3D thermal modelling of complex facades

1. Introduction

Today’s building enclosures face ever greater demands regarding their thermal performance. In cold climates, the required total U-value of the façade is often very low. With thicker insulation layers, the relative impact of thermal bridges increases.
Thermal bridging arises whenever there is a discontinuity in the envelope’s insulation layer. Repeating linear thermal bridges are typically considered an integral part of the building component and have mostly been considered in the U-value calculation and assessment per component required by building regulations (e.g. timber studs in timber frame walls).
Non-repeating linear thermal bridges, typically arising at junctions of different building components, and point thermal bridges, due to single penetrations of the insulation layer, have only more recently been recognised in building regulations, gradually having their influence on the building’s heat loss and energy performance assessment incorporated. In Europe, these regulatory efforts have generally started in the first decade of this century. For example, Belgian building regulations require an assessment of all thermal bridges since 2010. The regulatory efforts in the USA are more recent. ASHRAE 90.1 included prescriptive and performance requirements for thermal bridges in version 2022. The IECC will include similar requirements in its 2024 update.
Typically, building regulations are elaborated in the most detail for linear thermal bridges. The UK’s SAP v10.2 for example offers a set of 44 types of junction detail for which a default linear thermal transmittance value is proposed. For point thermal bridges, the guidance is limited to the notion that they should be taken into account when they are ‘significant’.
Point thermal bridges can have a significant impact on the envelope’s total heat loss. To illustrate this, consider the example of Figure 1. This example is of the façade of an office building designed and constructed in Belgium, where an additional cladding system in front of the curtain wall is supported by anchors attached to the concrete floor. The U-value of the curtain wall without this additional cladding system was calculated according to EN ISO 13789 to equal 0.59 W/m²K. The point thermal transmittance or χ-value of each anchor was however calculated to be 0.92 W/K, which meant that the corrected U-value to include the cladding system rises to 0.77 W/m²K.

In parallel with the regulatory evolution to assess thermal bridging, BIM (Building Information Modelling) and the proliferation of easily accessible modelling tools such as Rhino have made 3D modelling of building enclosures pervasive. The upside of this trend is that no additional engineering time is required to (re)build 2D or 3D models specifically for thermal analysis. Of course, this means that thermal analysis software has to be able to import these 2D or 3D geometric models. In addition, the calculation has to be able to cope with suboptimal models, typically larger than strictly necessary and/or containing thermally insignificant small details.

This paper will start by providing an overview and comparison of the inclusion of thermal bridges in Building Regulations in the UK, Belgium and the USA and the different paths to compliance offered in each. Next, SOLIDO is presented as a tool to calculate 2D and 3D heat transfer in envelopes, with special focus on the current and improved implementation of the meshing algorithm, specifically designed to allow importing suboptimal 3D geometries. Finally, a case study is discussed.

2. Comparison of thermal bridges in building regulations

Repeating thermal bridges are typically considered to be an inherent part of the building component and thus expected to be included in the component’s thermal transmittance or U-value. For example, EN ISO 6946, widely mandated in European building regulations, offers a simplified calculation (averaging the parallel path and the series path method) for the thermal resistance of non-homogeneous layers such as timber studs with insulation in between. Building regulations setting upper limits to a component’s U-value (including thus its repeating thermal bridges) are standard practice in Europe. In the USA, this is part of the prescriptive compliance path offered by ASHRAE 90.1 and the IECC. The alternative to the prescriptive compliance path in both ASHRAE 90.1 and the IECC is the trade-off compliance path, wherein the energy performance of the façade design and/or building design as a whole is compared to that of a base design.

Below, the different assessment methods and compliance paths for non-repeating thermal bridges set out in the building regulations in the UK, Belgium and the USA are discussed. The terminology for the thermal transmittance sometimes differs (ψ-value or psi-factor for linear, χ-value or chi-factor for point), we’ll use each source’s terminology.

2.1. SAP 10.2 (2023) and SBEM (2022)

In the UK, the government has issued the Standard Assessment Procedure (SAP) for assessing the energy performance of dwellings and has approved the Simplified Building Energy Model (SBEM) methodology for assessing the energy performance of non-dwellings. SAP and SBEM calculations are a requirement of Part L of the Building Regulations, and are required for all new-build buildings.

