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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.

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... It represents interesting strategies since they intend to compute solutions with a lower computational cost while preserving the whole complexity of the physical phenomena. Lumped models, such as lumped capacitance ones do have a lower computational cost but the predictions' reliability are questionable as remarked in [10,11]. In recent years, several works have been published to propose reduced-order models for the computation of diffusion problems in building physics [12]. ...

... In [20,21], the Proper Generalized Decomposition (PGD) is used to compute the heat and mass transfer in porous walls for one and two-dimensional problems. A Spectral reduced method is employed to solve similar problems in [10,22]. The well-known Proper Orthogonal Decomposition (POD) is also used in [23,24]. ...

... Before developing any numerical algorithm to compute the governing equations, a dimensionless model is formulated [10,11,41]. For each layer i, the dimensionless temperature is defined: ...

Within the environmental context, numerical modeling is a promising approach to assess the energy efficiency of building. Resilient buildings need to be designed, capable of adapting to future extreme heat. Simulations are required assuming a one-dimensional heat transfer problem through walls and a simulation horizon of several years (nearly 30). The computational cost associated with such modeling is quite significant and model reduction methods are worth investigating. The objective is to propose a reliable reduced-order model for such long-term simulations. For this, an alternative model reduction approach is investigated, assuming a known Proper Orthogonal Decomposition reduced basis for time, and not for space as usually. The model enables computing parametric solutions using basis interpolation on the tangent space of the Grassmann manifold. Three study cases are considered to verify the efficiency of the reduced-order model. Results highlight that the model has a satisfying accuracy of 10-3 compared to reference solutions. The last case study focuses on the wall energy efficiency design under climate change according to a four-dimensional parameter space. The latter is composed of the load material emissivity, heat capacity, thermal conductivity and thickness insulation layer. Simulations are carried over 30 years considering climate change. The solution minimizing the wall work rate is determined with a computational ratio of 0.1% compared to standard approaches.

... In the future, the data that will be used to compile the equations that can be solved using various equations are selected. The problem involves heat conduction, thermal conductivity, density and specific heat in a wall of thickness, can be formulated by the Fourier (or heat) equation: [5] (eq. 1) ...

... The moisture transfer happens under isothermal conditions in a wall of thickness, with a single material of permeability κ and moisture capacity, both depending on the vapor pressure: [5] (eq. 2) ...

In the global consumption of electric and thermal energy, buildings occupied a sizeable share of the consumption, as well as CO 2 emissions. Thermally Activated Buildings (TABS) are built in order to minimize such conditions, which are distributed in certain regions of Europe and Central America, mainly as exchange and store heat. In this regard, the design of simulation models is important for the study of TABS management. Response time and storage capacity is a challenge to monitor and control those systems. At the moment, many researchers in the field devote much time to this in their research, trying to make full use of the potential and increase the using of renewable energy. That research summarizes the different approaches of TABS.

... The data-driven lumped-parameter models represented by resistance-capacitance networks (RC models) provided a choice for inverse building modeling and load forecasting [19]. The lumped-parameters, to characterize the integrated thermal behavior of a building, are fitted identification based on actual operating data combined with the differential equations of heat transfer and mass transfer [20,21]. Actually, the thermal model applied in [22] is a one-order RC model, and because the thermal inertia of the building can be ignored, the model structure used is relatively simple. ...

The thermal transmittance plays a decisive role in building energy efficiency design, which determined by standard in-situ measurement are obviously not equivalent to the practical building load level. There is still a lack of post-evaluation method that can separately evaluate the energy saving effect provided by integrated building envelope, especially for the nearly-zero energy building (NZEB) whose thermal inertia cannot be neglected. This paper proposed a model-based integrated building envelope performance evaluation method, which contains a reverse parameters identification and a forward load calculation. The data-driven lumped-parameter model represented by resistance–capacitance network (RC-model) reflects the integrated envelope performance, and four different model structures were designed for different treatment of thermal mass were compared. The applicability of the selected 7R6C model in the simulation of NZEB is demonstrated by the accuracy verification, sensitivity analysis of initial values of temperature node and uncertainty assessment of internal heat gain. A renovation case achieved an annual heating demand assessment result of 14.0 kWh/(m²·a) under typical annual meteorological data, which is lower than the limit value of a NZEB in cold region of 15.0 kWh/(m²·a). The evaluation results show that the envelope renovation meets the requirements of energy efficiency design. On the way to low-carbon buildings, the corresponding post-evaluation method needs to be gradually improved.

