AS-74.3179 Modelling and Control of Processes Dependent on Time and Spatial Coordinates P
Weekly exercise 1 model answer – COMSOL Multiphysics Introduction
Questions: olli.haavisto(a)aalto.fi
Solutions to the tasks
1) What is the meaning of elliptic, parabolic and hyperbolic problems in a mathematical, partial differential equation sense? What are the properties of their respective variables, differentials of variables and parameters? (Use
e.g. [1] for help.) Find a physics topic within COMSOL Multiphysics that exemplifies each of the problem types
above (one example is enough, but it isn’t forbidden to do more). (1p.)
In the COMSOL Multiphysics context (a) elliptic, (b) parabolic and (c) hyperbolic problems denote the types of problems that can be described by the following partial differential equations:
(a)
∙
∙
(b)
∙
∙
(c)
∙
∙
In the equations u is the variable being solved for, t is the time and is the spatial differential operator, whose dimensions and coordinates depend on the geometry of the model/problem. The symbols c, e, g, b, a, d and f are parameters of
the model. The parameters are usually constant, but may also be e.g. time-dependent.
The model space can be described in orthogonal Cartesian coordinates, usually denoted by x, (x,y) or (x,y,z) in one, two
and three dimensions respectively. Alternatively one can use cylinder coordinates, denoted by (r,φ,z). Since the variable u
is being solved for in terms of spatial (e.g. x) and time coordinates (t), we call the variables x and t independent variables
and u the dependent variable. The model space, depending on its dimensionality, is usually given by the symbol Ω, its
edges as Ω, and the extreme points or ”edges of the edges” (if they exist) as Ω and Ω.
Equations (a), (b) and (c) are different with regards to the time differential. Equation (a) has no time differential at all and
it is called the static equation. A physical example of these kinds of systems is an electric field, that always (in practice)
satisfies the Poisson equation ∙ σ
0, where σ is the conductivity of the medium (S/m) and theelectricpoten‐
tial V anda=f=0.
Equation (b) contains a first order time differential, i.e. this equation can be used to describe dynamic phenomena. Its
solutions are always given relative to some point in time (initial time) and some state (initial state), and the timedependent solver should be used. A physical example of (b) is conductive heat transfer, where u in this case is the temperature (often denoted T) in Kelvins, the time scaling parameter d=1, c is the thermal conductivity (often denoted k) in
W/mK and a=f=0.
Equation (c) contains a second order time differential, so this is also a dynamic equation, but its solution principles are
often substantially different from equation (b). Instead of solving the equation explicitly as a set of solutions relative to
the time t=0, equation (c) can be solved by a time-harmonic analysis, as if it were stationary (more on this later). A physical example of (c) is a sourceless sound wave equation, where u is the pressure (often p) in Pa, d=1/( cs2) kg/m/s2 and
c=1/ m3/kg.Thephysicalparameters andcs are the density and sound speed of the medium. Again we have a=f=0,
from which the name ”sourceless” is derived – a and f are often so-called source term coefficients.
Regarding the dimensionality of the variables and parameters of the equations, note that they can be scalars, vectors, matrices or even tensors. However, when programming multidimensional systems into COMSOL to be solved, they must be
disassembled into a series of scalar systems.
In addition to the equations listed above, COMSOL can solve a number of other types of equations. If the model is described by some other equation, it should not be classified into any of the aforementioned problem classes. A common
way of classifying models is dividing them into linear and nonlinear problems; nonlinear models are (among others)
those, whose parameters depend on independent variables, e.g. time.
2) Describe the (generalized) Dirichlet and Neumann boundary conditions i) mathematically ii) physically. Mention some practical physical examples of both Dirichlet and Neumann conditions (use COMSOL for help). (1p.)
In task 1) the listed equations describe the relevant physical phenomenon as it occurs in continuous space. To be able to
solve the equations, one should additionally define one or more of the following:
i)
Boundary conditions (for a stationary problem)
ii)
Initial conditions (for a time-dependent problem).
The most typical two boundary conditions are the (d) Dirichlet and (e) Neumann conditions:
(d)
(e)
∙
In these equations u is once again being solved for, with c and as in task 1). The symbols h, r, q and g are parameters
and n is the (outward-facing) normal vector of the boundary in question. In practice the difference between (d) and (e) is
that the former explicitly defines the value of u on the boundary Ω (u = r/h) and the latter defines the flux of u over the
boundary Ω. Note that calculating the differential
over an infinitely thin edge is not in itself meaningful, but this
condition can well be defined and when solved together with one of the equations (a)-(c) the solution as a system is meaningful.
One physical example of Dirichlet boundary conditions is the flow velocity at some wall boundary in a flow problem; at
the wall the flow velocity is zero (u=0). Similarly one physical example of Neumann boundary conditions is a heating
plate, whose power and area is known; the heat flux through the plate surface is the power divided by the area.
3) Can COMSOL be used to solve ordinary differential equations (ODEs)? If no, why not? If yes, in which ways?
(1p.)
Yes it can. COMSOL has a number of different ways of solving ODEs. One way is to solve equation (b) in onedimensional space, s.t. c=a=0 and f/d is constant, which gives us the solution of the first order time differential of u. A
second order ODE can be formed in a similar way using (a) in one-dimensional space, s.t. a=0 and f/c is constant, etc.
COMSOL also has the Global Equations menu that is used in time-dependent models and enables the global solution of
equations of parameters. Here the term “global” means that the equation variables and solution are usable everywhere in
the model. It is (only) possible to insert a time differential equation in standard form into the Global Equations menu and
COMSOL will solve it globally.
Moreover, COMSOL has the Integral Coupling Variables feature, which enables the integration of variables over points,
boundaries and regions both in time and space. This is also solving an ODE; a separable ODE is solved by integrating on
both sides.
4) Solve the heat transfer problem described below in three technically different ways, e.g. in one and/or two dimensions, using COMSOL's different Application Modes and/or using simplifications of different degrees. List
the assumptions made for the different solutions in your report. Furthermore, describe in your report the system
of equations that is to be solved, along with the required variables and parameters (once is enough). Use each
solution to find:
a. The temperature of the equilibrium state at the point A.
b. The temperature at A when t = 0 - 10 s.
c. The total heat flux gone through the protective surface S at the time t = 10s.
d. The total heat flux gone through S at t = 0 - 10s.
e. At what time equilibrium is reached.
If there are differences in the solutions, think about what the cause might be. (2p.)
Appended to the solution are four different models. Their differences are the solution technique (stationary, timedependent), dimensionality (1D or 2D) and the elaborateness of the model. The numerical answers below are collected
from these four models.
The solutions contain distinct differences. The 1D models predict the temperature of the stationary state to be a lot higher
than the 2D models. Likewise the heat flux through the protective plate and the subsequent waste heat is a lot higher in
the 1D models compared to the 2D models. The latter difference can partly be attributed to the difficulty in determining a
surface area for the protective plating. The surface area must be estimated manually in the 1D case, while it is calculable
in the 2D model.
The difference in the equilibrium temperature in the two 2D models is also notable. This can be assumed to be a consequence of the heat sink configured into model 4, which makes the cooling surface area larger than in model 3.
Model
Eq. temp. in A [K]
Heat through S when t=10 [J]
Eq. time [s]
1
431,5
1790
10...15
2
438,8
1825
20...30
3
410,6
762
10...20
4
347,4
780
10...20
Below is also given the heat energy value in part (c) for the time interval 0…10s as well as the equilibrium temperature
solution of model 4. From the graph it is evident that the errors in the 1D models are systematic and can therefore be assumed to be caused by the faulty estimation of the protective plate’s surface area.