| One of the main advantages of R-function method is
the possibility to describe geometrical domains changing
in time. Moreover RFM solution structures satisfy exactly
the boundary conditions given on the moving geometry. The following example illustrates this advantage of
RFM on transient heat transfer problem in time-varying
domain. Figure 1(a) shows a simplified
model of heat transfer in the engine combustion chamber.
For the sake of simplicity we assume heat flux inside
the combustion chamber is known and it is constant during
the piston's motion. On the external walls of the combustion
chamber we assume the known temperature. Thus there
are two different boundary conditions prescribed on
the inner wall and on the outer wall of the combustion
chamber (Figure 1(b)). In order to construct a normalized
function we parametrize the geometry of the combustion
chamber (Figure 2(a)) and choose
a set of geometrical primitives (Figure
2(b)). Combining the equations for each primitive
halfspace by R-functions
we obtain two implicit functions describing the geometry
of internal and external walls of the combustion chamber
(Figure 3). All geometrical parameters
shown in Figure 2(a) came into equations
for the normalized implicit functions shown in Figure
3. That is why the motion of the piston can be modeled
by the change of the parameter H according to
the motion of a slider-crank mechanism. Figure
3 shows the normalized implicit functions for two
different positions of the piston. The constructed normalized implicit functions, boundary
conditions and basis functions are combined in the RFM
solution structure which represents a temperature
fiels inside the engine construction. The basis functions
are bicubic B-splines given on non-conforming uniform
grid (Figure 4). The same grid
of B-splines is used for all computations. Figure
5 represents a plot of isolines of the temperature
field inside the engine construction.
For more information see the reference
[3].
Figure 1: (a) A simplified
model of heat transfer in the engine combustion chamber;
(b) boundaries of the combustion chamber correspond
to two different types of boundary conditions on the
chamber
Figure 2: (a) Parametrization
of the engine combustion chamber; (b) geometrical
primitives used to describe the engine combustion
chamber by normalized
implicit functions
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(a)
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(b)
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(c)
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(d)
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Figure 3: The two rows
correspond to the two piston positions: (a,c) a normalized
implicit function describing the boundary where temperature
Toil is prescribed; (b,d) a normalized implicit
function describing the boundary where heat flux q
is prescribed
Figure 4: The same rectangular grid of B-splines may be
used to compute approximate solutions over any geometric
domain
Figure 5: Computed quasi-steady temperature field at
the particular position of the piston
(to see the animation click on the picture)
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