VECFEM3 Reference Manual: equation

NAME

PURPOSE

In the data set equation the user defines the functional equation to be solved by VECFEM. Using the parameters defined in the resource file it is transformed into a FORTRAN source code for the solution of the problem, see vembuild. The equation may be steady or nonsteady.

In the case of a steady problem, the NK-valued solution u=(u<i>) is selected from a space H of smooth, NK-valued functions. The equation can be given in the formulation of a steady functional equation:

[1] for all v in H with v(i)|D<i>=0 for all i=1,..,NK:

        F{u}(v)=0

    and for all i=1,...,NK : u<i>|D<i>=b<i>

[2]     F{u}=minimal

    over all u in H with u<i>|D<i>=b<i> for all i=1,..,NK.

In the case of a nonsteady problem, the NK-valued solution u=(u<i>) depends on the time t, t0<=t<=tend, and for all t it is selected from a space H of smooth, NK-valued functions. The equation can be given in the formulation of a nonsteady functional equation:

[3] for all v in H with v(i)|D<i>=0 for all i=1,..,NK:

        F{t,ut,u}(v)=0 for all t0<t<tend,

    for all i=1,...,NK : u<i>(t)|D<i>=b<i>(t)  for all t0<t<tend

    and for all i=1,...,NK : u<i>(t0)=u0<i>.

SYNOPSIS

The script for the definition of the functional equation has three sections. In the first section parameters are defined, in the second one the Dirichlet conditions are specified and in the third one the functional F is described. Lines starting with '#' are comment lines. Spaces are insignificant. Upper- and lower-case letters are equivalent.

Allowed expressions are real constants, reserved symbols, the parameters declared in the parameter section and operator evaluations. The following operators are available:

parameter

<parameter>=<expr>

Initial Values

u0<i>=<expr0>

Dirichlet Conditions

u<i>=<exprd>

Functional Equations

The functional equations [1] and [3] are described by the statement:

volume{<exprv>}+area{<expra>}+line{<exprl>}+point{<exprp>}=0

volume{<exprv>}+area{<expra>}+line{<exprl>}+point{<exprp>}=minimize

For the 3-dimensional case DIM=3, the integration kernels <exprv>, <expra>, <exprl> and <exprp> are allowed expressions. After the insertion of the parameters, the kernels must only depend on real constants, the available operators and the following symbols:

For the 2-dimensional case DIM=2, the integration kernels <expra>, <exprl> and <exprp> are allowed expressions. The volume integration is not allowed. After the insertion of the parameters, the kernels must only depend on real constants, the available operators and the following symbols:

For the 1-dimensional case DIM=1, the integration kernels <exprl> and <exprp> are allowed expressions. The volume and area integrations are not allowed. After the insertion of the parameters, the kernels must only depend on real constants, the available operators and the following symbols:

DISCRETIZATION

        line            quadrilateral             triangle

                       4------7------3         3 
                       |             |         | \
                       |             |         |   \
                       | ^           |         | ^   \
                       8 |           6         6 |     5       
                       | tau<.>2     |         | tau<.>2 \
                       | |           |         | |         \
     --tau<.>-->       | --tau<.>1-> |         | --tau<.>1-> \
   1------3------2     1------5------2         1-------4------2              

EXAMPLES

#  
#  chr=0 for x1<=1 and chr=1 x1>1
#
 chr=1./(1.+exp(-100*(x1-1.))

#
#     / 900.          x1<=1.
#  k=|         for           = thermal conductivity
#     \   1.          x1>1.
#
    k= chr * 900. + (1-chr) * 1.
#
#  no heat generation:
#
    qb=0
#
#  convection boundary condition:
#
    qs= (u1 - 20) ^ 2 
#
#  the prescribed temperature is given by the input file:
#
    u1=prevalue
#
#  and here is the variational problem:
#
   volume { k/2 * ( u1x1^2 + u1x2^2 ) - qb*u1 } +
   area { qs/2 } = minimal

#
#   the Reynolds number:
#
     Re=10.
#
#   the input and output profile is a parabolic profile : 
#
     u1=(1.-x1)*x1
     u2=0.
     u3=0.
     u4=1.
#
#   the initial solution :
#
     u01=0.
     u02=0.
     u03=0.
     u04=1.
#
    volume{
#
#        the first momentum equation
#
     v1*u1t+v1x1*u1x1+v1x2*u1x2+v1x3*u1x3 + 
            Re*v1*(u1*u1x1+u2*u1x2+u3*u1x3)+
            v1x1*u4 +
#
#        the second momentum equation
#
     v2*u2t+v2x1*u2x1+ v2x2*u2x2+ v2x3*u2x3 + 
            Re*v2*(u1*u2x1+u2*u2x2+u3*u2x3)+
            v2x2*u4 +
#
#        the third momentum equation
#
     v3*u3t+v3x1*u3x1+ v3x2*u3x2+ v3x3*u3x3 + 
            Re*v3*(u1*u3x1+u2*u3x2+u3*u3x3)+
            v3x3*u4 +
#
#        the equation of continuity
#
            -v4*(u1x1+u2x2+u3x3)   }=0

REFERENCES

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