IEEE Circuits and Systems Magazine - Q1 2023 - 51

neighbors along the three axes. As an example, the following
code shows the loops required for connection
along the vertical axis.
for j in 1:M loop
for k in 1:P loop
for i in 1:N-1 loop
connect(vol[i,j,k].bo,vol[i+1,j,k].to);
end for;
end for;
end for;
In principle the presented modeling strategy should
be handled efficiently by Modelica compilers, however
to date Modelica tools are not able to infer arrays of
variables from arrays of components that contain those
variables and thus perform optimisations across components.
Naïve models such as the one just presented
thus generate a large number of unnecessary algebraic
equations arising from the interconnections of conductances
at the faces of the finite volumes, thus resulting
in a significant simulation overhead.
This issue can be overcome by structuring the 3D
discretisation model as a single component containing a
3D array of state variables-temperatures in this case-
and writing exchange equations referring directly to the
state variables without involving the temperatures at
the faces of the finite volumes. This approach has the
joint effect of a priori reducing the total number of equations
in the model and avoiding algebraic equations,
both factors contributing to improved simulation speed.
The trade-off is the complexity of having to handle
the solid boundaries explicitly. In fact, while the naïve
finite volume model handled unconnected ports automatically,
in this case we need to handle separately all
possible cases where one or more of the volume faces is
facing outward of the solid.
A graphical representation of all the different cases is
shown in Fig. 8, where the different element colors evidence
the need for seven groups of for loops, plus eight
individual equations for the solid vertices (that do not require
loops). For example, denoting by N,M,P the number
of elements along the three axes and by T the 3-dimensional
array of volume temperatures, the equations for the
inner volumes (green elements in Fig. 8) in Modelica read.
for i in 2:N-1 loop
for j in 2:M-1 loop
for k in 2:P-1 loop
der(T[i,j,k]) = 1/C*(
Gx*(T[i-1,j,k]-2*T[i,j,k]+T[i+1,j,k])
+Gy*(T[i,j-1,k]-2*T[i,j,k]+T[i,j+1,k])
+Gz*(T[i,j,k-1]-2*T[i,j,k]+T[i,j,k+1]));
end for;
end for;
end for;
FIRST QUARTER 2023
Figure 8. Volume decomposition for optimized 3D spatial discretisation
via uniform grid.
As said, a current research line in the field of Modelica
compilers aims to make them array-aware (see the research
presented in [16]), so next-generation Modelica
compilers will be capable of handling the human-readable
alternative equally efficient, but until then, manual
optimization is required.
VIII. Interfacing With 3D-ICE
3D-ICE is a free and open source thermal simulator specifically
designed for ICs. Its purpose is to model a silicon
die, together with an optional heat spreader, through finite
volume discretisation. The model is configurable in
terms of material properties, dimensions, finite volume
sizes and number of layers. Numerical integration of the
resulting differential equations is performed using the
implicit Euler method, optimized to efficiently produce
high-resolution transient temperature maps.
To model arbitrary heat dissipation systems, we recently
introduced in 3D-ICE support for co-simulation
[9] with Modelica models through the Functional Mockup
Interface (FMI) standard [12]; FMI provides a cosimulation
interface that allows Modelica models to be
exported as shared libraries, to be loaded by the 3D-ICE
pluggable heat sink infrastructure.
A typical co-simulation workflow begins with the
modeling of a heat dissipation system through the
graphical user interface of a Modelica environment
such as OpenModelica. Modeling is assisted by the
use of Modelica libraries of components such as those
provided as part of the 3D-ICE distribution, the library
we are presenting as part of this article, as well as the
Modelica standard library. In order to be suitable for
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