Implicit solutions with consistent additive and multiplicative components

Abstract

a b s t r a c tThis 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 constituents 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.

The extended unsymmetric frontal solution for multiple-point constraints

“…Since the frontal solution method is purely clique-based, no need for assembling exists (not even symbolic assembling) and therefore a considerable part of our previous derivation is not required. In addition, since the role of DOF in the frontal solution method is limited to the elimination order, DOF contraction is not required, simplifying the MPC pre-processing required in Areias et al (2012 and the amount of indirect addressing. A topological ordering is performed, similarly to our recent developments but with considerable simplifications.…”

“…As can be observed in Figure 2, there are no repetitions [7] in the processing of the sequences of DOFs if the order of the Hasse diagram is followed. Multiplication of transformation matrices benefit from this procedure Areias et al, 2012. Non-slave nodes have unit T * -coefficients whereas slave nodes' T * -coefficients depend on the specific constraint imposed.…”

A Dimensional Reduction Algorithm and Software for Acyclically Dependent Constraints

“…Specific formulations of many of such constituents are provided in the book by Belytschko et al [1] and in many papers, see e.g. [2]. Details concerning the solution of problems resulting from systematic creation and combination of new constituents (made possible with tools such as Mathematica [3] with the AceGen add-on [4]), has not been shown with Algorithmic depth in the literature.…”

Modeling of dynamic fracture based on the cracking particles method

“…Artificial fracture has been one concern in modeling fracture and fragmentation in early meshfree methods but those issues have been resolved now [24][25][26]. Meshfree methods for discrete crack approaches have been proposed for example by [27][28][29], alternative approaches are for instances based on phase fields [30,31], remeshing based on edge rotation [32][33][34][35][36][37][38], or multiscale approaches [39][40][41][42][43] or others [44][45][46][47]. A very promising approach for fragmentation is the cracking particles method (CPM) developed by [48].…”