# Which are the benchmark problems for a numerical method ?

Hello mechanicians,

Which are the benchmark problems must be tested for a new numerical method in 1D, 2D, and 3D ?

If there are two methods, with what criterions we can say one is better than other in such a particular problem? in all problems ?

And who supposed above rules ?

I look forward to your replies.

Happy new year to all !

Quoc-Duan

### a very open-ended question

Quoc,

I could provide a very long response to this question.  There are many many numerical methods out there, and there are many many problems researchers are trying to solve.  It would help if you could be more specific.  What problems are you interested in?  What methods would you like to use?

Accepted benchmarks are very much a function of the class of problems being studied and the methods being employed.  It is rare that one person or group of people come up with the "rules", but rather these tend to grow more organically from a community.

### Can you give an example of a benchmark problem?

Dear John: Your comment now has raised my curiosity. Can you give an example of a well known benchmark problem for a class of well known numerical methods? How does this benchmark grow organically? What does it test? Do people really take it seriously? This is not a burning question, so please take your time or just ignore it. I'm just curious how you computational people work together.

### FEM Benchmarks

If you are developing new finite element formulations you pretty much have to compare your element with the appropriate NAFEMS or MacNeal-Harder benchmark problems to be taken seriously. The MacNeal-Harder problems are just for linear FEA but NAFEMS has expanded their set of benchmarks considerably beyond this.

Bill

### more on benchmarks

Zhigang,

Can you give an example of a well known benchmark problem for a class of well known numerical methods?

Bill has listed a few above for finite elements.  For thin-shell finite elements, there is a well-known "obstacle course" of benchmark problems described in a paper by Belytschko and coworkers around 1985 (I'll see if I can get the exact reference).  The basic idea is that these are problems with varying degrees of difficulty that demonstrate the accuracy and robustness of the method.

There are many other examples.  Zalesak's disk is a widely-used benchmark to examine the fidelity of interface-tracking/capturing methods.

How does this benchmark grow organically?

It's rare that a benchmark is proposed the first time a new class of methods is introduced.  For example, shell finite elements had been in existence for a long time before the obstacle course was proposed.  Typically researchers discover a problem that a method may have a difficult time handling.  If they then develop a solution (i.e. an improved method), and other researchers start to also use this problem to test the fidelity of their methods/improvements, the problem may become a benchmark.  It is in this sense that the process is organic.

What does it test?

A benchmark problem can test a variety of things.  It may test the ability of the method to reproduce a known analytical solution, for example.  This is perhaps the most common.  However, there also exist many benchmark problems for which analytical solutions are not available.  In these instances, experimental results or widely-accepted numerical results may be the basis for comparison.

Do people really take it seriously?

They certainly do. Many reviewers (myself included) will often insist that a new approach pass the accepted benchmarks before that method is accepted for publication in a top journal.

### Benchmarks for a meshfree method ?

Thank you very much for your replies, professors !

They are very helpful for me.

Dear Prof. John E.Dolbow,

I am dealing with a truly Kriging meshless method and there are two problems in my work.

The Kriging approximation is an interpolator, so Kriging shape functions always satisfy the kronecker property. However, local Kriging shape functions (interpolate on local support domains) do not partition the unity and their derivatives may not vanish at given nodes. These results are belong to correlation parameters, and I need benchmarks and criterions to test for this parameter.

The second is an integration technique. I use a new truly meshless integration technique for my problem with moving quadratures ( I posted on the blog of Prof.Sukumar). Which benchmarks and criterions are suitable for this test ?

In my opinions, the more benchmarks are tested, the more reliable the method can be. But, with a limited time, which ones are best for this kind of problems in 1D, 2D and 3D ?

Thanks again and look forward to your help.

Quoc-Duan

### a benchmark for meshfree methods

Quoc-Duan,

I would suggest you begin with a two-dimensional benchmark problem. A nice problem to investigate is the Laplace problem described in Fernandez-Mendez and Huerta, "Imposing essential boundary conditions in mesh-free methods," Computer Methods in Applied Mechanics and Engineering, vol. 193, pp. 1257-1275, 2004.

If you can't obtain this paper, let me know. We also describe the test in one of our papers and I am happy to send you a reprint.

### Thank you for your advices, Prof J.E.Dolbow

Thank you very much for your advices, prof.J.Dolbow !

It is very lucky for me if you can send me those papers to my mail below.

Thanks again,

Quoc-Duan