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Von Neumann stability analysis of Biot's general two‐dimensional theory of consolidation
Author(s) -
Miga Michael I.,
Paulsen Keith D.,
Kennedy Francis E.
Publication year - 1998
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/(sici)1097-0207(19981115)43:5<955::aid-nme452>3.0.co;2-y
Subject(s) - biot number , discretization , mathematics , neumann boundary condition , galerkin method , mathematical analysis , finite element method , boundary value problem , physics , mechanics , thermodynamics
Abstract Von Neumann stability analysis is performed for a Galerkin finite element formulation of Biot's consolidation equations on two‐dimensional bilinear elements. Two dimensionless groups—the Time Factor and Void Factor—are identified and these quantities, along with the time‐integration weighting, are used to explore the stability implications for variations in physical property and discretization parameters. The results show that the presence and persistence of stable spurious oscillations in the pore pressure are influenced by the ratio of time‐step size to the square of the space‐step for fixed time‐integration weightings and physical property selections. In general, increasing the time‐step or decreasing the mesh spacing has a smoothing effect on the discrete solution, however, special cases exist that violate this generality which can be readily identified through the Von Neumann approach. The analysis also reveals that explicitly dominated schemes are not stable for saturated media and only become possible through a decoupling of the equilibrium and continuity equations. In the case of unsaturated media, a break down in the Von Neumann results has been shown to occur due to the influence of boundary conditions on stability. © 1998 John Wiley & Sons, Ltd.