Case Studies in Thermal Engineering 25 (2021) 100936
Contents lists available at ScienceDirect
Case Studies in Thermal Engineering
journal homepage: http://www.elsevier.com/locate/csite
Detailed and fast calculation of wall surface temperatures near
thermal bridge area
Jae-Sol Choi a, Chang-Min Kim b, Hyang-In Jang b, Eui-Jong Kim a, *
a
b
Department of Architectural Engineering, Inha University, Incheon, 22212, South Korea
Institute of Green Building and New Technology, Mirae Environment Plan, Seoul, 01905, South Korea
A R T I C L E I N F O
A B S T R A C T
Keywords:
Numerical model
State-space model
Reduction technique
FEM
The popularity of digital-twin technology has increased the demand for a fast and accurate model
for prompt analysis. This work proposes a low-order but accurate numerical model for the
thermal analysis of wall-window joint surfaces. A simple method to develop such a model is
presented. An ABAQUS-based extraction method was developed to quasi-automatically define a
state-space model for the target cases used, which may reduce the amount of elaborate pro­
gramming work required. Then, an order reduction technique was applied to the state-space
model. The results showed that the state-space model obtained from ABAQUS describes almost
the same thermal responses as the reference ABAQUS simulation model. After reduction, the
proposed model with an order of 10, equivalent to 10 equations, sufficiently described the dy­
namics of temperature variations, with an error of less than 1%. The model conversion to statespace formulism and reduction technique significantly decreased the CPU time by more than
30,000 times (from 79.5 s to 0.002 s).
1. Introduction
The development of digital-twin technology, which depicts real phenomena through virtual simulations, has increased the demand
for fast and accurate models [1,2]. This demand is particularly evident for large and complex objects, such as building elements
consisting of various materials and exposed to dynamic fluctuations in boundary temperature [3,4]. For buildings, detailed
three-dimensional heat transfer analyses are required for the surface area surrounding joints of building elements to quantify the heat
loss through the area and to evaluate surface condensation. Such analyses are often conducted using commercial software based on the
finite element method (FEM). This method has the advantage of high accuracy but requires considerable computational resources. This
limitation conflicts with the perspective of existing digital-twin technology, which requires prompt analysis. Moreover, it is better to
extract information from a numerical model with a simple formula for easy implementation in the calculation platform.
Several studies have used commercial software to conduct the thermal analysis of a wall component [5–8]. Another method is to
directly program the calculation domain with a matrix formula. For instance, Ascione et al. [9] compared their numerical model with
in-situ measurements to verify the results of numerical thermal bridge analysis. The numerical model used in their study was a
state-space (SS) model to evaluate the corner flux and temperatures. However, this process requires elaborate programming.
The aim of this study can be summarized as follows. First, an attempt is made to quasi-automatically transform a spatially
* Corresponding author.
E-mail addresses: sol1213@inha.edu (J.-S. Choi), kcm@mrplan.co.kr (C.-M. Kim), hijang@mrplan.co.kr (H.-I. Jang), ejkim@inha.ac.kr
(E.-J. Kim).
https://doi.org/10.1016/j.csite.2021.100936
Received 24 August 2020; Received in revised form 17 December 2020; Accepted 9 March 2021
Available online 13 March 2021
2214-157X/© 2021 The Authors.
Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Case Studies in Thermal Engineering 25 (2021) 100936
J.-S. Choi et al.
Fig. 1. Window-wall mesh example: (a) concrete wall, (b) window frame, and (c) glass.
Table 1
Material properties.
Materials
Conductivity (W/m∙◦ C)
Density (kg/m3)
Specific heat (J/kg∙◦ C)
Concrete wall (a)
Window frame (b)
Window glass (c)
1.8
0.16
1.0
2,400
500
2,500
1,000
1,600
840
Fig. 2. Modification of an input file to extract matrices in ABAQUS.
discretized domain into a simple matrix formula, i.e., the SS model, which includes the material properties, shape information, and
boundaries. This may facilitate the generation of a numerical model for structures with highly complex shapes or properties. Second, a
fast calculation scheme is applied to the SS model to enable its application in various engineering processes at the measurement site.