In SAP, (non-repeating) linear thermal bridges are taken into account when calculating the building’s heat losses through the envelope with one of three possibilities:

• Use of a global factor (‘y-value’): an average additional heat loss of 0.20 W/m²K caused by thermal bridges is taken into account for the total area of external elements. This factor has increased from 0.15 W/m²K in SAP 2012, in effect incentivizing to move away from this method.
• Use of default linear thermal transmittance or ψ-values, defined for a set of 44 junction types (Figure 2).
• Use of user-supplied ψ-values: these can come from a suitable database or be calculated according to BR 497.
  

Point thermal bridges should be taken into account when they are significant, such as metal components bridging insulation layers (e.g. balcony supports), in which case their point thermal transmittance or χ-value should be calculated according to EN ISO 10211 or taken from tables or catalogues prepared in accordance with EN ISO 14683, as is outlined in BR 443.

BR 497 outlines the conventions for calculating the linear thermal transmittance of thermal bridges, summarizing and contextualizing the procedures of standards EN ISO 10211, EN ISO 6946, EN ISO 13370 and EN ISO 10077.

In SBEM, (non-repeating) linear thermal bridges are taken into account either by using the default ψ-values (Figure 3) or by entering a user-supplied ψ-value. Point thermal bridges are assumed to be taken into account in the U-value of building components and are not treated separately.

2.2. Belgium: EPBD regulations (2022)

The Building Energy Performance Regulations in Belgium require to take the additional heat losses due to thermal bridging into account when calculation the energy performance, using one of two options:

• User supplied linear and point thermal transmittances, either calculated according to a guideline document or from a table of default values (see Table 1). The guideline document outlines the conventions for calculating thermal transmittance of thermal bridges, summarizing and contextualizing the procedures of standards EN ISO 10211 and EN ISO 10077.
• Use of a global additional heat loss (in W/m²K) due to thermal bridging, the value of which depends on if the thermal bridges are compliant. Two methods to assess compliance are offered:
  • Following the prescriptive rules per building detail, e.g. how to guarantee the insulation layer continuity.
  • Having the calculated thermal transmittance not exceed a tabulated limit value (Table 2).

Table 1: Default thermal transmittances – based on external dimensions (Flemish EPBD, 2022)

Table 2: Limit thermal transmittances - based on external dimensions (Flemish EPBD, 2022)

2.3. ASHRAE 90.1 (2022) and Commercial IECC (2024 draft)

ASHRAE standard 90.1 underwent a major change in its 2022 version, including prescriptive and performance requirements to account for and minimize thermal bridges.

In the prescriptive compliance path for the envelope, 2 ways to comply are offered:

• Either follow prescriptive rules set out for each thermal bridge type (see Table 3) separately, e.g. how to guarantee the insulation layer continuity;
• Or have a psi-factor or chi-factor not exceeding the tabulated default or mitigated values (see Table 3 and Table 4) The psi-factors and chi-factors can be determined by testing or performing numerical calculations according to ISO 10211, or based on ISO 14683, which offers a set of default values, or the option to use another catalogue with values based on calculations with ISO 10211.
  

Table 3: Default (mitigated) and unmitigated linear thermal transmittances – based on external dimensions (ASHRAE 90.1, 2022)

Table 4: Default (mitigated) and unmitigated point thermal transmittances for “other elements and assembly intersections” – based on external dimensions (ASHRAE 90.1, 2022)

Default and unmitigated chi-factors for point thermal bridges of “other elements and building assembly intersections” or given in Table 4. Note however that point thermal bridges need only be accounted for if their cross-sectional area is larger than 3 in.² (0.0019 m²) for carbon steel, 9 in.² (0.0058 m²) for stainless steel and 65 in.² (0.042 m²) for concrete and masonry.

In the trade-off compliance path for the enveloped, either the determined psi-factor or chi-factor is to be considered, or the tabulated values of Table 3 and Table 4 (default values if the prescriptive rules are met, unmitigated values otherwise).

Next to ASHRAE 90.1, the International Energy Conservation Codes (IECC) is an energy code adopted by many states in the USA.

In the draft version of the Commercial IECC 2024, guidance on thermal bridging in exterior walls is adopted for the first time. This new language intends to reduce thermal bridges, which will mitigate heat loss through the building’s envelope.

In a very similar process to ASHRAE 90.1, a prescriptive compliance path is offered, with prescriptive guidelines for each type of thermal bridge (see Table 5 and Table 6). Alternatively, the trade-off compliance path is offered, requiring to take the influence of thermal bridges on the total envelope’s heat loss into account. Either their determined psi-factor or chi-factor is to be considered, or the tabulated values of Table 5 and Table 6, selected depending on if the thermal bridge design is compliant with the prescriptive rules.

