Flexible multibody dynamic analysis of shells with an edge center-based strain smoothing MITC method

Abstract

In this study, a dynamic analysis method for flexible multibody systems using Reissner–Mindlin shells was developed, and an edge center-based strain smoothing mixed interpolation of tensorial components (MITC) method that described the deformation of flexible shells was presented. Three primary achievements were completed. First, a new dynamics equation for Reissner–Mindlin shells based on the floating frame of reference formulation was modeled, in which the coupling of the bending and membrane deformations with rigid motion was considered comprehensively. Second, an edge center-based strain smoothing (ECSS) method that effectively improved the membrane and bending behavior for MITC3 elements was proposed. Third, the ECSS method was designed to construct a linear strain field within an element to ensure the strain consistency at the junction center of the element, resulting in a smoother stress field. The superior performance of the proposed method was verified by convergence analyses, and its advantages for flexible multibody dynamics analyses were also highlighted numerically.

Appendices

Appendix A

1.1 Integral of the element mass matrix

After the structure is discretized, the symmetrical mass matrix can be expressed as follows:

$$ {\mathbf{M}}^{I} = \sum\limits_{e} {\int_{{\Omega_{e}^{I} }} {\rho^{I} \left( {{\mathbf{L}}^{I} } \right)^{{\text{T}}} {\mathbf{L}}^{I} {\text{d}}\Omega^{I} } } = \sum\limits_{e} {{\mathbf{M}}_{\mathrm{e}}^{I} } $$

where ${\mathbf{M}}_{\mathrm{e}}^{I}$ represents the element mass matrix, which is symmetric. It can be decomposed into the following block matrices, which represent the rigid translation, rigid rotation, flexible deformation and the coupling among them, respectively.

$$ {\mathbf{M}}_{\mathrm{e}}^{I} = \sum\limits_{e} {\int_{{\Omega_{e}^{I} }} {\rho^{I} \left( {{\mathbf{L}}^{I} } \right)^{{\text{T}}} {\mathbf{L}}^{I} {\text{d}}\Omega^{I} } } = \left[ {\begin{array}{*{20}l} {{\mathbf{M}}_{\mathrm{e}RR}^{I} } & {{\mathbf{M}}_{\mathrm{e}R\theta }^{I} } & {{\mathbf{M}}_{\mathrm{e}Rf}^{I} } \\ {\left( {{\mathbf{M}}_{\mathrm{e}R\theta }^{I} } \right)^{{\text{T}}} } & {{\mathbf{M}}_{\mathrm{e}\theta \theta }^{I} } & {{\mathbf{M}}_{\mathrm{e}\theta f}^{I} } \\ {\left( {{\mathbf{M}}_{\mathrm{e}Rf}^{I} } \right)^{{\text{T}}} } & {\left( {{\mathbf{M}}_{\mathrm{e}\theta f}^{I} } \right)^{{\text{T}}} } & {{\mathbf{M}}_{\mathrm{eff}}^{I} } \\ \end{array} } \right] $$

Based on the formula of ${\mathbf{L}}^{I} = \left[ {\begin{array}{*{20}l} {\mathbf{I}} & { - {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }} } & {{\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} {\overline{\mathbf{S}}}} \\ \end{array} } \right]$ derived in Sect. 2, the formula of each block matrix in ${\mathbf{M}}_{\mathrm{e}}^{I}$ is presented in detail below.

(1) ${\mathbf{M}}_{\mathrm{e}RR}^{I} = \int_{{\Omega_{e}^{I} }} {\rho^{I} {\mathbf{I}}{\text{d}}\Omega^{I} } = m_{\mathrm{e}}^{I} {\mathbf{I}}$where $m_{\mathrm{e}}^{I} = \int_{{\Omega_{e}^{I} }} {\rho^{I} {\text{d}}\Omega^{I} }$ denotes the element mass.

(2) ${\mathbf{M}}_{\mathrm{e}R\theta }^{I} = - \int_{{\Omega_{e}^{I} }} {\rho^{I} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }} {\text{d}}\Omega^{I} }$.

Considering Eq. (24):

$$ \begin{aligned} {\mathbf{M}}_{\mathrm{e}R\theta }^{I} &= - \int_{{\Omega_{e}^{I} }} {\rho^{I} \left( {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{0} + {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{m} + \hat{z}{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{b} } \right){\text{d}}\Omega^{I} } \\ & = - \int_{{A_{e} }} {\int_{{ - \frac{h}{2}}}^{\frac{h}{2}} {\rho^{I} \left( {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{0} + {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{m} + \hat{z}{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{b} } \right)} {\text{d}}\hat{z}{\text{d}}A} \\ & = - \rho^{I} h^{I} \int_{{A_{e} }} {\left( {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{0} + {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{m} } \right){\text{d}}A} \end{aligned} $$

where $h^{I}$ is the element thickness.