... First, once we operate the numbers of the same magnitude, we minimize the rounding numerical errors [46]. Second, it enables a general investigation of the model behaviour regardless of the unit used to measure variable [47,48]. Finally, it simplifies equations by reducing the number of variables. ...

Within the framework of building energy assessment, this article proposes to use a derivative based sensitivity analysis of heat transfer models in a building envelope. Two, global and local, estimators are obtained at low computational cost, to evaluate the influence of the parameters on the model outputs. Ranking of these estimators values allows to reduce the number of model unknown parameters by excluding non-significant parameters. A comparison with variance and regression-based methods is carried out and the results highlight the satisfactory accuracy of the continuous-based approach. Moreover, for the carried investigations the approach is $100$ times faster compared to the variance-based methods. A case study applies the method to a real-world building wall. The sensitivity of the thermal loads to local or global variations of the wall thermal is investigated. Additionally, a case study of wall with window is analyzed.

... Since the floating point numbers have the highest density within the interval ( 0, 1 ) , it is wise to carry numerical analysis with dimensionless equations. This lack of accuracy has been illustrated in [35] where the computation of the equation of heat or mass transfer in physical and dimensionless forms are compared. When solving the equation in its physical dimension, the significant digits accuracy of the solution can lose one order. ...

Within the environmental context, several tools based on simulations have been proposed to analyze the physical phenomena of heat and mass transfer in porous materials. However, it is still an open challenge to propose tools that do not require to perform computations to catch the dominant processes. Thus, this article proposes to explore advantages of using a dimensionless analysis by scaling the governing equations of heat and mass transfer. Proposed methodology introduces dimensionless numbers and their nonlinear distortions. The relevant investigation enables to enhance the preponderant phenomena to \emph{(i)} compare different categories of materials, \emph{(ii)} evaluate the competition between heat and mass transfer for each material or \emph{(iii)} describe the transfer in multi-layered wall configurations. It also permits to define hygrothermal kinetic, geometric and dynamic similarities among different physical materials. Equivalent systems can be characterized in the framework of experimental or wall designs. Three cases are presented for similarity studies in terms of \emph{(i)} equivalent material length, \emph{(ii)} time of heat and mass transfer and \emph{(iii)} experimental configurations. All these advantages are illustrated in the given article considering $49$ building materials separated in $7$ categories.

... The governing equation (1) together with boundary conditions is solved numerically in a dimensionless form. The solution in the dimensionless formulation has advantages, such as the application to a class of problems sharing the same scaling parameters (e.g., Fourier and Biot numbers), 17,18] simplification of a problem using asymptotic methods, 19 and restriction of round-off errors. For these purposes, the following dimensionless quantities are defined: ...

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.

... First, once we operate the numbers of the same magnitude, we minimize the rounding numerical errors (Kahan and Palmer 1979). Second, it enables a general investigation of the model behaviour regardless of the unit used to measure variable (Berger et al. 2020;Trabelsi et al. 2018). Finally, it simplifies equations by reducing the number of variables. ...

Within the framework of building energy assessment, this article proposes to use a derivative based sensitivity analysis of heat transfer models in a building envelope. Two, global and local, estimators are obtained at low computational cost, to evaluate the influence of the parameters on the model outputs. Ranking of these estimators values allows to reduce the number of model unknown parameters by excluding non-significant parameters. A comparison with variance and regression-based methods is carried out and the results highlight the satisfactory accuracy of the continuous-based approach. Moreover, for the carried investigations the approach is 100 times faster compared to the variance-based methods. A case study applies the method to a real-world building wall. The sensitivity of the thermal loads to local or global variations of the wall thermal properties is investigated. Additionally, a case study of wall with window is analyzed.

... The connectivity sub-matrices representing the walls, door and the indoor air are denoted, [ wall ], [ door ] and [ air ], respectively. The expressions of these sub-matrices are given in equations (25), (26) and (27), as shown at the bottom of this page, respectively. In this notation, the ternary elements, ''−1'', ''0'' and ''1'' corresponds to the coefficients of the branch flux in function of mesh fluxes with respect the second Kirchoff circuit node's law. ...

