Meshfree methods: An efficient advanced computing approach for Bio-medical problems (original) (raw)
Related papers
2015
A review is presented for the analysis of heat transfer and fluid flow problems in engineering and science, with the use of different meshfree methods. The success of the meshfree methods lay in the local nature, as well as higher order continuity, of the trial function approximations and a low cost to prepare input data for numerical analyses, since the creation of a finite element mesh is not required. There is a broad variety of meshless methods available today; however the focus is placed on the meshless local Petrov- Galerkin (MLPG) method, in this paper.
A review is presented for the analysis of heat transfer and fluid flow problems in engineering and science, with the use of different meshfree methods. The success of the meshfree methods lay in the local nature, as well as higher order continuity, of the trial function approximations and a low cost to prepare input data for numerical analyses, since the creation of a finite element mesh is not required. There is a broad variety of meshless methods available today; however the focus is placed on the meshless local Petrov- Galerkin (MLPG) method, in this paper.
Smoothing and accelerated computations in the element free Galerkin method
Journal of Computational and Applied Mathematics, 1996
Two topics in the formulation and implementation of meshless methods are considered: the smoothing of the approximating functions at concave boundaries and the speedup of the calculation of the approximating functions and their derivatives. These techniques are described in the context of the element free Galerkin method, but they are applicable to other meshless methods. Results are presented for some elastostatic problems which show a moderate improvement in the accuracy of the smoothed interpolant. The speedup in calculating the shape functions is about a factor of two.
In the present study, a numerical analysis is carried out to investigate unsteady MHD (magneto-hydrodynamic) flow and heat transfer of a non-Newtonian second grade viscoelastic fluid over an oscillatory stretching sheet. The flow is induced due to an infinite elastic sheet which is stretched oscillatory (back and forth) in its own plane. Effect of viscous dissipation and joule heating are taken into account. The non-linear differential equations governing the problem are transformed into system of non-dimensional differential equations using similarity transformations. A newly developed meshfree numerical technique Element free Galerkin method (EFGM) is employed to solve the coupled non linear differential equations. The results illustrating the effect of various parameters like viscoelastic parameter, Hartman number, relative frequency amplitude of the oscillatory sheet to the stretching rate and Eckert number on velocity and temperature field are reported in terms of graphs and tables. The present model finds its application in polymer extrusion, drawing of plastic films and wires, glass, fiber and paper production etc.
Non-Linear Steady State Heat Conduction using Element-Free Galerkin Method
Research Journal of Applied Sciences, Engineering and Technology, 2021
This study aims to interpret the non-linear steady-state heat conduction for temperature-dependent thermal conductivity (k (T)) using Element-Free Galerkin (EFG) method. In this present study, a one-dimensional heat conduction problem with uniform heat generation was explicated. Moving Least Squares (MLS) approximants were applied to estimate the unknown function of temperature T (x) with T h (x) using linear basis and weight functions. The variational method has been used to develop discrete equations. Essential boundary conditions are enforced by using the Penalty method. The results have been obtained for the one-dimensional model using essential MATLAB codes. The results obtained by the EFG method are compared with the analytical and finite-element method results. The results are also studied by increasing the number of nodes to study the convergence which indicated that EFG has good convergence behavior. The results have also been obtained for different values of the scaling parameter (αs) and any values of αs between 1.8 and 2.0 were found suitable for providing better results in the EFG method.
An Element-Free Galerkin Scaled Boundary Method for Steady-State Heat Transfer Problems
Numerical Heat Transfer, Part B: Fundamentals, 2013
for solving steady-state heat transfer problems. The SBM weakens the governing differential equations along the circumferential coordinate direction and solves analytically in the radial direction. Unlike the conventional scaled boundary finite element method (SBFEM), an EFG approach is used in the circumferential direction in this paper. The proposed method is verified via numerical examples including the problems with thermal singularity and unbounded domain, and satisfactory results are obtained in comparison with analytical and SBFEM solutions.
An extended meshfree method for boundary value problems
2004
An extended meshfree method is presented for treating boundary value problems, where the total solution is expressed by a combination of particular and homogeneous solutions, each of which is assigned a specific role. The particular solution is any analytical (or numerical) expression satisfying the governing differential equation containing the source term but not necessarily the boundary conditions. A general method is presented for constructing this solution. Thus, the problem is reduced to a homogeneous equation where the original boundary conditions are modified by the particular solution. Herein, the meshfree method is used to solve this homogeneous equation, which involves a lower order behavior so that a relatively coarse discretization is acceptable. Several boundary value problems from potential theory as well as from shear deformable plate theory are solved, where linear exactness [Int. J. Numer. Methods Engrg. 50 435; Int. J. Numer. Methods Engrg. 53 (2002) 2587] and bending exactness [Comput. Methods Appl. Mech. Engrg. 193 (2004) 1065], respectively, are imposed in their meshfree approximation fields. Numerical results demonstrate that this extended meshfree approach significantly improves the solution accuracy with commensurately less computational effort compared to the conventional meshfree formulation.
Numerical Heat Transfer Part B-fundamentals, 2019
The localized radial basis function collocation meshless method (LRBFCMM), also known as radial basis function generated finite differences (RBF-FD) meshless method, is employed to solve time-dependent, twodimensional (2D) incompressible fluid flow problems with heat transfer using multiquadric RBFs. A projection approach is employed to decouple the continuity and momentum equations for which a fully implicit scheme is adopted for the time integration. The node distributions are characterized by non-Cartesian node arrangements and large sizes, i.e., in the order of 10 5 nodes, while nodal refinement is employed where large gradients are expected, i.e., near the walls. Particular attention is given to the accurate and efficient solution of unsteady flows at high Reynolds or Rayleigh numbers, in order to assess the capability of this specific meshless approach to deal with practical problems. Three benchmark test cases are considered: a lid-driven cavity, a differentially heated cavity and a flow past a circular cylinder between parallel walls. The obtained numerical results compare very favorably with literature references for each of the considered cases. It is concluded that the presented numerical approach can be employed for the efficient simulation of fluid-flow problems of engineering relevance over complex-shaped domains.
Numerical Simulation of Fluid Flow Using Meshfree Approach
Abstract. Meshfree fluid flow simulation has achieved large popularity in the last years. Meshfree Galerkin Methods and Smooth Particle Hydrodynamics are typical examples of meshfree techniques, whose ability to handle complex problems has motivated the CFD community. In this work we present a new meshfree approach that uses moving least square (MLS) to discretize the model equations.
Numerical Solutions to Wave Propagation and Heat Transfer Non-Linear PDEs by Using a Meshless Method
Mathematics, 2022
The applications of the Eikonal and stationary heat transfer equations in broad fields of science and engineering are the motivation to present an implementation, not only valid for structured domains but also for completely irregular domains, of the meshless Generalized Finite Difference Method (GFDM). In this paper, the fully non-linear Eikonal equation and the stationary heat transfer equation with variable thermal conductivity and source term are solved in 2D. The explicit formulae for derivatives are developed and applied to the equations in order to obtain the numerical schemes to be used. Moreover, the numerical values that approximate the functions for the considered domain are obtained. Numerous examples for both equations on irregular 2D domains are exposed to underline the effectiveness and practicality of the method.