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 andj 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