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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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... 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. ...

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.

... The value of m was determined empirically according to numerical investigations, which is considered as m = N + 5 in our computations (Gasparin et al., 2019a). ...

... In the first part of the work with the Spectral method Solving nonlinear diffusion problems in buildings by means of a Spectral Reduced-Order Model (Gasparin et al., 2019a) we simulate the moisture diffusion in one-dimension through porous materials as a Reduced-Order Model (ROM). The efficiency of the Spectral approach is demonstrated for simple and multilayered domains with highly nonlinear properties with sharp boundary conditions and profiles of solutions. ...

... However, if the problem is linear, the number of modes required can be even further reduced. In Gasparin et al. (2019a), in the linear case, 6 modes were enough to compose a sufficiently accurate solution. The other factor influencing the Spectral solution is the tolerance of the solver. ...

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. 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. ...

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.

... 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. ...

... The value of m is approximately the same as the number of modes, as discussed in [20]. ...

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. ...

... The value of m is approximately the same as the number of modes, as discussed in [20]. ...

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.

... Therefore, the scope of this work is limited to the application of two advanced a priori reduced-order model techniques, the PGD (Berger et al. 2015) and the Spectral-ROM (Gasparin et al. 2017a) techniques, which have been successfully implemented for coupled heat and moisture diffusion problems. The comparison is carried out for moisture transfer phenomena providing an important primary evaluation for anyone intending to build a reduced-order model for diffusion problem. ...

... The scope of this work is limited to the two a priori reduced-order model techniques, the PGD (Berger et al. 2015) and the Spectral-ROM (Gasparin et al. 2017a) techniques, which have been successfully implemented in building physics to reduce computational cost while maintaining high-fidelity solutions. Both techniques assume separated tensorial representation of the solution by a finite sum of function products. ...

... In recent works, researchers have implemented Spectral methods for solving heat and moisture transfer in food engineering (Pasban et al. 2017) and on fluid flows (Motsa 2015). According to the authors' knowledge, there are no results in the literature regarding the application of Spectral methods for solving diffusive moisture transfer in building physics applications other than (Gasparin et al. 2017a). ...

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.

... We investigate here the use of a polynomial basis like Chebyshev or Legendre. Out of the approximation theory [5], those basis have proven to be very efficient at solving partial differential equations using spectral methods [6,7]. ...

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.

... where t t ( − ) 2 1 is equal to 1 month. The governing equation (1) together with boundary conditions is solved numerically in a dimensionless form. ...

The design of numerical tools to model the behavior of building materials is a challenging task. The crucial point is to save computational costs and maintain the high accuracy of predictions. There are two main limitations on the time scale choice, which places an obstacle to solving the above issues. The first one is the numerical restriction. A number of research studies are dedicated to overcome this limitation and it is shown that it can be relaxed with innovative numerical schemes. The second one is the physical restriction. It is imposed by the properties of a material, the phenomena itself, and the corresponding boundary conditions. This study is focused on the study of a methodology that enables to overcome the physical restriction on the time grid; a so-called average reduced model is suggested. It is based on smoothing the time-dependent boundary conditions. Besides this, the approximate solution is decomposed into average and fluctuating components. The primer is obtained by integrating the equations over time, whereas the latter is a user-defined EM. The methodology is investigated for both heat diffusion and coupled heat and mass transfer. It is demonstrated that the signal core of the boundary conditions is preserved and the physical restriction can be relaxed. The model proved to be reliable, accurate, and efficient also in comparison with the experimental data of 2 years. The implementation of the scarce time-step of 1 h is justified. It is shown, that by maintaining the tolerable error, it is possible to cut computational effort up to almost four times in comparison with the complete model with the same time grid.

... Moisture-based problems have been one of the critical issues as long-term effects on walls in terms of energy efficiency, indoor environmental quality, and durability. Hygrothermal performance has had direct impacts on the users, material durability, and energy efficiency of a building [1]. A well-performing building façade is based on two critical issues: thermal resistance and vapor permeability [2]. ...

... In order to overcome the limitation of calculation time, which is a disadvantage of the detailed heat and moisture transfer model, various numerical analysis methods have been studied. Gasparin et al. [12] showed a study that drastically reduces the simulation time while guaranteeing accuracy compared to both classical EULER implicit and CRANK-NICOLSON scheme in the analysis of heat and humidity movement using a spectral reduced-order model (Spectral ROM). Explicit models require very fine time discretization for stability conditions. ...

Several building energy simulation programs have been developed to evaluate the indoor conditions and energy performance of buildings. As a fundamental component of heating, ventilating, and air conditioning loads, each building energy modeling tool calculates the heat and moisture exchange among the outdoor environment, building envelope, and indoor environments. This paper presents a simplified heat and moisture transfer model of the building envelope, and case studies for building performance obtained by different heat and moisture transfer models are conducted to investigate the contribution of the proposed steady-state moisture flux (SSMF) method. For the analysis, three representative humid locations in the United States are considered: Miami, Atlanta, and Chicago. The results show that the SSMF model effectively complements the latent heat transfer calculation in conduction transfer function (CTF) and effective moisture penetration depth (EMPD) models during the cooling season. In addition, it is found that the ceiling part of a building largely constitutes the latent heat generated by the SSMF model.

... 36 We investigate here the use of a polynomial basis like Chebyshev or Legendre. Out of the approximation 37 theory [5], those basis have proven to be very efficient at solving partial differential equations using spectral 38 methods [6, 7]. ...

