Article

# A mixed POD-PGD approach to parametric thermal impervious soil modeling: Application to canyon streets

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

## Abstract

Numerical simulation is a powerful tool for assessing the causes of an Urban Heat Island (UHI) effect or quantifying the impact of mitigation solutions on local climatic conditions. However, the numerical cost associated with such a tool is quite significant at the scale of an entire district. Today, the main challenge consists of achieving both a proper representation of the physical phenomena and a critical reduction in the numerical costs of running simulations. This paper presents a combined parametric urban soil model that accurately reproduces thermal heat flux exchanges between the soil and the urban environment with a reduced computational time. For this purpose, the use of a combination of two reduced-order methods is proposed herein: the Proper Orthogonal Decomposition method, and the Proper Generalized Decomposition method. The developed model is applied to two case studies in order to establish a practical evaluation: an open area independent of the influences of the surrounding surface, and a theoretical urban scene with two canyon streets. The error due to the model reduction remains below 0.2 °C on the mean surface temperature for a reduced computational cost of 80%. Compared to in situ measurements the error remains bellow 1.24 °C at the surface.

## No full-text available

... Applied to urban soil heat transfer modeling, this model reduction method has shown its efficiency [3]. A cut computational cost of 80% was observed for a mean surface temperature error below 0.52 • C. Applied to building wall heat transfer modeling, the PGD parametric model computes the solution 100 times faster than a classical numerical method [4]. ...
... The enrichment process of the PGD basis stops when the ǫ criterion, defined by the user, is reached [11]. Details on the alternating directions strategy equations and algorithms for a similar problem can be found on [3]. For further details on the method and its developments, the interested reader may refer to [11,15]. ...
... As in the theoretical example, the POD basis is the most accurate one for N ∈ [2,8], if the full data-set is used for the training period. However if only a part of the data is available, the Chebyshev and Legendre approximation basis are more efficient for N ∈ [2,3]. The POD basis trained with half of the cycles seems to be as efficient as the one built with the full training data-set for N ∈ [4,8]. ...
Preprint
Full-text available
Estimating the temperature field of a building envelope could be a time-consuming task. The use of a reduced-order method is then proposed: the Proper Generalized Decomposition method. The solution of the transient heat equation is then re-written as a function of its parameters: the boundary conditions, the initial condition, etc. To avoid a tremendous number of parameters, the initial condition is parameterized. This is usually done by using the Proper Orthogonal Decomposition method to provide an optimal basis. Building this basis requires data and a learning strategy. As an alternative, the use of orthogonal polynomials (Chebyshev, Legendre) is here proposed.
... The heat equation needs then to be solved for each set of boundary conditions. To reduce the computational time, a reduced parametric model has been proposed in Azam et al. (2018). It combines the use of the Proper Generalized Decomposition (PGD) method to generate a parametric solution of the previously described problem and the use of the Proper Orthogonal Decomposition (POD) to reduce the number of parameters involved in the parametric PGD model. ...
... As described in Azam et al. (2018), the soil model is combined with several one-dimensional models using a co-simulation approach to assess the thermal behavior of an urban scene. During a time interval t ∈ [t n , t n+1 ], each model computes the field of interest, which consists of the temperature of each surface. ...
... For this reason, the snapshots must be representative of the problem (boundary values, initial conditions, materials properties). For more details on the development of the specific combined parametric model, readers can see Azam et al. (2018). A complete description of the POD and PGD method can be found in Chinesta et al. (2013). ...
Conference Paper
A parametric soil model has been developed to improve the computational time of microclimate simulation tools. It combines the use of two methods: the Proper Generalized Decomposition and the Proper Orthogonal Decomposition. Offline, a learning process is required to build the model, before its use on-line. A methodology to select a short and representative learning process needs to be developed. The k-means clustering method is used to build a training climate made of 24 days representative of a full climate. The offline computation costs are reduced by 94.4% for an error of 0.8%.
... In [26], the wall energy efficiency is computed according to the thickness and thermal diffusivity of the layers. A parametric solution is proposed in [27] to compute the heat transfer in soil within the urban environment. In [28], the two-dimensional heat transfer is solved for different climatic boundary conditions. ...
... In the works related to the POD [23,24,28], the solution is not parametric per se. Recent works [27,28] investigate the accuracy of the POD basis by using the same reduced basis for space for computations with different parameters. However, the accuracy of this approach can be very low in some cases as noted in [23]. ...