(3) ${\mathbf{M}}_{\mathrm{e}Rf}^{I} = - \int_{{\Omega_{e}^{I} }} {\rho^{I} {\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} {\overline{\mathbf{S}}}{\text{d}}\Omega^{I} }$.

Considering Eq. (16):

$$ \begin{aligned} {\mathbf{M}}_{\mathrm{e}Rf}^{I} &= - \rho^{I} {\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} \int_{{\Omega_{e}^{I} }} {{\overline{\mathbf{S}}}{\text{d}}\Omega^{I} } \\ & = - \rho^{I} {\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} \int_{{A_{e} }} {\int_{{ - \frac{h}{2}}}^{\frac{h}{2}} {\left( {{\overline{\mathbf{S}}}_{m} + \hat{z}{\overline{\mathbf{S}}}_{b} } \right){\text{d}}\hat{z}} {\text{d}}A} \\ & = - \rho^{I} h^{I} {\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} \int_{{A_{e} }} {{\overline{\mathbf{S}}}_{m} {\text{d}}A} \\ \end{aligned} $$

(4) ${\mathbf{M}}_{\mathrm{e}\theta \theta }^{I} = \int_{{\Omega_{e}^{I} }} {\rho^{I} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{T} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }} {\text{d}}\Omega^{I} } = - \int_{{\Omega_{e}^{I} }} {\rho^{I} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }} {\text{d}}\Omega^{I} }$.

Considering Eq. (24):

$$ {\mathbf{M}}_{\mathrm{e}\theta \theta }^{I} = \left\{ \begin{gathered} - \rho^{I} h^{I} \int_{{A_{e} }} {\left( {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{0} + {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{m} } \right)\left( {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{0} + {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{m} } \right){\text{d}}A} \hfill \\ - \rho^{I} \frac{{\left( {h^{I} } \right)^{3} }}{12}\int_{{A_{e} }} {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{b} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{b} {\text{d}}A} \hfill \\ \end{gathered} \right\} $$

(5) ${\mathbf{M}}_{\mathrm{e}\theta f}^{I} = - \int_{{\Omega_{e}^{I} }} {\rho^{I} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{{\text{T}}} {\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} {\overline{\mathbf{S}}}{\text{d}}} \Omega^{I} = \int_{{\Omega_{e}^{I} }} {\rho^{I} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }} {\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} {\overline{\mathbf{S}}}{\text{d}}\Omega^{I} }$.

Similarly:

$$ \begin{aligned} {\mathbf{M}}_{\mathrm{e}\theta f}^{I} &= \int_{{A_{e} }} {\int_{{ - \frac{{h^{I} }}{2}}}^{{\frac{{h^{I} }}{2}}} {\rho^{I} \left( {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{0} + {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{m} + \hat{z}{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{b} } \right){\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} \left( {{\overline{\mathbf{S}}}_{m} + \hat{z}{\overline{\mathbf{S}}}_{b} } \right){\text{d}}\hat{z}{\text{d}}A} } \hfill \\ & = \left\{ \begin{gathered} \rho^{I} h^{I} \int_{{A_{e} }} {\left( {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{0} + {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{m} } \right){\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{I} {\overline{\mathbf{S}}}_{m} {\text{d}}A} \hfill \\ + \rho^{I} \frac{{\left( {h^{I} } \right)^{3} }}{12}\int_{{\Omega^{e} }} {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{b} {\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} {\overline{\mathbf{S}}}_{b} {\text{d}}\Omega^{e} } \hfill \\ \end{gathered} \right\} \\ \end{aligned} $$

(6) ${\mathbf{M}}_{\mathrm{eff}}^{I} = \int_{{\Omega_{e}^{I} }} {\rho^{I} \left( {{\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} {\overline{\mathbf{S}}}} \right)^{{\text{T}}} \left( {{\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} {\overline{\mathbf{S}}}} \right){\text{d}}\Omega^{I} }$.

