Hello,
This is a math question about modal shapes and orthogonal / orthonormal base
Suppose I have some modal functions which form an orthonormal base. Let's call them W. By definition < Wi // Wj >= Kronekeri,j. It is possible to calculate an upper right triangular matrix with all the values such that the diagonal values are close to 1 and the non-diagonal values are close to 0. The under left triangular matrix is a mirror of the upper right triangular matrix so there is no use calculating that. (< Wi // Wj >=< Wj // Wi >)
Suppose also that for high modes (i or j > 8 for example) there is an approximation that yields better results because it some numerical errors are avoided. This new base is called W' and we observe that
< Wi // Wj > > < W'i // W'j > for i=9..inf and j=9..inf
With that said I decides to mix the two basis and create the following:
W_mix= [W1,..,W8,W'9,...W'inf]
Unfortunately, I observed that < W1 // W'100 > for example doesn't give a result close to 0.
Is there a way to create a mixed base that gives better results?
Any good resources to share?
Thanks,
Samy
If W'9, ... , W'inf are not an orthonormal basis, orthonormalize them, e.g. using Gram-Schmidt, to obtain new W''9,...,W''inf. The old basis W9,...,Winf is also orthonormal. Thus, you can relate these two 9,...,inf-Bases by an orthogonal matrix S such that W''j=sum_k Sjk Wk. for j,k in [9,inf). ( If your bases are complex orthonormal you have S unitary instead). It is then guaranteed that the complete basis
W1,...W8,W''9,...,W''inf
is orthonormal.
This is probably just a somewhat more detailed construction for the procedure as proposed by Demetris.
Why don't you mix the two basis and build a new one Z by using the well known method of Gram-Schmidt? Then the new basis will be orthonormal by its construction.
If W'9, ... , W'inf are not an orthonormal basis, orthonormalize them, e.g. using Gram-Schmidt, to obtain new W''9,...,W''inf. The old basis W9,...,Winf is also orthonormal. Thus, you can relate these two 9,...,inf-Bases by an orthogonal matrix S such that W''j=sum_k Sjk Wk. for j,k in [9,inf). ( If your bases are complex orthonormal you have S unitary instead). It is then guaranteed that the complete basis
W1,...W8,W''9,...,W''inf
is orthonormal.
This is probably just a somewhat more detailed construction for the procedure as proposed by Demetris.
Thanks for your answers.
The thing is that I need to to have expressions as closer as possible to the W modal shapes because I am using them in an approximate method to solve my problem in order to ensure the highest possible convergence rate.
I guess that by using the gram-Schmidt orthogonalization method, the result could work but the modal shapes won't be close to W.
Another thing is that Theoretically W1..Winf forms and orthogonal basis. When implemented in a computer it's not the case anymore as from W9 as given in the previous example. So I guess that theoretically W1,...W8,W''9,...,W''inf is an orthogonal basis but when it will be implemented on the combuter it won't be orthogonal anymore because of the same numerical errors.
That is why the approximation was introduced: As from W'9 and on it forms an orthogonal basis and it does not have numerical errors for high modes and the approximation is supposed to be valid for high modes
So we have the following situation (numerically verified):
- W1..W8 is orthogonal
- W'9...W'200 is orthogonal
- W1 and W'100 are not orthogonal (???) W'100 is veeeery close to W100!
Dear Samy, A set of orthogonal polynomials is characterized by 2 conditions: A weight function and an interval. So, I can not see any method to combine the orthogonal polynomials.
However, in some cases, we might be interested in a weight function that gives more importance to a part of the interval and not the other part, then we can adjust our weight function accordingly.
Hello Mr. Rababah,
The modal shapes in question are not polynomials. It a weighed sum of a sine, a cosine, a hyperbolic sine and a hyperbolic cosine.
Does that change anything?
Samy
Dear @Samy, not at all, sine and cosine are polynomials, while hyperbolic sine and hyperbolic cosine are exponential ex .and e-x
The functions are also orthonormal?
- W1..W8 is orthogonal
- W'9...W'200 is orthogonal
What is the Matrix S then? (Theoretically and/or numerically)
Did you try to orthonormalize the W9..W200 numerically? By the way, numerical orthogonalization is numerically tricky quite often, as is known for the Lanczos method.
How do you measure 'very close' for W100 and W'100 ? What is the numerical deviation in that measure?
How do you calculate exactly < W1 | W'100 > ? In what precision (number of digits) was the calculation done and was the result numerically? Is there a trend for the numerical results for for increasing k?
Is is necessary for some practical reason that W1 and W'100 are not numerically orthogonal?
Although the approximation is simple, it would be to complicated to express it here. I will refer you to the following article.
https://www.researchgate.net/publication/256799972_Numerical_evaluation_of_high-order_modes_of_vibration_in_uniform_EulerBernoulli_beams?ev=srch_pub&_sg=t6W3xsgHk32YrNmQzHrMp1DGejDFqRf5NiQcpYC2hOIt1KBeNwQrxvIEwHWexf48_utXMElj35ed2AT3eA14unQGfYlgDWUjJPRhk%2Fm7HDgDxWrt0qMoxdawbcv63dHCK_IkkZhJByy8mowzYrK4hQmoJAEVgRh7Oo1PPoC%2B%2BYPVtmjcIqH0ptV2Onf4RS2Q6w
There is a necessity for the modal shapes to be orthogonal between them because with the approximate resolution method I'm using ( Vibration problem ). After having resolved the Eigen problem I need to resolve a system of coupled equations. The orthogonality is necessary in order to uncouple them.
The resolution is of 10 digits so it's not a very high resolution and that is what I am trying to avoid. I want the code to be as fast as possible with a relatively low resolution. If you read the article you will understand that the numerical errors for high modes comes from the fact that the hyperbolic sine tends to infinity and since the computer can't represent infinity, the is a numerical error/illness. The approximation is used to get rid of that illness.
Article Numerical evaluation of high-order modes of vibration in uni...
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