HHT-$\alpha$ and TR-BDF2 schemes for dynamic contact problems

Abstract

This work focuses on the numerical performance of HHT-$\alpha$ and TR-BDF2 schemes for dynamic frictionless unilateral contact problems between an elastic body and a rigid obstacle. Nitsche’s method, the penalty method, and the augmented Lagrangian method are considered to handle unilateral contact conditions. Analysis of the convergence of an opposed value of the parameter $\tilde{\alpha }$ for the HHT-$\alpha$ method is achieved. The mass redistribution method has also been tested and compared with the standard mass matrix. Numerical results for 1D and 3D benchmarks show the functionality of the combinations of schemes and methods used.

Appendices

Appendices

The first appendix details the local truncation error for the HHT-alpha scheme. The second appendix provides the complete amplification matrix for HHT-alpha. The third one provides an introduction to the Nitsche-Hybrid scheme.

1.1 A.1 Local truncation error for HHT-$\alpha$ scheme

Applying the local truncation error (27) to the SDOF system (25), the following holds:

$$\begin{aligned}{} & {} \Delta tT_n \nonumber \\{} & {} \quad {=} \frac{{-} \tilde{\alpha }\omega ^2\Delta t^2 {+} \frac{1}{\tilde{\beta }}}{\omega ^2\Delta t^2 \left( 1 {-} \tilde{\alpha }\right) {+} \frac{1}{\tilde{\beta }}}u(t^n) {+} \frac{\frac{1}{\tilde{\beta }}}{\omega ^2\Delta t^2 \left( 1 {-} \tilde{\alpha }\right) {+} \frac{1}{\tilde{\beta }}}\Delta t{\dot{u}}(t^n) \nonumber \\{} & {} \quad \quad {+} \frac{1 {-} 2 \tilde{\beta }}{2 \tilde{\beta }\left( \omega ^2\Delta t^2 \left( 1 - \tilde{\alpha }\right) {+} \frac{1}{\tilde{\beta }}\right) }\Delta t^2\ddot{u}(t^n) {-} u(t^n) {-} \Delta t{\dot{u}}(t^n) \nonumber \\{} & {} \quad - \frac{1}{2} \Delta t^2\ddot{u}(t^n) + O(\Delta t^3) \nonumber \\{} & {} {=} \frac{{-}\omega ^2\Delta t^2}{\omega ^2\Delta t^2 \left( 1 {-} \tilde{\alpha }\right) {+} \frac{1}{\tilde{\beta }}}u(t^n) {+} \frac{\left( 1-\tilde{\alpha }\right) \omega ^2\Delta t^2}{\omega ^2\Delta t^2 \left( 1 {-} \tilde{\alpha }\right) {+} \frac{1}{\tilde{\beta }}}\Delta t{\dot{u}}(t^n) \nonumber \\{} & {} \quad + \frac{1}{2}\left( \frac{(1-2\tilde{\beta })\frac{1}{\tilde{\beta }}}{\omega ^2\Delta t^2 \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}} - 1\right) \Delta t^2\ddot{u}(t^n) + O(\Delta t^3), \nonumber \\{} & {} = \frac{1}{\omega ^2\Delta t^2 \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}}\left( -\omega ^2\Delta t^2 u(t^n)\right. \nonumber \\{} & {} \quad \left. -\left( 1+\frac{\omega ^2\Delta t^2\left( 1-\tilde{\alpha }\right) }{2}\right) \Delta t^2\ddot{u}(t^n)\right) \nonumber \\{} & {} \quad + \frac{\left( 1-\tilde{\alpha }\right) \omega ^2\Delta t^2}{\omega ^2\Delta t^2 \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}}\Delta t{\dot{u}}(t^n) + O(\Delta t^3) \nonumber \\{} & {} = \frac{1}{\omega ^2\Delta t^2 \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}}\left( -\left( \omega ^2 u(t^n) + \ddot{u}(t^n)\right) \Delta t^2 \right. \nonumber \\{} & {} \left. \quad - \frac{\omega ^2\Delta t^2\left( 1-\tilde{\alpha }\right) }{2}\Delta t^4\ddot{u}(t^n)\right) \nonumber \\{} & {} \quad + \frac{\left( 1-\tilde{\alpha }\right) \omega ^2}{\omega ^2\Delta t^2 \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}}\Delta t^3{\dot{u}}(t^n) + O(\Delta t^3) \nonumber \\{} & {} \le \frac{1}{\omega ^2\Delta t^2 \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}}\left| \omega ^2 u(t^n) + \ddot{u}(t^n)\right| \Delta t^2 \nonumber \\{} & {} \quad + \frac{\left( 1-\tilde{\alpha }\right) \omega ^2}{\omega ^2\Delta t^2 \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}}\left| {\dot{u}}(t^n)\right| \Delta t^3 + O(\Delta t^3), \nonumber \\ \end{aligned}$$

(36)

which implies the boundness of the local truncation error.