Since $\left( {{\mathbf{A}}^{I} } \right)^{{\text{T}}} {\mathbf{A}}^{I} = \left( {{{\varvec{\uplambda}}}^{e} } \right)^{{\text{T}}} {{\varvec{\uplambda}}}^{e} = {\mathbf{I}}$:

$$ \begin{aligned} {\mathbf{M}}_{\mathrm{eff}}^{I} &= \int_{{\Omega_{e}^{I} }} {\rho^{I} {\overline{\mathbf{S}}}^{{\text{T}}} {\overline{\mathbf{S}}}{\text{d}}\Omega^{I} } \\ & = \int_{{A_{e} }} {\int_{{ - \frac{{h^{I} }}{2}}}^{{\frac{{h^{I} }}{2}}} {\rho^{I} \left( {{\overline{\mathbf{S}}}_{m} + \hat{z}{\overline{\mathbf{S}}}_{b} } \right)^{{\text{T}}} \left( {{\overline{\mathbf{S}}}_{m} + \hat{z}{\overline{\mathbf{S}}}_{b} } \right){\text{d}}\hat{z}{\text{d}}A} } \hfill \\ &= \rho^{I} h^{I} \int_{{A_{e} }} {{\overline{\mathbf{S}}}_{m}^{{\text{T}}} {\overline{\mathbf{S}}}_{m} {\text{d}}A} + \rho^{I} \frac{{\left( {h^{I} } \right)^{3} }}{12}\int_{{A_{e} }} {{\overline{\mathbf{S}}}_{b}^{{\text{T}}} {\overline{\mathbf{S}}}_{b} {\text{d}}A} \\ \end{aligned} $$

1.2 Integral of the element velocity coupling vector

The specific expression for ${\mathbf{Q}}_{\mathrm{v}}^{I}$ can be similarly decomposed:

$$ {\mathbf{Q}}_{\mathrm{v}}^{I} = \sum\limits_{e} {\int_{{\Omega_{e}^{I} }} {\rho^{I} \left( {{\mathbf{L}}^{I} } \right)^{{\text{T}}} {\mathbf{a}}_{v}^{I} {\text{d}}\Omega^{I} } } = \sum\limits_{e} {\left[ {\begin{array}{*{20}l} {{\mathbf{Q}}_{evR}^{I} } \\ {{\mathbf{Q}}_{ev\theta }^{I} } \\ {{\mathbf{Q}}_{evf}^{I} } \\ \end{array} } \right]} $$

Similarly, consider Eqs. (16), (24) and (25) to integrate the above vectors:

$$ \begin{aligned} {\mathbf{Q}}_{evR}^{I} &= \int_{{\Omega_{e}^{I} }} {\rho^{I} {\mathbf{a}}_{v}^{I} {\text{d}}\Omega^{I} } \\ & = \rho^{I} h^{I} \int_{{A_{e} }} {{\mathbf{a}}_{v}^{I0} + {\mathbf{a}}_{v}^{Im} {\text{d}}A} \\ \end{aligned} $$

$$ \begin{aligned} {\mathbf{Q}}_{ev\theta }^{I} &= - \int_{{\Omega_{e}^{I} }} {\rho^{I} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{{\text{T}}} {\mathbf{a}}_{v}^{I} {\text{d}}\Omega^{I} } \\ & = \int_{{\Omega_{e}^{I} }} {\rho^{I} {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }} {\mathbf{a}}_{v}^{I} {\text{d}}\Omega^{I} } \\ & = \left\{ \begin{gathered} \rho^{I} h^{I} \int_{{A_{e} }} {\left( {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{0} + {\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{m} } \right)\left( {{\mathbf{a}}_{v}^{I0} + {\mathbf{a}}_{v}^{Im} } \right){\text{d}}A} \hfill \\ + \rho^{I} \frac{{\left( {h^{I} } \right)^{3} }}{12}\int_{{A_{e} }} {{\tilde{\mathbf{r}}}_{{\overline{O}^{I} P_{f} }}^{b} {\mathbf{a}}_{v}^{Ib} {\text{d}}A} \hfill \\ \end{gathered} \right\} \\ \end{aligned} $$