Table 5: Compliant and non-compliant linear thermal transmittances – based on external dimensions (Commercial IECC, 2024)

Table 6: Compliant and non-compliant point thermal transmittances – based on external dimensions (Commercial IECC, 2024)

2.4. Comparing default values

A quantitative comparison can be made of the ψ-values of mitigated or compliant linear thermal bridges, since the methodology to calculate ψ-values is universal. This comparison unfortunately excludes the UK values, as the definition of a ψ-value there is based on using internal dimensions, which makes it impossible to compare actual values.

In Figure 4, the limit psi-values for compliance from ASHRAE 90.1 (2022), the Commercial IECC 2024 draft and the Belgian EPBD building regulations are compared for a set of building junctions. While one should of course be cautious to compare values one by one given the differences in building practice, regulatory framework and climates between countries, it is nevertheless clear that the limit values in Belgium are much stricter than those in the USA. The default values of the draft IECC are in line with those in ASHRAE 90.1. It is worth noting that typical building practice in Belgium would be classified as ‘mass (interior)’.

What also stands out is the large difference in default ψ-values between lintels on the hand and sills and jambs on the other hand in the UK building regulations, which doesn’t occur in the USA or Belgian building regulations. The specific treatment of an intermediate floor balcony in contact with vertical fenestration, assigned a very large unmitigated psi-factor, in ASHRAE 90.1 is also remarkable.

In terms of χ-values, the unmitigated values of ASHRAE 90.1 and the IECC 20214 draft in the USA can be quantitively compared to the default values in the Belgian EPBD regulations:

• The χ-value of ASHRAE 90.1 for steel framed buildings (0.91 W/K) is in line with that of the IECC 2024 draft for carbon steel columns and beams (1.05 W/K), which translate to metal elements with a circumscribed square with side 0.19 m and 0.22 m, respectively.
• The χ-value of ASHRAE 90.1 for construction walls with mass (0.48 W/K) is in line with that of the IECC 2024 draft for concrete columns (0.53 W/K), which translate to non-metal elements with a cross-section of 0.10 m² and 0.11 m², respectively.

3. SOLIDO: 3D thermal analysis tool

Physibel produces and distributes software tools aimed specifically at complex building façade (hygro-) thermal analysis. Two tools are for thermal analysis of steady-state heat transfer in three-dimensional (3D) objects: TRISCO and SOLIDO. TRISCO is limited to rectangular objects and includes a built-in 3D geometric modeler, but can also extrude a 3D geometry based on a (2D) DXF file import. SOLIDO can be considered an extension of TRISCO, in that it allows thermal analysis of 3D objects of any shape. SOLIDO also carries the option to import 3D geometries from STL files. Both TRISCO and SOLIDO automatically mesh a 3D geometric object and set up a system of linear equations based on the finite volume method, assuming a linear temperature profile between the nodes. Both TRISCO and SOLIDO are fully validated according to EN ISO 10211.

3.1. Current meshing in TRISCO and SOLIDO

Currently, both TRISCO and SOLIDO employ a structured grid made up entirely of hexahedrons, more precisely rectangular prisms or ‘boxes’, with faces parallel to the coordinate planes. This means that each system node has exactly 6 neighbors (except at the model edges). The mesh size of this grid does not need to be identical everywhere, which allows for the gradual refinement of the grid towards model zones where non-linear heat transfer is expected (see Figure 5 left).

In TRISCO, this rectangular calculation grid is directly defined by the user in the built-in geometry editor. The procedure in SOLIDO is more complex: first a triangulation of the model’s surface is made (based on imported STL-files if applicable), then this triangulated object is rasterized (creating a rectangular calculation grid, similar to TRISCO) and finally the surface grid nodes are fitted by projection to the triangulated object resulting in the calculation grid.

Because of the use of the structured grid for setting up the system of equations, this system is reliably and accurately solved in TRISCO and SOLIDO.

In practice, geometric façade models often contain a single small element with a complex, detailed geometry – typically the reason for calculating the (point) thermal bridge in the first place, see the case study further in this paper for an example. Locally, a very fine calculation grid is necessary to capture the strongly non-linear heat transfer. However, because the calculation grid is a structured grid, this local refinement ‘fans out’ to zones of the model where a calculation grid with such a fine resolution is superfluous from a physical standpoint (see Figure 5). This leads to a large number of additional system nodes, and thus to longer calculation times and higher use of the computer’s working memory.