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

... This coupled system of differential equations can be solved numerically or analytically be using a method to get the exponential of the 4 × 4 matrix in Equation (5) as illustrated in [51] and applied in [52][53][54]. By applying Newton interpolation method [51] for getting the matrix exponential, which states that for a matrix A with eigenvalues λ j , (j = 1, 2, .., n), n is the dimension of the matrix, ...

An open quantum bipartite system consisting of two independent two-level atoms interacting nonlinearly with a two-mode electromagnetic cavity field is investigated by proposing a suitable non-Hermitian generalization of the Hamiltonian. The mathematical procedure of obtaining the corresponding wave function of the system is clearly given. Pancharatnam phase is studied to give a precise information about the required initial system state, which is related to artificial phase jumps, to control the degree of entanglement (DEM) and get the highest concurrence. We discuss the effect of time-variation coupling, and dissipation of both atoms and cavity. The effect of the time-variation function appears as frequency modulation (FM) effect in the radio waves. Concurrence rapidly reaches the disentangled state (death of entanglement) by increasing the effect of field decay. On the contrary, the atomic decay has no effect.

... This coupled system of differential equations can be solved numerically or analytically be using a method to get the exponential of the 4 × 4 matrix in Equation (5) as illustrated in [51] and applied in [52][53][54]. By applying Newton interpolation method [51] for getting the matrix exponential, which states that for a matrix A with eigenvalues λ j , (j = 1, 2, .., n), n is the dimension of the matrix, ...

An open quantum bipartite system consisting of two independent two-level atoms interacting non-linearly with a two-mode electromagnetic cavity field is investigated by proposing a suitable non-Hermitian generalization of Hamiltonian. The mathematical procedure of obtaining the corresponding wave function of the system is clearly given. Panchartnam phase is studied to give a precise information about the required initial system state, which is related to artificial phase jumps, to control the Degree of Entanglement (DEM) and get the highest Concurrence. We discuss the effect of time-variation coupling, and dissipation of both atoms and cavity. The effect of the time-variation function appears as frequency modulation (FM) effect in the radio waves. Concurrence rapidly reaches the disentangled state (death of entanglement) by increasing the effect of field decay. On the contrary, the atomic decay has no effect.

... Furthermore, some work is underway to efficiently combine Domus with CFD (Computational Fluid Dynamics) tools Mazuroski et al. 2018), which could strengthen the Domus use potential in the country for natural ventilation and HVAC analyses. In addition, Domus heat and moisture diffusion models are planned to incorporate non-traditional numerical methods as presented in (Mendes et al. 2016;Gasparin et al. 2018aGasparin et al. , 2018b to improve accuracy, rapidness and inter-software synchronicity. However, the more complex the model gets -e.g. ...

This article describes the development of a new instructional design (ISD) to promote building energy simulation (BES) education. The study is based upon education fundamentals combined with computer-based learning and hypermedia to enable the development of a BES-based distance learning system. Some cognitive tools are established such as: (i) an interdisciplinary knowledge tree of BES that can be used by professionals with different backgrounds; (ii) a hypermedia navigational aid to understand the simulation software, called the BES tool graphic organizer; (iii) a concept map with an overview of building energy performance and (iv) a cooperative problem-based learning (CPBL) environment. Furthermore, the paper also brings an analysis of the students’ comprehension – from a course applied across Brazil – by means of concept network graphs from text mining provided by the CPBL environment, showing a significant potential to develop interdisciplinary e-learning related to building energy efficiency.

... Some works in the context of heat and moisture transfer in building physics application presented different techniques aiming at replace the standard techniques, such as the Proper Generalized Decomposition (PGD) [2], the improved explicit scheme of Dufort-Frankel [13] and spectral methods [14]. Therefore, this paper aims at presenting an innovative numerical method, the Method of Horizontal Lines to solve the nonlinear heat and moisture transfer through porous materials. ...

This work presents an alternative view on the numerical simulation of diffusion processes applied to the heat and moisture transfer through multilayered porous building materials. Traditionally, by using the finite-difference approach, the discretization follows the Method Of Lines (MOL), when the problem is first discretized in space to obtain a large system of coupled Ordinary Differential Equations (ODEs). This paper proposes to change this
viewpoint. First, we discretize in time to obtain a small system of coupled ODEs, which means instead of having a Cauchy (Initial Value) Problem (IVP), we have a Boundary Value Problem (BVP). Fortunately, BVPs can be solved efficiently today using adaptive collocation finite-difference methods of high order. To demonstrate the benefits of this new approach, three case studies are presented. The first one considers nonlinear heat and moisture transfer through one material layer. The second case includes the rain effect, while the last one considers two material layers. Results show how the nonlinearities and the interface between materials are easily treated, by reasonably using a fourth-order adaptative method. In our numerical simulations, we use adaptive methods of the fourth order which in most practical situations is more than enough.

In this work a stabilised and reduced Galerkin projection of the incompressible unsteady Navier-Stokes equations for moderate Reynolds number is presented. The full-order model, on which the Galerkin projection is applied, is based on a finite volumes approximation. The reduced basis spaces are constructed with a POD approach. Two different pressure stabilisation strategies are proposed and compared: the former one is based on the supremizer enrichment of the velocity space, and the latter one is based on a pressure Poisson equation approach.

