accurately calculate moisture distribution with a reduced compu-
tational effort. In this way, model reduction methods are promising approaches to bring a
solution to this issue since they do not degrade the physical model and provide a significant
reduction of computational cost. Therefore, this article explores in details the capabilities
of two model reduction techniques—the Spectral reduced-order model and the proper gen-
eralized decomposition—to numerically solve moisture diffusive transfer through porous
materials. Both approaches are applied to three different problems to provide clear exam-
ples of the construction and use of these reduced-order models. The methodology of both
approaches is explained extensively so that the article can be used as a numerical benchmark
by anyone interested in building a reduced-order model for diffusion problems in porous
materials. Linear and nonlinear unsteady behaviors of unidimensional moisture diffusion are
investigated. The last case focuses on solving a parametric problem in which the solution
depends on space, time and the diffusivity properties. Results have highlighted that both
methods provide accurate solutions and enable to reduce significantly the order of the model
around 10 times lower than the large original model. It also allows an efficient computation
of the physical phenomena with an error lower than 102when compared to a reference
solution.
Keywords Reduced-order modeling ·Moisture diffusion ·Numerical methods ·Spectral
methods ·Proper generalized decomposition (PGD)
BSuelen Gasparin
suelen.gasparin@pucpr.edu.br
1LST, Thermal Systems Laboratory, Mechanical Engineering Graduate Program, Pontifical Catholic
University of Paraná, Rua Imaculada Conceição, 1155, Curitiba, Paraná CEP: 80215-901, Brazil
2CNRS, LOCIE, Université Savoie Mont Blanc, 73000 Chambéry, France
3LAMA, Laboratory of Mathematics, UMR 5127 CNRS, University of Savoie Mont Blanc, Campus
Scientifique, 73376 Le Bourget-du-Lac, France
123
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... Moreover, in A comparative study of two reduced order models for moisture diffusion problems in building physics (Gasparin et al., 2018d), we explore in details the capabilities of two model-reduction techniques -the Spectral Reduced-Order Model (Spectral-ROM) and the Proper Generalised Decomposition (PGD) -to numerically solve moisture diffusion transfer through porous materials. Both approaches are applied to three different problems to provide clear examples of the construction and use of these reduced-order models. ...
... The speed of calculation is higher for the spectral-parametric approach as reported in Table 3.11. These results are also commented in (Gasparin et al., 2018d). Indeed, the number of modes in the Spectral-parametric approach increases by considering a new parameter as another coordinate, making the integration in time slower than computing the solution for each parameter. ...
... For strong reduction of the computational cost with a very satisfactory accuracy of the solution, the Spectral has been shown as the best option (Gasparin et al., 2018e). It is particularly efficient to treat parametric problems where one aims to compute the solution depending on an input parameter (Gasparin et al., 2018d). Furthermore, for 2D transport phenomena the Spectral method is also an interesting ap-proach. ...
Thesis
Building energy consumption is directly impacted by weather parameters such as temperature, solar radiation, atmospheric pressure, relative humidity and wind velocity. The knowledge of the building hygrothermal performance enables the design of energy efficient buildings and the prediction of overall durability and sustainability of envelopes. Therefore, designers and builders are interested in modeling the long-term performance of the envelopes by means of accurate, reliable and fast simulation tools.Several numerical models have been proposed in the literature to study the heat and moisture transfer in building materials. In general, this problem is solved by traditional methods, such as finite-difference and finite-volume methods, using mainly implicit schemes. Nevertheless, these methods impose costly sub-iterations to treat the nonlinearities and very fine discretization, which increase substantially the simulation computational cost. Therefore, this research has been focused on the development and analyses of numerical methods for efficiently simulate the problem of heat and mass transfer through porous materials.In the first part of this thesis, improved schemes of the traditional numerical methods have been developed to eliminate costly sub-iterations to treat nonlinearities, to improve the order of accuracy and to save computer run time. Despite the great progress with the new numerical schemes, the conclusion of the first part shows that we still have to deal with large systems of equations, particularly when treating multi-dimensional transfer problems. For this reason, to reduce even more the computational burden and the size of the system, a reduced-order model, based on spectral methods is proposed in the sequence to provide an accurate description of the physical phenomena. The degrees of freedom of the solution is strongly decreased while maintaining the model fidelity. It ensures a computational cost much lower than the complete original model.All these methods are applied to problems related to building physics, such as single and multilayer nonlinear transfer, the determination of optimum insulation thickness, the process of moisture buffer effects and transfer in one- or two-zone building models. In conclusion, we show how to build efficient numerical models, in terms of computational cost and accuracy, to investigate the heat and mass transfer in porous materials.