The methodology can be divided into three parts. The first part consists of three-dimensional modeling using an FEM commercial
program. The second part is the extraction of matrices from the FEM software and definition of the SS model. The third part deals with
the order reduction of the SS model for fast calculation.
2. Matrix information extraction from FEM model
2.1. FEM modeling with ABAQUS
In this study, ABAQUS [10] was used to define an FEM model that describes a wall element with a window. To the best of the
authors’ knowledge, ABAQUS is the only commercial program that explicitly allows matrix extraction. A three-dimensional numerical
model was created in ABAQUS, and the properties of the materials and corresponding boundary conditions were set under adequate
mesh sizes. In the analysis phase, a single time-step was simulated to generate the required calculation matrices. Fig. 1 shows the
component meshes defined in ABAQUS, and the window-wall model consisting of three parts: (a) wall, (b) window frame, and (c)
window glass. In this study, only the thermal properties of the window pane were considered, as the focus is on the effectiveness of the
2
Case Studies in Thermal Engineering 25 (2021) 100936
J.-S. Choi et al.
Fig. 3. Example of extracted matrix files (damping, stiffness, and load from the left).
proposed methods and thermal analysis of the corner area. Table 1 lists the conductivity, density, and specific heat of the materials; the
boundary conditions for the indoor and outdoor sides are also given. The boundaries were set using the air temperatures and corre­
sponding convective heat transfer coefficients. These coefficients are the assumed constants of 5 W/m2 ◦ C and 20 W/m2 ◦ C for the
indoor and outdoor sides, respectively. The indoor and outdoor temperatures periodically changed within a 24 h cycle.
2.2. Matrix extraction and modification
Manually setting the calculation matrices is a difficult task. A mesh generator describes the shape of the system, and proper thermal
properties are allocated for each area with respect to the mesh numbering protocols. Then, the governing equation must be applied to
each calculation node formed by the mesh. An easy-to-use user interface in ABAQUS can define all the processes in a straightforward
manner, and a simple modification of the input file generated after the procedures is required to extract the matrices. The input file can
be generated by performing a simulation run under the initial conditions. After entering the text command in the input file as shown in
Fig. 2, a simulation with the modified input file can write three matrices: damping (D), stiffness (K), and load (L), on separate text files,
as shown in Fig. 3. These files extracted from ABAQUS contain whole nonzero matrix terms with the value location. In the damping
matrix, the first line indicates the term value of 5.00e+02 for the first column and row of the matrix.
The matrices are defined by the following spatially discretized heat transfer equation with continuous time:
∫
∫
∫
∂N N
∂T
N N ρU̇dV = −
(1)
· k · dV + N N qdS
∂x
∂x
V
V
S
where T is the temperature field, NN are the finite element interpolation functions, U̇ is the time derivative of the internal energy, k is
the conductivity, and S is the surface on which the heat flux per unit area q is either directly prescribed or specified by film and ra­
diation conditions.
3. Order reduction of the state-space model
3.1. State-space model
The damping, stiffness, and load matrices directly correspond to the matrices of the SS formalism, as given in Eq. (2).
(2)
CṪ(t) = AT(t) + BU(t)
3
Case Studies in Thermal Engineering 25 (2021) 100936
J.-S. Choi et al.
Fig. 4. Target surface area for model outputs.
