In order to simplify the FE Model, we would calculate the stiffness of some components separately. For example a T-Section or Elbow in Piping System.
The stiffness matrix obtained by the Inverse Matrix of displacement matrix.
We would utilize this stiffness matrix and introduce it as the property of an element (e.g. Link element)
Now the issue is the original T-Section will be having different orientations in the main FE Model. The software is not cable of changing the Local Axis of the Link Element, so we would need to rotate the stiffness matrix.
The Stiffness Matrix is 6X6 and I am looking for the 6X6 Matrix transformation (Rotation)
Since your Stiffness matrix is Invertible, the following rotation matrix can be used.... this method is related to Lowdin's orthogonalization :
SVD of the Invertible matrix A( n by n) :
A ( n by n) = U ( n by n) *S( n by n) *VH( n by n)
here S ( n by n) is a diagonal matrix of the singular values, VH denotes the hermitian conjugate of the matrix V( n by n), the matrices U and V are unitary matrices.
The Rotation matrix R( n by n) is given as:
R( n by n) = U ( n by n) *VH( n by n)
If your Invertible stiffness matrix is also hermitian, it is a Positive definite matrix and can be diagonalized by Unitary similarity transformation:
A( n by n) = Z( n by n) "D ( n by n) *ZH ( n by n)
here D is the diagonal matrix of eigen values of A, which in this case are all strictly positive, Z is a Unitary matrix, this matrix Z can be used as the rotation matrix to transform the matrix A into it's diagonal similar form matrix D:
ZH( n by n) *A( n by n) *Z( n by n) = D ( n by n)
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