Nearly all physical phenomena occurring in nature can be described by differential equations and boundary conditions. In the solution of these boundary value problems we aim to determine a response to given boundary conditions. For example we may be interested in determining the response of the rock mass due to the excavation of a tunnel, or the response of a structure to dynamic excitations of its foundations (caused by an earthquake). Analytical solutions of boundary value problems, i.e. solutions that satisfy both the differential equations (DE) and the boundary conditions (BCs), can only be obtained for few problems with very simple boundary conditions. For example, analytical solutions exist for the excavation of a circular tunnel in a homogeneous rock mass, not really a realistic scenario for practical tunnelling. To be able to solve real life problems, the engineer must revert to approximate solutions. Two approaches can be taken: instead of satisfying both the DE and the BCs, one can attempt to satisfy only one of the two and minimise the error in satisfying the other one. In the first approach (based on the original idea of
) solutions are proposed that satisfy the boundary conditions exactly. The error in satisfying the differential equation is then minimised. This is the well known Finite Element Method. In the alternative (proposed by
), the assumed functions satisfy the DE exactly and the error in the satisfaction of the boundary conditions is minimised.