T(t) is the state vector (dimension: n × 1), which is the temperature node, and represents all the temperature nodes defined under
spatial discretization. Matrix C is the same as the damping matrix D (dimension: n × 1) and is related to the thermal inertia repre­
senting the capacitance of the structure. Matrix A (dimension: n × n) is a symmetric matrix and is equal to the stiffness matrix K, but
with a negative sign (A = − K). It explains the relationship between the nodes represented by the state vector T(t). U(t) (dimension: p
× 1) is the input vector, which includes the boundary conditions (i.e., indoor and outdoor temperatures). Matrix B represents the
physical reaction between the boundary condition and system (dimension: n, p). The load matrix L is the same as the product of B and
U. Therefore, matrix B must be reshaped from L, dividing the terms of L by the initial boundary condition values.
It is necessary to define an output vector to measure the time-variable temperatures of a specific surface or points where the thermal
bridge occurs, such as joints of different materials. Here, Y in Eq. (3) is the output vector, which represents the mean temperatures of a
specific area. Matrix J is set by the number of nodes in the target area, which can be obtained by an additional flux boundary setting. D
is the zero matrix, where only mean temperatures are measured.
(3)
Y(t) = JT(t) + DU(t)
Fig. 4 shows three target surfaces whose mean temperatures were measured. The first target surface was set in the middle of the
wall but near the window. The second and third target surfaces were located at the edges that encountered distinct materials: glass and
frame for surface 2, and frame and wall for surface 3. These areas may be affected by thermal bridge effects. Thus, Y contains these
values ({Ts1 , Ts2 , Ts3 } ∈ Y). Most of the terms in J are zeros except the location related to the surface nodes. As explained above, the
surfaces were set as additional boundaries to identify the number of related nodes and to define J.
3.2. Order reduction with Moore’s method
Model order reduction methods were developed using the simple truncation method [11]. A number of studies have been con­
ducted on building component applications [12–14], and Moore’s method [15] is regarded as one of the most efficient and accurate
methods. Moore’s method is also called the balanced truncation method because it is based on a balancing approach between the
controllability and observability of state variables [15,16]. This means that it transforms state variables into controllable and
observable ones simultaneously and with similar degrees and eliminates uncontrollable or unobservable variables [17,18].
In the method, the controllable Gramian matrix Wc and observability Gramian matrix Wo are used, and they can be defined as:
∫∞
∫∞
At
Wc =
T AT t
e BB e
0
(4)
T
eA t CT CeAt dt
dt, Wo =
0
4
Case Studies in Thermal Engineering 25 (2021) 100936
J.-S. Choi et al.
Fig. 5. Mesh independence test results.
Table 2
Boundary conditions.
Boundaries
Value
Convective heat transfer coefficient (W/m2∙◦ C)
Temperature (◦ C)
Indoor
Outdoor
Indoor
Outdoor
5
20
20 + sin(2πt/24h)
10 + 5sin(2πt/24h)
Initial temperature of material (◦ C)
0
Using these matrices, the initial system (Eqs. (2) and (3)) can be transformed into a new basis X that balances both controllability
and observability. A linear transformation of T can be achieved by T=MX, where M is the matrix of eigen vectors of the balanced
Gramian matrices:
(5)
M− 1 Wc M− T = MT Wo M
After the basis change with M, Eqs. (2) and (3) becomes the following equation:
⎧
− 1 − 1
⎪
Ẋ(t) = ⏟̅̅̅̅̅̅̅̅⏞⏞̅̅̅̅̅̅̅̅⏟
M− 1 C− 1 AMX(t) + M
C BU(t)
⎪
⏟̅̅̅̅̅̅⏞⏞̅̅̅̅̅̅⏟
⎨
Ω
Π
Y(t) = ⏟⏞⏞⏟
JM X(t) + DU(t)
⎪
⎪
⎩
(6)
Н
Then, the reduced model with a desired order (m) can be directly obtained by eliminating so-called uncontrollable or unobservable
variables as:
⎧
)
(
)
(
⎪
⎨ Ẋm (t) = Ωm,m − Ωm,n− m Ω−n−1m,n− m Ωn− m,m X(t) + Πm − Ωm,n− m Ω−n−1m,n− m Πn− m U(t)
(
)
(
)
(7)
⎪
− 1
̃ = Нm − Нn− m Ω− 1
⎩ Y(t)
n− m,n− m Ωn− m,m X(t) + D − Нn− m Ωn− m,n− m Πn− m U(t)
where, the vectors and matrices with subscripts represent truncated ones, and the subscripts indicate the dimension and term infor­
mation. For instance, the dimension of Ωn− m,m is (n-m) × m, and the terms are identical with the rows of m+1 to n and the columns of 1
to m of the original matrix Ω.