3.2. New meshing algorithm for SOLIDO

An internal research project at Physibel had been launched with the goal of developing a new meshing algorithm. The goal was a meshing algorithm that avoids zones of superfluous refinement in the calculation grid, without jeopardizing the solving reliability and efficiency that comes with having a structured grid. The result is a meshing algorithm using tetrahedrons. In this new meshing algorithm, the entire model is first rasterized with a large mesh size. Subsequentially, the raster is gradually refined in locations where non-linear heat transfer is expected, i.e. at the boundary between materials with different thermal conductivities and around corners and edges. The resulting raster of rectangular prisms then inevitably contains so-called ‘hanging nodes’, meaning the mesh is non-conforming and not – necessarily – solvable. This is remedied by filling the raster with tetrahedrons, where care must be taken to avoid computationally problematic tetrahedron shapes (so-called ‘slivers’, containing very small or very small dihedral angles). In the following section, the performance of this new meshing algorithm is compared to the current approach in TRISCO and SOLIDO.

4. Case study

4.1. Input data

To compare the performance of the different meshing techniques, we’ll consider the case of a balcony with a thermally broken slab. The external wall is a reinforced concrete wall (0.2 m thickness, λ = 2.5 W /mK) with an exterior insulation (0.1 m thickness, λ = 0.03 W/mK) protected by a cement sand render (0.0125 m thickness, λ = 1.0 W/mK). The wall’s U-value is thus 0.278 W/m²K. The reinforced concrete floor slab (λ = 2.5 W/mK) has a thickness of 0.2 m. It interrupts the external wall’s insulation layer, but is thermally broken with an insulation element (λ = 0.035 W/mK) of 0.07 m thickness.

Structurally, the balcony slab is connected to floor slab by stainless steel (λ = 17.0 W/mK) connectors (see Figure 7). The straight connectors have a diameter of 7 mm and a length 0.32 m, while the bended connectors have a diameter of 6 mm and a total length of 0.34 m. A pair of each is placed each 50 mm. An STL file of these connectors is made available by the manufacturer. This file contains the triangulated 3D surface description of the connectors (see Figure 6).

The goal is to calculate the additional heat loss due to balcony. We’ll express this result both as a ‘total’ linear thermal transmittance ψ_total (combining the influence of the balcony, thermal break and connectors) and as a combination of a linear thermal transmittance ψ_slab and an additional point thermal transmittance χ, purely due to the connectors.

4.2. Thermal models

This case is clearly a 3D heat transfer problem. Given the repeating nature of the connectors, all constructed models will have a length of only 50 mm to contain one pair of connectors. This assumes that pairs of connectors do not influence each other thermally, which is a conservative assumption from the point of view of heat loss assessment. In the other 2 dimensions, the models are large enough to ensure 1D heat transfer at the model’s edges, so that we are sure the models captures the non-linear heat transfer of this building junction fully. 1 m of wall, floor, and balcony slab is modelled in each direction away from the studied junction. First, a model is created without the connectors to perform a preliminary calculation to assess the influence of the discontinuity in the wall due to the balcony slab and thermal break only (ψ_slab). Since this is a perfectly rectangular geometry, the model is created in TRISCO (model 0). This model is meshed with a minimum mesh size of 1 mm and the mesh size is allowed to increase away from non-linear heat transfer locations with a ratio of 1.5.

Next, 3 thermal models are constructed of the entire set-up:

• Model 1 is constructed in TRISCO, meaning that the geometry of the connectors needs to be simplified to be able to be represented in the rectangular grid. The connectors are modelled as 2 box shaped rods with a cross-section equal to that of the actual connectors, to ensure that the amount of conductive material bridging the insulation layer is identical. A calculation grid is again made with a minimum mesh size of 1 mm and the mesh size is allowed to increase away from non-linear heat transfer locations with a ratio of 1.5.
• Model 2 is constructed in SOLIDO by importing the STL file of the connectors containing the actual triangulated geometry (see Figure 10). This is the current SOLIDO, so a hexahedral calculation grid is constructed (again with a minimum mesh size of 1 mm and the mesh size is allowed to increase away from non-linear heat transfer locations with a ratio of 1.5).
• Model 3 is again constructed in SOLIDO by importing the STL file of the connectors containing the actual triangulated geometry. This time, the new meshing algorithm is used to construct the calculation grid (see Figure 11).

In every model, an ambient temperature of 20°C and a surface heat transfer coefficient of 7.7 W/m²K are imposed on the inside. On the outside, an ambient temperature of 0°C and a surface heat transfer coefficient of 25 W/m²K are imposed.

4.3. Results and discussion

A comparison of all calculation results is given in Table 7.