Article
Full-text available
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.
... With the computational gain, other studies of parametric problems using reduced-order methods are found in the literature. In this way, in Azam et al. (2018) they have combined the POD and PGD approaches to modelling the soil for application in the study of canyon streets and in Azam et al. (2021) to assess efficiently the building's envelope thermal performance. ...
Article
In this paper, the spectral method is developed as a reduced-order model for the solution of parametric problems within the building refurbishment framework. We propose to use the spectral reduced-order method to solve parametric problems in an innovative way, integrating the unknown parameter as one of the coordinates of the decomposition. The residual is minimized combining the Tau–Galerkin method with the Collocation approach. The developed method is evaluated in terms of accuracy and reduction of the computational time in three different cases. The dynamic behaviour of unidimensional moisture diffusion is investigated. The cases focus on solving parametric problems in which the solution depends on space, time, diffusivity and material thickness. Results highlight that the parametric spectral reduced-order method provides accurate solutions and can reduce 10 times the degree of freedom of the solution. It allows efficient computation of the physical phenomena with a lower error when compared to traditional approaches.
... Second, the porous structure of the PAP surface layer has a larger specific surface area in contact with solar radiation and air temperature than that of the DAP [23]. The porous structure can also reflect solar radiation to the interior voids several times. ...
Article
Permeable asphalt pavement (PAP) has many advantages such as reducing surface runoff and improving driving safety on rainy days; hence, it has attracted much attention from researchers. PAP is greatly affected by environmental factors due to its porous structure and the temperature characteristics are different from those of dense asphalt pavement (DAP). This study aims to understand the influences of environmental conditions on the patterns of temperature distribution in PAP and DAP. Weather station and temperature sensors were installed while paving PAP and DAP. Meteorological data and temperature data of each structural layer of PAP and DAP were collected during 2019–2021. The results demonstrate that temperatures in the depth range of 4 ∼ 10 cm below the PAP surface were 1 ∼ 6℃ higher than DAP on rainless days. The heat storage and dissipation efficiency of PAP within the depth range of 4 ∼ 40 cm were better than that of DAP. The maximum influence depth of PAP that heavy rainfall can affect is 60 cm, the maximum cooling extent of PAP is 5 ∼ 6℃, and the minimum cooling duration of PAP is 10 h on rainy days. This study provides a reliable foundation for evaluating the extent of the environmental impacts of PAP.
Chapter
The SOLENE-microclimat model has been developed to investigate the consequences of urban context on local microclimate and indoor thermal conditions. It is dedicated to modeling urban microclimate and building thermal behavior at the district scale. The modeling approach is based on the coupling of several modules: radiative, thermal, and CFD models. The model can simulate a large range of cases encountered in urban projects: modeling of vegetation, water ponds, soils, building energy simulation, and techniques such as cool paints and surface water aspersion. It offers a way for enhancing the knowledge and constitutes a decision-making support system for establishing effective urban environmental policies. For each module included in SOLENE-microclimat, the validation steps that have been carried out to check the model’s ability to accurately represent phenomena are presented. Two case studies are presented to show how this tool has been used to assess and compare climate adaptation measures based on vegetation (trees, lawns, green walls, and roofs), water use (aspersion), and cool materials. The advantage of SOLENE-microclimat is its ability to model all the fluxes at both the building and the district scales. This helps in understanding the direct and indirect effects of various adaptation measures and their relative impact on building energy demand and indoor/outdoor thermal comfort.
Article
Estimating the temperature field of a building envelope could be a time-consuming task. The use of a reduced-order method is then proposed: the Proper Generalized Decomposition method. The solution of the transient heat equation is then re-written as a function of its parameters: the boundary conditions, the initial condition, etc. To avoid a tremendous number of parameters, the initial condition is parameterized. This is usually done by using the Proper Orthogonal Decomposition method to provide an optimal basis. Building this basis requires data and a learning strategy. As an alternative, the use of orthogonal polynomials (Chebyshev, Legendre) is here proposed. Highlights • Chebyshev and Legendre polynomials are used to approximate the initial condition • Performance of Chebyshev and Legendre polynomials are compared to the POD basis • Each basis combined with the PGD model is compared to laboratory measurements • The influence of four different parameters on the accuracy of the basis is studied • For each approximation basis, CPU calculation times are evaluated and compared
Article
Full-text available
The elevated air temperature of a city, urban heat island (UHI), increases the heat and pollution-related mortality, reduces the habitats’ comfort and elevates the mean and peak energy demand of buildings. To countermeasure this unwanted phenomenon, a series of strategies and policies have been proposed and adapted to the cities. Various types of models are developed to evaluate the effectiveness of such strategies in addition to predict the UHI. This paper explains the compatibility of each type of model suitable for various objectives and scales of UHI studies. The recent studies, mainly from 2013 to 2015, are further categorized and summarized in accordance with their context of study.