An original thermal model of a single room structure is developed by using the tensorial
network-based Kron's method. The modelling principle is using the equivalent RC-network of wall, door and air constituting the house. For a better understanding, the temperature propagation was assumed only in a 1-D horizontal direction. The problem geometrization is defined in function of rectangular approximation meshing. After the determination of the equivalent thermal resistor and thermal capacitor, the innovative thermal circuit representing the room is elaborated. The methodology of the Kron's formalism, implicitly
described with the different action steps is introduced. The thermal room Kron's method is implemented from the branch to mesh spaces before the expression of the problem metric. The thermal transfer functions (TTFs) at three cases of indoor points, situated near, middle and far of the door are established from the Kron's problem metric. The feasibility of the room thermal Kron's TTF model is validated with SPICE TTF simulations in both frequency and time domains. The thermal cut-off frequencies are verified with very
good correlation between the established TTF model and simulation. An excellent prediction of transient responses with unit-step and arbitrary waveform temperature signals with a minimal and maximal amplitude of about 20°C and 40°C is proposed.

... Thus, simplifying building thermal models was taken into consideration to improve simulation efficiency. Firstly, in the research of envelope simulation, the heat transfer process can be described as a three-dimensional model (Gao et al. 2004;Kong et al. 2017;Berger et al. 2020), or simplified to a one-dimensional problem (Bertagnolio and Lebrun 2008;Zhou et al. 2010). Gao et al. (2014) established a complex three-dimensional model and then reduced it to an 8-order model. ...

When representing building thermal characteristics for a large-scale district heating (DH) system, the traditional 1-order model is easy to solve but has a nonnegligible delay when conditions change. High-order models are widely used in building simulation, but their complexity and high time cost prevent them from being applied in DH simulation. The purpose of this paper is to find out an appropriate building thermal model for heating accident simulation. An unoccupied residential building located in northeast China was tested for 72 h heating outage and subsequent 32 h recovery. Based on the lumped parameter method and state space model, a high order was established by 4240 mass points to describe the tested building. By integrating mass points, the low-order models reduced the orders to eight, three, and one respectively. Frequency domain analysis revealed the similarities and differences among the models. The 1-order model showed a large deviation from the other three models. The subsequent time domain analysis of the 1-order model also simulated a root mean square error (RMSE) of 1.69, a delay of 6 h, and the lowest temperature drop during a heating outage. In both frequency and time domain analyses, the 8-order and 3-order models showed similar results approaching the high-order model, with an RMSE of approximately 0.50, a delay of 1 h, and small deviation. The simpler 3-order model could be used to estimate the indoor air temperature during a heating accident in large-scale DH systems. The further study on integrating the 3-order model and system identification method may overcome the shortcomings of model reduction.

... However, to our knowledge, no works have been proposed to evaluate the reliability of the two mathematical models. A complementary work [12] investigates the fidelity of the two approaches to predict the physical phenomena with comparison to experimental observations. As a second step, this work intends to appraise their reliability to estimate unknown parameter from 1 The word "mathematical" is used because the mathematical language is used to write the model. ...

The fidelity of a model relies both on its accuracy to predict the physical phenomena and its capability to estimate unknown parameters using observations. This article focuses on this second aspect by analyzing the reliability of two mathematical models proposed in the literature for the simulation of heat losses through building walls. The first one, named DuFort-Frankel (DF), is the classical heat diffusion equation combined with the DuFort-Frankel numerical scheme. The second is the so-called RC lumped approach, based on a simple ordinary differential equation to compute the temperature within the wall. The reliability is evaluated following a two stages method. First, samples of observations are generated using a pseudo-spectral numerical model for the heat diffusion equation with known input parameters. The results are then modified by adding a noise to simulate experimental measurements. Then, for each sample of observation, the parameter estimation problem is solved using one of the two mathematical models. The reliability is assessed based on the accuracy of the approach to recover the unknown parameter. Three case studies are considered for the estimation of (i) the heat capacity, (ii) the thermal conductivity or (iii) the heat transfer coefficient at the interface between the wall and the ambient air. For all cases, the DF mathematical model has a very satisfactory reliability to estimate the unknown parameters without any bias. However, the RC model lacks of fidelity and reliability. The error on the estimated parameter can reach 40% for the heat capacity, 80% for the thermal conductivity and 450% for the heat transfer coefficient.