In this study, the balred MATLAB function was used, which is easily applied to the SS model, as shown in Eqs. (2) and (3). When m is
set in the function option, balancing and eliminating processes are automatically achieved with the function. Generally, the larger the
final order, the more accurate the output values. Theoretically, when the order is set to be the same as that of the SS model (m = n), the
results of the SS model and the reduced model (RM) obtained by the balred function are the same.
4. Results
4.1. Mesh independence
A mesh independence test was conducted. Five different mesh sizes with 16,899, 3,270, 1,566, 350, and 232 nodes (n) were tested.
Fig. 5 shows the test results obtained under the boundary conditions given in Table 2 for a period of 192 h. These results were obtained
from surface 1 of Fig. 3 because the temperature was the most sensitive and pattern variations were the highest. The figure shows the
node temperatures and the error compared with the case of the largest nodes (ref) as given in Eq. (8). The first two models (n = 3,270
and 1,566) had mean errors of 0.023◦ C and 0.024◦ C, corresponding to 0.27% and 0.28%, respectively. Therefore, the mesh model with
3,270 nodes was selected as the reference simulation mesh model in this study.
⃒
⃒
⃒Tref − Ttest ⃒
Error =
× 100(% )
(8)
Tref
5
Case Studies in Thermal Engineering 25 (2021) 100936
J.-S. Choi et al.
Fig. 6. Mean surface temperature comparison between the ABAQUS model and CM.
Fig. 7. Mean surface temperatures of target surface 1, 2, 3 with different RMs.
4.2. Model validation
The SS version of the reference model of 3,270 nodes is called the complete model (CM). The CM must be validated via comparison
with the initial ABAQUS model because processes such as boundary and property setting, matrix extraction, and the SS definition were
proposed without step-by-step verification. Fig. 6 shows the comparison of the CM and initial ABAQUS model with 16,899 nodes. As
the MATLAB function has a memory limit of approximately 5,000 nodes, a CM will less nodes was used for the validation.
The results show that the CM provides similar fluctuation patterns as the ABAQUS model, with an error of 0.08◦ C, corresponding to
0.75% for the entire time period. However, as the simulation proceeded, the error decreased. The results for the last three days (last 72
h) showed an error of 0.13◦ C, corresponding to 0.85%.
6
Case Studies in Thermal Engineering 25 (2021) 100936
J.-S. Choi et al.
Table 3
Maximum errors according to different orders of the reduced model.
Model order
Max. Error (%)
1
2
10
46
Target surface 1
Target surface 2
Target surface 3
4.35
2.59
6e-04
2e-11
3.28
1.48
4e-05
6e-13
3.54
1.86
6e-04
1e-11
Table 4
Calculation time comparison.
Simulation tool
Total CPU time (sec)
ABAQUS simulation
ABAQUS simulation
Complete model (CM)
Reduced model (RM)
(n = 16,899)
(n = 3,270)
(n = 3,270)
(m = 10)
21,387
79.5
0.013
0.002172
4.3. Comparison between state-space complete model and reduced model
Different orders of RMs were tested to select a final version. As shown in Fig. 7, the mean temperatures of the three surfaces were
measured with the test RMs. Cases with m = 46, 10, 2, and 1 were tested, and the results were compared with those of the CM. If m = 1,
then only a single equation was calculated as the dimension of A in Eq. (2) is 1 × 1. In fact, an RM is defined in another SS model, where
A is transformed into another matrix via the change of basis.
