### Implicit solutions with consistent additive and multiplicative components

P. Areias, T. Rabczuk, D. Dias-da-Costa, E.B. Pires
2012 Finite elements in analysis and design
This work describes an algorithm and corresponding software for incorporating general nonlinear multiple-point equality constraints in a implicit sparse direct solver. It is shown that direct addressing of sparse matrices is possible in general circumstances, circumventing the traditional linear or binary search for introducing (generalized) constituents to a sparse matrix. Nested and arbitrarily interconnected multiple-point constraints are introduced by processing of multiplicative
more » ... s with a built-in topological ordering of the resulting directed graph. A classification of discretization methods is performed and some re-classified problems are described and solved under this proposed perspective. The dependence relations between solution methods, algorithms and constituents becomes apparent. Fracture algorithms can be naturally casted in this framework. Solutions based on control equations are also directly incorporated as equality constraints. We show that arbitrary constituents can be used as long as the resulting directed graph is acyclic. It is also shown that graph partitions and orderings should be performed in the innermost part of the algorithm, a fact with some peculiar consequences. The core of our implicit code is described, specifically new algorithms for direct access of sparse matrices (by means of the clique structure) and general constituent processing. It is demonstrated that the graph structure of the second derivatives of the equality constraints are cliques (or pseudoelements) and are naturally included as such. A complete algorithm is presented which allows a complete automation of equality constraints, avoiding the need of pre-sorting. Verification applications in four distinct areas are shown: single and multiple rigid body dynamics, solution control and computational fracture. store(iel,ndofiel,lnods,ltyps,efor,emat) where: J iel is the global element number. J ndofiel is the number of degrees-of-freedom of element iel. J lnods (size ndofiel) is the list of global nodes corresponding to each degree-of-freedom. J ltyps (size ndofiel) is the list of global types corresponding to each degree-of-freedom. J efor (size ndofiel) is the element "force" vector f . J emat (size ndofiel ndofiel) is the element "stiffness" matrix K.