... Although the appearance of digital computers started in the fifties and the rapid and progressive hardware evolution since the seventies, this RC method is still used in many algorithms to solve the partial differential equation of heat transfer as for instance in Fraisse et al. (2002); Naveros and Ghiaus (2015); Roels et al. (2017). The third method is more advanced spectral method which was recently applied for the solution of diffusion problems through porous building elements Gasparin et al. (2017Gasparin et al. ( , 2018. ...
... The main advantage of the spectral method is that n ≪ p, where p is the number of degrees of freedom needed to solve problem (5) by means of conventional methods such as finitedifferences, finite-volume or finite-element methods. For these reasons, the spectral method is also denoted as the spectral-Reduced Order Method (spectral-ROM) Gasparin et al. (2017Gasparin et al. ( , 2018. ...
... where u 0 ( x ), is the dimensionless initial condition. Interested readers may refer to Gasparin et al. (2017Gasparin et al. ( , 2018 for further details on the spectral method. ...
Preprint
Predictions of physical phenomena in buildings are carried out by using physical models formulated as a mathematical problem and solved by means of numerical methods, aiming at evaluating, for instance, the building thermal or hygrothermal performance by calculating distributions and fluxes of heat and moisture transfer. Therefore, the choice of the numerical method is crucial since it is a compromise among (i) the solution accuracy, (ii) the computational cost to obtain the solution and (iii) the complexity of the method implementation. An efficient numerical method enables to compute an accurate solution with a minimum computational run time (CPU). On that account, this article brings an investigation on the performance of three numerical methods. The first one is the standard and widely used finite-difference approach, while the second one is the so-called RC approach, which is a particular method brought to the building physics area by means of an analogy of electric circuits. The third numerical method is the spectral one, which has been recently proposed to solve nonlinear diffusive problems in building physics. The three methods are evaluated in terms of accuracy on the assessment of the dependent variable (temperature or vapor pressure) or of density of fluxes for three different cases: i) heat diffusion through a concrete slab, ii) moisture diffusion through an aerated concrete slab and iii) heat diffusion using measured temperatures as boundary conditions. Results highlight the spectral approach as the most accurate method. The RC based model with a few number of resistances does not provide accurate results for temperature and vapor pressure distributions neither to flux densities nor conduction loads.
... Although the appearance of digital computers started in the fifties and the rapid and progressive hardware evolution since the seventies, this RC method is still used in many algorithms to solve the partial differential equation of heat transfer as for instance in Fraisse et al. (2002); Naveros and Ghiaus (2015); Roels et al. (2017). The third method is more advanced spectral method which was recently applied for the solution of diffusion problems through porous building elements (Gasparin et al. 2018(Gasparin et al. , 2019. ...
... The main advantage of the spectral method is that n  p, where p is the number of degrees of freedom needed to solve problem (5) by means of conventional methods such as finite-differences, finitevolume or finite-element methods. For these reasons, the spectral method is also denoted as the spectral-Reduced Order Method (spectral-ROM) (Gasparin et al. 2018(Gasparin et al. , 2019. The derivatives are written as follows: ...
... where u 0 (x), is the dimensionless initial condition. Interested readers may refer to Gasparin et al. (2018Gasparin et al. ( , 2019 for further details on the spectral method. ...
Article
Predictions of physical phenomena in buildings are carried out by using physical models formulated as a mathematical problem and solved by means of numerical methods, aiming at evaluating, for instance, the building thermal or hygrothermal performance by calculating distributions and fluxes of heat and moisture transfer. Therefore, the choice of the numerical method is crucial since it is a compromise among (i) the solution accuracy, (ii) the computational cost to obtain the solution and (iii) the complexity of the method implementation. An efficient numerical method enables to compute an accurate solution with a minimum computational run time (CPU). On that account, this article brings an investigation on the performance of three numerical methods. The first one is the standard and widely used finite-difference approach, while the second one is the so-called RC approach, which is a particular method brought to the building physics area by means of an analogy of electric circuits. The third numerical method is the spectral one, which has been recently proposed to solve nonlinear diffusive problems in building physics. The three methods are evaluated in terms of accuracy on the assessment of the dependent variable (temperature or vapor pressure) or of density of fluxes for three different cases: i) heat diffusion through a concrete slab, ii) moisture diffusion through an aerated concrete slab and iii) heat diffusion using measured temperatures as boundary conditions. Results highlight the spectral approach as the most accurate method. The RC based model with a few number of resistances does not provide accurate results for temperature and vapor pressure distributions neither to flux densities nor conduction loads.
