Computer Methods in Applied Mechanics and Engineering

We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB 'deficiency' of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.

Stabilization of low-order mixed finite elements for the stokes equations.

Thank you very much for downloading mathematical foundations of elasticity. As you may know, people have search numerous times for their favorite books like this mathematical foundations of elasticity, but end up in infectious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they are facing with some infectious bugs inside their laptop. mathematical foundations of elasticity is available in our book collection an online access to it is set as public so you can download it instantly. Our digital library saves in multiple countries, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the mathematical foundations of elasticity is universally compatible with any devices to read.

Mathematical Foundations Of Elasticity

Numerical simulation of undrained insertion problems in geotechnical engineering with the Particle Finite Element Method (PFEM)

The particle finite element method (PFEM) is a powerful and robust numerical tool for the simulation of multi-physics problems in evolving domains. The PFEM exploits the Lagrangian framework to automatically identify and follow interfaces between different materials (e.g. fluid–fluid, fluid–solid or free surfaces). The method solves the governing equations with the standard finite element method and overcomes mesh distortion issues using a fast and efficient remeshing procedure. The flexibility and robustness of the method together with its capability for dealing with large topological variations of the computational domains, explain its success for solving a wide range of industrial and engineering problems. This paper provides an extended overview of the theory and applications of the method, giving the tools required to understand the PFEM from its basic ideas to the more advanced applications. Moreover, this work aims to confirm the flexibility and robustness of the PFEM for a broad range of engineering applications. Furthermore, presenting the advantages and disadvantages of the method, this overview can be the starting point for improvements of PFEM technology and for widening its application fields.

/pdf/a-state-of-the-art-review-of-the-particle-finite-element-4yttz7zq9k.pdf

A State of the Art Review of the Particle Finite Element Method (PFEM)

Coupled effective stress analysis of insertion problems in geotechnics with the Particle Finite Element Method

This paper presents a computational framework for the numerical analysis of fluid-saturated porous media at large strains. The proposal relies, on one hand, on the particle finite element method (PFEM), known for its capability to tackle large deformations and rapid changing boundaries, and, on the other hand, on constitutive descriptions well established in current geotechnical analyses (Darcy’s law; Modified Cam Clay; Houlsby hyperelasticity). An important feature of this kind of problem is that incompressibility may arise either from undrained conditions or as a consequence of material behaviour; incompressibility may lead to volumetric locking of the low-order elements that are typically used in PFEM. In this work, two different three-field mixed formulations for the coupled hydromechanical problem are presented, in which either the effective pressure or the Jacobian are considered as nodal variables, in addition to the solid skeleton displacement and water pressure. Additionally, several mixed formulations are described for the simplified single-phase problem due to its formal similitude to the poromechanical case and its relevance in geotechnics, since it may approximate the saturated soil behaviour under undrained conditions. In order to use equal-order interpolants in displacements and scalar fields, stabilization techniques are used in the mass conservation equation of the biphasic medium and in the rest of scalar equations. Finally, all mixed formulations are assessed in some benchmark problems and their performances are compared. It is found that mixed formulations that have the Jacobian as a nodal variable perform better.

/pdf/performance-of-mixed-formulations-for-the-particle-finite-1x1r81u7i2.pdf

Performance of mixed formulations for the particle finite element method in soil mechanics problems

The mechanics of fully saturated porous media is relevant for many engineering applications. A fundamental aspect of that mechanics is the description of couplings between the solid and fluid phases, for which several theoretical models have been proposed. One of the most successful proposals has been that by Biot.1, who accepted the effective stress concept proposed by Terzaghi, and extended its reach from quasi-static one-dimensional problems to a dynamic three-dimensional setting. The Biot equations allow for several choices of primary variables. For solutions using the finite element method, Zienkiewicz et al2,3 discuss alternative formulations. Some of these, denoted as u−ẇ−pw, u−U or u−U−pw are able to represent the whole set of Biot equations (full Biot). Another, known as the u− pw or reduced formulation is valid only when fluid accelerations are far smaller than those of the solid phase. Of those variables u and U represent, respectively, the absolute displacement vectors of the solid skeleton and the fluid phase, w the relative displacement between the phases and pw the fluid pressure. The reduced u− pw formulationis not applicable to high-frequency loading, such as that resulting from impact events3. On the other hand, it is used quite often, since many seismic applications fall within its remit and the formulation is simpler and, having less degrees of freedom per node, computationally less demanding. This widespread use has brought much attention to potential numerical problems that may arise during its solution and, particularly, the development of stable formulations. Unstable elements manifest themselves in the numerical solution as over-stiffening of mechanical response and spurious high spatial variability of the water pressure field, eventually leading to instability, non-uniqueness and mesh-dependence of the solutions. This pathology appears when the problem conditions lead to incompressibility. For water-saturated soils this is approximately the case in undrained conditions. The system of discretized equations for the u − pw formulation, has then similar structure to that found when using a mixed u−p formulation of Solid Mechanics problems4. As the problem approaches zero permeability and given the quasi-incompressilibity of the soil and water constituents, it may become ill-posed from a mathematical point of view. This numerical pathology is due to an improper finite-dimensional space in the finite element discretization and may be identified using analytical tools such as the inf-sup condition5 or the patch-test. More importantly, it can be also avoided or

/pdf/low-order-stabilized-finite-element-for-the-full-biot-4sv07baez5.pdf

Low-order stabilized finite element for the full Biot formulation in soil mechanics at finite strain

https://discovery.dundee.ac.uk/ws/files/37185552/POSTPRINT_Pfem_Structure_LMV_MC_JMC_MA_AG_2019_CG.pdf

Lluís Monforte

Papers

Numerical simulation of undrained insertion problems in geotechnical engineering with the Particle Finite Element Method (PFEM)

Coupled effective stress analysis of insertion problems in geotechnics with the Particle Finite Element Method

Performance of mixed formulations for the particle finite element method in soil mechanics problems

Low-order stabilized finite element for the full Biot formulation in soil mechanics at finite strain

A stable mesh-independent approach for numerical modelling of structured soils at large strains