... However, to our knowledge, no works have been proposed to evaluate the reliability of the two mathematical models. A complementary work [12] investigates the fidelity of the two approaches to predict the physical phenomena with comparison to experimental observations. As a second step, this work intends to appraise their reliability to estimate unknown parameter from 1 The word "mathematical" is used because the mathematical language is used to write the model. ...

The fidelity of a model relies both on its accuracy to predict the physical phenomena
and its capability to estimate unknown parameters using observations. This article focuses on this second aspect by analyzing the reliability of two mathematical models proposed in the literature for the simulation of heat losses through building walls. The first one, named DF, is the classical heat diffusion equation combined with the DuFort-Frankel numerical scheme. The second is the so-called RC lumped approach, based on a simple ordinary differential equation to compute the temperature within the wall. The reliability is evaluated following a two stages method. First, samples of observations are generated using a pseudospectral numerical model for the heat diffusion equation with known input parameters. The results are then modified by adding a noise to simulate experimental measurements. Then, for each sample of observation, the parameter estimation problem is solved using one of the two mathematical models. The reliability is assessed based on the accuracy of the approach to recover the unknown parameter. Three case studies are considered for the estimation of (i) the heat capacity, (ii) the thermal conductivity or (iii) the heat transfer coefficient at the interface between the wall and the ambient air. For all cases, the DF mathematical model has a very satisfactory reliability to estimate the unknown parameters without any bias. However, the RC model lacks of fidelity and reliability. The error on the estimated parameter can reach 40% for the heat capacity, 80% for the thermal conductivity and 450% for the heat transfer coefficient.

This work deals with an inverse two-dimensional nonlinear heat conduction problem to determine the top and lateral surface transfer coefficients. For this, the \textsc{B}ayesian framework with the \textsc{M}arkov Chain \textsc{M}onte \textsc{C}arlo algorithm is used to determine the posterior distribution of unknown parameters. To handle the computational burden, a lumped one-dimensional model is proposed. The lumped model approximations are considered within the parameter estimation procedure thanks to the Approximation Error Model. The experiments are carried out for several configurations of chamber ventilator speed. Experimental observations are obtained through a complete measurement uncertainty propagation. By solving the inverse problem, accurate probability distributions are determined. Additional investigations are performed to demonstrate the reliability of the lumped model, in terms of accuracy and computational gains.

Comprehensive studies of detailed dynamic building models, which take into consideration both the envelope and the connected systems, yield more precise results compared with simplified ones, but at considerable computational expense. Aside from classical approaches that work on the model itself to accelerate the simulation process such as model reduction or metamodels, this paper focuses on the concept of applying reduced simulation sequences directly to detailed models to calculate annual results. The objective is to quickly and precisely reproduce the integrated annual profiles of predefined criteria of a computationally expensive reference model. After presenting and analyzing methods used in the literature to reduce weather data, we categorize the methods based on the type of data used and the nature of the process for selecting the typical days. Analysis of these methods led to the development of a new iterative approach with an embedded grouping algorithm. The method creates and iteratively enhances a short simulation sequence of typical days based on data reflecting the integrated annual profiles calculated using the detailed model. The reduced sequence led to much faster simulations while achieving profiles highly correlated with the reference integrated annual profiles. In addition, the last annual value, i.e., final annual sum, of each criterion extrapolated from a typical 12-day simulation differs little from the reference values (errors less than 1%). Moreover, the method was compared to two other clustering methods based on different types of selection data and an iterative method used in the literature. The results show that the classical method of day selection based only on weather data, typically used to generate Short Reference Years (SRYs), is in fact unable to accurately reproduce the annual reference profiles. Finally, the approach was also efficient when generalized, demonstrating its applicability to future optimization studies.

Within the environmental context, several tools based on simulations have been proposed to analyze the physical phenomena of heat and mass transfer in porous materials. However, it is still an open challenge to propose tools that do not require to perform computations to catch the dominant processes. Thus, this article proposes to explore advantages of using a dimensionless analysis by scaling the governing equations of heat and mass transfer. Proposed methodology introduces dimensionless numbers and their nonlinear distortirons. The relevant investigation enables to enhance the preponderant phenomena to (i) compare different categories of materials, (ii) evaluate the competition between heat and mass transfer for each material or (iii) describe the transfer in multi-layered wall configurations. It also permits to define hygrothermal kinetic, geometric and dynamic similarities among different physical materials. Equivalent systems can be characterized in the framework of experimental or wall designs. Three cases are presented for similarity studies in terms of (i) equivalent material length, (ii) time of heat and mass transfer and (iii) experimental configurations. All these advantages are illustrated in the given article considering 49 building materials separated in 7 categories.

