Linear Combination Boundary Conditions¶

By linear combination boundary conditions (LCBCs) we mean conditions of the following type:

(1)¶ $\sum_{i = 1}^{L} a_i u_i(\ul{x}) = 0 \;, \quad \forall \ul{x} \in \omega \;,$

where $a_i$ are given coefficients, $u_i(\ul{x})$ are some components of unknown fields evaluated point-wise in points $\ul{x} \in \omega$ , and $\omega$ is a subset of the entire domain $\Omega$ (e.g. a part of its boundary). Note that the coefficients $a_i$ can also depend on $\ul{x}$ .

A typical example is the no penetration condition $\ul{u}(\ul{x}) \cdot \ul{n}(\ul{x}) = 0$ , where $\ul{u}$ are the velocity (or displacement) components, and $\ul{n}$ 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 $j$ such that $a_j \neq 0$ and substitute

(2)¶ $u_j(\ul{x}) = - \frac{1}{a_j} \sum_{i = 1, i \neq j}^{L} a_i u_i(\ul{x}) \;, \quad \forall \ul{x} \in \omega \;,$

into the equations. This is done, however, after the discretization by the finite element method, as explained below.

Let us denote $c_i = \frac{a_i}{a_j}$ ( $j$ is fixed). Then

(3)¶ $u_j(\ul{x}) = - \sum_{i = 1, i \neq j}^{L} c_i u_i(\ul{x}) \;, \quad \forall \ul{x} \in \omega \;.$

Weak Formulation¶

We multiply (3) by a test function $v_j$ and integrate the equation over $\omega$ to obtain

(4)¶ $\int_{\omega} v_j u_j(\ul{x}) = - \int_{\omega} v_j \sum_{i = 1, i \neq j}^{L} c_i u_i(\ul{x}) \;, \quad \forall v_j \in H(\omega) \;,$

where $H(\omega)$ is some suitable function space (e.g. the same space which $u_j$ belongs to).

Finite Element Approximation¶

On a finite element $T_K$ (facet or cell) we have $u_i(\ul{x}) = \sum_{k=1}^{N} u_i^k \phi^k (\ul{x})$ , where $\phi^k$ are the local (element) base functions. Using the more compact matrix notation $\ub_i = [u_i^1, \dots, u_i^N]$ , $\vphib = [\vphib^1, \dots, \vphib^N]^T$ we have $u_i(\ul{x}) = \vphib(\ul{x}) \ub_i$ and similarly $v_i(\ul{x}) = \vphib(\ul{x}) \vb_i$ .

The relation (3), restricted to $T_K$ , can be then written (we omit the $\ul{x}$ arguments) as

(5)¶ $\int_{T_K} \vb_j^T \vphib^T \vphib \ub_j = - \int_{T_K} \vb_j^T \vphib^T\sum_{i = 1, i \neq j}^{L} c_i \vphib\ub_i \;, \quad \forall \vb_j$

As (5) holds for any $\vb_j$ , we have a linear system to solve. After denoting the “mass” matrices $\Mb = \int_{T_K} \vphib^T \vphib$ , $\Mb_i = \int_{T_K} c_i \vphib^T \vphib$ the linear system is

(6)¶ $\Mb \ub_j = - \sum_{i = 1, i \neq j}^{L} \Mb_i \ub_i \;.$

Then the individual coefficients $\ub_j$ can be expressed as

(7)¶ $\ub_j = - \Mb^{-1} \sum_{i = 1, i \neq j}^{L} \Mb_i \ub_i \;.$

Implementation¶

Above is the general treatment. The code uses its somewhat simplified version described here. If the coefficients $c_i$ are constant in the element $T_K$ , i.e. $c_i(\ul{x}) = \bar c_i$ for $x \in T_K$ , we can readily see that $\Mb_i = \bar c_i \Mb$ . The relation (7) then reduces to

(8)¶ $\ub_j = - \Mb^{-1} \sum_{i = 1, i \neq j}^{L} \bar c_i \Mb \ub_i = \sum_{i = 1, i \neq j}^{L} \bar c_i \ub_i \;,$

hence we can work with the individual components of the coefficient vectors (= degrees of freedom) only, as the above relation means, that $u_j^k = \bar c_i u_i^k$ for $k = 1, \dots, N$ .