Implementation of Essential Boundary Conditions¶

The essential boundary conditions can be applied in several ways. Here we describe the implementation used in SfePy.

Motivation¶

Let us solve a linear system $A x = b$ with $n \times n$ matrix $A$ with $n_f$ values in the $x$ vector known. The known values can be for example EBC values on a boundary, if $A$ comes from a PDE discretization. If we put the known fixed values into a vector $x_f$ , that has the same size as $x$ , and has zeros in positions that are not fixed, we can easily construct a $n \times n_r$ matrix $T$ that maps the reduced vector $x_r$ of size $n_r = n - n_f$ , where the fixed values are removed, to the full vector $x$ :

$x = T x_r + x_f \;.$

With that the reduced linear system with a $n_r \times n_r$ can be formed:

$T^T A T x_r = T^T (b - A x_f)$

that can be solved by a linear solver. We can see, that the (non-zero) known values are now on the right-hand side of the linear system. When the known values are all zero, we have simply

$T^T A T x_r = T^T b \;,$

which is convenient, as it allows simply throwing away the A and b entries corresponding to the known values already during the finite element assembling.

Implementation¶

All PDEs in SfePy are solved in a uniform way as a system of non-linear equations

$f(u) = 0 \;,$

where $f$ is the nonlinear function and $u$ the vector of unknown DOFs. This system is solved iteratively by the Newton method

$u^{new} = u^{old} - (\tdiff{f}{u^{old}})^{-1} f(u^{old})$

until a convergence criterion is met. Each iteration involves solution of the system of linear equations

$K \Delta u = r \;,$

where the tangent matrix $K$ and the residual $r$ are

$K \equiv \tdiff{f}{u^{old}} \;, r \equiv f(u^{old}) \;.$

Then

$u^{new} = u^{old} - \Delta u \;.$

If the initial (old) vector $u^{old}$ contains the values of EBCs at correct positions, the increment $\Delta u$ is zero at those positions. This allows us to assemble directly the reduced matrix $T^T K T$ , the right-hand side $T^T r$ , and ignore the values of EBCs during assembling. The EBCs are satisfied automatically by applying them to the initial guess $u^{0}$ , that is given to the Newton solver.

Linear Problems¶

For linear problems we have

$f(u) \equiv A u - b = 0 \;, \tdiff{f}{u} = A \;,$

and so the Newton method converges in a single iteration:

$u^{new} = u^{old} - A^{-1} (A u^{old} - b) = A^{-1} b \;.$

Evaluation of Residual and Tangent Matrix¶

The evaluation of the residual $f$ as well as the tangent matrix $K$ within the Newton solver proceeds in the following steps:

The EBCs are applied to the full DOF vector $u$ .
The reduced vector $u_r$ is passed to the Newton solver.
Newton iteration loop:
- Evaluation of $f_r$ or $K_r$ :
  1. $u$ is reconstructed from $u_r$ ;
  2. local element contributions are evaluated using $u$ ;
  3. local element contributions are assembled into $f_r$ or $K_r$ - values corresponding to fixed DOF positions are thrown away.
- The reduced system $K_r \Delta u_r = r_r$ is solved.
- Solution is updated: $u_r \leftarrow u_r - \Delta u_r$ .
- The loop is terminated if a stopping condition is satisfied, the solver returns the final $u_r$ .
The final $u$ is reconstructed from $u_r$ .