6.3. Moving Coordinate System and Moving Point Methods
175
The simple one-dimensional problem discussed above shows that Lagrangian methods are powerful in handling advection-dominated problems.
Unfortunately, this kind of moving coordinate system method is not easily
extended to multidimensional and other complex practical problems.
6.3.2 Element Deformation Methods
The idea of this method is that the finite element nodes can move within a
fixed coordinate system. If the nodes are concentrated around the front and
can automatically move with the front of migration, the local Peclet number
will become smaller and the difficulties of numerical solutions can be
overcome.
Assume that the spatial distribution of the nodes at time t k is {X k } and
their concentration distribution is known to be Ck(X k ). We wish to find the
distribution of nodes {Xk+l} at time tk+l = t k + Ilt and the corresponding
concentration distribution Ck+l (Xk+l). The outline of this method is given as
follows:
1. Predict the position of the steep front at time tk+l using the common
Galerkin FEM.
2. Rearrange the nodes so as to increase the density near the steep front and
form the distribution {Xk+d.
3. Solve C k (Xk+l) from the known Ck(X k ) as the initial values of the solution
by means of interpolation.
4. Reform the coefficients of the finite element equations according to the
positions of new nodes {Xk+l} and the given boundary conditions.
5. Solve the finite element equations to obtain Ck+l(Xk+d.
As shown by the steps, the positions of the nodes change with time, so the
basis functions and the coefficients of the equations all depend on time. Since
the numerical system has to be updated each time step, the computational
effort is significantly increased.
O'Neill (1981) studied the one-dimensional element deformation method
combined with moving coordinates. The position of each moving point x at
time t can be expressed as
x = x(xo,t)
(6.3.8)
where Xo is the initial position of the moving point. Because C(x, t) =
C(x(xo, t), t), we have
( aC) (aC) oc (ox)
at Xo = at x + ox at xo'
(6.3.9)
where (ox/ot)xo = dx/dt represents the velocity of moving point x, and
(oC;ot)x indicates the change of concentration with time. Then Eq. (6.3.1)
175
The simple one-dimensional problem discussed above shows that Lagrangian methods are powerful in handling advection-dominated problems.
Unfortunately, this kind of moving coordinate system method is not easily
extended to multidimensional and other complex practical problems.
6.3.2 Element Deformation Methods
The idea of this method is that the finite element nodes can move within a
fixed coordinate system. If the nodes are concentrated around the front and
can automatically move with the front of migration, the local Peclet number
will become smaller and the difficulties of numerical solutions can be
overcome.
Assume that the spatial distribution of the nodes at time t k is {X k } and
their concentration distribution is known to be Ck(X k ). We wish to find the
distribution of nodes {Xk+l} at time tk+l = t k + Ilt and the corresponding
concentration distribution Ck+l (Xk+l). The outline of this method is given as
follows:
1. Predict the position of the steep front at time tk+l using the common
Galerkin FEM.
2. Rearrange the nodes so as to increase the density near the steep front and
form the distribution {Xk+d.
3. Solve C k (Xk+l) from the known Ck(X k ) as the initial values of the solution
by means of interpolation.
4. Reform the coefficients of the finite element equations according to the
positions of new nodes {Xk+l} and the given boundary conditions.
5. Solve the finite element equations to obtain Ck+l(Xk+d.
As shown by the steps, the positions of the nodes change with time, so the
basis functions and the coefficients of the equations all depend on time. Since
the numerical system has to be updated each time step, the computational
effort is significantly increased.
O'Neill (1981) studied the one-dimensional element deformation method
combined with moving coordinates. The position of each moving point x at
time t can be expressed as
x = x(xo,t)
(6.3.8)
where Xo is the initial position of the moving point. Because C(x, t) =
C(x(xo, t), t), we have
( aC) (aC) oc (ox)
at Xo = at x + ox at xo'
(6.3.9)
where (ox/ot)xo = dx/dt represents the velocity of moving point x, and
(oC;ot)x indicates the change of concentration with time. Then Eq. (6.3.1)