Article

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

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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.

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... 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]. ...
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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]. ...
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... 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. ...
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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
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