Conservation of Forces and Total Work at the Interface Using the Internodes Method

Abstract

The Internodes method is a general purpose method to deal with non-conforming discretizations of partial differential equations on 2D and 3D regions partitioned into disjoint subdomains. In this paper we are interested in measuring how much the Internodes method is conservative across the interface. If hp-fem discretizations are employed, we prove that both the total force and total work generated by the numerical solution at the interface of the decomposition vanish in an optimal way when the mesh size tends to zero, i.e., like $\mathcal {O}(h^{p})$ O ( h p ) , where p is the local polynomial degree and h the mesh-size. This is the same as the error decay in the H1-broken norm. We observe that the conservation properties of a method are intrinsic to the method itself because they depend on the way the interface conditions are enforced rather then on the problem we are called to approximate. For this reason, in this paper, we focus on second-order elliptic PDEs, although we use the terminology (of forces and works) proper of linear elasticity. Two and three dimensional numerical experiments corroborate the theoretical findings, also by comparing Internodes with Mortar and WACA methods.

The INTERNODES method for applications in contact mechanics and dedicated preconditioning techniques

“…It is a flexible interpolation based method which overcomes many of the implementation issues of the mortar method and was shown to be at least as accurate [5]. In a recent paper [6], the authors have shown that INTERNODES, like the Mortar method, allows to approximate the conservation of specific quantities, namely that both total force and total work generated by the numerical solution at the interface of the decomposition vanish in an optimal way when the mesh size tends to zero. Preliminary work in computational contact mechanics showed that the INTERN-ODES method could be successfully applied but also revealed several challenges in efficiently solving the sequence of linear systems arising from the method [7].…”