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.

Reliable in-situ thermal characterisation allows to study the actual thermal performance of building components rather than the theoretical performance calculated from thermal properties of the constituent material layers. The most generally accepted method for in-situ thermal characterisation is the average method as described in ISO 9869. However, due to steady-state assumptions, the method's applicability can require long measurement periods and is often seasonally bounded. A correction for storage effects might shorten the required measurement time spans for the average method, but will not eliminate the seasonally bounded limitations. More advanced dynamic data analysis methods, such as regression modelling, ARX-modelling or stochastic grey-box modelling, can be used to overcome these difficulties. In this paper, a comparison between several semi-stationary and dynamic data analysis methods typically used for the thermal characterisation of building components from on-site measurements is made. Thereby, special attention is given to the reliability of the methods thermal resistance estimates when confronted with data sets of limited measurement time spans and different seasonal boundary conditions. First, the methods’ performances are assessed for simulated measurements of a south-facing insulated cavity wall in a moderate European climate. Subsequently, the performances are examined for actual measurement data of a similar test wall.

Evaluating how much heat is lost through external walls is a key requirement for building energy simulators and is necessary for quality assurance and successful decision making in policy making and building design, construction and refurbishment. Heat loss can be estimated using the temperature differences between the inside and outside air and an estimate of the thermal transmittance (U-value) of the wall. Unfortunately the actual U-value may be different from those values obtained using assumptions about the materials, their properties and the structure of the wall after a cursory visual inspection.
In-situ monitoring using thermometers and heat flux plates enables more accurate characterisation of the thermal properties of walls in their context. However, standard practices require that the measurements are carried out in winter over a two-week period to significantly reduce the dynamic effects of the wall's thermal mass from the data.
A novel combination of a lumped thermal mass model, together with Bayesian statistical analysis is presented to derive estimates of the U-value and effective thermal mass. The method needs only a few days of measurements, provides an estimate of the effective thermal mass and could potentially be used in summer.

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.

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.

Many of the popular building energy simulation programs around the world are reaching maturity — some use simulation methods (and even code) that originated in the 1960s. For more than two decades, the US government supported development of two hourly building energy simulation programs, BLAST and DOE-2. Designed in the days of mainframe computers, expanding their capabilities further has become difficult, time-consuming, and expensive. At the same time, the 30 years have seen significant advances in analysis and computational methods and power — providing an opportunity for significant improvement in these tools.In 1996, a US federal agency began developing a new building energy simulation tool, EnergyPlus, building on development experience with two existing programs: DOE-2 and BLAST. EnergyPlus includes a number of innovative simulation features — such as variable time steps, user-configurable modular systems that are integrated with a heat and mass balance-based zone simulation — and input and output data structures tailored to facilitate third party module and interface development. Other planned simulation capabilities include multizone airflow, and electric power and solar thermal and photovoltaic simulation. Beta testing of EnergyPlus began in late 1999 and the first release is scheduled for early 2001.

. 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 is the first book on constructive methods for, and applications of orthogonal polynomials, and the first available collection of relevant Matlab codes. The book begins with a concise introduction to the theory of polynomials orthogonal on the real line (or a portion thereof), relative to a positive measure of integration. Topics which are particularly relevant to computation are emphasized. The second chapter develops computational methods for generating the coefficients in the basic three-term recurrence relation. The methods are of two kinds: moment-based methods and discretization methods. The former are provided with a detailed sensitivity analysis. Other topics addressed concern Cauchy integrals of orthogonal polynomials and their computation, a new discussion of modification algorithms, and the generation of Sobolev orthogonal polynomials. The final chapter deals with selected applications: the numerical evaluation of integrals, especially by Gauss-type quadrature methods, polynomial least squares approximation, moment-preserving spline approximation, and the summation of slowly convergent series. Detailed historic and bibliographic notes are appended to each chapter. The book will be of interest not only to mathematicians and numerical analysts, but also to a wide clientele of scientists and engineers who perceive a need for applying orthogonal polynomials.

