Linear Combination Boundary Conditions¶
By linear combination boundary conditions (LCBCs) we mean conditions of the following type:
(1)¶
where are given coefficients, are some components of unknown fields evaluated point-wise in points , and is a subset of the entire domain (e.g. a part of its boundary). Note that the coefficients can also depend on .
A typical example is the no penetration condition , where are the velocity (or displacement) components, and is the unit normal outward to the domain boundary.
Enforcing LCBCs¶
There are several methods to enforce the conditions:
penalty method
substitution method
We use the substitution method, e.i. we choose such that and substitute
(2)¶
into the equations. This is done, however, after the discretization by the finite element method, as explained below.
Let us denote ( is fixed). Then
(3)¶
Weak Formulation¶
We multiply (3) by a test function and integrate the equation over to obtain
(4)¶
where is some suitable function space (e.g. the same space which belongs to).
Finite Element Approximation¶
On a finite element (facet or cell) we have , where are the local (element) base functions. Using the more compact matrix notation , we have and similarly .
The relation (3), restricted to , can be then written (we omit the arguments) as
(5)¶
As (5) holds for any , we have a linear system to solve. After denoting the “mass” matrices , the linear system is
(6)¶
Then the individual coefficients can be expressed as
(7)¶
Implementation¶
Above is the general treatment. The code uses its somewhat simplified version described here. If the coefficients are constant in the element , i.e. for , we can readily see that . The relation (7) then reduces to
(8)¶
hence we can work with the individual components of the coefficient vectors (= degrees of freedom) only, as the above relation means, that for .