Article
Full-text available
EnglishIn the framework of a research program on the role of vegetation in the sustainable urban development, the objective of FluxSAP is to quantify the importance of vegetation in the sensible heat and water vapour fluxes from a mixed district. The 2010 campaign took place around the permanent hydrometeorological survey site of IRSTV (Institut de Recherche en Sciences et Techniques de la Ville). It was principally devoted to the feasibility of making measurements over a very heterogeneous area, by using in parallel five sets of sensors wich all allow to measure the fluxes. This creates the possibility by evaluating their origin, of separating the contributions of the various land cover modes. francaisDans le cadre d'un programme de recherche sur le role de la vegetation dans le developpement urbain durable, FluxSAP a pour objectif de quantifier l'importance de la vegetation dans les flux de chaleur et de vapeur d'eau d'un quartier mixte. La campagne 2010, realisee autour du site d'observation hydrometeorologique permanente de l'Institut de recherche en sciences et techniques de la ville (IRSTV), etait principalement orientee vers la faisabilite de la mesure au-dessus d'un sol tres heterogene, en mettant en oeuvre en parallele cinq jeux de capteurs permettant potentiellement de mesurer ces flux et en evaluant leur origine, en vue de separer les contributions des differents modes d'occupation des sols.
Article
Full-text available
Dynamic Data-Driven Application Systems—DDDAS—appear as a new paradigm in the field of applied sciences and engineering, and in particular in Simulation-based Engineering Sciences. By DDDAS we mean a set of techniques that allow the linkage of simulation tools with measurement devices for real-time control of systems and processes. One essential feature of DDDAS is the ability to dynamically incorporate additional data into an executing application, and in reverse, the ability of an application to dynamically control the measurement process. DDDAS need accurate and fast simulation tools using if possible off-line computations to limit as much as possible the on-line computations. With this aim, efficient solvers can be constructed by introducing all the sources of variability as extra-coordinates in order to solve the model off-line only once. This way, its most general solution is obtained and therefore it can be then considered in on-line purposes. So to speak, we introduce a physics-based meta-modeling technique without the need for prior computer experiments. However, such models, that must be solved off-line, are defined in highly multidimensional spaces suffering the so-called curse of dimensionality. We proposed recently a technique, the Proper Generalized Decomposition—PGD—able to circumvent the redoubtable curse of dimensionality. The marriage of DDDAS concepts and tools and PGD off-line computations could open unimaginable possibilities in the field of dynamic data-driven application systems. In this work we explore some possibilities in the context of on-line parameter estimation.
Article
Full-text available
A study is made of the number of dimensions needed to specify chaotic Rayleigh–Bénard convection, over a range of Rayleigh numbers (γ = Ra/Rac < 102). This is based on the calculation of Lyapunov dimension over the range, as well as the notion of Karhunen–Loéve dimension. An argument suggesting a universal relation between these estimates and supporting numerical evidence is presented. Numerical evidence is also presented that the reciprocal of the largest Lyapunov exponent and the correlation time are of the same order of magnitude. Several other universal features are suggested. In particular it is suggested that the intrinsic attractor dimension is $O(Ra^{\frac{2}{3}})$, which is sharper than previous results.
