Time discontinuous finite element method for transient response analysis of linear time-varying structures

Abstract

In this paper, a mixed form of Hamilton’s law of variable mass system is proposed, and then a time discontinuous finite element method for transient response analysis of linear time-varying structures is developed based on the law. As these time-varying parameters are degraded into time-invariant ones, the time discontinuous finite element method for linear time-varying structures is degraded into an unconditionally stable higher-order accurate time integration method for linear time-invariant structures. The performance of the proposed time integration method has been verified and assessed extensively through many numerical examples, including the single-degree-of-freedom system with a time-varying mass and the string and beam structure with a moving mass. Numerical results demonstrate that the proposed time finite element method for linear time-varying structures performs extremely well compare with the Newmark method, the existing time continuous finite element method for linear time-varying structures as well as the combination of linear time-invariant time integration method and time frozen technique.

Appendices

Appendix 1: Compound matrix multiplication

If all elements of a matrix are row (column) vectors with the same order, then this matrix is defined as a row (column) compound matrix. Assume that $\varvec{A} = \left[ {\varvec{a}_{ij} } \right]_{m \times n}$ is a $m \times n$ row compound matrix and its elements $\varvec{a}_{ij}$ are all k-order row vectors, and also assume that $\varvec{B} = \left[ {\varvec{b}_{rs} } \right]_{p \times q}$ is a $p \times q$ column compound matrix and its elements $\varvec{b}_{rs}$ are all k-order column vectors. The compound matrix product ‘$\times$’ of $\varvec{A}$ and $\varvec{B}$ is defined as

$$\varvec{A} \times \varvec{B} = \varvec{D} = \left[ {\begin{array}{*{20}c} {\varvec{D}_{11} } & \cdots & {\varvec{D}_{1j} } & \cdots & {\varvec{D}_{1n} } \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ {\varvec{D}_{i1} } & \cdots & {\varvec{D}_{ij} } & \cdots & {\varvec{D}_{in} } \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ {\varvec{D}_{m1} } & \cdots & {\varvec{D}_{mj} } & \cdots & {\varvec{D}_{mn} } \\ \end{array} } \right]$$

(102)

where the submatrix $\varvec{D}_{ij}$ is given by

$$\varvec{D}_{ij} = \left[ {\begin{array}{*{20}c} {\varvec{a}_{ij} \cdot \varvec{b}_{11} } & \cdots & {\varvec{a}_{ij} \cdot \varvec{b}_{1s} } & \cdots & {\varvec{a}_{ij} \cdot \varvec{b}_{1q} } \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ {\varvec{a}_{ij} \cdot \varvec{b}_{r1} } & \cdots & {\varvec{a}_{ij} \cdot \varvec{b}_{rs} } & \cdots & {\varvec{a}_{ij} \cdot \varvec{b}_{rq} } \\ \vdots & \ddots & \vdots & \ddots & \vdots \\ {\varvec{a}_{ij} \cdot \varvec{b}_{p1} } & \cdots & {\varvec{a}_{ij} \cdot \varvec{b}_{ps} } & \cdots & {\varvec{a}_{ij} \cdot \varvec{b}_{pq} } \\ \end{array} } \right]$$

(103)

in which ‘$\cdot$’ denotes the vector dot product. As the order of the elements k is reduced to one (namely $\varvec{a}_{ij}$ and $\varvec{b}_{rs}$ are both scalars), the compound matrix product ‘$\times$’ is reduced to Kronecker product ‘$\otimes$’.

Appendix 2: Coefficient matrixes of time discontinuous finite element methods

The differential coefficient matrixes of the proposed time discontinuous finite element methods for transient response analysis of time-varying and time-invariant structures are given here. The coefficient matrixes of TFEM-1121 method are given by

(104)

(105)

$$\varvec{D}_{F} = \frac{{\Delta t}}{6}\left[ {\begin{array}{*{20}c} { - 1} & { - 2} & 0 \\ 0 & { - 2} & { - 1} \\ \end{array} } \right]$$

(106)

The coefficient matrixes of TFEM-2242 method are given by

(107)

(108)

$$\varvec{D}_{F} = \frac{{\Delta t}}{630}\left[ {\begin{array}{*{20}c} { - 44} & { - 104} & {30} & 8 & 5 \\ { - 10} & { - 128} & { - 144} & { - 128} & { - 10} \\ 5 & 8 & {30} & { - 104} & { - 44} \\ \end{array} } \right]$$

(109)

The coefficient matrixes of TFEM-1120 method are given by

$$\bar{\varvec{D}}_{11} = \bar{\varvec{D}}_{22}^{M} = \frac{1}{2}\left[ {\begin{array}{*{20}c} { - 1} & { - 1} \\ 1 & { - 1} \\ \end{array} } \right]$$

(110)

$$\bar{\varvec{D}}_{12} = - \bar{\varvec{D}}_{21} = - \bar{\varvec{D}}_{22}^{C} = \frac{{\Delta t}}{6}\left[ {\begin{array}{*{20}c} 2 & 1 \\ 1 & 2 \\ \end{array} } \right]$$

(111)

$$\varvec{D}_{F} = \frac{{\Delta t}}{6}\left[ {\begin{array}{*{20}c} { - 1} & { - 2} & 0 \\ 0 & { - 2} & { - 1} \\ \end{array} } \right]$$

(112)

The coefficient matrixes of TFEM-2240 method are given by

$$\bar{\varvec{D}}_{11} = \bar{\varvec{D}}_{22}^{M} = \frac{1}{6}\left[ {\begin{array}{*{20}c} { - 3} & { - 4} & 1 \\ 4 & 0 & { - 4} \\ { - 1} & 4 & { - 3} \\ \end{array} } \right]$$

(113)

$$\bar{\varvec{D}}_{12} = - \bar{\varvec{D}}_{21} = - \bar{\varvec{D}}_{22}^{C} = \frac{{\Delta t}}{30}\left[ {\begin{array}{*{20}c} 4 & 2 & { - 1} \\ 2 & {16} & 2 \\ { - 1} & 2 & 4 \\ \end{array} } \right]$$

(114)

$$\varvec{D}_{F} = \frac{{\Delta t}}{630}\left[ {\begin{array}{*{20}c} { - 44} & { - 104} & {30} & 8 & 5 \\ { - 10} & { - 128} & { - 144} & { - 128} & { - 10} \\ 5 & 8 & {30} & { - 104} & { - 44} \\ \end{array} } \right]$$

(115)