... As a matter of fact, in several cases, the advection moisture transfer phenomenon should also be considered in the context of porous media. Moisture advection is the transport of water vapor by the air and is included as an additional term in the equation of the process (4) (details are not reported here, the reader can see [28,29] as recent references). Berger et al. [28] show the inclusion of an advective term in the model and results in the context of moisture transfer in porous building elements, whereas Gasparin et al. [29] discuss numerically efficient methods for moisture diffusion, with a reduced computational effort. ...
... Moisture advection is the transport of water vapor by the air and is included as an additional term in the equation of the process (4) (details are not reported here, the reader can see [28,29] as recent references). Berger et al. [28] show the inclusion of an advective term in the model and results in the context of moisture transfer in porous building elements, whereas Gasparin et al. [29] discuss numerically efficient methods for moisture diffusion, with a reduced computational effort. Azeem et al. [30] focused recently on a review on the liquid moisture transport behavior of fabric, stating that transportation of liquid water in the fabric cannot be defined at only one condition, but a range of conditions that should be measured regarding the ability to transport liquid moisture. ...
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The aim is to develop soft sensors (SSs) to provide an estimation of the laundry moisture of clothes introduced in a household Heat Pump Washer–Dryer (WD-HP) appliance. The developed SS represents a cost-effective alternative to physical sensors, and it aims at improving the WD-HP performance in terms of drying process efficiency of the automatic drying cycle. To this end, we make use of appropriate Machine Learning models, which are derived by means of Regularization and Symbolic Regression methods. These methods connect easy-to-measure variables with the laundry moisture content, which is a difficult and costly to measure variable. Thanks to the use of SSs, the laundry moisture estimation during the drying process is effectively available. The proposed models have been tested by exploiting real data through an experimental test campaign on household drying machines.
... In recent works, researchers have implemented spectral methods for solving heat and moisture transfer in food engineering [38] and on fluid flow [36]. Recently, in [20], the authors have studied the moisture transfer in porous building materials considering layered domains, and in [18], they have compared the Spectral method to others ROMs, applied to parametric problems of the building physics field. ...
... Therefore, the scope of this work is to continue the investigations presented in [20] and [18], extending it to the coupled heat and mass transfer. Here, the Spectral method is used to compute one-dimensional heat and moisture diffusion transfer in porous materials, which is validated against experimental data from the literature. ...
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This work presents an efficient numerical method based on spectral expansions for simulation of heat and moisture diffusive transfers through multilayered porous materials. Traditionally, by using the finite-difference approach, the problem is discretized in time and space domains (Method of lines) to obtain a large system of coupled Ordinary Differential Equations (ODEs), which is computationally expensive. To avoid such a cost, this paper proposes a reduced-order method that is faster and accurate, using a much smaller system of ODEs. To demonstrate the benefits of this approach, tree case studies are presented. The first one considers nonlinear heat and moisture transfer through one material layer. The second case - highly nonlinear - imposes a high moisture content gradient - simulating a rain like condition - over a two-layered domain, while the last one compares the numerical prediction against experimental data for validation purposes. Results show how the nonlinearities and the interface between materials are easily and naturally treated with the spectral reduced-order method. Concerning the reliability part, predictions show a good agreement with experimental results, which confirm robustness, calculation efficiency and high accuracy of the proposed approach for predicting the coupled heat and moisture transfer through porous materials.
... In recent works, researchers have implemented spectral methods for solving heat and moisture transfer in food engineering [38] and on fluid flow [36]. Recently, in [20], the authors have studied the moisture transfer in porous building materials considering layered domains, and in [18], they have compared the Spectral method to others ROMs, applied to parametric problems of the building physics field. ...
... Therefore, the scope of this work is to continue the investigations presented in [20] and [18], extending it to the coupled heat and mass transfer. Here, the Spectral method is used to compute one-dimensional heat and moisture diffusion transfer in porous materials, which is validated against experimental data from the literature. ...