Article
Full-text available
An urban surface scheme for atmospheric mesoscale models ispresented. A generalization of local canyon geometry isdefined instead of the usual bare soil formulation currently usedto represent cities in atmospheric models. This allows refinement ofthe radiative budgets as well as momentum, turbulent heat and ground fluxes.The scheme is aimed to be as general as possible, in order to representany city in the world, for any time or weather condition(heat island cooling by night, urban wake, water evaporation after rainfalland snow effects). Two main parts of the scheme are validated against published data.Firstly, it is shown that the evolution of the model-predictedfluxes during a night with calm winds is satisfactory, considering both the longwave budget and the surface temperatures. Secondly, the original shortwave scheme is tested off-line and compared to the effective albedoof a canyon scale model. These two validations show that the radiative energy input to the urban surface model is realistic. Sensitivity tests of the model are performed for one-yearsimulation periods, for both oceanic and continental climates. The scheme has the ability to retrieve, without ad hoc assumptions, the diurnal hysteresis between the turbulent heat flux and ground heat flux. It reproduces the damping of the daytime turbulent heat flux by the heat storage flux observed in city centres. The latent heat flux is negligible on average,but can be large when short time scales are considered (especially afterrainfall). It also suggests that in densely built areas, domesticheating can overwhelm the net radiation, and supply a continuous turbulentheat flux towards the atmosphere. This becomes very important inwinter for continental climates. Finally, a comparison with a vegetation scheme shows that the suburban environment can be represented with a bare soil formulation for large temporal or spatial averages (typical of globalclimatic studies), but that a surface scheme dedicated to the urban surface is necessary when smaller scales are considered: town meteorological forecasts, mesoscale or local studies.
Thesis
Full-text available
Le contrôle en temps réel des locaux climatisés requiert la connaissance de modèles de la distribution de température et du champ de vitesse. Des modèles complets, basés sur les codes CFD, donnent accès à ces informations, mais sont incompatibles avec des applications en temps réel. Ainsi, un modèle d’ordre réduit est nécessaire. Cette étude propose de réduire la taille d’un modèle CFD en adoptant, dans un premier temps, l’hypothèse d’un champ de vitesse fixé, ce qui permet de ne résoudre que l’équation de conservation de l’énergie. Puis, mis sous forme d’un système d’état, la décomposition orthogonale aux valeurs propres est utilisée pour réduire son ordre. Cette méthode est appliquée au cas d’un local équipé d’un ventilo-convecteur. La forme de système d’état du modèle réduit ainsi obtenu permet, par application de la théorie moderne du contrôle, d’estimer la température dans la zone d’occupation sans mesure directe, et ainsi de l’intégrer dans une boucle fermée de contrôle. Plusieurs contrôleurs sont comparés avec pour point commun d’être synthétisés sur le principe du modèle interne basé sur la connaissance d’un modèle d’ordre réduit de l’air du local.
Article
In a dense urban area, pavement watering could be a solution to mitigate the Urban Heat Island. So far, mainly experimental studies have been used to evaluate watering techniques. In this study, a soil model dedicated to pavement watering has been developed within the urban climate model SOLENE-microclimat. This watering model is presented and evaluated via a measurement campaign performed on an asphalt car park during warm days. The measurement campaign reveals that the surface cooling is mainly due to evaporation (80%). However, under warm conditions, the heat flux exchanged between the runoff water and the surface should also be modelled. Indeed, watering events are modelled through a runoff convective heat flux and a latent heat flux. The mean daily RMSE between estimated and observed surface temperature is 1.04°C, 0.86°C, 0.66°C, 0.35°C and 0.21°C respectively at the surface, 5 cm-, 10 cm-, 34 cm-and 50 cm-depths.
Article
The main purpose of this study is to evaluate an urban soil model that will accurately reproduce the heat flux into urban soil, which has an influence on the urban heat island effect, for typical urban land use such as a car park. After a complete literature review, a sensitivity study is carried out on a large number of parameters: material properties, layer size, deep boundary condition, and convective heat transfer coefficient. The model's ability to reproduce heat conduction transfer is validated via a measurement campaign performed on an asphalt car park during hot days. The mean daily RMSE between estimated and observed surface temperature is 0.86°C, and 0.72°C, 0.58°C, 0.26°C and 0.13°C respectively at 5. cm-, 10. cm-, 34. cm- and 50. cm-depths. Performances obtained using different node distributions are discussed and compared with results from the literature. The model is more efficient than most of the other models applied under similar conditions. Finally, application of the proposed model on a yearly basis demonstrates that the accuracy loss caused by the decrease in the number of nodes is higher for clear and sunny days.