This paper deals with model-order reduction of parametric partial differential equations (PPDE). More specifically, we consider the problem of finding a good approximation subspace of the solution manifold of the PPDE when only partial information on the latter is available. We assume that two sources of information are available: i) a “rough” prior knowledge, taking the form of a manifold containing the target solution manifold; ii) partial linear measurements of the solutions of the PPDE (the term partial refers to the fact that observation operator cannot be inverted). We provide and study several tools to derive good approximation subspaces from these two sources of information. We first identify the best worst-case performance achievable in this setup and propose simple procedures to approximate the corresponding optimal approximation subspace. We then provide, in a simplified setup, a theoretical analysis relating the achievable reduction performance to the choice of the observation operator and the prior knowledge available on the solution manifold. This article is protected by copyright. All rights reserved.

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.

The multidomain Legendre-Galerkin Chebyshev-collocation method is considered to solve one-dimensional linear evolution equations with two nonhomogeneous jump conditions. The scheme treats the first jump condition essentially and the second one naturally. We adopt appropriate base functions to deal with interfaces. The proposed method can be implemented in parallel. Error analysis shows that the approach has an optimal convergence rate. The proposed method is also applied to computing the one-dimensional Maxwell equation and the one-dimensional two phase Stefan problem, respectively. Numerical examples are given to confirm the theoretical analysis.

In this paper, we present a high-order accurate method for two-dimensional semilinear parabolic equations. The method is based on a Galerkin-Chebyshev spectral method for discretizing spatial derivatives and a block boundary value methods of fourth-order for temporal discretization. Our formulation has high-order accurate in both space and time. Optimal a priori error bound is derived in the weighted Lω2$L^{2}_{\omega }$-norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence properties of the method.

Simulation models for moisture transfer in building materials are highly incongruent with respect to the moisture potential used. Often the relatively better numerical efficiency and accuracy of a certain moisture potential is put forward as motivation. Various claims are made in that respect, but factual evidence is typically lacking. This paper aims at providing such support by assessing simulation efficiency and accuracy for capillary pressure, relative humidity and -log(-capillary pressure). To that goal, a suite of benchmark simulations are performed with those three potentials and performances are compared, based on deviations from reference solutions and on numbers of iterations required. The study initially reveals mixed results, showing no consistent advantages for either potential. Further analysis uncovers though that -log(-capillary pressure) suffers from a strongly nonlinear moisture capacity near saturation. This finally results in a decision in favour of capillary pressure and relative humidity, at least for general-purpose moisture transfer simulation.

This paper presents the Umidus program which has been developed to model coupled heat and moisture transfer within porous media, in order to analyze higrothemal performance of building elements when subjected to any kind of climate conditions. Both diffusion and capillary regimes are taken into account, that is the transfer of water in the vapor and liquid phases through the material can be analyzed. The model predicts moisture and temperature profiles within multi-layer walls and low-slope roofs for any time step and calculates heat and mass transfer. Umidus has been built in an OOP language to be a fast and precise easy-to-use software.

A whole building hygrothermal model has been developed on the basis of an existing detailed model for thermal simula- tion of buildings. The thermal model is a well-proven transient tool for hour-by-hour simulation of the thermal conditions in multizone buildings. The model has been expanded with new capabilities for transient simulation of indoor humidity condi- tions, taking into account the moisture buffer capacity of build- ing components and furnishings and the supply of humidity from indoor activities. Also integrated in the model are tran- sient calculations of the moisture conditions in the layers of all the external building envelope components. The advantage of the new model is that both the boundary conditions for the envelope and the capacity of building mate- rials to buffer the indoor humidity are considered in the same calculation. The model considers the latent heat effect asso- ciated with the absorption or evaporation of moisture, and it takes into account the way in which moisture in the building materials affects their thermal conductivity. The paper presents the principles for the model and some applications and calculation results. The model is validated against experimental data from a full-scale test cell. In the test cell, it is possible to control the release or withdrawal of humidity from the indoor space and measure the response in humidity of the air and the moisture content of building materials in the room. A sequence of exper- iments has been conducted using different interior materials to provide source data for the effect of moisture absorption and release. Examples of comparisons between simulated and measured data are presented.

In principle, the exponential of a matrix could be computed in many ways. Methods involv-ing approximation theory, differential equations, the matrix eigenvalues, and the matrix characteristic polynomial have been proposed. In practice, consideration of computational stability and efficiency indicates that some of the methods are preferable to others but that none are completely satisfactory. Most of this paper was originally published in 1978. An update, with a separate bibliography, describes a few recent developments.

In this first section we present a high level discussion on computational science, and the need for compact models of phenomena
observed in nature and industry. We argue that much more complex problems can be addressed by making use of current computing
technology and advanced algorithms, but that there is a need for model order reduction in order to cope with even more complex
problems. We also go into somewhat more detail about the question as to what model order reduction is.

Humidity of indoor air is an important factor influencing the air quality and energy consumption of buildings as well as durability
of building components. Indoor humidity depends on several factors, such as moisture sources, air change, sorption in materials
and possible condensation. Since all these phenomena are strongly dependent on each other, numerical predictions of indoor
humidity need to be integrated into combined heat and airflow simulation tools. The purpose of a recent international collaborative
project, IEA ECBCS Annex 41, has been to advance development in modelling the integral heat, air and moisture transfer processes
that take place in “whole buildings” by considering all relevant parts of its constituents. It is believed that full understanding
of these processes for the whole building is absolutely crucial for future energy optimization of buildings, as this cannot
take place without a coherent and complete description of all hygrothermal processes. This paper will illustrate some of the
modelling work that has taken place within the project and present some of the simulation tools used.

