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Compared with complete 3-D approaches, the multi-harmonic Fourier technique is a powerful method to perform coupled linear and non-linear analyses for initially axisymmetric structure undergoing asymmetrical boundary conditions or made of material that are not uniformly distributed. In a few words, this technique uses a 2-D model together with Fourier series developments about the rotational axis to analyze full 3-D axisymmetric structures.
Asymmetrical responses come often from asymmetrical boundary conditions (loading or fixations). They can also be due to unbalanced orthotropic or anisotropic materials of the axisymmetric structure although under symmetrical boundary conditions. Finally, some dynamic or buckling responses are by default of asymmetrical nature even under symmetrical boundary conditions and with balanced material.
The family of multi-harmonic Fourier elements includes four types:
- shell element,
- quadrangular element,
- triangular element and
- a link element that makes possible the connection of a 3-D model to the 2-D multi-harmonic one.
A multi-layer version is available for all these elements. A wide range of material behavior laws can be used.
The interest of this technique lies mainly in the nonlinear coupled multi-harmonic analyses offered by the SAMCEF Mecano, and also in some SAMCEF Mecano related chaining analyses using the standard procedure in SAMCEF like incremental stability, substructuring, etc. In the linear case, the computational cost can be reduced by considering uncoupled harmonics (or harmonics coupled by pairs). | 