This book is the second edition of Numerical methods for diffusion phenomena in building physics: a practical introduction originally published by PUCPRESS (2016). It intends to stimulate research in simulation of diffusion problems in building physics, by providing an overview of mathematical models and numerical techniques such as the finite difference and finite-element methods traditionally used in building simulation tools. 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. In this reviewed edition, an innovative way to simulate energy and hydrothermal performance are presented, bringing some light on innovative approaches in the field.

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.

Several studies have shown that the actual thermal performance of buildings after construction may deviate significantly from its performance anticipated at design stage. As a result, there is growing interest in on site testing as a means to assess real performance. The IEA EBC Annex 58-project ‘Reliable Building Energy Performance Characterisation Based on Full Scale Dynamic Measurements’ focused on on site testing and dynamic data analysis methods that can be used to characterise the actual thermal performance and energy efficiency of building components and whole buildings. The research within this project was driven by case studies. The current paper describes one of them: the thermal characterisation of a round robin test box. This test box can be seen as a scale model of a building, and was built by one of the participants. During the project, its fabric properties remained unknown to all other participants. Full scale measurements have been performed on the test box in different countries under real climatic conditions. The obtained dynamic data has been distributed to all participants who had to characterise the thermal performance of the test box’s fabric based on the provided data. The paper compares the result of different techniques, ranging from a simple quasi-stationary analysis to advanced dynamic data analysis methods, which can be used to characterise the thermal performance based on on-site collected data.

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

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.

In this paper, the use of Bayesian inference is explored for estimating both the thermal conductivity and the internal convective heat transfer coefficient of an old historic building wall. The room air temperature, as well as the temperatures at the surface and within the wall have been monitored during one year and then used to solve the identification problem. With Bayesian inference, the posterior distributions of the unknown parameters are explored based on their prior distributions and on the likelihood function that models the measurement errors. In this work, the Markov Chain Monte Carlo method is used to explore the posterior distribution. The error of the inadequacy of mathematical model are considered using the approximation error model. The distribution of the estimated parameters have a small standard deviation, which illustrates the accuracy of the method. The parameters have been compared to the standard values from the French thermal regulations. The heat flux at the internal surface has been calculated with the estimated parameters and the standard values. It is shown that the standard values underestimate the heat flux of an order by 10%. This study also illustrates the importance of the preliminary diagnosis of a building with the estimation of the thermal properties of the wall for model calibration.

Thermal resistor-capacitor networks are a popular method for control-oriented building modeling. A basic assumption underlying this method is that the continuous temperature distribution in a wall or window is well-approximated by a small number of lumped capacitances. In this paper, we explore the accuracy of this approximation when a single capacitance is used. We derive conditions on the dimensionless parameters that characterize the problem, called Biot numbers, that lead to small errors in approximating a wall or window's surface heat fluxes and internal energy. The lumped capacitance approximation can be surprisingly accurate for Biot numbers much larger than the conventional upper bound of 0.1. In particular, the approximation is nearly exact for window panes, and is often acceptable for uniform walls. A large Biot number at an indoor wall surface, however, leads to large lumped capacitance approximation errors.

Excessive levels of moisture in buildings lead to building pathologies. Moisture also has an impact on the indoor air quality and the hygrothermal comfort of the building's occupants. A comprehensive list of the possible types of damage caused by moisture in buildings is discussed in the present paper. Damage is classified into four types: damage due to the direct action of moisture, damage activated by moisture, damage that occurred in a moist environment and deterioration of the indoor environment. Since moisture pathologies strongly depend on the hygrothermal fields in buildings, integrating these factors into a global model combining heat air and mass transfers and building energy simulation is important. Therefore, the list of moisture damage types is completed with a proposal of factors governing the risk of occurrence of each type of damage. The methodology is experimented on a simple test case combining hygrothermal simulations with the assessment of possible moisture disorders.

This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.

