M. Pastor

A new stabilized enhanced strain element with equal order of interpolation for soil consolidation problems

Abstract In order to accurately model the behaviour of geostructures it is usually not possible to neglect the interaction between the soil skeleton and the pore fluid. Classical finite element models taking into account this interaction are formulated in terms of the displacement and pore pressure fields and are based on the assumption that the fluid acceleration relative to the soil skeleton is negligible. This type of mixed problems is similar to others found in solid and fluid mechanics and might give rise to numerical instabilities unless certain requirements are met. There are two classical approaches to this problem. The first is usually known as the Zienkiewicz–Taylor patch test for mixed formulations. As a consequence of this test the interpolation degree of the displacement field is required to be higher than the corresponding one of the pressure field. Mathematically speaking this is a necessary condition for stability. The second approach, mathematically more involved, is usually known as the Babŭska–Brezzi inf–sup condition and constitutes a sufficient condition for stability. However it is possible to obtain stable formulations circumventing the interpolation degree requirement through the so-called stabilization techniques. These techniques were initially applied in the context of fluid mechanics and later extended to solid mechanics. This article presents a new formulation in which stabilization is achieved through an approach based on the Simo–Rifai enhanced strain element.

An iterative stabilized fractional step algorithm for finite element analysis in saturated soil dynamics

The discretization of the u–p model for saturated porous media results in the semi-discrete system of mixed type in displacements and pressures. The Babuska–Brezzi condition or the simpler patch test proposed by Zienkiewicz and Taylor precludes the use of elements with the equal low order of interpolation for u and p, unless special stabilization techniques, such as the fractional step algorithms, are used. The purpose of this paper is to present a modified version of the fractional step algorithm which allows much larger time step sizes than those for the existing ones. The method is based on introducing an iteration algorithm. The numerical experiments demonstrate the effectiveness and improved performance of the proposed modified version of the fractional step algorithm.

Stabilized low-order finite elements for failure and localization problems in undrained soils and foundations

Geomaterials in general, and soils in particular, are highly nonlinear materials presenting a very strong coupling between solid skeleton and intersticial water. In the limit of zero compressibility of water and soil grains and zero permeability (which correspond to the classical ‘undrained’ assumption of Soil Mechanics), the functions used to interpolate displacements and pressures must fulfill either the Babuska-Brezzi conditions or the much simpler patch test proposed by Zienkiewicz and Taylor. These requirements exclude the use of elements with equal order interpolation for pressures and displacements, for which spurious oscillations may appear. The simplest elements with continuous pressures which can be used in 2D are the quadratic triangle and quadrilateral with linear and bilinear pressures, respectively. The purpose of this paper is to present a stabilization technique allowing the use of both linear triangles for displacements and pressures (T3P3) and bilinear quadrilaterals (Q4P4). The proposed element will be applied to obtain limit loads and failure surfaces in simple boundary value problems for which analytical solutions exist.

A mixed displacement-pressure formulation for numerical analysis of plastic failure

Abstract Finite elements used for analysis of plastic failure must admit near incompressible behaviour for meaningful results whenever the plastic flow rule is isochoric. Most of the classical elements based on displacement formulations present problems to a greater or lesser extent when obtaining limit loads and failure mechanisms. Mixed pressure-displacement formulations are a suitable alternative, but not every combination of interpolation functions for pressures and displacements is allowed, since they have to satisfy Babuska-Brezzi conditions or pass a patch test for convergence. This excludes interpolations of the same order for both fields unless special techniques are used. This paper describes the formulation of mixed displacement-pressure elements with the same order of interpolation, and presents application to elastic and contained viscoplastic failure problems. It is shown how such different elements, like the T3P3 (linear displacements and pressures triangle) or the T6P6 (quadratic triangle in both variables), avoid volumetric locking and circumvent above mentioned restrictions. Indeed the formulation allows arbitrary interpolations as all restrictions are removed. However, the proposed formulation does not improve other aspects of element behaviour such as bending, and, therefore, the linear triangle is not recommended. Finally, performance of the quadratic T6P6 triangle is assessed using boundary value problems for which analytical solutions are available. Robustness against distortion is checked, and a good overall performance of this element is found

A fractional step algorithm allowing equal order of interpolation for coupled analysis of saturated soil problems

