Point Thermal Transmittance of Rib Intersections in Concrete Sandwich Wall Panels

ABSTRACT Concrete sandwich panels are building elements made by two concrete wythes separated by a layer of lightweight material: the central layer is inhomogeneous due to the presence of concrete ribs which tie the external wythes and act as thermal bridges. This paper deals with the problem of determining point thermal transmittance associated with rib intersections. Together with previous results by the authors, it allows accurate calculation of thermal transmittance of sandwich panels according to current International Standards. A dataset of 1080 point transmittance values is obtained upon use of a spectral element method, varying systematically material conductivities and thickness of panel layers, for the most common pairs of rib widths in current panel production. To limit the computational cost, a solution strategy based on the use of low-order polynomials on three grids of increasing refinement, coupled with Richardson extrapolation is adopted. Finally, a power law correlation is proposed that allows to estimate point transmittance within a relative error of 10%.


Introduction
Precast concrete sandwich wall panels are building elements that allow fast and economical constructions of buildings such as factories, warehouses, and malls: these panels are designed to span from the foundation to the roof and can be used as load bearing walls, or exterior claddings, or both. Since panels are usually produced far from the construction site, weight becomes a crucial point with regard to handling, transportation, and installation issues. The easiest way to meet both the requirements of structure robustness and weight containment, is to have a frame of concrete and fill the empty zones with lightweight materials like expanded polystyrene. In what follows, this kind of panel will be referred to as precast concrete lightened sandwich wall panel (LSP). It is worth underlining that the use of insulating slabs in LSP, which is aimed to reduce weight, does reduce the average thermal transmittance of the panel at the same time. There are other types of sandwich panels much more thermal efficient than LSP because of the presence of a continuous layer of homogeneous insulations between the concrete wythes (insulated sandwich wall panel): they are not considered here since the computation of their thermal transmittance is straightforward.
In a LSP, two prestressed concrete wythes are separated by a heterogeneous layer made by lightweight slabs and concrete ribs: so there are panel regions made of solid concrete which act as thermal bridges. Building designers in order to fulfill the requirements of the European directives on the energy performance of buildings (Directives 2002/91/EC and 2010/31/EU) as implemented by the EU member states, need reliable estimates of the thermal transmittance of building elements, including the effects of thermal bridges. In principle, computation of the thermal transmittance U of LSPs is not a critical issue: ISO 6946, ISO 14683, and ISO 10211 [1]- [3] describe accurate methods to do that. These methods require the knowledge of linear and point thermal transmittances associated with thermal bridges in LSPs. Their values can be computed upon numerical simulations performed as described in ISO 10211 [3]. However, most panel manufacturers are small-medium enterprises (SMEs) and their technical staff does not have either the know-how or the time to numerically compute values necessary to determine the transmittance U of their product range: they consider much more convenient the usage of transmittance catalogs or correlations easily implemented in a spreadsheet or in an in-house code.
Other recent studies [4], [5] pursue a goal similar to this study, that is to find correlations to evaluate thermal properties of building components which satisfy the following requirements: they must reproduce with good approximation the results of standardized methods; in addition, they must be suited for routine use in industry. Tenpierik et al. [4] propose a closed analytical model for calculating linear transmittances in vacuum insulation panels that is much more accurate than models based on electric circuits, but less laborious than numerical simulations. Buratti et al. [5] develop an artificial neural network (ANN) for estimating U of wooden framed windows. Based on 278 experimental and numerical data obtained in accordance with standards, the ANN returns an estimate of window transmittance as a function of 10 simple parameters like window typology, wood kind, frame thickness, etc.
In a companion paper [6], the problem of finding an accurate correlation for prediction of linear thermal transmittance values of LSPs was addressed. The goal was reached through the combined use of a fast and accurate spectral element method (SEM) and of an ANN. The SEM allowed the authors to obtain a large dataset of linear transmittance values and then they used it to train the ANN. In this work, the study investigating the point thermal bridges in LSPs and determining the associated point thermal transmittance was completed.