1.2 A.2 Amplification matrix for HHT-$\alpha$ scheme

Applying the HHT-$\alpha$ scheme (24) to the SDOF system (25), by replacing the operators $\textbf{M}$ and $\textbf{B}$ by m and k respectively, the amplification matrix for the HHT-$\alpha$ scheme reads below:

$$\begin{aligned} \left[ \begin{array}{lll} \frac{- \tilde{\alpha }\tilde{\omega }^{2} + \frac{1}{\tilde{\beta }}}{\tilde{\omega }^{2} \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}} &{} \frac{1}{\tilde{\beta }\left( \tilde{\omega }^{2} \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}\right) } &{} \frac{1 - 2 \tilde{\beta }}{2 \tilde{\beta }\left( \tilde{\omega }^{2} \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}\right) }\\ \tilde{\gamma }\left( \frac{- \tilde{\alpha }\tilde{\omega }^{2} + \frac{1}{\tilde{\beta }}}{\tilde{\beta }\left( \tilde{\omega }^{2} \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}\right) } - \frac{1}{\tilde{\beta }}\right) &{} \tilde{\gamma }\left( - \frac{1}{\tilde{\beta }} + \frac{1}{\tilde{\beta }^{2} \left( \tilde{\omega }^{2} \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}\right) }\right) + 1 &{} \tilde{\gamma }\left( \frac{2 \tilde{\beta }- 1}{2 \tilde{\beta }} + \frac{1 - 2 \tilde{\beta }}{2 \tilde{\beta }^{2} \left( \tilde{\omega }^{2} \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}\right) }\right) - \tilde{\gamma }+ 1\\ \frac{- \tilde{\alpha }\tilde{\omega }^{2} + \frac{1}{\tilde{\beta }}}{\tilde{\beta }\left( \tilde{\omega }^{2} \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}\right) } - \frac{1}{\tilde{\beta }} &{} - \frac{1}{\tilde{\beta }} + \frac{1}{\tilde{\beta }^{2} \left( \tilde{\omega }^{2} \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}\right) } &{} \frac{2 \tilde{\beta }- 1}{2 \tilde{\beta }} + \frac{1 - 2 \tilde{\beta }}{2 \tilde{\beta }^{2} \left( \tilde{\omega }^{2} \left( 1 - \tilde{\alpha }\right) + \frac{1}{\tilde{\beta }}\right) }\end{array}\right] . \end{aligned}$$

(37)

We recall that $\textbf{Y}^{n+1} = \textbf{A}\textbf{Y}^n$ with $\textbf{Y}^n = [u^n, \Delta t{\dot{u}}^n, \Delta t^2\ddot{u}^n]^T$ and $\tilde{\omega }:= \omega ^2\Delta t^2$.

1.3 A.3 Nitsche-Hybrid time scheme

The Nitsche-Hybrid scheme has been introduced in [22] and solved the following system at each time step:

$$\begin{aligned} \left\{ \begin{aligned}&\text {Seek }\textbf{u}^{h, n+1}, {\dot{\textbf{u}}}^{h, n+1}, \ddot{\textbf{u}}^{h, n+1} \in \textbf{V}^h \text { s.t. }\\&\textbf{u}^{h, n+1} = \textbf{u}^{h, n} + \frac{\Delta t}{2}({\dot{\textbf{u}}}^{h, n} + {\dot{\textbf{u}}}^{h, n+1}), \\&{\dot{\textbf{u}}}^{h, n+1} = {\dot{\textbf{u}}}^{h, n} + \frac{\Delta t}{2}(\ddot{\textbf{u}}^{h, n} + \ddot{\textbf{u}}^{h, n+1}), \\&\frac{\textbf{M}}{2}(\ddot{\textbf{u}}^{h, n} {+} \ddot{\textbf{u}}^{h, n{+}1}) {+} \textbf{B}_{Nh}(\textbf{u}^{h, n}, \textbf{u}^{h, n{+}1}) {=} \frac{1}{2}(\textbf{L}^{n} {+} \textbf{L}^{n{+}1}). \end{aligned} \right. \end{aligned}$$

(38)

where non-linear operator $\textbf{B}_{Nh}$ is defined by integration:

$$\begin{aligned} \begin{aligned}&\left( \textbf{B}_{Nh}(\textbf{u}^{h, n}, \textbf{u}^{h, n+1}), \textbf{w}^h\right) _{\Omega } \\&= \frac{1}{2}\left( a_{\gamma _N}(\textbf{u}^{h, n}, \textbf{w}^h) + a_{\gamma _N}(\textbf{u}^{h, n+1}, \textbf{w}^h)\right) \\&\quad {+} \left( \frac{1}{\gamma _N}H(P_N(\textbf{u}^{h, n}))\left[ \frac{1}{2}(P_N(\textbf{u}^{h, n}{+}\textbf{u}^{h, n{+}1}))\right] _{{\mathbb {R}}^{-}}, \textbf{w}^h\right) _{\Gamma _C} \\&\quad + \left( \frac{1}{\gamma _N}H(-P_N(\textbf{u}^{h, n}))\frac{1}{2}\left( \left[ P_N(\textbf{u}^{h, n})\right] _{{\mathbb {R}}^-} \right. \right. \\&\quad \left. \left. + \left[ P_N(\textbf{u}^{h, n+1})\right] _{{\mathbb {R}}^-}\right) , \textbf{w}^h\right) _{\Gamma _C}, \end{aligned}\nonumber \\ \end{aligned}$$

(39)

where $H(\cdot )$ is the Heaviside function. This scheme is unconditionally stable for the symmetric version of Nitsche’s method. The main idea is to combine two different schemes with one during contact phases and one during non-contact phases. Hence, this scheme conserves the mechanical energy on the linear regime (when the contact is not active) as its equivalence with the mid-point scheme but it dissipates during contact phases. Moreover, this scheme is more nonlinear than the others and a bit more difficult to implement.