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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.

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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]. ...

... 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: ...

... due to the orthogonality of the basis vectors. Equations (10), (11) and (12) enable to compute the spatial coefficients b i n N n=1 for each layer i . With this approach, the problem to solve scale with N χ × N . ...

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.

... Lumped model such as RC approaches have been proposed due to their very small computational cost. However, their reliability to estimate accurately the thermal diffusivity is questioned in [13]. The model has to be based on detailed heat diffusion process. ...

... The final time is t f = 24 h and the wall of length L = 0.5 m . The points of interest are the same as defined in Eq. (13). The discretisation parameters is ∆t = 30 s for both models. ...

... The issue is to estimate the global thermal diffusivity α of the wall by assuming an equivalent homogenous model [14]. Note that the structural identifiability of this parameter has been demonstrated in [13]. For this aim, the wall are monitored using five intrusive calibrated sensors HOBO TMC-6-HA as shown in Figures 8(a) to 8(c). ...

The estimation of wall thermal properties by \emph{in situ} measurement enables to increase the reliability of the model predictions for building energy efficiency. Nevertheless, retrieving the unknown parameters has an important computational cost. Indeed, several computations of the heat transfer problem are required to identify these thermal properties. To handle this drawback, an innovative approach is investigated. The first step is to search the optimal experiment design among the sequence of observation of several months. A reduced sequence of observations of three days is identified which guarantees to estimate the parameter with the maximum accuracy. Moreover, the inverse problem is only solved for this short sequence. To decrease further the computational efforts, a reduced order model based on the modal identification method is employed. This \emph{a posteriori} model reduction method approximates the solution with a lower degree of freedom. The whole methodology is illustrated to estimate the thermal diffusivity of an historical building that has been monitored with temperature sensors for several months. The computational efforts is cut by five. The estimated parameter improves the reliability of the predictions of the wall thermal efficiency.

... This may be very relevant within the context of determining materials properties using experimental observations of the field. In [33], the unknown parameters to estimate are reduced from four to two between the dimensionless and dimensional models. Then, from a numerical point of view, solving one dimensionless problem is equivalent to solving a whole class of dimensionless problems sharing the same scaling parameters. ...

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.

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.

The estimation of wall thermal properties by in situ measurement enables to increase the reliability of the model predictions for building energy efficiency. Nevertheless, retrieving the unknown parameters has an important computational cost. Indeed, several computations of the heat transfer problem are required to identify these thermal properties. To handle this drawback, an innovative approach is investigated. The first step is to search the optimal experiment design among the sequence of observation of several months. A reduced sequence of observations of three days is identified which guarantees to estimate the parameter with the maximum accuracy. Moreover, the inverse problem is only solved for this short sequence. To decrease further the computational efforts, a reduced order model based on the modal identification method is employed. This a posteriori model reduction method approximates the solution with a lower degree of freedom. The whole methodology is illustrated to estimate the thermal diffusivity of an historical building that has been monitored with temperature sensors for several months. The computational efforts is cut by five. The estimated parameter improves the reliability of the predictions of the wall thermal efficiency.

Comparisons of experimental observation of heat and moisture transfer through porous building materials with numerical results have been presented in numerous studies reported in the literature. However, some discrepancies have been observed, highlighting underestimation of sorption process and overestimation of desorption process. Some studies intend to explain the discrepancies by analysing the importance of hysteresis effects as well as carrying out the sensitivity analysis on the input parameters as convective transfer coefficients. This article intends to investigate the accuracy and efficiency of the coupled solution by adding advective transfer of both heat and moisture in the physical model. The efficient Scharfetter-Gummel numerical scheme is proposed to solve the system of advection-diffusion equations, which has the advantages of being well-balanced and asymptotically preserving. Moreover, the scheme is particularly efficient in terms of accuracy and reduction of computational time when using large spatial discretization parameters. Several linear and nonlinear cases are studied to validate the method and highlight its specific features. At the end, an experimental benchmark from the literature is considered. The numerical results are compared with the experimental data for a purely diffusive model and also for the proposed model. The latter presents better agreement with the experimental data. The influence of the hysteresis effects on the moisture capacity is also studied, by adding a third differential equation.

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.