Point thermal bridges in LSPs coincide with concrete rib intersections, like in the four panel corners. ISO 14683 [2,Clause 5.3.2] states that, in general, the effect of point thermal bridges "insofar they result from the intersection of linear thermal bridges, " can be neglected. In past years, in a few real cases point thermal bridge contribution in LSPs was evaluated by the authors and it was found that it accounted for up to 2%, approximately. Besides, the associated point transmittances were negative: neglecting them implied overestimating panel transmittance U. In this work, the correctness and the generality of these conclusions were checked, on the basis of a systematic study of point transmittance as a function of concrete and lightweight material conductivities and panel geometrical parameters.
Evaluation of point thermal transmittance requires three dimensional (3D) numerical simulations, besides the knowledge of linear transmittance associated with the intersecting concrete ribs. As in [6], a conformal quadrilateral SEM is used: the computational effort required to approximate point transmittances is much larger than that needed to estimate linear ones. That forces the use of coarse meshes and to approximate the temperature field in each mesh element with low-order polynomials, but with the major drawback of loss of accuracy. Here, a solution strategy that allows to by-pass this problem is presented and discussed. The numerical problem is solved on three grids of increasing refinement -although overall rather coarse -upon use of low-order polynomials (p = 4): results are extrapolated by Richardson method [7], [8]. This procedure assures a good trade-off between accuracy, as required by International Standards, and computational cost.
A dataset of point transmittance values is obtained varying systematically material conductivities and thickness of external and central layers, for the most frequent pairs of rib widths in current panel production. A simple power law correlation in terms of a new variable depending on linear transmittances and concrete wythes thickness has been proposed in this study. This correlation allows to estimate point transmittances within a relative error of ±10% which is intermediate between the typical accuracy of numerical calculation of linear thermal transmittance (±5%) and that of linear thermal bridge catalogs (±20%), as stated in ISO 14683 [2,Clause 6.4].
In the literature there are only a few other studies on point thermal bridges in precast concrete panels: all of them consider insulated sandwich wall panels and not LSPs. An insulated sandwich wall panel is characterized by a continuous layer of insulation between the outer concrete wythe and the inner one. The two wythes are tied together by connectors (usually metal ones). Since the connectors pass through the insulation layer, they act as point thermal bridges. Studies [9]- [11] are concerned with the thermal effect of metal connectors used in insulated sandwich wall panels.
Lee and Pessiki [9] propose a modified algebraic method to estimate point transmittance associated to wythe connectors made of steel wires. Their results are verified by comparison with 3D finite element solutions.
Willems and Hellinger [10] propose a correlation for point transmittance of metal connectors. Their results are based on numerical computations and apply to four types of connectors produced by German manufacturers.
Kim and Allard [11] present an experimental and finite element study of three types of steel wire connectors, considering the effect of shape and spacing.
To the authors' knowledge, therefore, this paper is the only one that tries and proposes a practical correlation for calculation of point transmittance associated with rib intersections in LSP.

Average thermal transmittance of a lightened sandwich panel
The heat flow rate q through a wall panel can be usually written as where T is the temperature difference between the internal and the external environments separated by the panel, A is the panel area, and U is the panel average thermal transmittance. In Eq. (1), which defines U, the four panel edges are considered adiabatic.
Although q could be calculated upon numerical solution of the conduction equation for the entire panel, International Standards [2], [3] suggest a more efficient method based on analytical and numerical solutions for a limited number of parts of the panel. Following this approach, q is written as In Eq. (2) A i and U i are area and thermal transmittance of the i-th section of the panel; l j and ψ j are length and linear transmittance of the j-th linear thermal bridge; n k and χ k are number and point transmittance of the kth point thermal bridge. Following ISO 6946 [1, Clause 6.7.2.2], here section denotes a panel part made of thermally homogeneous layers. As clearly shown in Figure 1, a LSP is made of two sections: the solid concrete part, corresponding to ribs (section a); and the three-layer lightened part, made of the concrete wythes and the lightened layer that separates them (section b).
