This project motivation is to explore the configuration-dependent interpolation as a novel, unorthodox and remarkably promising expansion of the framework within which the non-linear finite-element method has been traditionally contained. The basic idea underlying the project stems from an apparent disparity between the rather advanced extensions of the traditional linear finite-element principles to non-linear problems and the fact that the key finite-element concept – that of interpolation of the unknown functions – is surprisingly kept mostly constant, i.e. configuration-independent. Enabling the finite-element approximation to become configuration-dependent is motivated by the existing need to improve the current non-linear finite-element procedures, in particular for mechanical problems defined on non-linear manifolds. This principle is presented as the general concept providing viable novel development paradigm with obvious benefits for a wider class of mechanical problems.

The majority of problems in mechanics and structural engineering are defined by differential equations of equilibrium or motion for which analytical solutions are not known, owing to either the complexities of the equations themselves or the complex boundary conditions imposed. Also, in those cases where the solution is known, it cannot usually be expressed in a closed form involving a finite number of parameters, but only in the form of a convergent infinite series. In these situations, the problem has to be solved approximately, normally using a suitable numerical technique. From a variety of numerical methods available nowadays, the finite-element method has been used with general success and continually increasing frequency for over half a century. In contrast to some other numerical techniques, the finite element method is applicable to non-linear problems in an algorithmically simple, general and precisely defined manner.