Article

# On the comparison of three numerical methods applied to building simulation: Finite-differences, RC circuit approximation and a spectral method

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## Abstract

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: ...
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... 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) ...
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... 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. ...
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... 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. ...
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... 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. ...
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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: ...
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... 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. ...
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... 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. ...
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... 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. ...
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... 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. ...
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... 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. ...
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This paper proposes the use of a Spectral method to simulate diffusive moisture transfer through porous materials as a Reduced-Order Model (ROM). The Spectral approach is an a priori method assuming a separated representation of the solution. The method is compared with both classical Euler implicit and Crank-Nicolson schemes, considered as large original models. Their performance - in terms of accuracy, complexity reduction and CPU time reduction - are discussed for linear and nonlinear cases of moisture diffusive transfer through single and multi-layered one-dimensional domains, considering highly moisture-dependent properties. Results show that the Spectral reduced-order model approach enables to simulate accurately the field of interest. Furthermore, numerical gains become particularly interesting for nonlinear cases since the proposed method can drastically reduce the computer run time, by a factor of 100, when compared to the traditional Crank-Nicolson scheme for one-dimensional applications.
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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.
Article
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.
Article
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.
Article
This paper describes mathematical and software developments for a suite of programs for solving ordinary differential equations in MATLAB.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Book
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.
Article
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.
Article
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.
Article
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.
Article
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