Linear Combination Boundary Conditions

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

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where

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are given coefficients,

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are some components of unknown fields evaluated point-wise in points

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, and

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is a subset of the entire domain

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(e.g. a part of its boundary). Note that the coefficients

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can also depend on

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.

A typical example is the no penetration condition

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, where

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are the velocity (or displacement) components, and

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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

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such that

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and substitute

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

Let us denote

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(

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is fixed). Then

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Weak Formulation

We multiply (3) by a test function

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and integrate the equation over

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to obtain

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where

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is some suitable function space (e.g. the same space which

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belongs to).

Finite Element Approximation

On a finite element

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(surface or volume) we have

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, where

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are the local (element) base functions. Using the more compact matrix notation

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,

System Message: WARNING/2 (\vphib = [\vphib^1, \dots, \vphib^N]^T)

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we have

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and similarly

System Message: WARNING/2 (v_i(\ul{x}) = \vphib(\ul{x}) \vb_i)

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.

The relation (3), restricted to

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, can be then written (we omit the

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arguments) as

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As (5) holds for any

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, we have a linear system to solve. After denoting the “mass” matrices

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,

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the linear system is

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Then the individual coefficients

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can be expressed as

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Implementation

Above is the general treatment. The code uses its somewhat simplified version described here. If the coefficients

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are constant in the element

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, i.e.

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for

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, we can readily see that

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. The relation (7) then reduces to

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hence we can work with the individual components of the coefficient vectors (= degrees of freedom) only, as the above relation means, that

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for

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.