Dear Colleagues,
I am moving into a territory outside my field of expertise, so please be tolerant if my questions are naïve.
Consider a matrix: an eigen-solver generates a set of eigen-values and eigen-vectors, and their combination allows to recalculate back the initial matrix, reversing the eigen-problem evaluation. The main limitation is the number of eigen-components one is ready to include; with a limited number the reverse solution is an approximation, of course.
Now, suppose that instead of a matrix, one has a tensor equation. For instance for simulating normal modes of a solid body, which is essentially the same eigen problem. An arbitrarily shaped solid can be evaluated by a corresponding FEM module, which treats this as a tensor equation: a tensor of stresses is equal to a tensor product of elastic tensor and the tensor of strains. One still gets both the eigen-values (natural modes frequencies) and eigen-vectors (the modes' shapes). For an isotropic material, when the elasticity tensor is reduced to two independent values, the the eigen-problem still has to be formulated as tensor equations for the tensors of stresses and strains, due to the complicity of the geometry.
The questions are:
- Can, in principle, the mathematically reverse problem be solved: can one, starting from the body's geometry and natural modes' frequencies (eigen-values) and shapes (eigen-vectors), recalculate the elasticity tensor that is responsible for those modes?
- I suppose the answer is "yes" for the isotropic and simple geometry that yields analytics for eigen-problem - but how about the isotropic case with complex geometry?
- If "yes", is there a program/algorithm which performs such "inversion"?
- Is the problem somewhat similar to building a 3D model from Fourier maps of X-ray scatter of proteins and the like?
- If "not", could you point to a mathematically similar problem that has been solved?
Although your queries are very interesting, however., sorry to say that this, need complete new literature review. I hope you can do it very well.
Looks like an inverse eigenvalue problem. In case of an isotropic material you will have two unknowns, i.e. Young's modulus and Poisson ratio \nu. I assume, that the density \rho is known.
You might try an optimization approach
\min \sum\limits_{i=1}^r (f_i - \hat{f}_i)^2
s.t. K x = \lambda M x, \lambda = \omega^2 , \omega = 2 \pi f_i
where \hat{f}_i is the i-th target natural frequency
It is a kind of ill-posed problem. It would be difficult to hope on an accurate solution, since the variation of elastic constants in broad interval slightly changes the spectrum.
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