Fig. 7 and Table 3 show the comparison between the results of the RMs at mean surface temperatures. All three results were similar
in the cases of m = 10, 46, and n = 3,270 (CM). On the contrary, the lowest orders of m = 1 and 2 showed different results. These results
agreed with those of previous studies on reduction techniques using Moore’s method. An order of more than 10 is sufficient to simulate
the building envelopes [19]. Thus, the order of 10 can sufficiently describe the indoor surface temperatures.
4.4. Calculation time
The advantage of RM is that it is accurate even with a lower order. A lower order can also exponentially reduce the calculation time.
Table 4 lists the CPU time measured with a computer equipped with an Intel Core i9-9900 3.60 GHz. For the ABAQUS simulation, when
the number of nodes increased five-fold from n = 3270 to 16,899, the calculation time drastically increased by 270 times. With the
same number of nodes (n = 3,270), the calculation time for the CM, converted into SS formalism, was less than 1 s. This demonstrates
the efficiency of the SS model for use in a real-time evaluation model. For the proposed RM, a shorter CPU time was required, i.e.,
0.002 s. It should be noted that the size of the RM is much smaller than that of the CM; hence, it is easy to implement the RM in a virtual
simulator.
5. Conclusion
In this paper, a low-order SS model of a wall-window component is proposed, which can be easily developed without elaborate
programming. The ABAQUS-based model extraction method was used, and a procedure to define an SS model from the extracted files
was developed. The proposed procedure not only reduced development time and human errors in programming, but also the calcu­
lation time, even for the same model size. Moore’s reduction technique was applied to the obtained SS model to reduce the order of the
model to 10. The reduced model showed accurate results compared with the initial ABAQUS model with significantly reduced
calculation time and model size. More complex and detailed cases could not be investigated because the reduction function used has a
size limit for matrices. Further work is required to resolve this limit through domain decomposition and coupling methods.
CRediT authorship contribution statement
Jae-Sol Choi: Conceptualization, Methodology, simulation, Validation, Data curation, Writing – original draft, preparation.
Chang-Min Kim: Methodology, Validation, Writing – review & editing. Hyang-In Jang: Methodology, Validation, Writing – review &
editing. Eui-Jong Kim: Conceptualization, Methodology, simulation, Data curation, Writing – review & editing, Conceptualization,
Methodology, simulation, Data curation, Writing – review & editing.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
7
Case Studies in Thermal Engineering 25 (2021) 100936
J.-S. Choi et al.
Acknowledgements
This work was supported by the Korea Agency for Infrastructure Technology Advancement (KAIA) grant funded by the Ministry of
Land, Infrastructure and Transport (Grant 20CTAP-C152248-02).
Nomenclature
A,B,C,J,D Initial state model matrices
M
State-transforming matrix
Wc
Controllability Gramian
Observability Gramian
Wo
D, K, L Modified input matrices (damping, stiffness, load)
k
Conductivity (W/m∙◦ C)
m
Reduced order
n
Complete order
NN
Finite element interpolation function
q
Film and radiation (W/m2∙◦ C)
r
Heat flux per unit volume (W/m3)
S
Surface (m2)
T
Temperature field
T
State vector
Ts1 , Ts2 , Ts3 Target surface temperature (◦ C)
U̇
Time derivative of internal energy
U
Input vector
V
Volume (m3)
X
Transformed-basis state vector
Y
Output vector
̃
Y
Approximated output vector
Greek symbols
Density (kg/m3)
Н, Π, Ω transformed-basis matrices
ρ
Subscripts
m
Reduced order
n
Complete order
p
Number of solicitation
ref
Reference case
test
Test case
Abbreviation
CM
Complete model
FEM
Finite element method
RM
Reduced model
SS
State-space model
References
[1] G.P. Lydon, S. Caranovic, I. Hischier, A. Schlueter, Coupled simulation of thermally active building systems to support a digital twin, Energy Build. 202 (2019)
109298.
