Dear Hardik,

Solution with BEM are more accurate than those obtained with FEM. Also BEM could handle analytically unbounded domains (infinite and semi-infinite). Since BEM only mesh the surface it was used in some optimization schemes and also in fracture problems. The main drawmback in the BEM is the non-local interaction of the Green functions that leads to fully populated (and non-symmetric, in general). So, the FEM could represent larger problems since the matrices are sparse and self-adjoint (in linear problems).

Unbounded domains, in the FEM, could be handle with: absorbent boundaries, infinite elements or using the BEM. You can mix both methods trying to get advantages of both. Actually you can formulate BEM as a Ku=f problem and couple it in a FEM software.

I hope it answer your question.

Regards,

Nicolás

Only to complete the very good answer provided by Nicolás:

1) FEM handles multi-material and non-linear problems in a easier way than BEM;

2) I don´t know exactly what you mean by "implementation of deriving Green's utilizing FEM", but there are some implementations of local Green functions that follow a path that is similar to FEM. They are called Local Boundary Integral Equations. We have one such implementation for Electromagnetic problems at https://www.researchgate.net/publication/224122522_A_Local_Boundary_Integral_Equation_(LBIE)_Method_in_2D_electromagnetic_wave_scattering_and_a_meshless_discretization_approach

Regards,

Renato.

Conference Paper A Local Boundary Integral Equation (LBIE) Method in 2D Elect...

Renato,

This formulation (Local BIE) yields symmetric matrices?

Regards,

Nicolás

No, Nicolas. It is a variation of the meshless local Petrov-Galerkin formulation. As it uses a Petrov-Galerkin formulation, the shape functions are not equal to the weight functions and the resulting matrix is unsymmetric.

Regards,

Renato.

Yes there is a real interest by industry in this approach. The reality is that the majority of industrial problems involve the growth of 3D cracks small naturally occurring (3D) initial defects that are typically less than a 1mm deep/long. Most components also have a standard uncracked FE model. To explicitly model the crack requires a major remeshing exercise and is time consuming. Furthermore as the crack grows and changes shape then you need to repeat the exercise (re-meshing) and resolve. In most cases at least 20-50 increments in the crack shape are required meaning that the problems will need to be re meshed and re-analysed at least 20-50 times. A typical real life subcomponent can be expected to have of the order of 600,000 degrees of freedom. On current computers this takes approximately 5 minutes elapse time. This means a total solution time of between 1-4 hrs plus all the time spent in assessing the meshes to ensure that they are OK. This is just not a practical way forward. For academic problems the meshing problems are much simpler and you do not need the same mesh density so this is not a problem for university academics who generally study simple geometries with simple load distributions.

The best way forward is to run the baseline fem or BEM model and determine the stresses in the uncracked structure. Extract the stresses on the plane of the crack then use Green Functions to determine the K solution for all possible combinations of a (depth) and c (length). Next fit a 3D surface to these solutions to determine the solution space for the problem. Then use this solution space to grow the crack and to determine the life of the component. Determining the solution space typically takes 2-3 minutes even for the most complex problems and the subsequent lifing takes less than 1 minute so that the total elapse time is less than 10 minutes. In other words the engineer can start the analysis walk away, make himself/herself a cup of coffee, bring it cack to his desk and by that time he will have his answer.

This is what we do routinely.

The problem is thus, how do you determine the 3D Green’s function for the problem of interest ? We have done this using the 3D Trefftz solution. However, you could do it using f.e. I would be most interested in that since there are classes of problems for which when using the Trefftz approach it is very difficult to obtain the Green function.

I should end by saying that it is irrelevant whether you use FEM or BEM to obtain the stresses in the uncracked body. Similarly both FEM and BEM take to long to solve real life fatigue life problems and are thus of little practical use. Of course this is not true if you only want to see if a given crack size will fail. In which case only a single static analysis is needed. Hope that this helps.

Rhys

Dear Rhys,

do you have a paper reference on the approach you have described?

Thank you very much.

Yes lots, but the paper in the forthcoming 2012 Rail Conf puts it together very nicely and gives several non aerospace examples. Sorry the conversion to pdf has stuffed up a couple of equations.

Sorry, but I can't seem to work out how to add a pdf file. I have sent it to you by email. Any advice on how to add a file would be appreciated.

Rhys

Rhys, in the left bottom in "Add attachment" then select "File.

Nicolás

Dear fellows it was great to hear responses from you, I was happy to find your answer and discussions. I worked on a novel type of Green's functions which are self regularized during my PhD. My idea is to put that work forward by implementing the technique to FEM. This was the reason for asking the question above. The application that Rhys was interested in was not known to me and I should particularly thank him for educating me. Interestingly, my future work will be related to structural health monitoring which is also close to Rhys area of expertise.

I was please to read through the papers of Rhys and trying to understand the application of Green's functions in his case. Dear Rhys I have also requested few papers of yours through research gate. I am not sure how does this request go through.

In regards to the comments provided by Renato and Nicolas, there are certainly other ways to produce the Green's functions but they are not traditional in nature and have their own pros and cons this is what I concluded after a bit of research on the way fundamental definition of Green's functions is defined.

Regards

Hardik

Dear Hardik

I am not a big fan of SHM for cracks. Nothing flying or likely to fly in the forseeable future. There has been a concerted effort to monitor in flight environment with a view to assessing the effect of the environment on crack growth. However, the upshot is that we now know that agressive environments do not cause an increase in crack growth in service aircraft. This conclusion has been reached in both the US and Australia in two indpendent studies. Hence it has become clear that corrosion and structural integrity issues decouple so that corrosion maintenance issues can be addressed without worrying about their effect on airworthiness. The only possible link serious application to aircraft could have been msd in civil and military aircraft. However, the FAA have changed the regulations and msd is no longer a problem in transport aircraft. Similarly SHM has been proposed for composite repairs. However, this technology is no longer in use with the RAAF and small delamibnation growth can be routinely detected on ground. The question there is not so much how can I monitor delamination damage, but rather if I have found delaminations what is the life of the structure. The problem is that the prognosis tools are currently all Paris based and as such do not work for small naturally occuring defects in composites or adhesive bonds. Two of my 2012 papers one in J. Composite Structures and the other in FFEMS go into this in some detail and explain how to overcome the prognosis problems. There will be talks on some of these issues at the forthcoming 2012 Corrosion Conference. Similarly we now also know that corrosion in bridges does not drive crack growth. As such the missing link in bridges is monitoring the rate of corrosion and also determining the useage spectra.

If I could get the file attachment thing to even ask me to select a file to add then I would send files to you but clicking attachment and then add file does absolutely nothing, zip, nilch. Most annoying.

Rhys