Interpolation of rotation and motion

Abstract

In Cosserat solids such as shear deformable beams and shells, the displacement and rotation fields are independent. The finite element implementation of these structural components within the framework of flexible multibody dynamics requires the interpolation of rotation and motion fields. In general, the interpolation process does not preserve fundamental properties of the interpolated field. For instance, interpolation of an orthogonal rotation tensor does not yield an orthogonal tensor, and furthermore, does not preserve the tensorial nature of the rotation field. Consequently, many researchers have been reluctant to apply the classical interpolation tools used in finite element procedures to interpolate these fields. This paper presents a systematic study of interpolation algorithms for rotation and motion. All the algorithms presented here preserve the fundamental properties of the interpolated rotation and motion fields, and furthermore, preserve their tensorial nature. It is also shown that the interpolation of rotation and motion is as accurate as the interpolation of displacement, a widely accepted tool in the finite element method. The algorithms presented in this paper provide interpolation tools for rotation and motion that are accurate, easy to implement, and physically meaningful.

Appendices

Appendix A: Notational conventions

When dealing with unit quaternions, $\hat {e}$, the rotation tensor, $\underline {\underline {R}}$, is expressed by the following matrix of size 4×4:

(A.1)

The following matrices are used to manipulate quaternions:

(A.2)

When dealing with bi-quaternions, $\check {g}^{T} = \{ \hat {q}^{T}, \hat {e}^{T} \}$, the motion tensor, $\underline {\underline {\mathcal {C}}}$, is expressed by the following matrix of size 8×8:

(A.3)

The following matrix is used to manipulate bi-quaternions:

(A.4)

When dealing with motion, generalized skew-symmetric matrices are of the following form:

(A.5)

where $\widetilde {a}$ and $\widetilde {b}$ are 3×3 skew-symmetric matrices. Matrix $\underline {\underline {\mathcal {Z}}}$ is defined as follows:

(A.6)

where α and β are two arbitrary scalars.

Appendix B: Polar decomposition theorem

The polar decomposition theorem states that an invertible matrix, $\underline {\underline {A}}$, can be decomposed into the product of an orthogonal matrix, $\underline {\underline {R}}$, by a positive-definite, symmetric matrix, $\underline {\underline {U}}$,

$$ \underline {\underline {A}}= \underline {\underline {R}}\, \underline {\underline {U}}= \underline {\underline {V}}\, \underline {\underline {R}}. $$

(B.1)

The second equality provides an alternative statement of the polar decomposition theorem, where matrix $\underline {\underline {A}}$ is now decomposed as the product a positive-definite, symmetric matrix, $\underline {\underline {V}}$, by the same orthogonal matrix, $\underline {\underline {R}}$. Matrices $\underline {\underline {R}}$, $\underline {\underline {U}}$, and $\underline {\underline {V}}$ are uniquely defined.

The polar decomposition theorem will be generalized by the following statement:

$$ \underline {\underline {\mathcal {T}}}= \underline {\underline {\mathcal {C}}}\, \underline {\underline {\mathcal {U}}}= \underline {\underline {\mathcal {V}}}\, \underline {\underline {\mathcal {C}}}, $$

(B.2)

where $\underline {\underline {\mathcal {C}}}$ is the motion tensor defined by Eq. (4.1). Matrix $\underline {\underline {\mathcal {T}}}$ is an arbitrary matrix of the following form:

(B.3)

where $\underline {\underline {T}}$ is an invertible matrix and matrices $\underline {\underline {\mathcal {U}}}$ and $\underline {\underline {\mathcal {V}}}$ are

(B.4a)

(B.4b)

Matrices $\underline {\underline {U}}$ and $\underline {\underline {V}}$ are positive-definite and symmetric matrices; matrices $\underline {\underline {F}}$ and $\underline {\underline {G}}$ are symmetric.

The generalization of the polar decomposition theorem expressed by Eq. (B.2) is proved easily. The first equality of Eq. (B.2) implies $\underline {\underline {T}}= \underline {\underline {R}}\, \underline {\underline {U}}$; this means that matrices $\underline {\underline {R}}$ and $\underline {\underline {U}}$ correspond to the polar decomposition of $\underline {\underline {T}}$, see Eq. (B.1) and, therefore, matrix $\underline {\underline {R}}$ is orthogonal, matrix $\underline {\underline {U}}$ positive-definite and symmetric, and both are defined uniquely. Next, the first equality of Eq. (B.2) also implies $\underline {\underline {W}}= \underline {\underline {R}}\, \underline {\underline {F}}+ \widetilde {u}\underline {\underline {R}}\, \underline {\underline {U}}$, which can be recast as $\widetilde {u}^{*} \underline {\underline {U}}+ \underline {\underline {F}}= \underline {\underline {R}}^{T} \underline {\underline {W}}$, where $\underline {u}^{*} = \underline {\underline {R}}^{T} \underline {u}$. Because matrix $\underline {\underline {F}}$ is symmetric, $(\widetilde {u}^{*} \underline {\underline {U}}) - (\widetilde {u}^{*} \underline {\underline {U}})^{T} = (\underline {\underline {R}}^{T} \underline {\underline {W}}) - (\underline {\underline {R}}^{T} \underline {\underline {W}})^{T}$, which leads to a linear system of equations for $\underline {u}^{*}$,

$$ \bigl[ \mathrm {tr}(\underline {\underline {U}}) \underline {\underline {I}}- \underline {\underline {U}}\bigr] \underline {u}^{*}= 2 \mathrm {axial}\bigl(\underline {\underline {R}}^{T} \underline {\underline {W}}\bigr). $$

(B.5)

Because matrices $\underline {\underline {U}}$, $\underline {\underline {R}}$, and $\underline {\underline {W}}$ are known, a unique solution exists $\underline {u}^{*}$, leading to $\underline {u}= \underline {\underline {R}}\underline {u}^{*}$, provided that the determinant of the system does not vanish. It is shown easily that $\mathrm {det}[ \mathrm {tr}(\underline {\underline {U}}) \underline {\underline {I}}- \underline {\underline {U}}] = (\lambda_{2}^{2} + \lambda_{3}^{2}) (\lambda_{1}^{2} + \lambda_{3}^{2}) (\lambda_{1}^{2} + \lambda_{2}^{2})$, where $\lambda_{i}^{2}$, i=1,2,3 are the positive eigenvalues of positive-definite matrix $\underline {\underline {U}}$, and hence, a solution of system (B.5) always exists. Finally, matrix $\underline {\underline {F}}= \underline {\underline {R}}^{T} (\underline {\underline {W}}- \widetilde {u}\underline {\underline {T}})$, which is symmetric because $\underline {u}$ was determined by the solution of system (B.5). This proves that all the quantities appearing in decomposition (B.2) can be found unequivocally, hence proving the stated generalization of the polar decomposition theorem. The second equality in Eq. (B.2 can be proved in a similar manner.