Dirichlet boundary conditions¶

Introduction¶

In finite element problems, it is essential to incorporate the Dirichlet boundary conditions, i.e., prescribing a part of the solution to a fixed value. Let us consider the following:

$$ \boldsymbol{u} = \boldsymbol{g} \quad \text{on } \Gamma_\mathrm{D}. $$ Here \( \boldsymbol{u} \) is the solution field with a prescribed value \(\boldsymbol{g}\) on the boundary \(\Gamma_\mathrm{D}\).

For the discrete algebraic problem, this translates to fixing the values of \(u_i\) to \(g_i\) in the approximation \( \boldsymbol{u}^h = \sum_i N^i u_i \), where \( N^i \) is the \(i\)-th ansatz function.

The handling of such a function \(\boldsymbol{g}\) is done by the class Ikarus::DirichletValues.

Interface¶

The interface of Ikarus::DirichletValues is represented by the following code snippet:

Ikarus::DirichletValues dirichletValues2(basis); // (1)!
void fixBoundaryDOFs(f); // (2)!
void fixDOFs(f); // (3)!
const auto& basis() const; // (4)!
bool isConstrained(std::size_t i) const; // (5)!
auto fixedDOFsize() const; // (6)!
auto size() const ; // (7)! 

1. Create class by inserting a global basis, ¹ Chapter 10.
2. Accepts a functor to fix boundary degrees of freedom. f is a functor that will be called with the boolean vector of fixed boundary. degrees of freedom and the usual arguments of Dune::Functions::forEachBoundaryDOF, as defined on page 388 of the Dune¹ book.
3. A more general version of fixBoundaryDOFs. Here, a functor is to be provided that accepts a basis and the corresponding boolean vector considering the Dirichlet degrees of freedom.
4. Returns the underlying basis.
5. Indicates whether the degree of freedom i is fixed.
6. Returns the number of fixed degrees of freedom.
7. Returns the number of all dirichlet degrees of freedom.