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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)!
void fixIthDOF(i); // (4)!
const auto& basis() const; // (5)!
bool isConstrained(std::size_t i) const; // (6)!
auto fixedDOFsize() const; // (7)!
auto size() const ; // (8)!
  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 Dune 1 book.
  3. A more general version of fixBoundaryDOFs. Here, a functor is to be provided that accepts a basis and the corresponding boolean
  4. A function that helps to fix the \(i\)-th degree of freedom vector considering the Dirichlet degrees of freedom.
  5. Returns the underlying basis.
  6. Indicates whether the degree of freedom i is fixed.
  7. Returns the number of fixed degrees of freedom.
  8. Returns the number of all dirichlet degrees of freedom.

  1. Oliver Sander. DUNE—The Distributed and Unified Numerics Environment. Volume 140. Springer Nature, 2020. doi:10.1007/978-3-030-59702-3. ↩↩