Meshless Meshless Methods Meshless Meshless Methods Methods Methods
Contents • Introduction • Mess Free Methods Element Free Galerkin Method Element Free Galerkin Method Moving Particle Semi-Implicit Method • Conclusion C l i • Reference
Introduction • Numerical methods can be classified in two groups • (1) Mesh-based method like FEM, FD, BEM, FVM • (2) Mesh-less methods/Mesh Free Methods (MFM)
Difficulty in applying FEM • There are certain classes of problems for which the FEM Th t i l f bl f hi h th FEM is difficult, or even impossible to be applied. • The FEM usually requires remeshing in order to insure equality between finite element boundaries and the moving discontinuities . • The main objective for the development of meshless The main objective for the development of meshless methods(also called mesh free method, MFM) is making approximation based on nodes, not elements .
Some major advantages of MFMs (i) problems with moving discontinuities such as crack propagation. (ii) l (ii) large deformation can be handled more robustly, d f ti b h dl d b tl (iii) higher-order continuous shape functions, (iv) non-local interpolation character and (iv) non local interpolation character and (v) no mesh alignment sensitivity .
History of meshless methods • The Smoothed Particle Hydrodynamics (SPH) The advent of the mesh free idea dates back from 1977, with Monaghan , Gingold and Lucy developing a Lagrangian method based on the Kernel Estimates method to model astrophysics problems. • The Diffuse Element Method (DEM) DEM was introduced by Nayroles and Touzot in 1991. The idea behind the DEM was to replace the FEM interpolation within an element by the Moving Least Square (MLS) local interpolation. p
Continued • • The Element Free Galerkin (EFG) The Element-Free Galerkin (EFG) In 1994 Belytschko and colleagues introduced the Element-Free Galerkin Method (EFG), an extended version of Nayroles’s Method EFG is one of the most popular mesh-free methods and extended version of Nayroles s Method. EFG is one of the most popular mesh-free methods and its application has been extended to different classes of problems such as fracture and crack propagation, wave propagation ,acoustics and fluid flow. • Reproducing Kernel Particle Method In 1995 Liu proposed the (RKPM) in an attempt to construct a procedure to correct the lack of consistency in the SPH method. The RKPM has been successfully used in multiscale techniques, i i h SPH h d Th RKPM h b f ll d i l i l h i vibration analysis, fluid dynamics and many other applications.
Continued • Finite Point Method • Finite Point Method The Finite Point method was proposed by On˜ate and colleagues in 1996. It was originally introduced to model fluid flow problems and later applied to model many originally introduced to model fluid flow problems and later applied to model many other mechanics problems such as elasticity and plate bending. • Meshless Local Petrov-Galerkin The Meshless Local Petrov-Galerkin introduced by Atluri and Zhu in 1998 presents a different approach in constructing a mesh-free method different approach in constructing a mesh free method
Continued • Radial Basis F nctions (RBFs) • Radial Basis Functions (RBFs) RBF were first applied to solve partial differential equations in 1991 by Kansa, when a technique based on the direct Collocation method and the Multiquadric RBF was used to model fluid dynamics. • Point Interpolation Method The Point Interpolation method (PIM) uses the Polynomial Interpolation technique to construct the approximation. It was introduced by Liu in 2001 as an alternative to the Moving Least Square method Moving Least Square method. • Moving Particle Semi-Implicit Method (MPS) MPS method was developed by Koshizuka and applied to fluid flow problems.
Introduction to EFGM • An element free Galerkin method which is applicable to arbitrary shapes but requires only nodal data nodal data. • In this method moving least-square interpolants are used to construct the trial and test functions are used to construct the trial and test functions for the variational principle (weak form); the dependant variable and its gradient are continuous p g in the entire domain.
Continued • Imposing essential boundary conditions I i i l b d di i In the EFG formulation, Lagrange Multipliers are used in the weak form to enforce the essential used in the weak form to enforce the essential boundary conditions. • Process for Numerical Integration Process for Numerical Integration An auxiliary cell structure,is used in order to create a “structure” to define the quadrature q points.
Continued
A computational model for a A computational model for a meshless method
Mathematical Formulation Mathematical Formulation EFG method uses Moving least square method for approximation of the EFG th d M i l t th d f i ti f th unknown function u, which encompasses • a weighted function associated to each node • • a polynomial basis function (Interpolation function) a polynomial basis function (Interpolation function) • a set of coefficients which depend on the position The approximate function u h (x) is defined as • ---------------------------- (1) (1) m is the number of terms in the basis function • p i (x) are the basis function • a i (x) are the unknown coefficients which are functions of the spatial ( ) • th k ffi i t hi h f ti f th ti l coordinates x.
Continued • A common linear and quadratic bases in 1D and 2D space are: A li d d i b i 1 d 2 Linear Bases: p (m=2) p T (m=2) = {1 , x} {1 , x} 1 D 1-D p T (m=3) = {1 , x , y} 2-D Quadratic Bases: p T (m=3) = {1 , x , x 2 } 1-D p T (m=6) = {1 , x , y , x 2 , xy , y 2 } 2-D
Continued • • Define a function Define a function w(I) is a weighted function with compact support. n is the total number of nodes inside the domain of influence. i th t t l b f d i id th d i f i fl • Using matrix notation J = (P a-u) T W(x) (P a-u) J (P a u) W(x) (P a u)
Continued For the approximation u h (x) , very similar with the ordinary h ( ) F th i il ith th di i ti finite element method, we have u h (x) = u (x) Ø is the shape function given by ø = p T (x) A ‐ 1 (x) B(x). T ( ) A 1 ( ) B( ) ø
Continued
Continued • In the EFGM the integration is performed at I th EFGM th i t ti i f d t each integration point of a simple integration cell also called a bucket cell also called a bucket. • The domain of influence of each integration point is defined by the radius of influence r . point is defined by the radius of influence r .
Application of EFG method to Application of EFG method to Structural Engineering Problem • A b A bar of unit length subjected to a linear body force and fixed f i l h bj d li b d f d fi d at point x = 0.
Comparison between EFG & Exact Comparison between EFG & Exact method for 1D problem 0.35 0.3 0.25 ment 0.2 displacem EFG displacement Exact displacement 0.15 0.1 0 1 0.05 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 position (x)
Bar subjected to a unit force at the free end using i) EFG and ii) FEM i i) EFG d ii) FEM Displacement in EFG Displacement in FEM 0.0000 0.0000 0.2011 0.2000 0.3995 0.4000 0 6005 0.6005 0 6000 0.6000 0.7989 0.8000 1 0000 1.0000 1 0000 1.0000
Continued EFG Displacement FEM Displacement 1.4 1 0.9 1.2 0 8 0.8 1 0.7 0.6 e m e n t e m e n t 0.8 0.5 0 5 d is p la c e d is p la c 0.6 0.4 0.3 0.4 0 2 0.2 0.2 0.1 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x x
The initial and boundary conditions are = = = ⎫ ⎫ t t t u x y v x y at 0 0, ( ( , , 0 0 ) ) ( ( , , 0 0 ) ) 0 0 ⎪ = Γ = = x u t at the edge 0 (or ) 0 , 0 , ⎪ y 1 1 ⎪ = Γ = = y v t ⎬ at the edge, 0 (or ) 0 , 0 x 2 2 ⎪ ⎪ = Γ = = x L t t at the edge, (or ) 0 , 0 ⎪ x y 3 ⎪ = Γ = = y W u v v ⎭ at the edge, ( or ) 0 , 4 4 4 0
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