The accurate prediction of the behaviour of geostructures is based on the strong coupling between the pore fluid and the solid skeleton. If the relative acceleration of the fluid phase relative to the skeleton is neglected, the equations describing the problem can be written in terms of skeleton displacements (or velocities) and pore pressures. n n n nThis mixed problem is similar to others found in solid and fluid dynamics. In the limit case of zero permeability and incompressibility of the fluid phase, the restrictions on the shape functions used to approximate displacements and pressures imposed by Babuska–Brezzi conditions or the Zienkiewicz–Taylor patch test hold. n n n nAs a consequence, it is not possible to use directly elements with the same order of interpolation for the field variables. n n n nThis paper proposes a generalization of the fractional-step method introduced by Chorin for fluid dynamics problems, which allows to circumvent BB restrictions in the incompressibility limit, thus making it possible to use elements with the same order of interpolation. Copyright

A Taylor–Galerkin algorithm for shock wave propagation and strain localization failure of viscoplastic continua

Abstract This paper presents a Taylor–Galerkin algorithm formulated in terms of velocities and stresses which can be applied to solid dynamics problems requiring good resolution of small wavelengths, such as propagation of shocks. The proposed model is both fast, as it uses simple linear elements (triangles in 2D and tetrahedra in 3D), and accurate. It mitigates locking and mesh alignment problems, and therefore can be applied to localized failure computations without the limitations exhibited by the classical displacement formulations.

Stabilized finite elements with equal order of interpolation for soil dynamics problems

SummaryThe accurate prediction of the behaviour of geostructures is based on the strong coupling between the pore fluid and the solid skeleton. If the relative acceleration of the fluid phase to the skeleton is neglected, the equations describing the problem can be written in terms of skeleton displacements (or velocities) and pore pressures.This mixed problem is similar to others found in solid and fluid dynamics. In the limit case of zero permeability and incompressibility of the fluid phase, the restrictions on the shape functions used to approximate displacements and pressures imposed by Babuska-Brezzi conditions or the Zienkiewicz-Taylor patch test hold.As a consequence, it is not possible to use directly elements with the same order of interpolation for the field variables.This paper proposes two alternative methods allowing us to circumvent the BB restrictions in the incompressibility limit, making it possible to use elements with the same order of interpolation. The first consists on introducing the divergence of the momentum equation of the mixture as an stabilization term, the second is a generalization of the two step-projection method introduced by Chorin for fluid dynamics problems.

Stabilized Finite Elements for harmonic soil dynamics problems near the undrained-incompressible limit

Abstract Simple Finite Element models for soil dynamics and earthquake engineering problems in the frequency domain are a fast and valuable tool providing a first approximation before a full non-linear analysis in the time domain is performed. Quite often the problem concerns saturated soils with very small permeability and pore fluid of neglectable compressibility. In the limit, the permeability is assumed to be zero and the pore fluid incompressible. Here, engineers use standard finite element codes formulated in terms of displacements but incompressibility may result in volumetric locking of the mesh with a severe loss of accuracy. The purpose of this paper is to present a simple mixed finite element formulation in the frequency domain based on displacements and pore pressures as main variables. A suitable stabilization technique allowing for equal order interpolation of displacements and pressures has been introduced for incompressible and zero permeability limits. Of course, the range of application is limited to those problems in which the behaviour of the material can be approximated by linear models, and therefore modelling of phenomena such as liquefaction, cyclic mobility or cavitation occur is excluded. The paper shows as well an extremely simple way of coupling solid and water domains as it occurs for instance in quay walls under dynamic loading.

Modelling of Landslides: (II) Propagation

This paper presents a numerical model wich can be used to simulate phenomena such as flowslides, avalanches, mudflows and debris flows. The proposed approach is eulerian, and balance of mass and momentum equations are integrated on depht. Depending on the material involved, the model can implement rheological models such as Bingham or frictional fluids. A simple dissipation law allows the approximation of pore pressure dissipation in the sliding mass. Finally some applications are presented.

Modelling of Landslides: (I) Failure Mechanisms

This paper presents a theoretical and numerical framework to model the initiation mechanisms of catastrophic landslides. The equations describing the coupling between the solid skeleton and the pore fluids are presented following an eulerian approach based on the mixture theory that can provide a unified formulation for both initiation and propagation phases. The system of Partial Differential Equations is then discretized using the classical Galerkin Finite Element Method and neglecting the convective terms. Some applications to localized and diffuse failure will be presented.