The terms "inverse problems" and "ill-posed problems" have been steadily and surely gaining popularity in modern science since the middle of the 20th century. A little more than fifty years of studying problems of this kind have shown that a great number of problems from various branches of classical mathematics (computational algebra, differential and integral equations, partial differential equations, functional analysis) can be classified as inverse or ill-posed, and they are among the most complicated ones (since they are unstable and usually nonlinear). At the same time, inverse and ill-posed problems began to be studied and applied systematically in physics, geophysics, medicine, astronomy, and all other areas of knowledge where mathematical methods are used. The reason is that solutions to inverse problems describe important properties of media under study, such as density and velocity of wave propagation, elasticity parameters, conductivity, dielectric permittivity and magnetic permeability, and properties and location of inhomogeneities in inaccessible areas, etc. In this paper we consider definitions and classification of inverse and ill-posed problems and describe some approaches which have been proposed by outstanding Russian mathematicians A.

An iterative gradient descent method is applied to solve an inverse coefficient heat conduction problem with overdetermined
boundary conditions. Theoretical estimates are derived showing how the target functional varies with varying the coefficient.
These estimates are used to construct an approximation for a target functional gradient. In numerical experiments, iteration
convergence rates are compared for different descent parameters.
Key wordscoefficient identification–inverse heat conduction problem–gradient–adjoint problem–descent parameter

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.

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.

Implicit schemes require important sub-iterations when dealing with highly nonlinear problems such as the combined heat and moisture transfer through porous building elements. The computational cost rises significantly when the whole-building is simulated, especially when there is important coupling among the building elements themselves with neighbouring zones and with HVAC (Heating Ventilation and Air Conditioning) systems. On the other hand, the classical Euler explicit scheme is generally not used because its stability condition imposes very fine time discretisation. Hence, this paper explores the use of an improved explicit approach - the Dufort-Frankel scheme - to overcome the disadvantage of the classical explicit one and to bring benefits that cannot be obtained by implicit methods. The Dufort-Frankel approach is first compared to the classical Euler implicit and explicit schemes to compute the solution of nonlinear heat and moisture transfer through porous materials. Then, the analysis of the Dufort-Frankel unconditionally stable explicit scheme is extended to the coupled heat and moisture balances on the scale of a one- and a two-zone building models. The Dufort-Frankel scheme has the benefits of being unconditionally stable, second-order accurate in time O(dt^2) and to compute explicitly the solution at each time step, avoiding costly sub-iterations. This approach may reduce the computational cost by twenty, as well as it may enable perfect synchronism for whole-building simulation and co-simulation.

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.

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.

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.

The text demonstrates the methods for proving the existence (if at all) and finding of inverse and ill-posed problems solutions in linear algebra, integral and operator equations, integral geometry, spectral inverse problems, and inverse scattering problems. It is given comprehensive background material for linear ill-posed problems and for coefficient inverse problems for hyperbolic, parabolic, and elliptic equations. A lot of examples for inverse problems from physics, geophysics, biology, medicine, and other areas of application of mathematics are included.

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.

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.

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.

As part of a project to simulate the energy flows within urban districts, this paper describes a simplified model to simulate heat flows in buildings. Taking a standard two-node resistance–capacitance model as a starting point, we discuss developments to improve the handling of external surface radiant and convective energy exchanges, internal convective exchanges and the placement of capacitance in multi-layer walls. We also describe the extension of the model to solve for a building with an arbitrary number of zones, accounting for the capacitance of separating walls. Comparisons with ESP-r for a range of building configurations show that in general the model reproduces well both aggregate energy demands and their temporal characteristics (likewise with internal temperature). The paper closes by discussing the computational implementation of the model and its coupling with the integrated urban energy modelling tool.

Whenever a model is not identifiable, there exist several parameter vectors which yield the same input-output behavior, so that any further interprtation of is questionable. In this paper two methods which can be used to test state-space models for identifiability with a global approach are presented. While the first one is useful for nonlinear and/or time-varying models, the second one is suitable for linear time-invariant models. An appendix provides tools for the solution of he sets of polynomial equations involved in both methods.

It is shown that the Du Fort-Frankel method is unstable for the diffusion equation, if the usual central difference approximations are made to linear boundary conditions involving first order space derivatives. This is shown to be true even when the corresponding differential equation is stable. A modified boundary condition is presented which is proved to be stable provided the differential equation is stable.

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