IEEE Circuits and Systems Magazine - Q1 2023 - 39

cyber-physical simulation, so we first need to set out
terminology. In the field of ICs and their heat dissipation
systems as addressed herein, for model we mean
a mathematical description of a physical or cyberphysical
system, aimed at simulating its behavior
over time under prescribed initial conditions and
exogenous stimuli.
When one needs to stress that a model replicates the
transient behavior and not only the steady-state conditions
of a system, that model is called dynamic. In this
work all models are dynamic, however, so hereinafter
we drop the adjective.
A. First-Principle and Data-Based Models
A model is first-principle when written based on the
laws of physics, plus possibly some generally validated
empirical correlations. It is conversely data-based when
built by observing data recorded on the physical system,
and suitably correlating them over time so as to
replicate the input-to-output relationships in the modeled
object without any knowledge or hypothesis about
the physics in between. The parameters of a first-principle
model almost always have a direct physical meaning
(such as masses, specific heats, exchange coefficients
and so forth) while those of data-based models hardly
ever admit any such interpretation. Between the two extrema
just mentioned there are many " mixed " modeling
paradigms, for completeness, but a taxonomy of these
is not in the scope of this work; a concise discussion is
reported in [15].
When models are used to design some new physical
equipment there is obviously no recorded data yet.
Hence, one has to stick to the first-principle setting, as
we dominantly do in this work; we shall therefore drop
the " first-principle " attribute as well.
B. Declarative and Imperative Models
In our context as just sketched, a model starts out as a
set of dynamic balance equations-e.g., the derivative
with time of the mass in a volume equals at any instant
the sum of the flow rates through its boundary-and
algebraic physical correlations, like for example the
Colebrook one to relate the pressure drop across a
duct and the flow rate through it. Mathematically, the
result is a Differential and Algebraic Equations (DAE)
system, in general nonlinear and in some cases even
discontinuous.
We call such a model declarative, as it holds all the
information needed to simulate the modeled system
but cannot be used as is to compute its behavior-
strictly speaking unless the DAE system can be solved
FIRST QUARTER 2023
analytically, but for models of engineering interest this
is never the case.
A model that can be used for computing the system
behavior-i.e., with a slight simplification acceptable
here, a solution algorithm for the DAE system-is conversely
said to be imperative, as it can be unambiguously
turned into a set of computer instructions to run.
The Declarative-to-Imperative (D2I) translation of a
model is thus a necessary step for its simulation. The
D2I translation is in general a complex and potentially
critical process, however, hence it plays a prominent
role in the following discussion.
C. Monolithic and Modular Declarative Models
For simulating a system, its DAE model needs solving as
a whole. Fortunately, however, this does not mean that
the DAE must be written by the human as a single, comprehensive
system, that is, be monolithic.
On the contrary, to tame the complexity of most systems
it is far more convenient to build a (declarative)
model by assembling components, that in turn may be
the composition of other components in a hierarchical
manner. We call models built this way modular.
In a modular modeling context, components must
have interfaces so that the analyst can compose them
together.
D. Causal and A-Causal Declarative Models
We say that a model is causal when its interface with
the outside consists of inputs and outputs: knowing the
inputs (plus the internal states, as we deal only with dynamic
models) is sufficient to know the outputs.
It is important to avoid confusing " causal " with " imperative. "
For example, a continuous-time transfer function
model is causal-because it has an input and an output-
but is still declarative until the choice of a numeric integration
method turns it into an imperative procedure to
compute state and output. Causal models must be closed,
i.e., they must have as many equations as unknowns.
Conversely, a model is a-causal when its variables
are not inputs or outputs per se but assume either role
when the model is connected to others. For example, a
resistor of resistance R is ruled by Ohm's law v = Ri; if it
is connected in parallel to a voltage generator then voltage
v is prescribed (hence it is the input) and current i
is the output, while the opposite occurs if the resistor
is in series to a current generator. Acausal models need
not be closed: for example, the above resistor model has
two unknowns (v and i) and one equation. In an a-causal
model, the equations that specify its behavior independently
of how that model is connected to any other (in
the resistor case, v − Ri = 0 for maximum clarity) are
called the constitutive equations.
IEEE CIRCUITS AND SYSTEMS MAGAZINE
39
IEEE Circuits and Systems Magazine - Q1 2023

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