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How to implement Robin Boundary Conditions?

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Old   February 8, 2013, 18:36
Default How to implement Robin Boundary Conditions?
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Hi, I'm currently working with a code that requires to assign the Boundary conditions using the coefficients on the general form of Robin Bc (a*u + b * du/dn = g)

From my understanding I can assign Dirichlet conditions by setting a=1, b=0, and g the desired specified value.

Similarly for Neuman bc I need to set a=0, b=1, and g the desired value.
is this correct so far?

Now lets considered a quadrilateral element
1--------2
| |
| |
3--------4

where there is Neuman boundary defined over the element and dirichlet on the nodes 1 and 3

Since I'm doing groundwater modeling, lets say I have specified hydraulic head for nodes 1 and 3 H=10 m and recharge flux equal to 1e-4 m/day. Lets say that the area of the element is 1e6, then 100 m^3/day are coming into that element. Using FE discretization the flux will split into 25 m^2/day per node.

I assume that for the nodes 2 and 4 the coefficients have to be a=0, b=1, g=25.
But what are the values for the nodes 1 and 3?

Thank you

George
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Old   February 9, 2013, 04:52
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Quote:
Originally Posted by giorgk View Post
Hi, I'm currently working with a code that requires to assign the Boundary conditions using the coefficients on the general form of Robin Bc (a*u + b * du/dn = g)

From my understanding I can assign Dirichlet conditions by setting a=1, b=0, and g the desired specified value.

Similarly for Neuman bc I need to set a=0, b=1, and g the desired value.
is this correct so far?

Now lets considered a quadrilateral element
1--------2
| |
| |
3--------4

where there is Neuman boundary defined over the element and dirichlet on the nodes 1 and 3

Since I'm doing groundwater modeling, lets say I have specified hydraulic head for nodes 1 and 3 H=10 m and recharge flux equal to 1e-4 m/day. Lets say that the area of the element is 1e6, then 100 m^3/day are coming into that element. Using FE discretization the flux will split into 25 m^2/day per node.

I assume that for the nodes 2 and 4 the coefficients have to be a=0, b=1, g=25.
But what are the values for the nodes 1 and 3?

Thank you

George


Assume the point 3 is at x=y=0 coordinates. Assume a bi-linear approximation on 1-2-3-4:

u(x,y) = c0 + c1*x +c2*y +c3*xy

that over the boundary x=0 leads to

du/dx = c1+c3*y

Hence, your Robin Bc along edge 3-1 becomes

a*(c2*y) + b*(c1+c3*y)= g

where a,b, g must be known from your problem.
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