Table 7: Comparison of calculation results (on laptop with 16 GB RAM)

First, let’s review the results of model 0. The ψ_slab-value can be deduced from equation:

Φ = U_wall∙L∙H + ψ_slab∙L

With a total heat loss Φ of 0.6498 W, U_wall equal to 0.278 W/m²K, the model height H equal to 2.2 m and the model length equal to 0.05 m, the linear thermal transmittance ψ_slab equals 0.04 W/mK.

Next, we compare models 1 to 3. All are different ways to model the same physical reality. Comparing the calculated resulting total heat flows of all 3 models, we see that the difference between them is smaller than 1%. All 3 are thus valid representations of reality producing accurate results. Model 1 is much simpler and smaller than models 2 and 3. Constructing it did however necessitate engineering time and experience to simplify reality. Models 2 and 3 did not need this additional engineering time, but allowed to simply import the STL geometry of the connectors. The new meshing algorithm in model 3 is clearly able to reduce the number of nodes of the thermal model (a decrease of 55% for this case study) and thus the calculation time needed, without jeopardizing the accuracy.

Finally, the ψ_total-value can be deduced from the results of any of models 1-3:

Φ = U_wall∙L∙H + ψ_total∙L = U_wall∙L∙H + ψ_slab∙L + χ

The result is a total linear transmittance ψ_total of the balcony slab – including connectors – of 0.21 W/mK.

Alternatively, a point thermal transmittance χ of only the connectors can be deduced from comparing any of models 1-3 with model 0. The additional heat loss from the connectors equals 0.009 W/K.

5. Conclusions

This article presented first a comparison of the integration of thermal bridging in building regulations between the UK, Belgium and the USA. Different ways to comply are offered in different countries. The default values of thermal bridges due to building junctions as currently proposed in the European building regulations are much stricter than those in the USA – possibly because the regulatory efforts to include them in Europe are not as recent.

Secondly, a new meshing algorithm implemented in the SOLIDO software is presented. This algorithm is able to mesh imported 3D geometries more efficiently than the current meshing algorithm of SOLIDO without losing calculation accuracy and efficacy. This is a next step in enabling to assess the thermal performance of complex façade without spending excessive amounts of engineering or calculation time.

[1] ASHRAE Standard 90.1-2022—Energy Standard for Sites and Buildings Except Low-Rise Residential Buildings

[2] 2024 International Energy Conservation Code (IECC) - Commercial Provisions, 2022 draft version

[3] The Government’s Standard Assessment Procedure (SAP) for Energy Rating of Dwellings v10.2, 2022

[4] SBEM - Simplified Building Energy Model, BRE, 2022

[5] EN ISO 13789 : Thermal performance of buildings - Transmission and ventilation heat transfer coefficients - Calculation method (ISO 13789:2017).

[6] EN ISO 6946 : Building components and building elements - Thermal resistance and thermal transmittance - Calculation methods (ISO 6946:2017).

[7] EN ISO 14683 : Thermal bridges in building construction – Linear thermal transmittance – Simplified methods and default values (ISO 14683:2017).

[8] EN ISO 10211 : Thermal bridges in building construction - Heat flows and surface temperatures - Detailed calculations (ISO 10211:2017).

[9] EN ISO 13370 : Thermal performance of buildings - Heat transfer via the ground - Calculation methods (ISO 13370:2017).

[10] EN ISO 10077-2 : Thermal performance of windows, doors and shutters - Calculation of thermal transmittance - Part 2: Numerical method for frames (ISO 10077-2:2017).

[11] BR 443 – Conventions for U-value calculations, BRE, 2019

[12] BR 497 – Conventions for calculating linear thermal transmittance and temperature factors, second edition, BRE, 2016

[13] Energy Performance and Interior Climate, EPB Vlaanderen, 2022 (in Dutch)

[14] Building Envelope Thermal Bridging Guide v1.6, BC Hydro, 2021

[15] Janssens et al. Development of limits for the linear thermal transmittance of thermal bridges in buildings, Proceedings – Thermal performance for the exterior envelopes of whole buildings X, Clearwater Beach, Florida, USA, 2007.

[16] Shewchuk, Jonathan Richard. Unstructured mesh generation. Combinatorial Scientific Computing, 2012, 12.257: 2.

[17] Nico Pietroni, Marcel Campen, Alla Sheffer, Gianmarco Cherchi, David Bommes, Xifeng Gao, Riccardo Scateni, Franck Ledoux, Jean-Francois Remacle, and Marco Livesu. 2022. Hex-Mesh Generation and Processing: a Survey. https://arxiv.org/abs/2202.12670