The thermal transmittance of the two sections, U a and U b , is easily calculated in terms of surface resistances, R se and R si , and of thermal resistances of the homogeneous layers. Therefore, the first sum in Eq. (2), i A i U i represents the transmission heat coefficient through the panel as if the sections were thermally insulated one from the other, and the temperature field was 1D within each section.
The other two sums, j l j ψ j and k n k χ k , represent the corrections associated with linear and point thermal bridges, that is with the regions where the temperature field is two dimensional (2D) and 3D.
In LSPs, the temperature field is closely approximated by a 2D field near the interfaces between concrete ribs and lightweight slabs (thick lines in Figure 1), whereas it is fully 3D in a neighborhood of the intersections of ribs (open circles in Figure 1).
The linear thermal transmittances ψ j must be evaluated upon solving the conduction equation on 2D domains as described in [6]. Evaluation of the point thermal transmittances χ k -which is the goal of this study - requires solution of the conduction equation on proper 3D domains which represent panel parts centered around rib intersections. These 3D geometrical models must be identified in accordance with ISO 10211 [3].

Problem Description
In Figure 2, the 3D model used to determine point thermal transmittance values is sketched: it represents the part of a panel near the intersection of two ribs. The L-shaped region formed by the intersecting ribs, represents the solid concrete section a, whereas the three-layered region is the lightened section b.
The 3D geometrical model coincides with the parallelepiped of sides L x = L ax + L bx and L z = L az + L bz and height (panel thickness) d = d 1 + d 2 + d 3 . d 1 and d 3 denote the thickness of the external and internal wythe, respectively, whereas d 2 is the lightweight material thickness.
On the domain boundaries, the following conditions must be satisfied. On the external and internal surfaces ∂ e (y = 0) and ∂ i (y = d) Robin condition is imposed. The heat transfer from surface ∂ e to the external environment at temperature T e is characterized by a heat transfer coefficient α e and a surface resistance R se = 1/α e . Internal environment temperature T i , heat transfer coefficient α i , and surface resistance R si = 1/α i characterize heat transfer to the internal surface ∂ i .
Planes x = 0, x = L x , z = 0, and z = L z are cut-off planes as defined in ISO 10211 [3, Clause 7.2], which coincide with adiabatic surfaces: the union of all adiabatic lateral surfaces will be denoted ∂ a .
Boundaries x = L x and z = L z belong to either an adiabatic lateral surface of the panel (as in domains 1 and 2 in Figure 1) or a symmetry plane of an internal rib (as in domains 2 and 3 in Figure 1): therefore L ax and L az are either the width of a bounding rib or the half-width of an internal rib. Boundaries x = 0 and z = 0 are placed so far from the intersection that the temperature field on them is 2D for all practical purposes. According to ISO 10211, the distances L bx and L bz between the cut-off planes and the point thermal bridge must be larger or equal to max (1 m, 3 d), where d is the total thickness of the panel. Symmetry planes of lightweight slab could also be taken as cut-off plane x = 0 (z = 0, respectively) if L bx (L bz , respectively) is smaller than max (1 m, 3 d).
Denoting q the heat flux through the 3D domain , Eq. (2) simplifies as follows: where A a + A b is equal to the area of surfaces ∂ e and ∂ i . ψ x and ψ z are the linear transmittances associated with the interfaces between concrete and lightweight material that are highlighted in black and gray, respectively, in Figure 2. Upon numerical evaluation of q and ψ x and ψ z (see [6]) the point transmittance χ associated with the rib intersection modeled in Figure 2, can be computed as

Input data
The point transmittance χ as defined by Eq. (4) depends on several parameters: geometrical, L ax , L az , L bx , L bz , d 1 , d 2 , d 3 ; and thermophysical, λ co , λ lw , R se , R si . In this study, R se , R si , L bx , and L bz are fixed, and χ is computed for varying values of the other variables. The surface resistances are set equal to the conventional values prescribed by ISO 6946 [1, Clause 6.8]: R se = 0.04 m 2 K/W and R si = 0.13 m 2 K/W. As observed in [6], linear transmittance ψ associated with a concrete rib tends asymptotically to a constant value as the lightweight slab length increases (L bx and L bz in Figure 2). As a matter of fact the growth is rapid: for length larger than 0.25 m the value of ψ is hardly distinguishable from the asymptotic value. Since in real panels the lightweight slab length and width are rarely smaller than 0.5 m, the asymptotic value only is needed for all practical purposes. A similar behavior is expected to characterize the dependence of χ on L bx and L bz : therefore, following ISO 10211 [3], in all computations we set L bx = L bz = 1 m.
