Comparison between iterative solution of symmetric and non‐symmetric forms of Biot's FEM equations using the generalized Jacobi preconditioner

“…This symmetry property can be exploited by using solvers for the symmetric, indefinite problem 1 . As pointed out by Toh and Phoon , sparse direct solvers taking advantage of symmetries of the underlying formulation are up to 50% more efficient than non‐symmetric sparse direct solvers.…”

Section: A Variational Principle For the Evolution Problemmentioning

confidence: 99%

Formulation and numerical exploitation of mixed variational principles for coupled problems of Cahn–Hilliard‐type and standard diffusion in elastic solids

Miehé

Mauthe

Ulmer

2014

Numerical Meth Engineering

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SUMMARYThis work develops variational principles for the coupled problem of standard and extended Cahn–Hilliard‐type species diffusion in solids undergoing finite elastic deformations. It shows that the coupled problem of diffusion in deforming solids, accounting for phenomena like swelling, diffusion‐induced stress generation and possible phase segregation caused by the diffusing species, is related to an intrinsic mixed variational principle. It determines the rates of deformation and concentration along with the chemical potential, where the latter plays the role of a mixed variable. The principle characterizes a canonically compact model structure, where the three governing equations involved, that is, the mechanical equilibrium condition, the mass balance for the species content and a microforce balance that determines the chemical potential, appear as the Euler equation of a variational statement. The existence of the variational principle underlines an inherent symmetry of the coupled deformation–diffusion problem. This can be exploited in the numerical implementation by the construction of time‐discrete and space‐discrete incremental potentials, which fully determine the update problems of typical time stepping procedures. The mixed variational principles provide the most fundamental approach to the monolithic finite element solution of the coupled deformation–diffusion problem based on low‐order basis functions. They induce in a natural format the choice of symmetric solvers for Newton‐type iterative updates, providing a speedup and reduction of data storage when compared with non‐symmetric implementations. This is a strong argument for the use of the developed variational principles in the computational design of deformation–diffusion problems. Copyright © 2014 John Wiley & Sons, Ltd.

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“…In the previous chapters, we show that the Terzaghi theory of consolidation may be understood as a way of decoupling the Biot system of equations. A comparison between an iterative solution of symmetric and non-symmetric forms of Biot's FEM equations has been presented in Toh and Phoon (2007). A tremendous effort on developing an accurate and stable numerical scheme for the coupled system of equations has been performed (Lippmann et al 1976;Hicks et al 1996;Narasimhan and Witherspoon 1976;Aguilar et al 2008; Barbeiro and Wheeler 2008;Hörlin 2010).…”

Section: • a Solution Of The Biot Equationsmentioning

confidence: 99%