Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

Extended meshfree method for elastic and inelastic media

Abstract: In this paper, an extended meshfree method [9] for solving elastic boundary value problems is summarized, and its extension to the elasto-plasticity problem is presented In extended meshfree method, the solution is decomposed into particular solution and homogeneous solution The particular solution without satisfying boundary conditions is obtained analytically, while a homogeneous problem with auxiliary boundary conditions is then solved under a Galerkin framework with moving least-squares reproducing kernel approximation The proposed method for linear differential operator in Poisson, elasticity, and Mindlin-Reissner problems is first summarized The extension to differential equations with nonlinear self-adjoint differential operator is then introduced, and the application to elasto-plasticity problem is presented Numerical results of an elasto-plasticity problem demonstrate a significant accuracy gain in the solution of extended meshfree method compared to that of the conventional Galerkin meshfree approach

Custom integration scheme for patch test in MLS meshfree methods

Abstract: In the “patch test”, a linear elasticity problem must be exactly solved when the solution is a linear function. The obtained strains and stresses in this case should be constant. When a numerical method of solving the partial differential equations fulfills this condition, we say that it passes the “patch test”. This approach to test the numerical formulation and the code itself is standard in the finite element method. The MLS shape functions do not have the polynomial form. Therefore, the integration is not well performed by the classical Gauss-Legendre scheme. In this paper, we propose a custom quadrature scheme for MLS shape functions in order to ensure the properties needed for exact verification of the patch test.

Meshfree Least Square-based Finite Difference method in CFD applications

Abstract: Most commercial computational fluid dynamics (CFD) packages available today are based on the finite volume- or finite element method. Both of these methods have been proven robust, efficient and appropriate for complex geometries. However, due to their crucial dependence on a well constructed grid, extensive preliminary work have to be invested in order to obtain satisfying results. During the last decades, several so-called meshfree methods have been proposed with the intension of entirely eliminating the grid dependence. Instead of a grid, meshfree methods use the nodal coordinates directly in order to calculate the spatial derivatives. In this master thesis, the meshfree least square-based finite difference (LSFD) method has been considered. The method has initially been thoroughly derived and tested for a simple Poisson equation. With its promising numerical performance, it has further been applied to the full Navier- Stokes equations, describing fluid motions in a continuum media. Several numerical methods used to solve the incompressible Navier-Stokes equations have been proposed, and some of them have also been presented in this thesis. However, the temporal discretization has finally been done using a 1st order semi-implicit projection method, for which the primitive variables (velocity and pressure) are solved directly. In order to verify the developed meshfree LSFD code, in total four flow problems have been considered. All of these cases are well known due to their benchmarking relevance, and LSFD performs well compared to both earlier observations and theory. Even though the developed program in this thesis only supports two dimensional, incompressible and laminar flow regimes, the idea of meshfree LSFD is quite general and may very well be applied to more complex flows, including turbulence

Numerical simulation of aneurysms with Finite Element and Meshless Methods

Abstract: The purpose of this work is to study the level of principal stresses, equivalent von Mises stress and displacements in two 3D geometric models of an abdominal aortic aneurysm (AAA) and a non-aneurysmal aorta (NAA), under three different blood pressure conditions. In order to perform the numerical simulation, it was considered the Finite Element Method (FEM) and a meshless method - the Radial Point Interpolation Method (RPIM). The results obtained with both numerical methods were compared. The obtained results demonstrated that the stresses and displacements in the NAA model are uniformly distributed along the inner surface. Differently, in the AAA model, the higher values are located at the end of the aneurysmal neck, indicating sites of potential rupture. Both FEM and RPIM allow to obtain similar results, regardless the analysed model. Lastly, since in this work elastic material assumptions were considered, the increase of blood pressure increases in the same proportion the value of stresses and displacements.

Meshless Algorithms for Computational Biomechanics of the Brain

Abstract: In this chapter, we discuss meshless methods of computational biomechanics, which utilise computational grids in a form of clouds of points, in the context of computation of the brain deformations due to surgery and injury. We highlight that meshless discretisation may be regarded as a possible solution for overcoming the limitations of the finite element method. We advocate application of weak-form meshless methods that use background cells for spatial integration, explicit time stepping, and total Lagrangian formulation of continuum mechanics (where the derivatives with respect to the spatial coordinates can be pre-computed). We make specific recommendations regarding the following key aspects of the brain deformation computation using meshless methods: (1) shape functions that facilitate robust numerical solution for irregular node placement (a feature necessary to enable end-users who are not experts in computational mechanics to build patient-specific computational biomechanics models of the brain and other body organs), (2) ensuring the desired accuracy of Gaussian spatial integration for non-polynomial shape functions through adaptive integration schemes, and (3) determining a critical time step that ensures stability of the solution provided by explicit dynamics meshless algorithms. We also discuss soft tissue dissection simulation using a visibility criterion (where the model nodes located on the opposite sides of the dissection-induced crack cannot interact with each other) while leaving open the question about the method of choice for three-dimensional dissection simulation. We provide examples of verification of the discussed meshless algorithms against the reference solutions obtained using the well-established non-linear finite element procedures.