In [6], linear transmittance ψ is computed for several thousands of different combinations of (L a , d 1 , d 2 , d 3 , λ co , λ lw ). Due to the larger computational cost of 3D simulations, χ has been calculated for a subset of the dataset. Values investigated here are summarized in Table 1. In particular, only three values for both concrete and lightweight material conductivity have been The dependence of point transmittance on solid concrete section widths L ax and L az , is similar to the one on L bx and L bz : it attains its asymptotic value for L ax and L az , larger than 0.2 m, approximately. The three most frequent values of rib width or half-width: 0.05, 0.10, and 0.20 m have been studied. In addition, we compute also the asymptotic cases which can be approximated setting widths equal to 1 m.
With respect to Figure 2, for symmetry reasons the value of χ is invariant when L ax and L az commute, thus only combinations L ax ࣘ L az need to be considered.
In current production of LSPs, the concrete wythes have usually the same thickness, therefore analysis has been restricted to the symmetric case d 1 = d 3 . Three values of wythe thickness have been considered: 0.04, 0.06, and 0.08 m. They are the most frequent for panels of total thickness less than or equal to 0.24 m. Thickness of the lightened layer, d 2 , has been varied among: 0.04, 0.06, 0.08, 0.12, and 0.16 m. However, the following constraint on panel total thickness has been imposed: d ࣘ 0.24 m: only 12 combinations of values of d 1 and d 2 satisfy this condition.
One thousand and eighty numerical estimates of χ have been obtained upon varying (L ax , L az , d 1 , d 2 , λ co , λ lw ) as specified above.

Numerical solution
The conduction problem described in the previous section is solved through SEM. SEMs are high accurate methods designed to discretize partial differential equations. Their best performance (in terms of computational efficiency) is achieved when the differential problem is set on Cartesian geometries, exactly as in the problem we are dealing with, since SEMs exploit the tensorial structure of the basic functions. The accuracy of SEM is restricted by the regularity of the data: the thermal conductivity and the prescribed boundary conditions.
Examples of application of SEM in heat and fluid flows can be found in [12], [13]. A brief description of SEM as applied to transmittance computation in LSPs can be found in [6,Sec. 4], whereas for an in-depth description of SEM and of its use to approximate partial differential equations, we refer, e.g., to [14], [15]. Only features and terms necessary to understand the solution strategy adopted have been presented in the study.
With respect to Figure 2, in order to approximate the heat flux q through we compute the temperature T = T (x) inside the panel. It solves a homogeneous elliptic equation with discontinuous thermal conductivity λ such that λ| co = λ co , and λ| lw = λ lw , where the nonoverlapping regions co and lw correspond to the concrete and the lightweight material, respectively.
T is the solution of In SEM, problem (Eq. 5) is reformulated in a weak sense, and it can be proved that it admits a unique weak solution T ∈ H 1 ( ), being H 1 ( ) = {v ∈ L 2 ( ) : ∇v ∈ [L 2 ( )] 2 } the Sobolev space of order 1 (see, e.g., [16]). The low regularity of the solution is a consequence of the discontinuous thermal conductivity. Nevertheless, when restricted to the subregions where the thermal conductivity is constant, the temperature field is more regular, precisely it belongs to the Sobolev space of order 2.
To approximate the temperature field inside the panel the computational domain is partitioned in N nonoverlapping parallelepiped Q k (also named elements) of size h (typically h denotes the diagonal), such that two adjacent elements share a vertex, an edge, or a complete face. Such a partition will be denoted by Q h = N k=1 Q k . We accept that the elements Q k can have different size h k , in such a case we set h = max k h k .