[2] Q. Lu, X. Xie, A.K. Parlikad, J.M. Schooling, Digital twin-enabled anomaly detection for built asset monitoring in operation and maintenance, Autom. ConStruct.
118 (2020) 103277.
[3] P. Cui, H. Yang, Z. Fang, Numerical analysis and experimental validation of heat transfer in ground heat exchangers in alternative operation modes, Energy
Build. 40 (6) (2008) 1060–1066.
[4] D. Kim, J.E. Braun, Reduced-order building modeling for application to model-based predictive control, Proc. SimBuild 5 (1) (2012) 554–561.
[5] B. Hudobivnik, L. Pajek, R. Kunič, M. Košir, FEM thermal performance analysis of multi-layer external walls during typical summer conditions considering high
intensity passive cooling, Appl. Energy 178 (2016) 363–375.
[6] P. Miąsik, L. Lichołai, The influence of a thermal bridge in the corner of the walls on the possibility of water vapour condensation, in: E3S Web of Conferences,
vol. 49, EDP Sciences, 2018, 00072.
[7] P. Santos, M. Gonçalves, C. Martins, N. Soares, J.J. Costa, Thermal transmittance of lightweight steel framed walls: experimental versus numerical and analytical
approaches, J. Build Eng. 25 (2019) 100776.
8
Case Studies in Thermal Engineering 25 (2021) 100936
J.-S. Choi et al.
[8] B. Nagy, G. Stocker, Numerical analysis of thermal and moisture bridges in insulation filled masonry walls and corner joints, Period. Polytech. Civ. Eng. 63 (2)
(2019) 446–455.
[9] F. Ascione, N. Bianco, R.F. De Masi, G.M. Mauro, M. Musto, G.P. Vanoli, Experimental validation of a numerical code by thin film heat flux sensors for the
resolution of thermal bridges in dynamic conditions, Appl. Energy 124 (2014) 213–222.
[10] H. Hibbitt, B. Karlsson, P. Sorensen, Abaqus Analysis User’s Manual Version 6.10, Dassault Systèmes Simulia Corp., Providence, RI, USA, 2011.
[11] S.A. Marshall, An approximate method for reducing the order of a linear system, Control 10 (1966) 642–643.
[12] Y. Gao, J.J. Roux, C. Teodosiu, L.H. Zhao, Reduced linear state model of hollow blocks walls, validation using hot box measurements, Energy Build. 36 (11)
(2004) 1107–1115.
[13] Y. Gao, J.J. Roux, L.H. Zhao, Y. Jiang, Dynamical building simulation: a low order model for thermal bridges losses, Energy Build. 40 (12) (2008) 2236–2243.
[14] E.J. Kim, G. Plessis, J.L. Hubert, J.J. Roux, Urban energy simulation: simplification and reduction of building envelope models, Energy Build. 84 (2014)
193–202.
[15] B. Moore, Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Automat. Contr. 26 (1) (1981) 17–32.
[16] A. Laub, M.T. Heath, C. Paige, R. Ward, Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms,
IEEE Trans. Automat. Contr. 32 (2) (1987) 115–122.
[17] E.J. Kim, J.J. Roux, M.A. Bernier, O. Cauret, Three-dimensional numerical modeling of vertical ground heat exchangers: domain decomposition and state model
reduction, HVAC R Res. 17 (6) (2011) 912–927.
[18] E.J. Kim, X. He, J.J. Roux, K. Johannes, F. Kuznik, Is it possible to use a single reduced model for a number of buildings in urban energy simulation, in: 14th
Conference of International Building Performance, 2015, December.
[19] E.J. Kim, X. He, J.J. Roux, K. Johannes, F. Kuznik, Fast and accurate district heating and cooling energy demand and load calculations using reduced-order
modelling, Appl. Energy 238 (2019) 963–971.
9