While the transfer equations for moisture and heat in building components are currently undergoing standardisation, atmospheric boundary conditions, conservative modelling and numerical efficiency are not addressed. In a first part, this paper adds a comprehensive description of those boundary conditions, emphasising wind-driven rain and vapour exchange, the main moisture supply and removal mechanism, respectively. In the second part the numerical implementation is tackled, with specific attention to the monotony of the spatial discretisation, and to the mass and energy conservation of the temporal discretisation. Both issues are illustrated with exemplary hygrothermal simulations. Numerical efficiency is treated in two follow-up papers.

A new algorithm is developed for computing $e^{tA}B$, where $A$ is an $n\times n$ matrix and $B$ is $n\times n_0$ with $n_0 \ll n$. The algorithm works for any $A$, its computational cost is dominated by the formation of products of $A$ with $n\times n_0$ matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix $e^{tA}B$ or a sequence $e^{t_kA}B$ on an equally spaced grid of points $t_k$. It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the recent analysis of Al-Mohy and Higham [\emph{SIAM J. Matrix Anal.\ Appl.} 31 (2009), pp.\ 970--989], which provides sharp truncation error bounds expressed in terms of the quantities $\|A^k\|^{1/k}$ for a few values of $k$, where the norms are estimated using a matrix norm estimator. Shifting and balancing are used as preprocessing steps to reduce the cost of the algorithm. Numerical experiments show that the algorithm performs in a numerically stable fashion across a wide range of problems, and analysis of rounding errors and of the conditioning of the problem provides theoretical support. Experimental comparisons with MATLAB codes based on Krylov subspace, Chebyshev polynomial, and Laguerre polynomial methods show the new algorithm to be sometimes much superior in terms of computational cost and accuracy. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form $\sum_{k=0}^p \varphi_k(A)u_k$ that arise in exponential integrators, where the $\varphi_k$ are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension $n+p$ built by augmenting $A$ with additional rows and columns, and the algorithm of this paper can therefore be employed.

The standardised Glaser method for calculation, prediction and evaluation of moisture performance is considered as rarely applicable. The present state of knowledge, analytical as well as experimental, concerning heat, air and moisture demands updating of standards. This paper presents five numerical benchmark cases for the quality assessment of simulation models for one-dimensional heat, air and moisture (HAM) transfer. In one case, the analytical solution is known and excellent agreement between several solutions from different universities and institutes is obtained. In the remaining four cases, consensus solutions have been found, with good agreement between different HAM models. The work presented here is an outcome of the EU-initiated project for standardisation of HAM calculation methods (HAMSTAD WP2).

The scaling and squaring method for the matrix exponential is based on the approximation eA ≈ (rm(2-sA))2s, where rm(x) is the [m/m] Padé approximant to ex and the integers m and s are to be chosen. Several authors have identified a weakness of existing scaling and squaring algorithms termed overscaling, in which a value of s much larger than necessary is chosen, causing a loss of accuracy in floating point arithmetic. Building on the scaling and squaring algorithm of Higham [SIAM J. Matrix Anal. Appl., 26 (2005), pp. 1179-1193], which is used by the MATLAB function expm, we derive a new algorithm that alleviates the overscaling problem. Two key ideas are employed. The first, specific to triangular matrices, is to compute the diagonal elements in the squaring phase as exponentials instead of from powers of rm. The second idea is to base the backward error analysis that underlies the algorithm on members of the sequence {∥Aκ∥ 1/κ} instead of ∥A∥, since for nonnormal matrices it is possible that ∥Aκ∥ 1/κ is much smaller than ∥A∥, and indeed this is likely when overscaling occurs in existing algorithms. The terms ∥Aκ∥ 1/κ are estimated without computing powers of A by using a matrix 1-norm estimator in conjunction with a bound of the form ∥Aκ∥ 1/κ ≤ max(∥Aρ∥ 1/ρ, ∥Aq∥) 1/q∥ that holds for certain fixed p and q less than κ. The improvements to the truncation error bounds have to be balanced by the potential for a large ∥A∥ to cause inaccurate evaluation of rm in floating point arithmetic. We employ rigorous error bounds along with some heuristics to ensure that rounding errors are kept under control. Our numerical experiments show that the new algorithm generally provides accuracy at least as good as the existing algorithm of Higham at no higher cost, while for matrices that are triangular or cause overscaling it usually yields significant improvements in accuracy, cost, or both.

Contents PREFACE x Acknowledgments xiv Errata and Extended-Bibliography xvi 1 Introduction 1 1.1 Series expansions .................................. 1 1.2 First Example .................................... 2 1.3 Comparison with finite element methods .................... 4 1.4 Comparisons with Finite Differences ....................... 6 1.5 Parallel Computers ................................. 9 1.6 Choice of basis functions .............................. 9 1.7 Boundary conditions ................................ 10 1.8 Non-Interpolating and Pseudospectral ...................... 12 1.9 Nonlinearity ..................................... 13 1.10 Time-dependent problems ............................. 15 1.11 FAQ: Frequently Asked Questions ........................ 16 1.12 The Chrysalis .................................... 17 2 Chebyshev & Fourier Series 19 2.1 Introduction ..................................... 19 2.2 Fourier series ...........