Abstract Experimental identification of the dynamic models of heat transfer in walls is needed for optimal control and characterization of building energy performance. These models use the heat equation in time domain which can be put in matrix form and then, through state-space representation, transformed in a transfer function which is of infinite order. However, the model acts as a low-pass filter and needs to respond only to the frequency spectrum present in the measured inputs. Then, the order of the transfer function can be determined by using the frequency spectrum of the measured inputs and the accuracy of the sensors. The main idea is that from two models of different orders, the one with a lower order can be used in building parameter identification, when the difference between the outputs is negligible or lower than the output measurement error. A homogeneous light wall is used as an example for a detailed study and examples of homogeneous building elements with very high and very low time constants are given. The first order model is compared with a very high order model (hundreds of states) which can be considered almost continuous in space.

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 integration of buildings in a Smart Grid, enabling demand-side management and thermal storage, requires robust reduced-order building models that allow for the development and evaluation of demand-side management control strategies. To develop such models for existing buildings, with often unknown the thermal properties, data-driven system identification methods are proposed.
In this paper, system identification is carried out to identify suitable reduced-order models. Therefore, grey-box models of increasing complexity are identified on results from simulations with a detailed physical model, deployed in the integrated district energy assessment simulation (IDEAS) package in Modelica.
Firstly, the robustness of identified grey-box models for day-ahead predictions and simulations of the thermal response of a dwelling, as well as the physical interpretation of the identified parameters, are analyzed. The influence of the identification dataset is quantified, comparing the added value of dedicated identification experiments against identification on data from in use buildings.
Secondly, the influence of the data used for identification on model performance and the reliability of the parameter estimates is quantified. Both alternative measurements and the influence of noise on the data are considered.

Moisture plays a central role in the provision of healthy buildings, both in relation to indoor humidity levels, which impacts on air quality and thermal comfort, and in relation to interstitial/surface condensation leading to fabric deterioration and mould growth, which impacts on performance and occupant well-being. Integrated building performance simulation (IBPS) provides a means to ensure that due consideration is given to these aspects at the design stage as designers attempt to deploy new approaches to energy demand reduction and sustainable supply. This paper describes how building space and construction moisture flow is modelled within the ESP-r system in a manner that is appropriately coupled to other domain models representing the heat, power, air and light flows within building/plant systems of arbitrary complexity (but with the focus here only on those domains that impact directly on moisture flow). The purpose of the paper is to describe the role of moisture flow modelling within IBPS, the barriers that are likely to be encountered in practice and future development needs. The application of the integrated approach is summarized for the case of mould growth alleviation and the deployment of passive methods for moisture control.

The effects of moisture on sensible and latent conduction loads are shown by using a heat and mass transfer model with variable material properties, under varying boundary conditions. This model was then simplified to reduce calculation time and used to predict conduction peak load and yearly integrated wall conduction heat flux in three different cities: Singapore (hot/humid), Seattle (cold/humid) and Phoenix (hot/dry). The room air temperature and relative humidity were calculated with the building energy simulation program DOE-2.1E. The materials studied were aerated cellular concrete, brick, lime mortar and wood. It is shown that the effects of moisture can be very significant and that simplified mathematical models can reduce the calculation time with varying effects on accuracy.

The design of a machine for solving heat-conduction problems by their electrical analogy with the flow of current in series-resistance, shunt-capacity, networks, is described in this paper. Consideration is given to the representation of cooling by radiation and convection, and the detailed design of the circuits is discussed.

The modeling of the thermal behavior of buildings requires using relatively heavy data processing when it comes to reproducing reality accurately. Electrical analogy makes it possible to develop directly accurate and simplified models insofar as the temperatures within the walls are not required.We present the various stages, which led us to propose the global analogical model of the building. Initially, we have studied how to transform a multi-layer wall into a three resistances and four capacities model (3R4C). We will compare this model with other electrical models thanks to a time and frequency analysis. Then, we will deal with the principle of aggregation of several walls. In order to facilitate the iterative resolutions related to numerical simulations, the lumped model integrates a heating floor.

System identification is very useful for finding the thermal properties of building components from outdoors tests. Many factors, such as data analysis skills, quality of recorded data, performance of the analysis tool employed, approximations and hypotheses on the component and its boundary, etc., can influence the results. The work reported in this paper intended to estimate achievable agreement when different analysis approaches are applied to the same and different datasets to find the U value of a given building component. The spread in the results is analysed.To isolate and highlight the effects of the different analysis approaches and test conditions from the effects of other assumptions about the test component and its boundaries, a well-known, quite simple uniform opaque wall was analysed taking direct component level measurements into consideration.

