Local Maximum Entropy Shape Functions Based FE-EFGM Coupling

In this paper, a new method for coupling the finite element method (FEM)and the element-free Galerkin method (EFGM) is proposed for linear elastic and geometrically nonlinear problems using local maximum entropy shape functions in theEFG zone of the problem domain. These shape functions possess a weak Kroneckerdelta property at the boundaries which provides a natural way to couple the EFGand the FE regions as compared to the use of moving least square basis functions.In this new approach, there is no need for interface/transition elements between theEFG and the FE regions or any other special treatment for shape function continuity across the FE-EFG interface. One- and two-dimensional linear elastic and two-dimensional geometrically nonlinear benchmark numerical examples are solved by the new approach to demonstrate the implementation and performance of the current approach

On the use of Reuleaux plasticity for geometric non-linear analysis.

Three dimensional analyses including geometric and material non--linearity require robust, efficient constitutive models able to simulate engineering materials. However, many existing constitutive models have not gained widespread use due to their computational burden and lack of guidance on choosing appropriate material constants. Here we offer a simple cone-type elasto-plastic formulation with a new deviatoric yielding criterion based on a modified Reuleaux triangle. The perfect plasticity model may be thought of as a hybrid between Drucker-Prager (D-P) and Mohr-Coulomb (M-C) that provides control over the internal friction angle independent of the shape of the deviatoric section. This surface allows an analytical backward Euler stress integration on the curved surface and exact integration in the regions where singularities appear. The attraction of the proposed algorithm is the improved fit to deviatoric yielding and the one--step integration scheme, plus a fully defined consistent tangent. The constitutive model is implemented within a lean 3D geometrically non-linear finite-element program. By using an updated Lagrangian logarithmic strain--Kirchhoff stress implementation, existing infinitesimal constitutive models can be incorporated without modification

A review of the Material Point Method and its links to other computational methods.

There is considerable interest in development of solid mechanics modelling which can cope with both material and geometric nonlinearity, particularly in areas such as computational geotechnics, for applications such as slope failure and foundation installation. One such technique is the Material Point Method (MPM), which appears to provide an efficient way to model these problems. The MPM models a problem domain using particles at which state variables are kept and tracked. The particles have no restriction on movement, unlike in the Finite Element Method (FEM) where element distortion limits the level of mesh deformation. In the MPM, calculations are carried out on a regular background grid to which state variables are mapped from the particles. It is clear, however, that the MPM is actually closely related to existing techniques, such as ALE and in this paper we review the MPM for solid mechanics and demonstrate these links

Implicit essential boundaries in the Material Point Method

The Material Point Method (MPM) is a numerical boundary value problem (BVP) solver developed from particle- in-cell (PIC) methods that discretises the continuum into a set of material points. Information at these material points is mapped to a background Eulerian grid which is used to solve the governing equations. Once solved information is updated at the material points and these points are convected through the grid. The background grid is then reset, allowing the method to easily handle problems involving large deformations without mesh distortion. However, imposition of essential boundary conditions in the (MPM) is challenging when the physical domain does not conform to the background grid. In this research, an implicit boundary method (IBM), based on the work of Kumar et al. [1], is proposed to ensure that essential boundary conditions are satisfied in elastostatic MPM problems

70-line 3D finite deformation elastoplastic finite-element code.

Few freeware FE programs offer the capabilities to include 3D finite deformation inelastic continuum analysis; those that do are typically expressed in tens of thousands of lines. This paper offers for the first time compact MATLAB scripts forming a complete finite deformation elasto–plastic FE program. The key modifications required to an infinitesimal FE program in order to include geometric non–linearity are described and the entire code given

On the use of advanced material point methods for problems involving large rotational deformation