Article
Urban areas are subject to high human pressure and forthcoming enhanced hydrologic and climatic risks due to both city development and climate change. An asphalt concrete parking lot was instrumented in Nantes, France, to quantify the energy and hydrological responses of the surface to simulated rainfalls. The surface fluxes (precipitation, evaporation, radiation exchanges, sensible heat convection and conduction, runoff) were measured in situ and used to close the water budget with residual closure errors lower than 10%, depending on the surface evaporation retrieval method. The latent heat flux estimated from scintillometry measurements provided a better water budget closure than the direct eddy-correlation measurements. Runoff was the primary component of the water budget and represented around 80% of the total precipitation, compared to 17% for surface evaporation. The scintillometry method provided water evaporation time series at a 1-min time scale during the experiment. These series were used to characterize the rapid changes in the hydrological and energetic budgets of the asphalt surface after a precipitation event. During the drying phase the surface evaporation was significantly active, yielding 80% of the turbulent fluxes with a Bowen ratio of 0.25.
Article
This work is devoted to proposing a hybrid numerical–analytical method to address the problem of heat and moisture transfer in porous soils. Several numerical and analytical models have been used to study heat and moisture transfer. The complexity of the coupled transfer in soils is such that analytical solutions exist only for limited problems, while numerical solutions can deal with more realistic ones but at a higher computational cost. Therefore, we propose to implement analytical solutions where variations of temperature and moisture content are known to be almost nonvarying, while the numerical solution is implemented in the remaining region, near the boundaries. The coupling between solutions is performed assuming the continuity of both fields and fluxes at each interface. This strategy allows assuring the physical phenomenon occurring at the interface. Numerical experiments are performed, showing the accuracy, the efficiency, and the great potential of the method regarding applications in nonlinear soil problems.
Article
In this paper, an innovative method to minimise energy losses through building envelopes is presented, using the Proper Generalised Decomposition (PGD), written in terms of space x, time t, thermal diffusivity and envelope thickness L. The physical phenomenon is solved at once, contrarily to classical numerical methods that cannot create a parameter dependent model. First, the PGD solution is validated with an analytical solution to prove its accuracy. Then a complex case study of a multi-layer wall submitted to transient boundary conditions is investigated. The parametric solution is computed as a function of the space and time coordinates, as well as the thermal insulation thickness and the load material thermal diffusivity. Physical behaviour and conduction loads are analysed for 76 values of thermal insulation thickness and 100 types of load material properties. Furthermore, the reduced computational cost of the PGD is highlighted. The method computes the solution 100 times faster than standard numerical approaches. In addition, the PGD solution has a low storage cost, providing interesting development of parametric solutions for real-time applications of energy management in buildings.
Book
This book is intended to help researchers overcome the entrance barrier to Proper Generalized Decomposition (PGD), by providing a valuable tool to begin the programming task. Detailed Matlab Codes are included for every chapter in the book, in which the theory previously described is translated into practice. Examples include parametric problems, non-linear model order reduction and real-time simulation, among others. Proper Generalized Decomposition (PGD) is a method for numerical simulation in many fields of applied science and engineering. As a generalization of Proper Orthogonal Decomposition or Principal Component Analysis to an arbitrary number of dimensions, PGD is able to provide the analyst with very accurate solutions for problems defined in high dimensional spaces, parametric problems and even real-time simulation.
Article
A proper generalised decomposition for solving inverse heat conduction problems is proposed in this article as an innovative method offering important numerical savings. It is based on the solution of a parametric problem, considering the unknown parameter as a coordinate of the problem. Then, considering this solution, all sets of cost function can be computed as a function of the unknown parameter of the defined domain, identifying the argument that minimises the cost function. In order to illustrate the applicability, the method is used to solve a non-linear inverse heat conduction problem to determine a temperature-dependent thermal conductivity. Then, a comparison is carried out with the local sensitivity and the genetic algorithm methods. It is shown that the proper generalised decomposition method estimates the unknown parameter with the same accuracy as the other two methods. Due to its advantage in terms of reducing the complexity, the method was then used to solve a transient three-dimensional non-linear heat transfer inverse problem. The results have shown that the method is appropriate to determine the unknown parameter with a low computational cost. Furthermore, the main advantage of the technique is its low capacity for storage, which can be used, as an inverse method, for building energy management and extended to evaluate thermal bridges from on-site measurements.
