Medusa

Medusa, a novel library for implementation of non-particle strong form mesh-free methods, such as GFDM or RBF-FD, is described. We identify and present common parts and patterns among many such methods reported in the literature, such as node positioning, stencil selection, and stencil weight computation. Many different algorithms exist for each part and the possible combinations offer a plethora of possibilities for improvements of solution procedures that are far from fully understood. As a consequence there are still many unanswered questions in the mesh-free community resulting in vivid ongoing research in the field. Medusa implements the core mesh-free elements as independent blocks, which offers users great flexibility in experimenting with the method they are developing, as well as easily comparing it with other existing methods. The article describes the chosen abstractions and their usage, illustrates aspects of the philosophy and design, offers some executions time benchmarks and demonstrates the application of the library on cases from linear elasticity and fluid flow in irregular 2D and 3D domains.

The Influence of the Weight Function on the Results Accuracy in Structural Analysis by EFG Method

The use of mesh free methods, including the EFG method, raises a number of user issues. For the best possible result, the user must choose both the weight function and a series of parameters of the method used. This paper presents a part of the research undertaken by the authors to substantiate, on the basis of numerical experiments, some conclusions that will lead the user as quickly as possible to a more accurate result, which can be validated by analytical results and often by experiment. The numerical experiments performed by the authors are based on the use of their own program, made in Matlab, which allowed the use of many weighting functions, in different conditions established by choosing several values of the internodal distance and the size of the support domain. The numerical experiment is performed on a simple structure, widely used in the technical literature under the name Timoshenko beam, for which there are analytical (exact) solutions for both the displacements and stresses. The obtained results are presented in graphical and tabular form, with the highlighting of the errors compared to the exact solution. The final conclusions provide valuable information for practical work with the EFG method, which are also valid for the use of other meshless methods.

On the forward modelling of three-dimensional magnetotelluric data using a radial-basis-function-based mesh-free method

SUMMARY The investigation of using a novel radial-basis-function-based mesh-free method for forward modelling magnetotelluric data is presented. The mesh-free method, which can be termed as radial-basis-function-based finite difference (RBF-FD), uses only a cloud of unconnected points to obtain the numerical solution throughout the computational domain. Unlike mesh-based numerical methods (e.g. grid-based finite difference, finite volume and finite element), the mesh-free method has the unique feature that the discretization of the conductivity model can be decoupled from the discretization used for numerical computation, thus avoiding traditional expensive mesh generation and allowing complicated geometries of the model be easily represented. To accelerate the computation, unstructured point discretization with local refinements is employed. Maxwell’s equations in the frequency domain are re-formulated using $\mathbf {A}$-ψ potentials in conjunction with the Coulomb gauge condition, and are solved numerically with a direct solver to obtain magnetotelluric responses. A major obstacle in applying common mesh-free methods in modelling geophysical electromagnetic data is that they are incapable of reproducing discontinuous fields such as the discontinuous electric field over conductivity jumps, causing spurious solutions. The occurrence of spurious, or non-physical, solutions when applying standard mesh-free methods is removed here by proposing a novel mixed scheme of the RBF-FD and a Galerkin-type weak-form treatment in discretizing the equations. The RBF-FD is applied to the points in uniform conductivity regions, whereas the weak-form treatment is introduced to points located on the interfaces separating different homogeneous conductivity regions. The effectiveness of the proposed mesh-free method is validated with two numerical examples of modelling the magnetotelluric responses over 3-D conductivity models.

Particle-based modeling and meshless simulation of flows with Smoothed Particle Hydrodynamics

<p>Smoothed Particle Hydrodynamics (SPH) is a promising simulation technique in the family of Lagrangian mesh-free methods, especially for flows that undergo large deformations. Particle methods do not require a mesh (grid) for their implementation, in contrast to conventional Computational Fluid Dynamics (CFD) methods. Conventional CFD algorithms have reached a very good level of maturity and the limits of their applicability are now fairly well understood. In this paper we investigate the application of the SPH method in Poiseuille and transient Couette flow along with a free surface flow example. Algorithmically, the method is viewed within the framework of an atomic-scale method, Molecular Dynamics (MD). In this way, we make use of MD codes and computational tools for macroscale systems.</p>

Analysis of a new class of rational RBF expansions

We propose a new method, namely an eigen-rational kernel-based scheme, for multivariate interpolation via mesh-free methods. It consists of a fractional radial basis function (RBF) expansion, with the denominator depending on the eigenvector associated to the largest eigenvalue of the kernel matrix. Classical bounds in terms of Lebesgue constants and convergence rates with respect to the mesh size of the eigen-rational interpolant are indeed comparable with those of classical kernel-based methods. However, the proposed approach takes advantage of rescaling the classical RBF expansion providing more robust approximations. Theoretical analysis, numerical experiments and applications support our findings.