. This paper describes mathematical and software developments for a suite of programs for solving ordinary di#erential equations in Matlab. Key words. ordinary di#erential equations, sti# systems, BDF, Gear method, Rosenbrock method, non-sti# systems, Runge-Kutta method, Adams method, software AMS subject classifications. 65L06, 65L05, 65Y99, 34A65 1. Introduction. This paper presents mathematical and software developments that are the basis for a suite of programs for the solution of initial value problems y # = F (t, y) on a time interval [t 0 ,t f ], given initial values y(t 0 )=y 0 . The solvers for sti# problems allow the more general form M(t) y # = f(t, y) with a mass matrix M(t) that is non-singular and (usually) sparse. The programs have been developed for Matlab [29], a widely used environment for scientific computing. This influenced the choice of methods and how they were implemented. As in many environments, the typical problem in Matlab is solved interactively and...

This book intends to stimulate research in simulation of diffusion problems in building physics, by first providing an overview of mathematical models and numerical techniques such as the finite difference and finite-element methods traditionally used in building simulation tools. Then, nonconventional methods such as reduced order models, boundary integral approaches and spectral methods are presented, which might be considered in the next generation of building-energy-simulation tools. The advantage of these methods includes the improvement of the numerical solution of diffusion phenomena, especially in large domains relevant to building energy performance analysis.

In the present study, a numerical method is proposed for simulating the coupled three dimensional heat and mass transfer processes during convective drying of apple slices. Spectral collocation method (pseudospectral method) is applied for discretizing both the space and time variables based on the Jacobi Gauss Lobatto (JGL) interpolation points. Also operational matrices of differentiation are implemented for approximating the derivative of the spatial and temporal variables. The external flow and temperature fields were simulated through the Fluent CFD package. The convective heat transfer coefficient is calculated from the lumped system analysis and convective mass transfer coefficient is computed through the analogy between the thermal and concentration boundary layers. The model is validated against experimental data in a range of air temperatures from 60 °C up to 90 °C. The results illustrate a remarkable agreement between the numerical predictions and experimental results, which confirm robustness, computationally efficient and high accuracy of the proposed approach for predicting the simultaneous heat and mass transfer in apple slices.

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

A novel approach is presented to constrain reduced-order models (ROM) based on proper orthogonal decomposition (POD). The Karush-Kuhn-Tucker (KKT) conditions were applied to the traditional reduced-order model to constrain the solution to user-defined bounds. The constrained reduced-order model (C-ROM) was applied and validated against the analytical solution to the first-order wave equation. C-ROM was also applied to the analysis of fluidized beds. It was shown that the ROM and C-ROM produced accurate results and that C-ROM was less sensitive to error propagation through time than the ROM.

Implicit schemes have been extensively used in building physics to compute the solution of moisture diffusion problems in porous materials for improving stability conditions. Nevertheless, these schemes require important sub-iterations when treating nonlinear problems. To overcome this disadvantage, this paper explores the use of improved explicit schemes, such as Dufort–Frankel, Crank–Nicolson and hyperbolization approaches. A first case study has been considered with the hypothesis of linear transfer. The Dufort–Frankel, Crank–Nicolson and hyperbolization schemes were compared to the classical Euler explicit scheme and to a reference solution. Results have shown that the hyperbolization scheme has a stability condition higher than the standard Courant–Friedrichs–Lewy condition. The error of this schemes depends on the parameter τ representing the hyperbolicity magnitude added into the equation. The Dufort–Frankel scheme has the advantages of being unconditionally stable and is preferable for nonlinear transfer, which is the three others cases studies. Results have shown the error is proportional to O(dt). A modified Crank–Nicolson scheme has been also studied in order to avoid sub-iterations to treat the nonlinearities at each time step. The main advantages of the Dufort–Frankel scheme are (i) to be twice faster than the Crank–Nicolson approach; (ii) to compute explicitly the solution at each time step; (iii) to be unconditionally stable and (iv) easier to parallelize on high-performance computer systems. Although the approach is unconditionally stable, the choice of the time discretization remains an important issue to accurately represent the physical phenomena.

This paper presents a review of the use of model reduction techniques for building physics applications. The use of separated representations, the so called Proper Generalised Decomposition (PGD), is particularly investigated. This technique can be applied for efficient building physics modelling at different levels: the wall and multizone models, the whole-building (coupled envelope and air volumes) simulation and their practical applications. The PGD can be formulated as a space-time representation to provide fast and accurate solutions of 2- and 3-dimensional problems at the wall or the whole-building level. Furthermore, the PGD solution allows to treat efficiently parametric problems of practical building applications.

We consider model order reduction by proper orthogonal decomposition (POD) for parametrized partial differential equations, where the underlying snapshots are computed with adaptive finite elements. We address computational and theoretical issues arising from the fact that the snapshots are members of different finite element spaces. We propose a method to create a POD-Galerkin model without interpolating the snapshots onto their common finite element mesh. The error of the reduced-order solution is not necessarily Galerkin orthogonal to the reduced space created from space-adapted snapshot. We analyze how this influences the error assessment for POD-Galerkin models of linear elliptic boundary value problems. As a numerical example we consider a two-dimensional convection-diffusion equation with a parametrized convective direction. To illustrate the applicability of our techniques to non-linear time-dependent problems, we present a test case of a two-dimensional viscous Burgers equation with parametrized initial data.

