Finite element method — the natural approach

The paper gives a survey of past and recent achievements of the natural mode technique of finite elements. The discussion is pertinent to solid mechanics extending from linear elasticity through nonlinear phenomena with large displacements but small strains to problems with large strains in compressible and incompressible as well as elastoplastic and elastoviscoplastic media. Mathematically, by 'natural method' one means that the natural modes of deformation imply a contemplation of the history of deformation of the element via a convective system of axes. Consistent application of the natural method is intimately connected with the subelement concept. For application of the subelement method one defines the natural modes of the simplest possible finite element types, called 'urelements'. Closely associated with this aspect of the natural modes is a definition of stresses and strains which is in harmony with the element or subelement in questions. For example, in a TRIM 3 element the stresses and strains are selected as corresponding direct measures parallel to the sides of the triangle.