Given a partition Q h of we look for an approximation T h of T that is globally continuous on¯ and locally (that is in each element Q k ) is a polynomial of degree p with respect to each variable. If the surfaces of discontinuity of the thermal conductivity do not cut any element Q k , it can be proved that the discrete solution T h converges to the weak solution T of Eq. (5) when the mesh size h tends to zero and/or the polynomial degree p grows to infinity. More precisely, there exists positive constants c k independent of both h k and p such that where H 2 (Q k ) is the Sobolev space of order 2 [14]. Once the discrete temperature T h is available, the heat flux through surfaces ∂ i and ∂ e can be computed by the following formulas: q i,h and q e,h are the discrete approximation of the heat flux through , q, that is required to calculate the point transmittance χ by Eq. (4). In particular, since both |q i,h − q| and |q e,h − q| behave like T h − T H 1 ( ) when h → 0 and/or p → ∞, we obtain q i,h → q, and q e,h → q when h → 0 and/or p → ∞. The SEM has been implemented in MATLAB.

Discretization strategies
In order to get a good trade-off between accuracy and computational time, one has to choose properly the partition Q h and the polynomial degree p. Since p is the same in every Q k , in view of Eq. (6) the element size h k should suitably be chosen in order to balance T H 2 (Q k ) , the latter being larger next to the surfaces where the thermal conductivity is discontinuous and smaller far from it. For each set of data, and any fixed p, the discrete fluxes (Eq. 7) have been computed for three different meshes: Q h (named coarse), Q h/2 (medium), and Q h/4 (fine). Therefore, the corresponding point transmittances χ h,p , χ h/2,p , and χ h/4,p have been evaluated by using Eq. (4) in which q is replaced by its discrete counterpart q h = (|q i,h | + |q e,h |)/2.
Finally, χ h,p , χ h/2,p , and χ h/4,p are used to better estimate the point transmittance χ through the Richardson extrapolation technique (see, e.g., [8,Eq. 9.6]), that in our case (with data from three different meshes and the parameter h that is halved at each step) reads In view of the convergence estimate of Richardson extrapolation (see, e.g., [8,Eq. 9.35]), there exists a positive constant C, independent of h, but possibly depending on p such that |χ e,p − χ | ≤ C(p)(h/4) 3 . The Richardson extrapolation turns out to be very efficient for the purpose of this study. As a matter of fact, even if the SEM approximation error (Eq. 6) vanishes as h 1 when h → 0 in view of the low regularity of the thermal conductivity, Richardson extrapolation allows to gain third order accuracy with respect to h, i.e., h 3 , by using a set of three meshes of moderate sizes h, h/2, and h/4, instead of a unique mesh with a very small mesh-size, that would require very large computational effort. Table . At left, columns -, CPU-time in seconds needed to compute the temperature by SEM discretization. At right, columns -, the Richardson extrapolation of the point transmittances (in W/K) for different values of p, the relative error with respect to the best computed value χ e, 6 , and the total CPU-times (in s).
In our simulations we have chosen to use polynomial degree p = 4. This choice is motivated by several numerical tests, carried out to measure both the computational effort required to solve the linear system associated with the SEM discretization of the conduction problem (Eq. 5), and the accuracy of the computed point transmittance.
As test case to study discretization error and computational effort, we have chosen the case: L ax = L az = 0.05 m, d 1 = d 3 = 0.06 m, d 2 = 0.12 m, λ co = 2.0 W/mK, λ lw = 0.04 W/mK, and on the associated geometry we have designed the following partitions.
The coarse partition or mesh Q h is obtained by defining 6 × 4 × 6 (= 144) elements with side sizes h x = [0.6, 0.2, 0.1, 0.05, 0.05, | 0.05], h y = [0.06, | 0.06, 0.06, | 0.06], and h z = h x . The symbols | highlight where the thermal conductivity is discontinuous. The ratio between the maximum and the minimum size h k is about 10, with max k h x 0.85 and min k h x 0.09. The medium mesh Q h/2 is obtained by halving (along each direction) any element of Q h , therefore we have 12 × 8 × 12 (= 1,152) elements; while the fine mesh Q h/4 is obtained by halving (along each direction) any element of Q h/2 , here we have 24 × 16 × 24 (= 9,216) elements.