This paper proposes a full grid interval collocation method (FGICM) and a sparse grid interval collocation method (SGICM) to solve the uncertain heat convection-diffusion problem with interval input parameters in material properties, applied loads and boundary conditions. The Legendre polynomial series is adopted to approximate the functional dependency of temperature response with respect to the interval parameters. In the process of calculating the expansion coefficients, FGICM evaluates the deterministic solutions directly on the full tensor product grids, while the Smolyak sparse grids are reconstructed in SGICM to avoid the curse of dimensionality. The eventual lower and upper bounds of temperature responses are easily predicted based on the continuously-differentiable property of the approximate function. Comparing results with traditional Monte Carlo simulations and perturbation method, the numerical example evidences the remarkable accuracy and effectiveness of the proposed methods for interval temperature field prediction in engineering.

The focus of this paper was to examine the contribution of two key mechanisms—moisture convection and diffusion–on heated air and moisture transfer in porous building envelopes and to define the validity of the sub-models. A numerical simulation was performed and is focused on the one-dimensional problem for drying test boundary conditions. Thereafter, a detailed parametric analysis was carried out in order to investigate the influence of typical nondimensional parameters. Results show that convection is a prominent driving potential with respect to the diffusion process when the hygric state is stable between the environment and the envelope. 2016

In this paper, detailed numerical calculations of the spectral collocation method (SCM) to simulate natural convection in a square porous cavity using two different local thermal models, i.e., the local thermal equilibrium model and the local thermal non-equilibrium model, are presented and formulated. In this approach, stream function and temperature are approximated by the Chebyshev polynomials and the dimensionless governing equations are discretized by the Chebyshev–Gauss–Lobatto collocation points. The two step method, i.e., the matrix diagonalization method, is used to solve the discretized equations. The present results are compared with those reported in literature. It is found that these results are in excellent agreement. Furthermore, the exact solutions for local thermal equilibrium model have been constructed to test the accuracy of the method, and it can be observed from the test that the Chebyshev spectral collocation method can produce high accuracy. Meanwhile, the isotherms and the streamlines are analyzed to reveal the fluid flow and heat transfer for various parameters, i.e., the Rayleigh number, the inter-phase heat transfer coefficient, and the thermal conductivity ratio.

In recent years, a great deal of attention has been devoted to Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. The surge of interest was triggered by the pressing need for efficient numerical techniques for simulations of extremely large-scale dynamical systems arising from circuit simulation, structural dynamics, and microelectromechanical systems. In this paper, we begin with a tutorial of a Lanczos process based Krylov subspace technique for reduced-order modeling of linear dynamical systems, and then give an overview of the recent progress in other Krylov subspace techniques for a variety of dynamical systems, including second-order and nonlinear systems. Case studies arising from circuit simulation, structural dynamics and microelectromechanical systems are presented.

Abstract In this composition, we use a new spectral relaxation method (SRM) to investigate the effects of linear and non-linear stratification on mixed convective transport along a vertical surface embedded in a porous medium and it is viewed for the first time in both aiding and opposing buoyancy cases. The governing partial differential equations are transformed into ordinary differential equations using similarity transformation and then the resulting differential equations are solved numerically using SRM. A comparison is also made about the accuracy of SRM results in relation to the results obtained using the shooting method. We show that the proposed technique is an efficient numerical algorithm with assured convergence that serves as an alternative to common numerical methods for solving nonlinear boundary value problems. A parametric study of the physical parameters involved in the problem is conducted and a representative set of numerical results is illustrated, with accent on the comparison between linear and non-linear stratification. It is significant to notice that the separation of flow is found to be more in the absence of stratification whereas it is less in the presence of stratification. Finally, thermal and solutal stratifications significantly affect the heat and mass transfer rates, besides delay the boundary layer separation.

A Chebyshev collocation spectral method based on modified discrete ordinates method (SP-MDOM) is developed for radiative heat transfer problems to remedy the ray effect. The media analyzed are absorbing, emitting and isotropic or anisotropic scattering. The radiative transfer equation (RTE) is decomposed into two parts: the wall-related intensity and the medium-related intensity. The former is solved analytically. For the latter, the discrete ordinates method is used for angularly discretization with quadrature scheme and then solved by Chebyshev collocation spectral method (CSM). In this study, the 2-D rectangular enclosures as well as the partitioned domain with different boundary conditions and optical parameters are considered. The domain decomposition concept is used along with the SP-MDOM to handle the radiative transfer in partitioned domain. The accuracy of the results obtained by the SP-MDOM is assessed by comparing the predictions with those obtained by other researchers. The results confirm the capability of the SP-MDOM to remedy the nonphysical anomalies caused by the ray effect.

Heat and mass transfer between capillary-porous bodies and surrounding incompressible liquid accompanied by a change of phase is not only of theoretical interest but also of great practical importance for some technological processes. Heat and mass transfer inside a porous body (internal heat and mass transfer) also has its unique character. Even now the mechanism of heat and mass transfer in evaporation processes is scantily investigated, and analytical investigations do not, therefore, lead to reliable results. This chapter presents an experimental study of heat and mass transfer in evaporation processes. To elucidate peculiarities of heat transfer with simultaneous mass transfer, a dry body (pure heat transfer) and a moist body (heat transfer in the presence of mass transfer) are investigated. Such a comparison makes it possible to establish relations for interconnected heat and mass transfer processes. In order to describe quantitative relations it is necessary to have a method of analysis which makes it possible to consider the interaction of the heat and mass transfer processes. One such method is the thermodynamics of irreversible processes. The experimental data presented well confirm the mathematical theory of thermodynamics of irreversible transfer processes.

