Abstract

Mesh-free methods have since their early developments been blended to the finite element formulation in order to benefit from the advantages of both numerical techniques. In this paper, two recently proposed formulations to couple mesh-free and finite element methods are discussed and compared.

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

Locking in finite elements has been a major concern since its early developments and has been extensively studied. However, locking in mesh-free methods is still an open topic. Until now the remedies proposed in the literature are extensions of already developed methods for finite elements. Here a new approach is explored and an improved formulation that asymptotically suppresses volumetric locking for the EFG method is proposed. The 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 and fluids corroborate this issue.

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

Incompressible modeling in finite elements has been a major concern since its early developments and has been extensively studied. However, incompressibility in mesh-free methods is still an open topic. Thus, instabilities or locking can preclude the use of mesh-free approximations in such problems. Here, a novel mesh-free formulation is proposed for incompressible flow. It is based on defining a pseudo-divergence-free interpolation space. That is, the finite dimensional interpolation space approaches a divergence-free space when the discretization is refined. Note that such an interpolation does not include any overhead in the computations. The numerical evaluations are performed using the inf–sup numerical test and two well-known benchmark examples for Stokes flow.

Abstract

A novel approach for implicit residual-type error estimation in meshfree methods is presented. This allows to compute upper and lower bounds of the error in energy norm with the ultimate goal of obtaining bounds for outputs of interest. The proposed approach precludes the main drawbacks of standard residual type estimators circumventing the need of flux-equilibration and resulting in a simple implementation that avoids integrals on edges/sides of a domain decomposition (mesh). This is especially interesting for mesh-free methods.

Abstract

Incompressible modelling in finite elements has been a major concern since its early developments and has been extensively studied. However, incompressibility in mesh-free method is still an open topic. Thus, instabilities or locking can preclude the use of mesh-free approximations in such problems. Here, a novel mesh-free formulation is proposed for incompressible flow. It is based on defining a pseudo-divergence-free interpolation space. That is, the finite dimensional interpolation space approaches a divergence-free space when the discretization is refined. Note that such an interpolation does not include any overhead I the computations. The numerical evaluations are performed using the inf-sup numerical test and two well-known benchmark examples for Stokes flow.