2. Benchmark solutions#

2.1. Calculation method used for the reference solution#

  • Analytical solution:

An analytical solution to this problem exists. It is described in [1]:

In the present case, the solution by modal superposition of this problem is written as:

\(u(x,t)\mathrm{=}\frac{{P}_{0}}{\mathit{EA}}x\mathrm{-}\frac{8{P}_{0}I}{{\pi }^{2}\mathit{EA}}\underset{s\mathrm{=}1}{\overset{\mathrm{\infty }}{\Sigma }}\left[\frac{{(\mathrm{-}1)}^{s\mathrm{-}1}}{{(\mathrm{2s}\mathrm{-}1)}^{2}}\mathrm{sin}((\mathrm{2s}\mathrm{-}1)\frac{\pi x}{\mathrm{2I}})\mathrm{cos}((\mathrm{2s}\mathrm{-}1)\frac{\pi c}{\mathrm{2I}}t)\right]\)

with: \(c=\sqrt{\frac{\mathrm{EA}}{m}}\): wave propagation speed in the bar

  • Reference solution selected:

Since the analytical solution involves an infinite sum of modes, it is preferable for the reference solution and the projection solution to correspond to the same configuration, with the same number of modes.

In addition, to avoid problems related to the discretization of the numerical mesh, the reference solution adopted is the response provided by the direct calculation carried out with*Code_Aster* with the command DYNA_TRAN_MODAL. Displacement, speed, and stress are simulated at a few structural points. The simulated responses were put into a file in universal format (IDEAS).

2.2. Benchmark results#

For modeling A, the comparison of the results focuses on the displacements, speeds, speeds, accelerations, accelerations, deformations and stresses along the \(x\) axis, of the nodes \(\mathrm{N2}\) and \(\mathit{N4}\) at 3 different times. \(\mathrm{N4}\) corresponds to a measurement node and \(\mathrm{N2}\) is not a measurement node. It is also checked whether exactly the component of the measured field is obtained after expansion of the measurement on the numerical model. To do this, we compare the component of the extended field along the direction of observation (displacement \(\mathrm{N3}\) along DX, speed \(\mathit{N5}\) along DX and SIXX at node \(\mathrm{N4}\)).

For B modeling, the comparison of the results focuses on the constraints of nodes \(\mathit{N3}\) and \(\mathrm{N4}\) at 3 different times.

2.3. Uncertainty about the solution#

The selected reference makes it possible to eliminate the uncertainties associated with the discretization of the numerical mesh. The number of modes of the projection base is equal to the number of measurements, so the solution of the inversion is exact (as opposed to an approximate solution of a generalized inverse problem).

In the case where the projection is based on a concept of the [mode_meca] type, the modal bases of the reference solution and of the solution obtained by projection are identical; the displacement, deformation and stress responses obtained must therefore be similar to the reference responses. Some approximation errors may appear on the speeds and accelerations which are determined by a linear time pattern.

In the case where the projection is based on a concept of the [modal_basis] type, the modal bases of the reference solution and of the solution obtained by projection contain the same number of modes but are different. Since the reference calculation is not possible on a concept such as [modal_basis], the comparison of the results is based only on responses corresponding to the measurements provided.

2.4. Bibliography#

    1. GERADIN, D. RIXEN: Theory of vibrations - Application to structural dynamics - Edition MASSON 1993