5. Generalized coordinate calculation#

5.1. Formulation of the problem#

The calculation of the generalized coordinates \(\eta\) is carried out on the displacement matrix (respectively speeds, accelerations, deformations, constraints) restricted to the measured degrees of freedom, by solving the matrix system:

\({q}_{\text{exp}}=\stackrel{ˉ}{\Phi }\eta\)

The dimensions of the \(\stackrel{ˉ}{\Phi }\) matrix to be « inverted » are (\({N}_{\text{exp}},{N}_{\text{num}}\)).

We can see here that the calculation of the generalized coordinates is carried out in a restricted space: the dimension of the space generated by the base vectors is smaller than the dimension of the numerical model, we only use the information with the measured degrees of freedom.

5.2. Determination of a virtual solution#

To solve the opposite problem, 3 cases may arise:

\({N}_{\text{exp}}={N}_{\text{num}}\): the number of degrees of freedom measured is equal to the number of projection base vectors whose generalized coordinates we want to identify.

In this case, there is a unique solution to the reversal problem: \(\eta \text{=}{\stackrel{ˉ}{\Phi }}^{-1}{q}_{\text{exp}}\)

\({N}_{\text{exp}}>{N}_{\text{num}}\): the number of degrees of freedom measured is greater than the number of basic projection vectors of the numerical model whose generalized coordinates we want to identify.

In this case, there is no exact solution to the reversal problem. However, a virtual solution can be defined that minimizes the distance: \(\mid {q}_{\text{exp}}-\stackrel{ˉ}{\Phi }\eta \mid\). Formula \(\eta \text{=}{\left[{\stackrel{ˉ}{\Phi }}^{T}\stackrel{ˉ}{\Phi }\right]}^{+}{\stackrel{ˉ}{\Phi }}^{T}{q}_{\text{exp}}\) then provides the (unique) solution in the sense of least squares. In this expression, matrix \({\left[{\stackrel{ˉ}{\Phi }}^{T}\stackrel{ˉ}{\Phi }\right]}^{+}{\stackrel{ˉ}{\Phi }}^{T}\) refers to the generalized inverse matrix of \(\stackrel{ˉ}{\Phi }\). The calculation of the pseudo-inverse can be done using the LU decomposition or the decomposition into singular values (SVD).

\({N}_{\text{exp}}<{N}_{\text{num}}\): the number of degrees of freedom measured is less than the number of projection base vectors whose generalized coordinates we want to identify (which corresponds to the most common case).

In this case, there are an infinity of solutions to the reversal problem and the objective is to determine an acceptable solution by introducing an additional condition (minimum standard of the solution or application of so-called « regularization » methods).

5.3. Determination of a regularized inverse solution#

5.3.1. Principles of regularization methods#

The aim of the regularization methods [bib4], [bib5] is to propose an approximate and stable solution with respect to variations in the input data. The aim is no longer to solve the minimization equation resulting from the formulation: \({q}_{\text{exp}}=\stackrel{ˉ}{\Phi }\eta\), but to determine an approximate (or regularized) solution meeting two requirements:

it meets a proximity condition: we are looking for \({\eta }_{\delta }\) such as \(\mid {q}_{\text{exp}}-{\Phi }_{\text{num}}{\eta }_{\delta }\mid <\delta\),
it meets an additional condition called « prior information ».

Regularization methods therefore consist in completing the statement of the problem by introducing a priory information in order to extract, from the family of solutions that are compatible with experimental data, the one that best corresponds to the problem. This is done by merging into a single criterion a measure of the fidelity of the solution with respect to the experimental data and a measure of its fidelity to the a priory information [bib2].