$$ \begin{aligned} {\mathbf{Q}}_{evf}^{I} &= \int_{{\Omega_{e}^{I} }} {\rho^{I} \left( {{\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} {\overline{\mathbf{S}}}} \right)^{{\text{T}}} {\mathbf{a}}_{v}^{I} {\text{d}}\Omega^{I} } \\ & = \left\{ \begin{gathered} \rho^{I} h^{I} \int_{{A_{e} }} {\left[ {{\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} \left( {{\overline{\mathbf{S}}}_{0} + {\overline{\mathbf{S}}}_{m} } \right)} \right]^{{\text{T}}} \left( {{\mathbf{a}}_{v}^{I0} + {\mathbf{a}}_{v}^{Im} } \right){\text{d}}A} \hfill \\ + \rho^{I} \frac{{\left( {h^{I} } \right)^{3} }}{12}\int_{{A_{e} }} {\left( {{\mathbf{A}}^{I} {{\varvec{\uplambda}}}^{e} {\overline{\mathbf{S}}}_{b} } \right)^{{\text{T}}} {\mathbf{a}}_{v}^{Ib} {\text{d}}A} \hfill \\ \end{gathered} \right\} \\ \end{aligned} $$

Appendix B

By solving the generalized eigenvalue equation $\omega^{2} {\mathbf{M}}_{ff}^{I} {\mathbf{\varphi }}^{I} = {\mathbf{K}}_{ff}^{I} {\mathbf{\varphi }}^{I}$, the modal shape ${\mathbf{\varphi }}_{j}^{I}$ corresponding to the $j{{{\text{th}}}}$ natural frequency of flexible body $I$ can be obtained. The physical coordinates ${\overline{\mathbf{q}}}_{f}^{I}$ can be converted to the modal space by selecting the first M-order modes:

$$ {\overline{\mathbf{q}}}_{f}^{I} = {\mathbf{V}}_{M} {\mathbf{p}}^{I} $$

where ${\mathbf{p}}^{I}$ indicates the modal coordinates, and ${\mathbf{V}}_{M} = \left[ {\begin{array}{*{20}l} {{\mathbf{\varphi }}_{1}^{I} } & {{\mathbf{\varphi }}_{2}^{I} } & \cdots & {{\mathbf{\varphi }}_{M}^{I} } \\ \end{array} } \right]$ consists of the first M-order modes.

In general, the employed modal order M is far less than the DOFs of the discrete flexible body. Therefore, the original governing equation Eq. (87) can be reduced to the following equation:

$$ \left[ {\begin{array}{*{20}l} {{\mathbf{M}}_{RR}^{I} } & {{\mathbf{M}}_{R\theta }^{I} } & {{\tilde{\mathbf{M}}}_{Rf}^{I} } \\ {\left( {{\mathbf{M}}_{R\theta }^{I} } \right)^{{\text{T}}} } & {{\mathbf{M}}_{\theta \theta }^{I} } & {{\tilde{\mathbf{M}}}_{\theta f}^{I} } \\ {\left( {{\tilde{\mathbf{M}}}_{Rf}^{I} } \right)^{{\text{T}}} } & {\left( {{\tilde{\mathbf{M}}}_{\theta f}^{I} } \right)^{{\text{T}}} } & {{\tilde{\mathbf{M}}}_{ff}^{I} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {{\ddot{\mathbf{R}}}^{I} } \\ {{{\varvec{\upvarepsilon}}}^{I} } \\ {{\ddot{\mathbf{p}}}^{I} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}l} {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{0}} & {{\tilde{\mathbf{K}}}_{ff}^{I} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}l} {{\mathbf{R}}^{I} } \\ {{{\varvec{\uptheta}}}^{I} } \\ {{\mathbf{p}}^{I} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}l} {{\mathbf{Q}}_{R}^{I} } \\ {{\mathbf{Q}}_{\theta }^{I} } \\ {{\tilde{\mathbf{Q}}}_{f}^{I} } \\ \end{array} } \right] $$

where ${\tilde{\mathbf{M}}}_{Rf}^{I} = {\mathbf{M}}_{Rf}^{I} {\mathbf{V}}_{M}$, ${\tilde{\mathbf{M}}}_{\theta f}^{I} = {\mathbf{M}}_{\theta f}^{I} {\mathbf{V}}_{M}$, ${\tilde{\mathbf{M}}}_{ff}^{I} = {\mathbf{V}}_{M}^{{\text{T}}} {\mathbf{M}}_{ff}^{I} {\mathbf{V}}_{M}$, ${\tilde{\mathbf{K}}}_{ff}^{I} = {\mathbf{V}}_{M}^{{\text{T}}} {\mathbf{K}}_{ff}^{I} {\mathbf{V}}_{M}$, ${\tilde{\mathbf{Q}}}_{f}^{I} = {\mathbf{V}}_{M}^{{\text{T}}} {\mathbf{Q}}_{f}^{I}$.