document: We order the discrete unknowns so that the vector of unknowns, xPS=[X,L], contains the nx unknown nodal coordinates, followed by the nb unknown discrete Lagrange multipliers. The linear systems to be solved in the course of the Newton-based solution of Eq. (10), subject to the displacement constraint (9), then have saddle-point structure,(15)where E is the tangent stiffness matrix of the unconstrained pseudo-solid problem, and the two off-diagonal blocks Cxl and Clx=CxlT arise through the imposition of the displacement constraint by the Lagrange multipliers. We refer to [34] for the proof of the LBB stability of this discretisation; see also [35,36] for a discussion of the LBB stability of the Lagrange-multiplier-based imposition of Dirichlet boundary conditions in related problems. We note that during the first step of the Newton iteration, E is symmetric positive definite since it represents the tangent stiffness matrix relative to the system’s equilibrium configuration.
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extractive_keyphrases: cxl dirichlet boundary conditions discretisation displacement constraint equilibrium configuration lagrange multipliers lbb newton iteration nodal coordinates order the discrete unknowns tangent stiffness matrix vector of unknowns
abstractive_keyphrases: clx=cxlt lagrange-multiplier-based imposition newton-based solution saddle-point structure two off-diagonal blocks unconstrained pseudo-solid problem xps=[x,l]