Computational Mechanics Forum

Hello,

The aim of this writting is to give a brief introduction to the eXtended Finite Element Method (XFEM) and investigation of its practical applications.

Firstly introduced in 1999 by the work of Black and Belytschko, XFEM is a local partition of unity (PUM) enriched finite element method. By local, it means that only a region near the discontinuties such as cracks, holes, material interfaces are enriched. The most important concept in this method is "enrichment" which means that the displacement approximation is enriched (incorporated) by additional problem-specific functions. For example, for crack modelling, the Heaviside function is used to enrich nodes whose support cut by the crack face whereas the near tip asymptotic functions are used to model the crack tip singularity (nodes whose support containes the tip are enriched).

Ninth U.S. National Congress on computational mechanics
July 22 -26, 2007. San Francisco, California

A mini-symposium on

PARTITION OF UNITY FINITE ELEMENT AND MESHLESS METHODS: ADVANCES AND ENGINEERING APPLICATIONS (In honor of Prof. Tinsley Oden's 70th birthday).

Hello,

I would like to know about multi-scale modelling methods, for solids, for examples. I guess that, from the name,  both macroscopic and microscopic are modelled. It implies that, in order to perform a multi-scale simulation, one must know micromechanics. Is this true? :)

Then, what is the finite element procedure for such a modelling?

Is there any review papers on this topic?

How come overlapping domains in meshfree or MLPG methods do not  increase the stiffness of the structure?

Thanks,

Thi Dang

In meshfree (this is more in vogue than the term meshless) methods, two key steps need to be mentioned: (A) construction of the trial and test approximations; and (B) numerical evaluation of the weak form (Galerkin or Rayleigh-Ritz procedure) integrals, which lead to a linear system of equations (Kd = f). In meshfree Galerkin methods, the main departure from FEM is in (A): meshfree approximation schemes (linear combination of basis functions) are constructed independent of an underlying mesh (union of elements).

However, since a Galerkin method is typically used in solid mechanics applications, (B) arises and the weak form integrals need to be evaluated. Three main directions have been pursued to evaluate these integrals:

I am writing to ask about the state of the art in finite element simulation using nonlinear elasticity and explicit dynamics.

Consider, for instance, a 3-d simulation of a hyperelastic beam that's fixed on one end, then  twisted about its long axis by 360 degrees and released. If we apply no friction or viscosity, the sum of kinetic plus potential energy should remain constant as the material springs back and oscillates.

Which FEM codes do the best at conserving KE+PE for a simulation of this type?  If you drop the time step, can you get energy conservation to many significant figures, as one can with, for instance, molecular dynamics simulation? 

I'm curious because I've recently written my own 3-d nonlinear explicit dynamics code that provides very high precision energy conservation, and I'm wondering if it's any better or worse than the nonlinear explicit dynamics codes already available.

For someone with a background in solid mechanics and finite element methods, where should he go to read up on the elementary ideas of the meshfree methods?

In fact, this is a common misconception with meshfree methods. Shape functions that satisfy Kronecker-Delta take a value of one at the node, and vanish at every other node in the domain. Finite element shape functions, for example, are usually designed with this property. This makes the satisfaction of essential boundary conditions relatively simple: we just set or fix the degree of freedom at the node to what it should be on the boundary. Unfortunately, this is usually not sufficient to impose essential boundary conditions with meshfree methods.

The issue is that meshfree shape functions associated with nodes located on the interior of the domain do not typically vanish on the boundary. So, what happens between nodes is just as important as what happens at the nodes. An excellent paper discussing the various options for imposing essential boundary conditions with meshfree methods is provided by Fernandez-Mendez and Huerta, Computer Methods in Applied Mechanics and Engineering, 193, pp. 1257-1275, 2004. At present, Nitsche's method is accepted as being the most robust for essential boundary conditions with meshfree methods. It should also be noted that with Natural-Neighbor interpolants, this is not an issue and the boundary conditions can be imposed just like they are with finite elements.