The Global Geometry Definition

Figure 1. Illustration of in-plane and out-of-plane current profiles for catenary riser.

1. Previous limitations

The previous VIVANA model was limited to unidirectional current profiles in the plane or perpendicular to the plane of the structure. Figure 1 illustrates these situations, and also that the restriction was in fact violated to some extent. This was accepted for structures like catenary risers and free spanning pipelines, and could be defended since the relevant angles are small for these cases.

2. The 3D model

The new model can in principle accept any 3D shape of the slender structure and current profile, see Figure 2. A static shape of the structure will be found by a non-linear static finite element analysis. The solution procedure will ensure static equilibrium and that consistency is obtained between the shape of the structure and the current forces at each node for the actual shape.

Figure 2. Global coordinate system and local normal current component.

The current velocity perpendicular to the structure, \(\mathrm {U_N}\), will be calculated at each node, see Figure 2. This vector will be used to define local IL and CF directions at each node.

3. Definition of IL and CF eigenvectors

A sufficient number of solutions for the eigenvalue equation will be found. Each solution will satisfy the equation

\[(\boldsymbol{M}_0-\omega _i^2\boldsymbol{K}_0)\boldsymbol{\phi }_i=\boldsymbol{0}\]

where \(\mathrm {\boldsymbol{M}}\) and \(\mathrm {\boldsymbol{K}}\) are the mass and stiffness matrices, respectively. These are defined for the actual static configuration with respect to shape, tension and possible bottom contact.

The eigenfrequencies, \(\mathrm {\omega _i}\), and eigenvectors, \(\mathrm {\boldsymbol{\phi }_i}\), are computed in the global coordinate system. These are not directly applied in the dynamic analysis, but will serve as initial values in iterations for response frequencies and shapes.

The eigenvectors, \(\mathrm {\boldsymbol{\phi }_i}\), will be transformed to the local IL and CF coordinate system as indicated on Figure 3. The term \(\mathrm {\boldsymbol{\phi }_{i,j,IL}}\) means the component of eigenvector \(\mathrm {\boldsymbol{\phi }_i}\) at node \(\mathrm {j}\) in IL direction, which is found by a coordinate transformation from the global to the local system. Hence, the vectors \(\mathrm {\boldsymbol{\phi }_{i,IL}}\) and \(\mathrm {\boldsymbol{\phi }_{i,CF}}\) are not true (orthogonal) eigenvectors, but assembled eigenvector components in local IL and CF directions respectively. The original eigenvectors are normalized, but the IL and CF vectors are not. Hence, they can be used to classify the eigenvalue solutions in IL and CF dominated eigenfrequencies.

Figure 3. Definition of local coordinate system and eigenvector components.

\(\mathrm {\boldsymbol{\Psi}_{i,j,IL}}\) is co-linear to \(\mathrm {U_{N,j}}\) while \(\mathrm {\boldsymbol{\Psi}_{i,j,CF}}\) is perpendicular to \(\mathrm {U_{N,j}}\). The two components are perpendicular to the local structure axis.

The classification will be based on the norms

\[\begin{array}{lll}|CF|_i&=&\displaystyle \sum_j\boldsymbol{\Psi}_{i,j,CF}^T\boldsymbol{\Psi}_{i,j,CF}L_j\\\\|IL|_i&=&\displaystyle \sum_j\boldsymbol{\Psi}_{i,j,IL}^T\boldsymbol{\Psi}_{i,j,IL}L_j\end{array}\]

The \(\mathrm {\boldsymbol{\Psi}_{i,j}}\) vector is the local IL or CF vector at node \(\mathrm {j}\) from the transformed vectors \(\mathrm {\boldsymbol{\phi }_{i,IL}}\) and \(\mathrm {\boldsymbol{\phi }_{i,CF}}\), see Figure 3. \(\mathrm {L_j}\) is the sum of the half lengths of the elements that meet at node \(\mathrm {j}\).

An eigenvector \(\mathrm {\boldsymbol{\Phi}_i}\) can now be defined as an IL or CF eigenvector depending on the relative magnitude of the two norms. The IL and CF modes are subsets of normalized modes, \(\mathrm {\boldsymbol{\phi }_i}\). Grouping the vectors means that the associated eigenfrequencies also can be split into IL and CF bins. The result of this classification is sets of IL and CF eigenfrequencies, each linked to an estimate for the response vector.