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This work presents an efficient numerical method based on spectral expansions for simulation of heat and moisture diffusive transfers through multilayered porous materials. Traditionally, by using the finite-difference approach, the problem is discretized in time and space domains (Method of lines) to obtain a large system of coupled Ordinary Differential Equations (ODEs), which is computationally expensive. To avoid such a cost, this paper proposes a reduced-order method that is faster and accurate, using a much smaller system of ODEs. To demonstrate the benefits of this approach, tree case studies are presented. The first one considers nonlinear heat and moisture transfer through one material layer. The second case - highly nonlinear - imposes a high moisture content gradient - simulating a rain like condition - over a two-layered domain, while the last one compares the numerical prediction against experimental data for validation purposes. Results show how the nonlinearities and the interface between materials are easily and naturally treated with the spectral reduced-order method. Concerning the reliability part, predictions show a good agreement with experimental results, which confirm robustness, calculation efficiency and high accuracy of the proposed approach for predicting the coupled heat and moisture transfer through porous materials.
... With spectral methods [20], the PGD method is one of the unique methods that allows to create a complete parametric model without knowing a priori the solution of the problem. ...
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... These results are still encouraging for future work on the integration of the Spectral method to 2D problems and to the highly nonlinear coupled problem of heat and moisture transfer. In addition, the Part 2[15]of this work aims at comparing the Spectral approach to the Proper Generalised Decomposition (PGD) method, recently applied in building physics[6].Problem (2.6) is considered with g ⋆ l, L = g ⋆ l, R = 0 and the dimensionless properties of the material are equal to d ⋆ m = 1 and c ⋆ m = 8.6. The reference time is t 0 = 1 h, thus the final simulation time is fixed to τ ⋆ = 120. ...
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This paper proposes the use of a Spectral method to simulate diffusive moisture transfer through porous materials as a Reduced-Order Model (ROM). The Spectral approach is an a priori method assuming a separated representation of the solution. The method is compared with both classical Euler implicit and Crank-Nicolson schemes, considered as large original models. Their performance - in terms of accuracy, complexity reduction and CPU time reduction - are discussed for linear and nonlinear cases of moisture diffusive transfer through single and multi-layered one-dimensional domains, considering highly moisture-dependent properties. Results show that the Spectral reduced-order model approach enables to simulate accurately the field of interest. Furthermore, numerical gains become particularly interesting for nonlinear cases since the proposed method can drastically reduce the computer run time, by a factor of 100, when compared to the traditional Crank-Nicolson scheme for one-dimensional applications.
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Estimating the temperature field of a building envelope could be a time-consuming task. The use of a reduced-order method is then proposed: the Proper Generalized Decomposition method. The solution of the transient heat equation is then re-written as a function of its parameters: the boundary conditions, the initial condition, etc. To avoid a tremendous number of parameters, the initial condition is parameterized. This is usually done by using the Proper Orthogonal Decomposition method to provide an optimal basis. Building this basis requires data and a learning strategy. As an alternative, the use of orthogonal polynomials (Chebyshev, Legendre) is here proposed. Highlights • Chebyshev and Legendre polynomials are used to approximate the initial condition • Performance of Chebyshev and Legendre polynomials are compared to the POD basis • Each basis combined with the PGD model is compared to laboratory measurements • The influence of four different parameters on the accuracy of the basis is studied • For each approximation basis, CPU calculation times are evaluated and compared
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Introduction * Fundamentals of Spectral Methods * Fourier Method * Chebyshev Method * Time-Dependent Equations * Navier-Stokes Equations for Incompressible Fluids * Vorticity-Streamfunction Equations * Velocity-Streamfunction Equations * Velocity-Pressure Equations * Stiff and Singular Problems * Domain Decomposition Method* Appendices * References * Index
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This paper proposes the use of a Spectral method to simulate diffusive moisture transfer through porous materials as a Reduced-Order Model (ROM). The Spectral approach is an a priori method assuming a separated representation of the solution. The method is compared with both classical Euler implicit and Crank-Nicolson schemes, considered as large original models. Their performance - in terms of accuracy, complexity reduction and CPU time reduction - are discussed for linear and nonlinear cases of moisture diffusive transfer through single and multi-layered one-dimensional domains, considering highly moisture-dependent properties. Results show that the Spectral reduced-order model approach enables to simulate accurately the field of interest. Furthermore, numerical gains become particularly interesting for nonlinear cases since the proposed method can drastically reduce the computer run time, by a factor of 100, when compared to the traditional Crank-Nicolson scheme for one-dimensional applications.
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