The aim of this book is to teach you the essentials of spectral collocation methods with the aid of 40 short MATLAB® programs, or “M-files.”* The programs are available online at http://www.comlab.ox.ac.uk/oucl/work/nick.trefethen, and you will run them and modify them to solve all kinds of ordinary and partial differential equations (ODEs and PDEs) connected with problems in fluid mechanics, quantum mechanics, vibrations, linear and nonlinear waves, complex analysis, and other fields. Concerning prerequisites, it is assumed that the words just written have meaning for you, that you have some knowledge of numerical methods, and that you already know MATLAB.
If you like computing and numerical mathematics, you will enjoy working through this book, whether alone or in the classroom—and if you learn a few new tricks of MATLAB along the way, that's OK too!

Thermal radiation transfer in one-dimensional parallel plates and two-dimensional rectangle enclosures filled with non-gray gases, namely CO2, H2O, or their mixtures, is calculated using the spectral collocation method with full spectrum k-distribution model (SCM–FSK). The k-distribution model is a promising model for treating the spectral properties of an absorbing–emitting medium, representing an alternative to line-by-line calculations which reduces the number of radiative transfer equation (RTE) evaluations from the order of a million to the order of ten without any significant loss of accuracy. Moreover, the full spectrum k-distribution model, as a global model, can be easily combined with other numerical methods to solve the RTE. The spectral collocation method (SCM) which is used to solve partial differential equations is combined with FSK for the present problems. In order to test the accuracy and efficiency of SCM–FSK, several gas radiation transfer problems involving isothermal/non-isothermal and homogeneous/inhomogeneous non-gray gases are examined. The results obtained by SCM–FSK are assessed by comparing the predictions with line by line results. These comparisons indicate that SCM–FSK is sufficiently accurate and convenient for engineering calculations.

Water infiltration is known to play an important part in the degradation process of construction materials. Over time, microscopic and macroscopic cracks progressively develop under the effects of mechanical loading and sorption/desorption cycles: their influence is to be accounted for in long-term hygrothermal performance assessments of the building envelope. The present work aims at showing the potential consequences of cracking on the heat and moisture transfer across building facades, in order to justify the need for the identification of damage to prevent durability and thermal issues. Specific simulation cases of insulated and non-insulated building facades were defined, and submitted to atmospheric boundary conditions for simulation times of one month. Some of the simulation geometries included previous measurements of crack patterns in concrete. The comparison of fractured and non-fractured building facades showed the effects of cracks on the moisture accumulation and thermal performance of these wall configurations, thus giving an estimate of what these effects might be in real conditions. A methodology is thus proposed for the identification of renovation needs, which may be applied for the purpose of durability assessments as well.

There is a strong demand for accurate moisture modeling since moisture poses a risk for both the constructions and the indoor climate. This investigation has special focus on moisture modeling. This study describes a new model based on a CFD tool enhanced to include both detailed modeling of airflows in rooms and heat and moisture transfer in walls by applying them as fluid walls. The impacts of different boundary conditions and how these influence microclimates in rooms are investigated, in a 3D configuration. The studied microclimate is a piece of furniture placed near a cold exterior wall.

In this paper, we review the development of new reduced-order modeling techniques and discuss their applicability to various problems in computational physics. Emphasis is given to methods based on Volterra series representations, the proper orthogonal decomposition, and harmonic balance. Results are reported for different nonlinear systems to provide clear examples of the construction and use of reduced-order models (ROMs), particularly in the multi-disciplinary field of computational aeroelasticity. Unsteady aerodynamic and aeroelastic behaviors of two-dimensional and three-dimensional geometries are described. Large increases in computational efficiency are obtained through the use of ROMs, thereby justifying the initial computational expense of constructing these models and motivating their use for multi-disciplinary design analysis.

Since the publication of "Spectral Methods in Fluid Dynamics", spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. While retaining the tight integration between the theoretical and practical aspects of spectral methods that was the hallmark of the earlier book, Canuto et al. now incorporate the many improvements in the algorithms and the theory of spectral methods that have been made since 1988. The initial treatment Fundamentals in Single Domains discusses the fundamentals of the approximation of solutions to ordinary and partial differential equations on single domains by expansions in smooth, global basis functions. The first half of the book provides the algorithmic details of orthogonal expansions, transform methods, spectral discretization of differential equations plus their boundary conditions, and solution of the discretized equations by direct and iterative methods. The second half furnishes a comprehensive discussion of the mathematical theory of spectral methods on single domains, including approximation theory, stability and convergence, and illustrative applications of the theory to model boundary-value problems. Both the algorithmic and theoretical discussions cover spectral methods on tensor-product domains, triangles and tetrahedra. All chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is greatly expanded as are the set of numerical examples that illustrate the key properties of the various types of spectral approximations and the solution algorithms.
A companion book "Evolution to Complex Geometries and Applications to Fluid Dynamics" contains an extensive survey of the essential algorithmic and theoretical aspects of spectral methods for complex geometries and provides detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries.

