Automation of Finite Element Methods

This chapter contains a summary of the continuum mechanics background that is needed for the finite element formulation of solid mechanics and structural problems. In detail the kinematical relations and the balance laws with their weak forms are described in this chapter.

It is obvious that if we could find characters or signs suited for expressing all our thoughts as clearly and as exactly as arithmetic expresses numbers or geometry expresses lines, we could do in all matters insofar as they are subject to reasoning all that we can do in arithmetic and geometry.

In order to formulate nonlinear finite elements symbolically in a general but simple way, a clear mathematical formulation is needed at the highest abstract level possible.

Finite elements will be applied to discretize weak forms of the nonlinear continuum or structural problems. Within this approach the primary variables, like displacements or rotations etc., have to be approximated within a finite element by interpolation functions. The emphasis of this chapter is to describe how the automation technique introduced in Chap. 2 can be employed to generate finite element vectors and matrices automatically as well as providing efficient source code for different finite element programs. Hence this chapter is focused on the essential details needed to apply the systems AceGen to a standard nonlinear analysis of continua using finite elements. To achieve this goal, first the necessary interpolation functions and related mappings are discussed then two-dimensional continuum elements are derived using AceGen. The derivation starts from different configurations and also as well from the functional and weak forms presented in the previous chapter.

Constitutive equations have to be formulated in continuum mechanics that characterizes the material response of a solid body.

After a short introduction in the main targets of element development for continuum elements this chapter will consider different approaches ranging from standard up to special formulations.

Trusses, beams and shells belong to the most important structural elements in engineering practice. Many structures in civil engineering—like masts, domes, frames or cooling towers—consist of such structural elements.

Optimization of solid structures lead to better engineering designs and thus to a longer life time of e.g. technical components, engines and buildings. In an optimization process as well material parameters as shapes can be improved. However the computations needed to find optimal solutions are quite complex. Here a proper sensitivity analysis provides an efficient tool in order to compute complex derivations that are needed in the associated algorithms which finally leads to faster convergence of the underlying iterative procedures. Another applications where sensitivity analysis is needed is related to multi-scale analysis of coplex structures. Especially the so called FE\({}^2\)-schemes need an accurate computation of sensitivities. Here micro and macro levels are combined in the computations to account for complex material behaviour at micro level.