On the Relationship between Primal/Dual Cell Complexes of the Cell Method and Primal/Dual Vector Spaces: an Application to the Cantilever Elastic Beam with Elastic Inclusion

Abstract

The Cell Method (CM) is an algebraic numerical method based on the use of global variables: the configuration, source and energetic global variables. The configuration variables with their topological equations, on the one hand, and the source variables with their topological equations, on the other hand, define two vector spaces that are a bialgebra and its dual algebra. The operators of these topological equations are generated by the outer product of the geometric algebra, for the primal vector space, and by the dual product of the dual algebra, for the dual vector space. The topological equations in the primal cell complex are coboundary processes on even exterior discrete p−forms, whereas the topological equations in the dual cell complex are coboundary processes on odd exterior discrete p−forms. Being expressed by coboundary processes in two different vector spaces, compatibility and equilibrium can be enforced at the same time, with compatibility enforced on the primal cell complex and equilibrium enforced on the dual cell complex. By way of example, in the present paper compatibility and equilibrium are enforced on a cantilever elastic beam with elastic inclusion. In effect, the CM shows its maximum potentialities right in domains made of several materials, as, being an algebraic approach, can treat any kind of discontinuities of the domain easily.

DECM: A Discrete Element for Multiscale Modeling of Composite Materials Using the Cell Method

“…In the spirit of comparison, we will now consider a 6 4 × array generated by the discrete element with round inclusion in Figure 20 and subjected to uniaxial traction ( Figure 26). The comparison results are those obtained in [29] Figure 20. The plot in Figure 28 gives the normal stress field x σ generated in the elastic cantilever beam by a tensile stress Along the bi-material cross-sections in Figure 28, x σ increases where the local stiffness is higher and decreases where the local stiffness is lower.…”

“…). [29] In the 6 4 × array of Figure 26, where 2 10 y p p = = kN m , the discrete elements prevent the adjacent elements of the same row from shrinking or expanding freely in the transverse direction (Figure 30a). On the vertical inner sides in Figure 31a, this gives rise to positive and negative normal stresses x σ , which would not be present in the continuum in the absence of inclusions.…”

“…A further micro-modeling approach that has proved to be suitable for modeling matrix/inclusions interactions is the Cell Method (CM) [29][30][31][32][33][34][35]. As far as the modeling of URM structures is concerned, the CM treats the masonry as a bi-material consisting of mortar and bricks and provides descriptions up to the scale of the individual bricks [36].…”

“…Since the global variables involved in obtaining the direct algebraic formulation need not necessarily be differentiable functions, the range of applicability of the algebraic formulation has no restrictions, while that of the differential formulation is restricted to regions without material discontinuities or concentrated sources. In fact, global variables are continuous across the interface of two different media, while their variations-hence also the field variables-can be discontinuous [29].…”

“…Geometry and loading condition of the elastic cantilever beam with round inclusion [29]. Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 3 December 2019 doi:10.20944/preprints201912.0014.v1…”