Until recently, the testing of ODE/DAE software has been limited to simple comparisons and benchmarking. The process of developing software from a mathematically specified method is complex: it entails constructing control structures and objectives, selecting iterative methods and termination criteria, choosing norms and many more decisions. Most software constructors have taken a heuristic approach to these design choices, and as a consequence two different implementations of the same method may show significant differences in performance. Yet it is common to try to deduce from software comparisons that one method is better than another. Such conclusions are not warranted, however, unless the testing is carried out under true ceteris paribus conditions. Moreover, testing is an empirical science and as such requires a formal test protocol; without it conclusions are questionable, invalid or even false.We argue that ODE/DAE software can be constructed and analyzed by proven, “standard” scientific techniques instead of heuristics. The goals are computational stability, reproducibility, and improved software quality. We also focus on different error criteria and norms, and discuss modifications to DASPK and RADAU5. Finally, some basic principles of a test protocol are outlined and applied to testing these codes on a variety of problems.

A standard for binary floating-point arithmetic is being proposed and there is a very real possibility that it will be adopted by many manufacturers and implemented on a wide range of computers. This development matters to all of us concerned with numerical software. One of the principal motivations for the standard is to distribute more evenly the burden of portability between hardware and software. At present, any program intended to be portable must be designed for a mythical computer that enjoys no capability not supported by every computer on which the program will be run. That mythical computer is so much grubbier than almost any real computer that a portable program will frequently be denigrated as "suboptimal" and then supplanted by another program supposedly "optimal" for the real computer in question but often inferior in critical respects like reliability. A standard --- almost any reasonable standard --- will surely improve the situation. A standard environment for numerical programs will promote fair comparisons and sharing of numerical codes, thereby lowering costs and prices. Furthermore, we have chosen repeatedly to enrich that environment in order that applications programs be simpler and more reliable. Thus will the onus of portability be shared among hardware manufacturers and software producers.

The International Building Physics Toolbox (IBPT) is a software library developed originally for heat, air and moisture system analysis in building physics. The toolbox is constructed as a modular structure of standard building elements, using the graphical programming language Simulink. To enable development of the toolbox, a common modelling platform is defined: a set of unique communication signals, material database and documentation protocol. The IBPT is an open source and available on the Internet. Any user can utilize, expand and develop the contents of the toolbox. This paper presents structure and essence of the library. Potential applications of the toolbox are illustrated through examples.

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.

In many countries, there is a great number of old buildings with local architectural, patrimonial, aesthetic and historic interest. They are the products of the vernacular traditional architecture fully integrating the environmental, social and economic local constraints.Moreover, this built inheritance is more heterogeneous than the modern stock of existing buildings. The historical buildings were built with different architectural designs featuring local styles of construction, different techniques and historical expertise.By experience, the actors of the building sector know that the thermal behaviour of historical buildings are not those of modern buildings set up at the time of the industrial period. However, they do not have assessed these specific thermal characteristics of historical buildings.This paper describes the complexity of architectural designs of historical dwellings in France. A field investigation during one year highlights various thermal characteristics of 11 dwellings. It provides a new understanding of thermal behaviour of these historical dwellings. The results show the thermal characteristics of historical dwellings and their differences with modern architecture.

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.

Most building materials are porous, composed of solid matrix and pores. The time varying indoor and outdoor climatic conditions result heat, air and moisture (HAM) transfer across building enclosures. In this paper, a transient model that solves the coupled heat, air and moisture transfer through multilayered porous media is developed and benchmarked using internationally published analytical, numerical and experimental test cases. The good agreements obtained with the respective test cases suggest that the model can be used to assess the hygrothermal performance of building envelope components as well as to simulate the dynamic moisture absorption and release of moisture buffering materials.

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.

Annual Energy Outlook

- U E I Administration

Administration, U. E. I. (2015). Annual Energy Outlook 2015, with projections to 2040. EIA,
Washington. 2

Complex heat transfer solved by electrical analogy

- J Kirkpatrick
- S K Lee
- T Olive
- H Batters
- J Callaham
- N Farquhar
- L Pope

Kirkpatrick J, Lee SK, Olive T, Batters H, Callaham J, Farquhar N,
Pope L (1943). Complex heat transfer solved by electrical analogy.
Chemical and Metallurgical Engineering, 50: 111-113.