Article
This study developed pavement temperature predictive models based on the characterized thermophysical properties of different pavements to assess urban climates in the built environment. A database comprising of six pavement types including conventional and modified asphalt, and cement concrete mixtures was available with their thermophysical properties: specific heat capacity, thermal conductivity and material density. Models were developed to predict temperature at the surface, and at 40 mm depth using the measured thermophysical properties, and recorded climatological parameters: air temperature, wind speed, and relative humidity. The two predictive models were robust and rational depicted by low bias and high precision. An increase in heat capacity increased pavement surface temperature indicating that higher energy is required to raise the pavement temperature, and also be able to release as much energy as stored, which would be best suitable at different times of the day to counter urban heat island (UHI) effects. An increase in thermal conductivity decreased pavement temperature illustrating that the pavement would store more heat within the system for a longer duration, and may release this heat at a particular timeframe changing the urban climate at that moment. An increase in wind speed by about 1 m/s increased pavement temperature by 1 °C, and this may increase UHI if there is already higher temperature in the environment. Overall, based on rational correlations between model predictions and actual field measurements it is recommended that the pavement temperatures of the systems be comfortably predicted for pavements using the developed models.
Article
Innovative and efficient ways to carry out numerical simulations are worth of investigation to reduce the computational complexity of building models and make it possible to solve complex problems. This paper presents a reduced order model, based on Proper Generalised Decomposition (PGD), to assess 2-dimensional heat and moisture transfer in walls. This model is associated with the multizone model Domus using an indirect coupling method. Both models are co-simulated to perform whole-building hygrothermal simulation, considering 2D transfer in walls. The whole-building model is first validated with data from the IEA Annex 41. Then, a case study is considered taking into account a 2-zones building with an intermediary shared wall modelled in 2 dimensions to illustrate the importance of the technique to analyse the hygrothermal behaviour of the wall. It has been highlighted that the whole model enables to perform more precisely analyses such as mould growth on the internal surface. In addition, important theoretical numerical savings (90%) are observed when compared to the large original model. However, the effective numerical savings are not so important (40%) due to the limitations of the co-simulation method.
Article
Given the current need for a comprehensive understanding of the consequences of urban development to achieve the goal of sustainable cities, knowledge of urban microclimates is essential to control the UHI effect, conduct urban amenity studies and monitor energy consumption. Modeling offers a way for enhancing this knowledge and it also constitutes a decision-making support system for establishing effective urban environmental policies. We present the SOLENE-microclimat model dedicated to modeling urban microclimate and building thermal behavior. We first explain the modeling approach, based on the coupling of radiative, thermal and CFD models and show how it has been enhanced to handle a large range of cases now encountered in urban projects: modeling of vegetation, water ponds, bare ground, building energy simulation, and technics such as surface water aspersion. Several illustrations are given to show how this tool has been used to assess and compare climate adaptation and/or mitigation measures based on vegetation: trees, lawns, green walls and roofs. Having both the building and the district scales, as well as the detailed description of all the fluxes, we distinguish the direct and indirect effects of theses measures on building energy consumption and highlight the dominant effects.
Article
SUMMARYA new method is developed here for the real-time integration of the equations of solid dynamics based on the use of proper orthogonal decomposition (POD)–proper generalized decomposition (PGD) approaches and direct time integration. The method is based upon the formulation of solid dynamics equations as a parametric problem, depending on their initial conditions. A sort of black-box integrator that takes the resulting displacement field of the current time step as input and (via POD) provides the result for the subsequent time step at feedback rates on the order of 1 kHz is obtained. To avoid the so-called curse of dimensionality produced by the large amount of parameters in the formulation (one per degree of freedom of the full model), a combined POD–PGD strategy is implemented. Examples that show the promising results of this technique are included. Copyright © 2014 John Wiley & Sons, Ltd.
Article
Chaotic Rayleigh-Benard convection over moderate Rayleigh numbers Ra is parametrically studied under highly idealized conditions. A detailed analysis of the weakly turbulent flow found at relatively low Re is performed, and the gross features such as Nusselt number and Reynolds number are found to possess scalings. The Karhunen-Loeve (K-L) decomposition is used to elucidate the physics of these complex flows in a quantitative manner. The structure of the empirical eigenfunctions is examined, revealing important apparent scaling with Ra. The number of parameters or dimensions needed to describe turbulent thermal convection is studied. It is shown that a fixed relation appears to hold between the K-L dimension and the Lyapunov dimension at low values of Ra. Numerical evidence is given that the Lyapunov exponents have a uniform density on average. Extensive numerical calculations for the flows and for their Lyapunov exponents over a range of Ra are presented.
