Automatic differentiation based solution of strongly coupled problems in engineering

The doctoral thesis presents an approach for the formulation and solution of strongly coupled\ud engineering problems with the finite element method using the automatic differentiation technique that\ud the software tools AceGen and AceFEM enables. It has been shown that it is possible to transform arbitrarily weak form of differential equation of coupled problems into scalar function pseudo-potential. By using the automatic differentiation and appropriate exceptions in the differentiation procedure, the equations of the problem and the consistent tangent matrix of finite element can be automatically derived from the pseudo-potential, which ensures quadratic convergence of the Newton-Raphson iterative procedure. At the same time such formulation of the problem leads to an extremely fast and accurate finite element codes. With a large number of physical fields we are faced with the problem of increasing size of element software code with each added field. The problem was solved by additive split of the total problem to individual subproblems, for which the code is derived inside separate final element in a manner that preserves the quadratic convergence of the Newton-Raphson iteration. It has been shown on the numerical examples of the thermo-hydro-mechanical problems that separate formulation is suitably efficient compared to the unified formulation. The separate formulation of finite elements is a property of sequential approach. Therefore, we have shown on several examples that unified solution of the full system is more efficient than sequential solution procedure. Additionally, a new approach to the evaluation of matrix functions is presented. These are necessary for the formulation of non-linear mechanical problems, such as certain hyperelastic models (e.g. Hencky and Ogden models) and the exact evolution of plastic flow in the case of large deformations. A new method of automatic derivation of an arbitrary matrix function and its first and second derivatives of the matrix of dimensions 3 _ 3 with real eigenvalues is presented. The described method provides an alternative to the formulations based on the eigenvalues, because the generating function is stable and smooth compared to the eigenvalues. The generating function is a function\ud of the eigenvalues of matrix. Therefore it is expanded into power series in the vicinity of multiple\ud eigenvalues. A library of subroutines which calculate the standard matrix functions in closed form was\ud created. Thus, matrix functions can be considered as elementary functions when formulating problems. We have shown on individual matrices and various combinations of hyperelastic and elasto-plastic models that the derived matrix function and its derivatives are accurate and precise, and the formulation is efficient.\u