An approach that can easily be implemented in finite dimensions is regularization by optimization. To be compared to Tikhonov’s method of regularization [bib3], it consists in considering an a priory solution \({\eta }_{\text{priori}}\) of the minimization problem and in seeking the solution of the approximate system that is closest to this solution. The aim is then to minimize the following functional:

\({\mid {q}_{\text{exp}}-\stackrel{ˉ}{\Phi }\eta \mid }^{2}+a{\mid \eta -{\eta }_{\text{priori}}\mid }^{2}\)

The parameter \(a\) determines the weight assigned to the information a priory.

The solution to the minimization equation is given by:

\(\eta \text{=}{\left[{\stackrel{ˉ}{\Phi }}^{T}\stackrel{ˉ}{\Phi }+aI\right]}^{-1}({\stackrel{ˉ}{\Phi }}^{T}{q}_{\text{exp}}+a{\eta }_{\text{priori}})\)

or, by explicitly showing the difference with respect to the a priory solution:

\(\eta \text{=}{\eta }_{\text{priori}}+{\left[{\stackrel{ˉ}{\Phi }}^{T}\stackrel{ˉ}{\Phi }+aI\right]}^{-1}{\stackrel{ˉ}{\Phi }}^{T}({q}_{\text{exp}}-\stackrel{ˉ}{\Phi }{\eta }_{\text{priori}})\)

If we ask \({\eta }_{\text{priori}}=0\), this formulation consists in looking for the so-called « minimum standard » solution (or Tikhonov of order 0).

The role of the addition of the regularizing term linked to matrix \(aI\) is to shift the spectrum by \({\stackrel{ˉ}{\Phi }}^{T}\stackrel{ˉ}{\Phi }\) in order to ensure the matrix reversal step. This calculation approach therefore makes it possible to implement a better conditioned calculation procedure, which softens the effects of noise and which provides a physically acceptable solution.

Moreover, the choice of the values in the matrix \(aI\) results from a compromise between the stability of the solution sought and the confidence that can be given to the solution a prima facie.

5.3.2. Choice of information a priory#

In the case of regularization methods, the choice of prior information is a key step that determines the representativeness of the final results. This choice may be based on physical knowledge of the solution or on knowledge of its evolution as a function of the parameter selected. In the following, we provide an example applied to the determination by minimization of a time variable [bib1].

The minimization of a variable as a function of time can be achieved at each time step regardless of the previous time step. However, the introduction of a priory information makes it possible to enrich the functional by assuming a slow evolution of the variables determined:

\({\eta }_{\text{priori}}(t)=\eta (t-\text{dt})\)

This assumption is only acceptable when the sampling step is sufficiently low. In fact, the solution at a given moment is approximated by (Taylor development):

\(\eta (t)=\eta (t-\text{dt})+\text{dt}\dot{\eta }(t-\text{dt})+o(\text{dt})\)

The maximum response frequency of the structure is determined by the pulsation of the highest order mode \({\omega }_{\text{max}}\) taken in the modeling. So we have:

\(\mid \frac{\eta (t)-\eta (t-\text{dt})}{\eta (t-\text{dt})}\mid <{\omega }_{\text{max}}\text{dt}+\mid o(\text{dt})\mid\)

For the corrective term to be weak (and therefore for the a priory information to constitute a first-order approximation of the solution sought), the sampling step must verify:

\(\text{dt}\text{<<}\frac{1}{{\omega }_{\text{max}}}\)

At the initial moment (t=0), since no information is available on the a priory solution, the calculation is carried out by looking for the minimum standard solution. In order to avoid propagating the resulting error, it may be necessary to assign low confidence to the a priory information on the first steps of time (via the parameter) and to exploit the results only from the moment when it can be considered that the errors have been sufficiently attenuated. If necessary, additional studies will be carried out in order to determine the optimal parameters for using the functionality developed in Code_Aster.

In the frequency domain, numerous possibilities are offered to determine the information a priory. They are based either on a physical knowledge of the solution (experimental demonstration of resonances or forced responses), or on a formulation of generalized displacements as a function of frequency (type: gain functions), in which case minimization finally leads to the characterization of dynamic stresses.