Abimael F. D. Loula

Improved accuracy in finite element analysis of Biot's consolidation problem

Abstract Numerical analysis and error estimates of finite element approximations of Biots consolidation problem are presented. Initially different orders of interpolation are employed, leading to lower orders of convergence for the pore pressure compared to the displacements of the porous medium. Lower accuracy also occurs in the approximation of the effective stress tensor, whether it is calculated directly from the constitutive equation, or through a primal mixed stress formulation. To improve the rates of convergence of the pore pressure and effective stresses, a sequential Galerkin Petrov-Galerkin post-processing technique is proposed.

Asymptotic behavior of semidiscrete finite-element approximations of Biot's consolidation problem

Error estimates for spatially discrete Galerkin finite-element approximations of Biot’s model for consolidation of saturated porous media are presented. The short- and long-time behaviors of such approximations based on both stable and unstable combinations of finite-element spaces of displacement and pore pressure fields are discussed.

A new family of stable elements for nearly incompressible elasticity based on a mixed Petrov-Galerkin finite element formulation

SummaryAdding to the classical Hellinger Reissner formulation another residual form of the equilibrium equation, a new Petrov-Galerkin finite element method is derived. It fits within the framework of a mixed finite element method and is proved to be stable for rather general combinations of stress and displacement interpolations, including equal-order discontinuous stress and continuous displacement interpolations which are unstable within the Galerkin approach. Error estimates are presented using the Babuška-Brezzi theory and numerical results confirm these estimates as well as the good accuracy and stability of the method.

Mixed Petrov-Galerkin methods for the Timoshenko beam problem

Abstract A new mixed Petrov-Galerkin method is presented for the Timoshenko beam problem. The method has enhanced stability compared to the Galerkin formulation, allowing new combinations of interpolation, in particular, equal-order stress and displacement fields. The methodology is easily generalizable for multi-dimensional Hellinger-Reissner systems.

Stability, convergence, and accuracy of a new finite element method for the circular arch problem

Abstract The arch problem with shear deformation based upon the Hellinger-Reissner variational formulation is studied in a parameter-dependent form. A mixed Petrov-Galerkin method is used to construct a discrete approximation. Finite elements with equal-order discontinuous stress and continuous displacement interpolations, unstable in the Galerkin method, are proved to be stable in the new formulation. Error estimates indicate optimal rates of convergence for displacements and suboptimal rates, with gap one, for stresses. Numerical experiments confirm these estimates. The good accuracy of the mixed Petrov-Galerkin method is illustrated in some deep and shallow thin arch examples. No shear or membrane locking is present using full integration schemes.

Micromechanical computational modeling of secondary consolidation and hereditary creep in soils

Abstract A computational modeling of a fluid-saturated deformable porous media characterized by two levels of hydrodynamics (flow in micro and macropores) is proposed based on a micromechanical analysis of dual porosity systems, i.e., media locally characterized by a porous matrix composed of permeable cells containing micropores and the surrounding system of macropores, void spaces or bulk flow paths (e.g., fissured rock or aggregated soil). The homogenization technique is applied to upscale the constitutive and geometric information available in the fine structure to the field scale leading to a microstructure model of dual porosity type, wherein the poroelastic cells act as distributed sources/sinks of mass and momentum to the global macroscopic medium. The theory provides a rigorous derivation of some secondary compression and hereditary creep effects in soils due to the delayed drainage of the fluid within the micropores under consolidation. Application of the Greens function method reduces the dual porosity system to a single-porosity viscoelastic integrodifferential system of Volterra type in which the constitutive law for the macroscopic stress tensor is given in terms of an hereditary integral with memory. A two-level finite element method is proposed to solve the coupled micro–macro governing equations of dual porosity type. Numerical experiments are performed showing the strong potential of the proposed formulation in solving consolidation problems with microstructure.

Higher-order gradient post-processings for second-order elliptic problems

Abstract Global, element-by-element and macroelement post-processing recovery techniques based on least-square residuals of equilibrium equation and irrotationality condition are proposed for second-order elliptic problems. Improved accuracy for the flux finite element approximations is obtained with low computational cost and easy implementation. Error estimates are derived and numerical experiments are reported confirming the higher-order rates of convergence predicted in the analysis.

Finite element analysis of nonlinear creeping flows

Abstract Steady-state creep problems with monotone constitutive laws are studied. Finite element approximations are constructed based on mixed Petrov-Galerkin formulations for constrained problems. Stability, convergence and a priori error estimates are proved for discontinuous stress and continuous velocity interpolations of the same order. Numerical results are presented confirming the rates of convergence predicted in the analysis and the good performance of this formulation.

Petrov-Galerkin formulations of the Timoshenko beam problem

Abstract Petrov-Galerkin formulations of the Timoshenko beam problem are presented. They are shown to provide the best approximation property, optimal rate of convergence, and nodally exact solution for arbitrary loading, for all values of the thickness of the beam.

Miscible displacement simulation by finite element methods in distributed memory machines

Abstract Finite element methods taylored for large scale simulation of incompressible miscible displacement in porous media are presented. Employing same order Lagrangian interpolations for all variables, pressure is approximated by Galerkins method, global or local post-processing techniques are used to compute higher-order velocity approximations, and a stabilized Petrov—Galerkin formulation is applied to concentration equation. Error estimates are discussed as well as the parallel implementation on a distributed memory machine. Numerical simulations of tracer injection processes and miscible displacements with high adverse mobility ratios in two and three dimensions are reported.