Rayleigh-Ritz Implementation
Rayleigh-Ritz Implementation
(OP)
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(y) 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!
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(y) 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!





RE: Rayleigh-Ritz Implementation
RE: Rayleigh-Ritz Implementation
RE: Rayleigh-Ritz Implementation
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
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Rayleigh-Ritz Implementation
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.
RE: Rayleigh-Ritz Implementation
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Rayleigh-Ritz Implementation
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
RE: Rayleigh-Ritz Implementation
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.
RE: Rayleigh-Ritz Implementation
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(2B)+(2B)' ?
RE: Rayleigh-Ritz Implementation
what basis functions are you using?
RE: Rayleigh-Ritz Implementation
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.
RE: Rayleigh-Ritz Implementation
Cheers
Greg Locock
New here? Try reading these, they might help FAQ731-376: Eng-Tips.com Forum Policies http://eng-tips.com/market.cfm?
RE: Rayleigh-Ritz Implementation