The shape function is the function which interpolates the solution between the discrete values obtained at the mesh nodes. Therefore, appropriate functions have to be used and, as already mentioned, low order polynomials are typically chosen as shape functions. In this work linear shape functions are used.
For threedimensional finite element simulations it is convenient to discretize the simulation domain using tetrahedrons, as depicted in Figure 4.1. Thus, linear shape functions must be defined for each tetrahedron of the mesh, in order to apply the Galerkin method described in Section 4.1.1.

Consider a tetrahedron in a cartesian system as depicted in Figure 4.2(a). The linear shape function of the node has the form [153]

The calculations can be significantly simplified by carring out a coordinate transformation. A tetrahedron in a transformed coordinate system is shown in Figure 4.2(b). Each point of the tetrahedron in the original coordinate system can be mapped to a corresponding point in the transformed coordinate system [155]
These shape functions are rather simple, so that the derivatives and integrals required for the finite element formulation can be readily evaluated in the transformed coordinate system. Given a function , the gradient in the transformed coordinates is of the form
Performing such a coordinate transformation significantly simplifies the practical implementation of the FEM. The nodal shape functions in the transformed coordinates are fixed and known in advance, thus, it is not necessary to solve the system of equations formed by (4.15) and (4.16) for each element of the mesh. Only the Jacobian matrix has to be determined, and the required calculations for the finite element formulation can be easily evaluated.