Rayleigh-Ritz Implementation 1

[Jeroen52]

I am implementing a Ritz solution for static deflection of an orthotropic laminate square plate. The method is clear to me and actually, I am getting some very nice results.

However, it is often claimed that the Ritz method is highly efficient. I am finding that as I increase my number of basis functions for my assumed displacement field, the solution is taking incredibly long (matter of hours) for only about 7 functions in each direction (x,y).

Thinking about it, 7 functions each in x and y gives 7*7 = 42 linearly independent terms (the displacement field is a surface and thus of the form coeff * f(x) * g i.e. the basis functions in x and y are combined). A Mindlin plate has 5 displacement variables that completely define the displacement field, so that's 5*42 = 210 linearly independent terms describing the assumed displacement field.

This also means there are 210 terms defining the strain field. These must be squared in the potential energy expression; we are now talking about 44,100 terms!

Okay, it is at this point, with the assumed strain field substituted into the TPE expression, that the magic happens. We must take the first variation. The way I proceed is as often described; evaluate the TPE double (area) integral over the plate domain, then differentiate w.r.t. to each unknown coefficient, giving a system of 210 equations in 210 unknowns that can be solved - beautiful, the Ritz solution has been obtained.

However, the evaluation of the 44,100 term area integral is clearly what takes a crazy amount of time. This will also grow exponentially with more terms.

The question is then - why is the Ritz method considered to be so efficient? Is there an absolute need to evaluate the TPE integral (energy functional) or not? Is there a trick to doing this (e.g. Some sort of divergence theorem shortcut that everybody else uses to e.g. Change the area integral to a contour integral, I'm just postulating, not sure if this is possible) or does everyone use numerical integration without stating so in their papers/books?

I have seen a paper where someone used 50 terms... I would like to use at least 20. This problem must be circumventable.

Any ideas/thoughts much appreciated!

 

[SNORGY]

7x7=42?

 

[Jeroen52]

Evidently a minor slip up. However, that detail is not really important - the question remains the same...

 

[GregLocock]

I must admit I am only familiar with using RR as an aid to analysis by hand of mode shapes, and hence frequencies. When used like thatit is helpful because by mixing two possible mode shapes in varying proportions the resulting mode shape is more accurate, ie the associated frequency is lower.

It is still an upper bound solution as the additional deflections from the true mode shape are effectively adding constraints to the system, increasing its energy.

As such it is efficient because it gives a relatively accurate answer that can be improved very quickly, but if you are dealing with systems where the actual dynamic behaviour of the system is relatively easy to calculate then I can't see why it would be better.

Incidentally what are you using as an optimiser?

Cheers

Greg Locock


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[Jeroen52]

Often times a closed form solution is not possible in elasticity problems. For the case of a generally orthotropic, simply supported laminate, there is no exact solution (see Reddy). We must therefore use approximate solution methods such as the RR, Galerkin or FE method.

The problem I am considering is slightly different to yours, but the principles are the same. An assumed displacement field is substituted into the energy functional and the first variation is taken. This is effectively finding the extremum of the set of basis functions being considered (what you are referring to as optimizing).

You say you mix two mode shapes. I agree when superimposing two mode shapes (which I am calling basis functions) the method is efficient. However, it is rare to achieve convergence using only two terms in the assumed displacement series. I am suggesting the use of about 20 independent mode shapes to achieve the desired accuracy. In this case, computations can take days. So I repeat my question, why is the Ritz method considered efficient? Maybe this statement should be nuanced to say: this method is efficient if a low accuracy solution is acceptable. For high accuracy (i.e. many basis functions) the method is slow.

I would like to concede that if the computation is performed symbolically, then the integration of the energy functional must only be performed once, resulting in an explicit expression in the variables defining the problem (e.g. A, B, D, G matrices, width, length, ply thickness, layup). So any future analysis can be performed in a matter of seconds... Maybe this is what is meant by the high efficiency.

I guess what the problem boils down to is if the symbolic computation can be performed in a reasonable amount of time on a standard PC. As it must only be done once, maybe 100 hours of computation seems like a reasonable limit. I will test this tomorrow by benchmarking the difference in computation times between fully symbolic and partially symbolic integration of the TPE functional.

To answer your question, I am solving my elasticity problem in MATLAB. I am using the symbolic toolbox to perform the integration of the TPE double integral, and the subsequent derivatives that optimize the functional I.e. enforce minimum energy.

 

[GregLocock]

I am out of my depth here, sorry, I haven't got any sensible advice. The derivative of the energy function sounds a bit like a Jacobian, if it is you might want to look at non linear solvers such as MSC ADAMS. Having said that I see that Matlab has a built in jacobian function.






Cheers

Greg Locock


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[hacksaw]

Don't Rayleigh-Ritz was really intended for use with unlimited terms for routine calcs.
you are lucky some calculations can take a week..., but the calculation method is valid, no question there.

Probably better suited as an independent bench-marking of the model setup used in the finite element calculation

 

[Jeroen52]

The result should at least be converged. If we assume the number of basis functions m=4, calculate the answer, then for m=5 the max displacement changes by more than 5%, the result is quite inaccurate and large safety factors must be taken. Also, consider that the derivatives of the displacement field (i.e. strains, and thus stresses) magnify the errors incurred by using a finite number of terms.

Thanks for your input, however. I think it's an accuracy vs. performance trade off. I also think convergence is faster for eigen values (fundamental frequencies) than for displacements/strains.

 

[electricpete]

If we assume the number of basis functions m=4, calculate the answer, then for m=5 the max displacement changes by more than 5%, the result is quite inaccurate and large safety factors must be taken.

I have a basic question. I'm familiar with Raleigh Ritz to estimate first resonant mode shape e (as sum of basis functions) and frequency. It's not obvious to me how this translates into a calculation of displacement. Can you elaborate on that briefly?

=====================================
(2B)+(2B)' ?

 

[hacksaw]

Its always accuracy vs computational load, you include as many terms as you can afford

what basis functions are you using?

 

[Jeroen52]

There is a good explanation of the Raleigh-Ritz method in Structural Analysis of Laminated Composites by J.M. Whitney. The Raleigh-Ritz method can be used for either static, stability (buckling) and free vibration analysis. I am primarily interested in static analysis I.e. calculating the deflections of a laminated plate under quasi-static loading, then from this I can calculate the strain and hence stress. Then I can proceed with a failure analysis.

The basis functions I am using are Bernstein basis polynomials. Eventually i want to switch these to NURBS. These basis functions don't necessarily satisfy the essential boundary conditions, but can be forced to do so by the use of Lagrange multipliers in the TPE functional or, as I am doing, by using extremely stiff artificial springs between the plate and the "ground." As I mentioned before, my method is working and agrees within fractions of a percent with the FEM solution. However, to achieve this high accuracy many terms are required and the calculation takes longer the FEM solution, which defeats the purpose of what I'm trying to do. I was under the impression that the RR solution was more efficient. Maybe for buckling or free vibration. Not necessarily, it seems, for static stress analysis.

 

[GregLocock]

Yes... I think that is the point, as a quick and dirty approach it is computationally efficient, as the bit where you differentiate the weightings to get the optimum mode shape is easy, but at some point finessing an approximate method to a given accuracy must become more work than working the thing out properly.

Cheers

Greg Locock


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[hacksaw]

thats par for the course with polynomial expansions, the computational load is immense