Computation of the matrix Kh and the vector fh without taking the boundary conditions into account
Computation of the element sitfffness matrix :
On the element T(r) the element stiffness matrix K(r) is defined by :
with :
- is the
reference element and
the
coordinates on it.
- i and
j
are shape functions defined on the reference element.
- : the coefficients
of thermal conductivity of the material.
- is the number
of nodes per element.
- J(r) is the jacobian matrix of the transformation which turns
the reference element into
an element of the mesh T(r).
Computation of the element right-hand side :
The element right-hand side f(r) is computed using the same method as the element stiffness matrix K(r), i.e.
with :
: coordinates on the element T(r)
Taking the boundary conditions of first type into account :
To take these boundary conditions into account, we modify the right-hand size using the following method :
with :
- g1(xj) : the value of the Dirichlet boundary
condition at the point xj
- : containing
the numbers of the nodes of the element T(r) which do not have Dirichlet
boundary conditions.
- : containing
the numbers of the nodes of the element T(r) which have
Dirichlet boundary conditions.
We also modify the stiffness matrix with :
with
: containing the numbers of the nodes of the element T(r).
Assembling of the element stiffness matrices and the element right-hand sides :
The element stiffness matrix and the element right-hand side are assembled
using the following algorithm. We use this algorithm for every domain because
two different domains have different coefficients of thermal conductivity.
For each domain
For each element T(r) of the domain
Computation of K(r) and f(r)
Modification of f(r) and K(r) to take the boundary conditions of first type into account
For each node of the element, which has i as global number and k as local number
![]()
For each node of the element, which has j as global number and l as local number