The Schur–Fréchet method of evaluating matrix functions consists of putting the matrix in upper triangular form, computing the scalar function values along the main diagonal, and then using the Fréchet derivative of the function to evaluate the upper diagonals. This approach requires a reliable method of computing the Fréchet derivative. For the logarithm this can be done by using repeated square roots and a hyperbolic tangent form of the logarithmic Fréchet derivative. Padé approximations of the hyperbolic tangent lead to a Schur–Fréchet algorithm for the logarithm that avoids problems associated with the standard "inverse scaling and squaring" method. Inverting the order of evaluation in the logarithmic Fréchet derivative gives a method of evaluating the derivative of the exponential. The resulting Schur–Fréchet algorithm for the exponential gives superior results compared to standard methods on a set of test problems from the literature.

This paper describes the coupling of a model for heat and moisture transport in porous materials to a commercial Computational Fluid Dynamics (CFD) package. The combination of CFD and the material model makes it possible to assess the risk of moisture related damage in valuable objects for cases with large temperature or humidity gradients in the air. To couple both models the choice was made to integrate the porous material model into the CFD package. This requires the heat and moisture transport equations in the air and the porous material to be written down in function of the same transported variables. Validation with benchmark experiments proved the good functionality of the coupled model. A simulation study of a microclimate vitrine for paintings shows that phenomena observed in these vitrines are well predicted by the model and that data generated by the model provides additional insights in the physical mechanisms behind these phenomena.

In order to precisely predict ground heat transfer, room air temperature and humidity, a combined model has been developed and conceived to calculate both the coupled heat and moisture transfer in soil and floor and the psychrometrics condition of indoor air. The present methodology for the soil is based on the theory of Philip and De Vries, using variable thermophysical properties for different materials. The governing equations were discretized using the finite-volume method and a three-dimensional model for describing the physical phenomena of heat and mass transfer in unsaturated moist porous soils and floor. Additionally, a lumped transient approach for a building room and a finite-volume multi-layer model for the building envelope have been developed to integrate with the soil model. Results are presented in terms of temperature, humidity and heat flux at the interface between room air and the floor, showing the importance of the approach presented and the model robustness for long-term simulations with a high time step.

We develop a class of numerical methods for stiff systems, based on the method of exponential time differencing. We describe schemes with second- and higher-order accuracy, introduce new Runge–Kutta versions of these schemes, and extend the method to show how it may be applied to systems whose linear part is nondiagonal. We test the method against other common schemes, including integrating factor and linearly implicit methods, and show how it is more accurate in a number of applications. We apply the method to both dissipative and dispersive partial differential equations, after illustrating its behavior using forced ordinary differential equations with stiff linear parts.

A mathematical formulation applied to a numerically robust solver is presented, showing that moisture content gradients can be used as driving forces for heat and moisture transport calculation through the interface between porous materials with different pore size distribution functions. For comparison purposes, several boundary conditions are tested—in order to gradually increase the discontinuity effects—and a detailed analysis is undertaken for the temperature and moisture content distributions and sensible and latent heat fluxes, when the discontinuity on the moisture content profile is taken or not into account.

In this paper, the Chebyshev collocation spectral method for one-dimensional radiative heat transfer equation with participating media is presented; and sequentially the iterative and direct solvers are developed. Implementation of the new method shows its flexibility to complex problems: highly anisotropic and space-dependent scattering. The new solvers can provide exponential convergence in space and can capture large oscillations. Numerical results verified the high accuracy of the new method, and its competitive ability compared with other newly appeared methods.

This paper gives an onset to whole building hygrothermal modelling in which the interaction between interior and exterior climates via building enclosures is simulated under a moderately cold and humid climate. The focus is particularly on the impact of wind-driven rain (WDR) on the hygrothermal response, mould growth at interior wall surfaces, indoor climate and energy consumption. First the WDR load on the facades of a 4 m × 4 m × 10 m tower is determined. Then the hygrothermal behaviour of the brick walls is analysed on a horizontal slice through the tower. The simulations demonstrate that the impact of WDR loads on the moisture contents in the walls is much larger near the edges of the walls than at the centre. The obtained relative humidity and temperature at the interior wall surfaces are combined with isopleths of generalised spore germination time of fungus mould. The results show that WDR loads can have a significant impact on mould growth especially at the edges of the walls. Finally, for the case analysed, the WDR load causes a significant increase of indoor relative humidity and energy consumption for heating.

The study of moisture migration in materials and building elements is of great importance for the characterization of their behaviour, and affects their durability, waterproofing, degradation and thermal performance. The different mechanisms of moisture transport in building walls and the analysis of interface phenomena have been investigated. Based on the theory proposed by Luikov and Philip—De Vries, a computer program has been developed. The comparison of calculated and measured values obtained by using gamma-ray equipment to measure water content profiles, is considered satisfactory.

Simulation Program for the Calculation of Coupled Heat, Moisture, Air, Pollutant, and Salt Transport

- B Bauklimatik Dresden

Bauklimatik Dresden, B. 2018. "Simulation Program for the Calculation of Coupled Heat, Moisture, Air, Pollutant, and Salt
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http://www.bauklimatik-dresden.de/delphin/
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An Analysis of Moisture Accumulation in Walls Subjected to Hot and Humid Climates

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Burch, D. 1993. "An Analysis of Moisture Accumulation in
Walls Subjected to Hot and Humid Climates." ASHRAE
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