The linear system arising from the SEM discretization of Eq. (5) has dimension close to p 3 N and the matrix is very sparse, therefore it is mandatory to solve it by an iterative method, like, e.g., the Bi-CGStab method [10], preconditioned by an incomplete LU factorization.
The CPU-times (in seconds) needed to compute the temperature field on an Intel i5-3470 4 core, 64 bit, 3.6 GHz, and 8 GB of RAM, are reported in Table 2, (left), for p = 2, . . . , 6 and the three partitions Q h , Q h/2 , and Q h/4 . Least square approximation of the measured values provides CPU-time 10 −7 p 5.6 N 1.7 s. It can be concluded that a large computational effort is required to solve Eq. (5) when either moderate or large p is used.
To measure the accuracy of numerical results, the Richardson extrapolation of the point transmittance χ e,p was computed, for any p = 2, 4, 6, as well as the relative error with respect to χ e,6 (the best estimate), i.e., e p = |χ e,p − χ e,6 | |χ e,6 | The computed point transmittances χ e,p , the errors (9), and the CPU-times needed to estimate χ e,p (i.e., the total CPU-time needed to compute the discrete temperature on all three meshes) are shown in Table 2, (right). It can be concluded that the best compromise, obtained by minimizing both the CPU-time and the error is achieved for p = 4.

Results and discussion
One thousand and eighty numerical estimates of χ have been obtained varying six parameters: The interval spanned by χ over the set of input data ranges between −4.38 × 10 −2 and −0.48 × 10 −2 W/K: all values are negative. There is no evidence that the dependence of χ on some of the variables could be neglected because it is much weaker than others.
With respect to Figure 3, it was observed that χ is a decreasing function of d 1 (see Figure 4 also) and of λ co , whereas it is an increasing function of λ lw : the rate of change does depend on the other variables, but it is always significant. Dependence on d 2 may be considered less stronger, however that is only because χ distributes around a relative minimum centered at d 2 < 0.08 m (see, e.g., Figure 4). In other regions of the input data space investigated here, there is no evidence of the minimum and χ is an increasing function of d 2 (see, e.g., Figure 5).  Finally, with respect to Figure 5 it was observed that χ tends quite fast to an asymptotic value for L ax and L az tending to infinity, so that values for L ax = L az = 0.2 m almost coincide with those for L ax = L az = 1 m. However, when one or both rib widths are equal either to 0.05 or 0.1 m, χ attains a larger value.
In recent years, ANN have been used in heat transfer studies to obtain accurate correlations for either heat transfer coefficient or other thermal quantities, as in [17], [18]. Indeed, at the beginning of this study it was planned to develop an ANN for prediction of χ, as was done for ψ in [6]. However, in [6] to train an ANN able to model correctly the dependence on L a , values of ψ for more than 10 L a between 0.05 and 1 m had to be obtained. In this study, computational cost of tens of (L ax , L az ) pairs was not affordable. Nor it seemed useful to develop an ANN for each pair (L ax , L az ) investigated. A different approach was followed.
According to ISO 14683 [2], the point thermal bridge studied here can be considered as the intersection  of two linear bridges associated with transmittances ψ x (L ax , d 1 , d 2 , λ co , λ lw ) and ψ z (L az , d 1 , d 2 , λ co , λ lw ): variable d 3 is not listed since only the case d 3 = d 1 is being considered. It was thought whether the dependence of χ on the six variables could be captured to some extent by ψ x and ψ z , that is whether χ depend implicitly on (L ax , L az , d 1 , d 2 , λ co , λ lw ) through ψ x and ψ z : For symmetry reason in Eq. (11) ψ x and ψ z must commute, that is g has to depend on commutative functions of ψ x and ψ z , such as ψ x ψ z , ψ x + ψ z , etc.
