The three‐field formulation for elasticity problems

Abstract

MSC (2000) 65N12, 65N22, 65N30, 65N55The three-field decomposition method is particularly suited for decompositions with non matching grids. It corresponds to introduce an additional grid (usually uniform, or "easy") at the interface. The unknown is then represented independently in each subdomain and on the interface. The matching between its value in each subdomain and on the interface is provided by suitable Lagrange multipliers. Here we discuss the main features of the method for a linear three-dimensional elasticity problem, in the simplest case of two subdomains. An easy numerical test to check whether the inf-sup conditions (necessary for the stability) are satisfied is also presented.

Non-intrusive Coupling: Recent Advances and Scalable Nonlinear Domain Decomposition

“…• When domain decomposition is used as a parallel solver, interfaces are most of time mesh-conforming. Nevertheless, nothing prevent the use of non-conforming interfaces [2,17,45,71] especially when one is dealing with heterogeneous models. As shown before, the above presented non-intrusive domain coupling solver is also ready for incompatible meshes at interface.…”

Asynchronous multi‐domain variational integrators for non‐linear problems

“…We employ the method of local Lagrange multipliers, introduced by Park et al [26], where each domain is constrained to an intermediate interface with position w, as shown in Figure 2. This method eliminates over-constraints where more than two subdomains meet, and is advantageous for non-matching meshes, as shown in Brezzi and Marini [28] and Park et al [27]. For simplicity, we will consider two domains, labeled with superscripts a and b.…”

“…Park et al [26] developed a variant using local Lagrange multipliers that constrain domains to an intermediate interface rather than directly to each other. Such an intermediate interface is used in Park et al [27] and Brezzi and Marini [28] to handle domains with non-matching meshes, as illustrated in Figure 1. We will consider the case of PDEs with non-matching meshes in future work, and thus introduce an intermediate interface here in anticipation.…”

Asynchronous multi-domain variational integrators for nonlinear hyperelastic solids

“…Park et al [36] developed a variant using local Lagrange multipliers that constrains domains to an intermediate interface rather than directly to each other. Such an intermediate interface is used in Park et al [37] and Brezzi and Marini [9] to handle domains with non-matching meshes, as illustrated in Fig. 1.…”