Abstract

In this article the behavior of a shape function based on the maximum entropy principle (maxent) is analyzed in a meshless collocation method, compared with a traditional fixed weighted least square shape function (FWLS). The maxent shape function used in this work has certain properties that are desired in a meshless collocation method, for example the positivity, the smooth and uniform aspect for different discretizations. Further, in the boundary, the approximation not depends of the shape function of the interior nodes, this property is know as a reduction of the shape function on the boundary. To compare this type of function, it was developed examples that include the solution of eliptical second order equations in 1D and 2D. The numerical results shown a better behavior of the maxent shape function compared with the FWLS, particularly in terms of the convergence and stability of the meshless collocations method that result.

Abstract

Abstract

Locking in finite elements has been a major concern since its early developments. It appears because poor numerical interpolation leads to an over-constrained system. This paper proposes a new formulation that asymptotically suppresses locking for the Element Free Galerkin (EFG) method in incompressible limit, i.e. the so-called volumetric locking. Originally it was claimed that EFG did not present volumetric locking. However, recently, performing a modal analysis, the senior author has shown that EFG presents volumetric locking. In fact, it is concluded that an increase of the dilation parameter attenuates, but never suppresses, the volumetric locking and that, as in standard finite elements, an increase in the order of reproducibility (interpolation degree) reduces the relative number of locking modes. Here an improved formulation of the Element Free Galerkin method is proposed in order to a alleviate volumetric locking.

Diffuse derivatives are defined in the thesis of the second author and their convergence to the derivatives of the exact solution, when the radius of the exact solution, when the radius of the support goes to zero (for a fixed dilation parameter) it’s proved. Therefore, diffuse solution, converges to the exact divergence. Since the diffuse divergence-free condition can be imposed a priori, new interpolation functions are defined that asymptotically verify the incompressibility condition. Modal analysis and numerical results for classical benchmark test in solids corroborate this issue.

Abstract

Volumetric locking (locking in the incompressible limit) for linear elastic isotropic materials is studied in the context of the element-free Galerkin method. The modal analysis developed here shows that the number of non-physical locking modes is independent of the dilation parameter (support of the interpolation functions). Thus increasing the dilation parameter does not suppress locking. Nevertheless, an increase in the dilation parameter does reduce the energy associated with the non-physical locking modes; thus, in part, it alleviates the locking phenomena. This is shown for linear and quadratic orders of consistency. Moreover, the biquadratic order of consistency, as in finite elements, improves the locking behaviour. Although more locking modes are present in the element-free Galerkin method with quadratic consistency than with standard biquadratic finite elements. Finally, numerical examples are shown to validate the modal analysis. In particular, the conclusions of the modal analysis are also confirmed in an elastoplastic example.

Abstract

Locking in finite elements has been a major concern since its early developments. It appears because poor numerical interpolation leads to an over-constrained system. This paper proposes a new formulation that asymptotically suppresses locking for the element free Galerkin (EFG) method in incompressible limit, i.e. the so-called volumetric locking. Originally it was claimed that EFG did not present volumetric locking. However, recently, performing a modal analysis, the senior author has shown that EFG presents volumetric locking. In fact, it is concluded that an increase of the dilation parameter attenuates, but never suppresses, the volumetric locking and that, as in standard finite elements, an increase in the order of reproducibility (interpolation degree) reduces the relative number of locking modes. Here an improved formulation of the EFG method is proposed in order to alleviate volumetric locking.Diffuse derivatives are defined in the thesis of the second author and their convergence to the derivatives of the exact solution, when the radius of the support goes to zero (for a fixed dilation parameter), it is proved. Therefore, diffuse divergence converges to the exact divergence. Since the diffuse divergence-free condition can be imposed a priori, new interpolation functions are defined that asymptotically verify the incompressibility condition. Modal analysis and numerical results for classical benchmark tests in solids corroborate this issue.

Abstract

This paper proposes a methodology for the continuous blending of the finite element method and smooth particle hydrodynamics. The coupled approximation with finite elements and particles, and the discretization of the boundary value problem with a coupled integration, are described. An integration correction is also proposed to stabilize the solution. Some numerical examples demonstrate the applicability of the method.

Abstract

This work is part of the ProTechTion project funded by MSCA - EU Horizon 2020, and consists of an Incremental Updated Lagrangian Smooth Particle Hydrodynamics (SPH) framework, aimed at applications in the field of fast solid dynamics, with the following highlights: - Mixed-based system of first-order conservation laws - Incremental ULF (multiplicative decomposition of the deformation) - Anisotropic updated kernel function - Entropy-stable Riemann-based stabilisation scheme.

Abstract