The Material Point Method (MPM) is a quasi Eulerian-Lagrangian approach to solve solid mechanics problems involving large deformations. The standard MPM [1] discretises the physical domain using material points which are advected through a standard finite element background mesh. The method of mapping state variables back and forth between the material points and background mesh nodes in the MPM significantly influences the results. In the standard MPM (sMPM), a material point only influences its parent element (i.e. the background element in which it is located), which can cause spurious stress oscillations when material points cross between elements. The instability is due to the sudden transfer of stiffness between elements. It can also result in some elements having very little stiffness or some internal elements loosing all stiffness. Therefore, several extensions to the sMPM have been proposed, each of which replaces the material point with a deformable particle domain. The most notable of these extensions are the Generalised Interpolation Material Point (GIMP), the Convected Particle Domain Interpolation (CPDI1) and Second-order CPDI (CPDI2) methods [2]. In this paper, the sMPM, CPDI1 and CPDI2 approaches are unified for geometrically non-linear elasto-plastic problems using an implicit solver and their performance investigated for large rotational problems. This type of deformation is common in applications in the area of soil mechanics, for example the vane shear test and, specifically of interest here, the installation of screw piles. Screw piles are currently used as an onshore foundation solution and research being undertaken at Durham, Dundee and Southampton universities is exploring their use in the area of offshore renewables. The numerical modelling using the MPM aims to predict the installation torque and vertical force as well as understanding the “state” of the soil around the screw pile which is critical in understanding the long term performance of the foundation. In the analysis, the pile is assumed to be a rigid body and no-slip boundary condition is used at the pile-soil interface. The boundary condition is imposed using the moving mesh concept within an unstructured mesh fixed to the pile. It will be shown that the CPDI2 approach produces erroneous torque due to particle domain distortion, while the CPDI1 approach and sMPM predict physically realistic mechanical responses

Gradient elasticity with the material point method

The Material Point Method (MPM) is a method that allows solid mechanics problems with large deformation and non-linearity to be modelled using particles at which state variables are stored and tracked. Calculations are then carried out on a regular background grid to which state variables are mapped from the particles. There have been a selection of extensions to the MPM, for example, a problem in the original method that arises when a material point crosses the boundary between one background grid cell and another is addressed by the Generalised Interpolation Material Point (GIMP) method [2]. An area yet to be studied in as much depth in MPM is that conventional analysis techniques constructed in terms of stress and strain are unable to resolve structural instabilities such as necking or shear banding. This is due to the fact that they do not contain any measure of the length of the microstructure of the material analysed such as molecule size of grain structure. Gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher-order terms [3]. This use of length scales makes it possible to model finite thickness shear bands that is not possible with traditional methods. Much work has been done on including the effect of microstructure on a linear elastic solid and has previously been combined with the Finite Element Method and with the Particle In Cell Method. In this work the MPM will be developed to include gradient effects

An open-source Julia code for geotechnical MPM

There is considerable interest in the Material Point Method (MPM) in the computational geotechnics community since it can model problems involving large deformations, e.g. landslides, collapses etc. without being too far from the standard finite element method, which can struggle with large deformation problems. The open-source code AMPLE developed at Durham University in recent years is a compact set of MATLAB functions that “address the severe learning curve for researchers wishing to understand, and start using, the MPM”. It is well known that MATLAB can be very slow hence limiting its utility for major studies of large problems, so here we introduce an MPM code with the same aims as AMPLE but written in the relatively new language Julia, specifically for fast runtimes. We highlight areas where MATLAB code constructs are inefficient if just transferred to Julia and show that to unlock large speed gains with Julia, one needs to code in a different way and we demonstrate this on a geotechnical problem. While this paper is concerned with the MPM, the advice regarding coding using Julia is transferable to other computational geotechnics methods and tools

On the use of Reuleaux plasticity for geometric non-linear analysis

Three dimensional analyses including geometric and material non--linearity require robust, efficient constitutive models able to simulate engineering materials. However, many existing constitutive models have not gained widespread use due to their computational burden and lack of guidance on choosing appropriate material constants. Here we offer a simple cone-type elasto-plastic formulation with a new deviatoric yielding criterion based on a modified Reuleaux triangle. The perfect plasticity model may be thought of as a hybrid between Drucker-Prager (D-P) and Mohr-Coulomb (M-C) that provides control over the internal friction angle independent of the shape of the deviatoric section. This surface allows an analytical backward Euler stress integration on the curved surface and exact integration in the regions where singularities appear. The attraction of the proposed algorithm is the improved fit to deviatoric yielding and the one--step integration scheme, plus a fully defined consistent tangent. The constitutive model is implemented within a lean 3D geometrically non-linear finite-element program. By using an updated Lagrangian logarithmic strain--Kirchhoff stress implementation, existing infinitesimal constitutive models can be incorporated without modification