As a matter of fact, if computed values of χ are plotted versus ψ x ψ z data tend to fall within a smooth narrow region of increasing width for increasing ψ x ψ z . Data dispersion depends on the original variables, but dependence on d 1 seems stronger. After a few trials, the following new variable was obtained: ξ = ψ x ψ z 2d 1 (12) for which dispersion is substantially reduced (see Figure 6). Although factor 2 multiplying d 1 is clearly unnecessary, it was preferred to introduce it because it was believed that for the more general case . Upon fitting data with a power law, the following correlation for point transmittance has been obtained: (13) where all quantities are in SI units. For all practical purposes Eq. (13) supplies a rather good estimate of χ when the linear transmittances associated with the intersecting ribs are known. In Figure 7 the point transmittance estimated by correlation (Eq. 13), χ c , is plotted versus the computed value χ: 97% of estimates fall within ±10% band. Eq. (13) on average neither overpredicts nor underpredicts significantlyχ, since the mean relative deviation, MRD, is equal to 0.09%. Dispersion of predicted values is limited since the standard deviation, SD, is equal to 4.5%. Here, SD is defined in terms of relative deviation RD = (χ c − χ )/χ: As mentioned in the Introduction, the relative contribution of point thermal bridges to the average panel transmittance U (term k n k χ k in Eq. 2) is up to 2%, approximately. Therefore, the error introduced upon use of correlation (Eq. 13) is up to 0.2% of U. This error is of the same order of magnitude as the one introduced upon use of the ANN of [6] for determining linear bridges contribution (term j l j ψ j ).
Due to the small contribution of point thermal bridges in Eq. (2), one may wonder whether it would be worth to perform some 3D numerical simulations to determine them: indeed, ISO 14683 [2] does allow to neglect them. However, if one has already determined the linear transmittances of the panel ψ j to evaluate the second term in Eq. (2), it is straightforward use of correlation (Eq. 13) to determine all point thermal transmittances and to calculate a better estimate of U. Besides, since the point thermal transmittances are always negative, taking them into account prevents panel manufacturers from overestimating the transmittance of their products.

Conclusions
This paper deals with the problem of determining point thermal transmittance associated with rib intersections in LSPs. Together with results presented in [6] it allows accurate calculation − within ±1% − of nominal average thermal transmittance of LSPs according to current International Standards [1]- [3].
To reach this goal, a dataset of point thermal transmittance associated with rib intersections of LSPs has been built through numerical simulations. One thousand and eighty data have been obtained as a function of six parameters: rib widths, thickness of the concrete wythes and of the internal layer, concrete and lightweight material conductivity. The parameters span a range of values typical of current LSPs production. To the authors' knowledge these are the only systematic data of point transmittances in concrete wall panels in literature. In general, ISO 14683 allows to omit point thermal bridge contribution to LSPs transmittance. For the input data investigated here it is shown that this contribution is always negative: one stays on the safe side neglecting it when evaluating thermal performance of LSPs. Besides, point transmittance values are rather small, their order of magnitude being 10 −2 W/K.
Accurate calculation of such a small quantity through numerical solution of heat conduction equation in a 3D domain has been tricky and required a solution strategy based on Richardson extrapolation. A conformal quadrilateral SEM implemented in MATLAB has been used to solve the 3D conduction problem. To contain the computational effort required to solve such a problem, the unknown temperature field has been approximated using low-order polynomials, p = 4, with a consequent significant loss of accuracy. To bypass this problem and improve accuracy, each problem has been solved on three partitions of different degrees of refinement, starting from a very coarse partition: results have been improved through an iterated application of Richardson extrapolation. This procedure allowed to determine point transmittance values with a relative error which is about 10 −4 .
Finally, an explicit correlation is proposed for prediction of point transmittance. Although data show a significant dependence of χ on each of the six variables mentioned above, a simple power law correlation was found which allowed to calculate χ as a function of a single variable, ξ = ψ x ψ z √ 2d 1 . The correlation has standard deviation equal to 4.5% and predicts more than 97% of computed data within ±10%. It represents a good practical tool, easily implemented in a spreadsheet or in an in-house code for calculation of U.