Benchmark solution
=====================


Calculation method used for the reference solution
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The direct calculation defines an assembled vector for the spatial distribution of the force and applies the spectral force density :math:`{G}_{\mathrm{FF}}(\omega )` to this distribution (method 1).

The decomposed calculation defines the excitation as an interspectral matrix of dimension 3 (equal to the number of excited nodes) and applies, in force imposed on the nodes, the following interspectral matrix (method 2):

:math:`\left[\begin{array}{ccc}\frac{1}{4}& \frac{1}{2}& \frac{1}{4}\\ \frac{1}{2}& 1& \frac{1}{2}\\ \frac{1}{4}& \frac{1}{2}& \frac{1}{4}\end{array}\right]{G}_{\mathrm{FF}}(\omega )`

Both results should be the same without any approximation.


Benchmark results
----------------------

Node :math:`\mathrm{P3}` displacement power spectral density at frequencies: :math:`4.,6.,8.,10.` and :math:`12\mathrm{Hz}`.


Bibliographical references
---------------------------


1. C. DUVAL "Dynamic response under random excitation in *Code_Aster*: theoretical principles and examples of use" - Note HP-61/92.148