Article
This paper proposes a new method for solving the heat transfer equation based on a parallelisation in time of the computation. A parametric multidimensional model is solved within the context of the Proper Generalised Decomposition (PGD). The initial field of temperature and the boundary conditions of the problem are treated as extra-coordinates, similar to time and space. Two main approaches are exposed: a "full" parallelisation based on an off-line parallel computation and a "partial" parallelisation based on a decomposition of the original problem. Thanks to an optimised overlapping strategy, the reattachment of the local solutions at the interfaces of the time subdomains can be improved. For large problems, the parallel execution of the algorithm provides an interesting speedup and opens new perspectives regarding real-time simulation.
Article
A simple energy balance model which simulates the thermal regime of urban and rural surfaces under calm, cloudless conditions at night is used to assess the relative importance of the commonly stated causes of urban heat islands. Results show that the effects of street canyon geometry on radiation and of thermal properties on heat storage release, are the primary and almost equal causes on most occasions. In very cold conditions, space heating of buildings can become a dominant cause but this depends on wall insulation. The effects of the urban greenhouse and surface emissivity are relatively minor. The model confirms the importance of local control especially the relation between street geometry and the heat island and highlights the importance of rural thermal properties and their ability to produce seasonal variation in the heat island. A possible explanation for the small heat
Article
Observations show that the urban heat island in the atmospheric layer below roof level is most strongly developed during calm, cloudless conditions at night. This paper outlines two versions of a numerical model to describe the cooling of rural and street canyon surfaces under these conditions using surface thermal and radiative properties and the radiative geometry of the canyons. One version uses a full system of differential equations and the other the simpler force-restore approach. The two approaches are shown to be in general agreement and the output of the simpler model is shown to give a faithful representation of cooling of rural and urban surfaces, and therefore heat islands, when compared with field observations.
Article
Les architectes, les urbanistes et les ingénieurs sont fortement sollicités pour élaborer des méthodes de conception permettant de limiter l'impact environnemental de l'urbanisation. De nombreux travaux montrent que des phénomènes climatiques comme l'îlot de chaleur urbain sont à la fois les causes et les conséquences de l'augmentation de la consommation énergétique à l'échelle de la ville. Par ailleurs, l'expertise énergétique des bâtiments est possible avec des outils opérationnels qui ne prennent pas correctement en compte les conditions climatiques à petite échelle spatiale alors qu'il est démontré que leurs effets sont avérés. Souvent négligé, l'impact direct et indirect de l'aménagement constitue pourtant une piste intéressante pour la régulation énergétique passive. Pour étudier ces phénomènes, nous proposons dans cette thèse d'utiliser un outil de simulation microclimatique, reposant sur le couplage d'un modèle thermoradiatif et d'un code de mécanique des fluides numérique. Dans une première partie, nous développons un modèle de sol et un modèle thermique de bâtiment, ce dernier permettant le calcul des consommations énergétiques d'un bâtiment interagissant avec son environnement urbain. Nous les intégrons à l'outil de simulation thermoradiative (Solene), puis adaptons la procédure de couplage physique avec l'outil de simulation thermoaéraulique (Fluent). Dans une deuxième partie, nous caractérisons le comportement d'un bâtiment de référence en site isolé et décrit par des paramètres variables, en établissant des classes de consommations énergétiques à partir d'une méthode statistique d'étude de sensibilité multicritères. Enfin, nous réutilisons ces classes de bâtiments dans un contexte urbain réel, le projet Lyon Confluence, pour analyser l'impact de deux modes d'aménagement des îlots étudiés : un aménagement minéral et un aménagement végétal. Cette dernière partie fait ressortir deux résultats principaux à savoir l'écart important entre des consommations énergétiques simulées en contexte théorique isolé et simulées en site urbain, puis, l'économie potentielle d'énergie entre deux choix d'aménagement urbain pour un même projet.
Analytical solution x31b60t0, slab body with cosine-periodic fluid convection at x = 0 and zero temperature at x = l
• K D Cole
• J Krahn
Cole, K. D., & Krahn, J. (2015). Analytical solution x31b60t0, slab body with cosine-periodic fluid convection at x = 0 and zero temperature at x = l, exact analytical conduction toolbox.
Modelling coupled heat and airflow: Ping pong vs. onions. Document-Air Infiltration Centre AIC Proc
• J Hensen
Hensen, J. (1995). Modelling coupled heat and airflow: Ping pong vs. onions. Document-Air Infiltration Centre AIC Proc, Oscar Faber PLC, 253.
Proper Generalized Decompositions, SpringerBriefs in Applied Sciences and Technology, Springer International Publishing
• Cueto