Conditioning analysis of separate displacement preconditioners for some nonlinear elasticity systems

Quasi‐Newton variable preconditioning for nonlinear elasticity systems in 3D

“…Preconditioner 3. A next simplification of $$$ {B}_n^{(1)} $$$ is motivated by the so‐called "separate displacement preconditioning", elaborated in References 14‐16. Thereby proper block diagonal preconditioners for linear elasticity problems, acting as three independent scalar Laplacians, were defined.…”

“…Remark As summarized for example, in References 13,16, there are several partial results available about the exact or approximate value of Korn's constant in different situations. Some most frequently studied conditions are as follows: Zero Dirichlet boundary conditions (pure displacement): $$$ {\Gamma}_D=\mathrm{\partial \Omega } $$$, that is, $$$$ u=0\kern1em \mathrm{on}\kern0.3em \mathrm{\partial \Omega }.$$…”

Equivalent operator preconditioning for elliptic problems

“…when the Poisson ratio is less than 1/2, otherwise a mixed formulation can be proposed). The detailed development of the separate displacement method is given in [5,29] for linear elasticity and in [14] in the nonlinear case. We rely on [14] and refer the reader there whenever more details are required.…”

“…when the Poisson ratio is close to 1/2, for nearly incompressible materials), which corresponds to the locking phenomenon, common for typical discretizations. A possible remedy is using a suitable mixed formulation, for which an outer-inner iteration scheme can still preserve mesh independent condition numbers, see [14] for details.…”

“…A similar preconditioning approach is used in [90,109] where preconditioning by S is replaced by a related solution operator, for first order problems in [37] and for second order nonlinear problems in e.g. [99] and the authors' works [9,14,56,79].…”