A new heterogeneous asynchronous explicit–implicit time integrator for nonsmooth dynamics

Abstract

In computational structural dynamics, particularly in the presence of nonsmooth behavior, the choice of the time-step and the time integrator has a critical impact on the feasibility of the simulation. Furthermore, in some cases, as in the case of a bridge crane under seismic loading, multiple time-scales coexist in the same problem. In that case, the use of multi-time scale methods is suitable. Here, we propose a new explicit–implicit heterogeneous asynchronous time integrator (HATI) for nonsmooth transient dynamics with frictionless unilateral contacts and impacts. Furthermore, we present a new explicit time integrator for contact/impact problems where the contact constraints are enforced using a Lagrange multiplier method. In other words, the aim of this paper consists in using an explicit time integrator with a fine time scale in the contact area for reproducing high frequency phenomena, while an implicit time integrator is adopted in the other parts in order to reproduce much low frequency phenomena and to optimize the CPU time. In a first step, the explicit time integrator is tested on a one-dimensional example and compared to Moreau-Jean’s event-capturing schemes. The explicit algorithm is found to be very accurate and the scheme has generally a higher order of convergence than Moreau-Jean’s schemes and provides also an excellent energy behavior. Then, the two time scales explicit–implicit HATI is applied to the numerical example of a bridge crane under seismic loading. The results are validated in comparison to a fine scale full explicit computation. The energy dissipated in the implicit–explicit interface is well controlled and the computational time is lower than a full-explicit simulation.

Appendices

Appendix 1: CD-Lagrange contact/impact algorithm

The implementation of the time integrator (47) is illustrated in the following flow-chart (Algorithm 2). For sake of simplicity, no contact searching procedure is used here, as we assume the p prospective contact points to be known. Thus, the $(p \times n)$ boolean contact operator $\mathbf{L }_{c}$ is known at the beginning of the computation, where n is the number of DOFs of the problem. In this paper, this algorithm will be called CD-Lagrange time integrator.

Appendix 2: Discrete energy balance equation

1.1 Appendix 2.1: Explicit CD-Lagrange time integrator for contact/impact problems

In order to evaluate the energy properties of the proposed CD-Lagrange time integrator, we give the following discrete energy balance equation. It is obtained by multiplying the nonsmooth equation of motion (32) by $\dot{\mathbf{U }}^{T}$.

$$\begin{aligned} d\left( \frac{1}{2}\dot{\mathbf{U }}^{T}\mathbf{M }\dot{\mathbf{U }}\right) = \dot{\mathbf{U }}^{T}\mathbf{F }dt + \dot{\mathbf{U }}^{T} d\mathbf{I } \end{aligned}$$

(75)

where (assuming that $\mathbf{C }=0$ for sake of simplicity):

$$\begin{aligned} \mathbf{F } = \mathbf{F }_{{ ext}} - \mathbf{F }_{{ int}} \end{aligned}$$

(76)

Using the relation $d\mathbf{U }=\dot{\mathbf{U }}dt$, we get the following energy balance:

$$\begin{aligned} d\left( \frac{1}{2}\dot{\mathbf{U }}^{T}\mathbf{M }\dot{\mathbf{U }}\right) = d\mathbf{U }^{T} \mathbf{F } + \dot{\mathbf{U }}^{T} d\mathbf{I } \end{aligned}$$

(77)

The term in the parenthesis on the left hand side is the kinetic energy, and the two terms on the right hand side are the rate of internal/external work and the contact/impact impulse work. The discrete form of the energy balance equation is obtained by expressing the increment of the kinetic energy over the time interval $[t_{n},t_{n+1}]$:

$$\begin{aligned} \Big [\frac{1}{2}\dot{\mathbf{U }}^{T}\mathbf{M }\dot{\mathbf{U }}\Big ]^{n+1}_{n} = \langle \dot{\mathbf{U }}_{n} \rangle ^{T} \mathbf{M } [\dot{\mathbf{U }}_{n}] \end{aligned}$$

(78)

The notations [.] and $\langle . \rangle$ denote the increment and the mean value. They are defined as follows:

$$\begin{aligned} {[}X_n]= & {} (X_{n+1}-X_n) \end{aligned}$$

(79)

$$\begin{aligned} \langle X_n \rangle= & {} \frac{1}{2} (X_{n+1} + X_n) \end{aligned}$$

(80)

The path independent kinetic energy increment uses the expressions of means value $\langle . \rangle$ and increment [.] (see Eq. (78)). For the CD time integrator, from Eq. (44a) and (44b), we obtain the following expressions in terms of mean values and increments:

$$\begin{aligned} {[}\dot{\mathbf{U }}_{n}]\,=\,&\langle \mathbf{W }_n \rangle \end{aligned}$$

(81a)

$$\begin{aligned} {[}\mathbf{U }_{n}]\,=\,&\varDelta t \langle \dot{\mathbf{U }}_{n} \rangle - \frac{\varDelta t }{4}[\mathbf{W }_n] \end{aligned}$$

(81b)

Combining (78) and (81a) involves:

$$\begin{aligned} \Big [\frac{1}{2}\dot{\mathbf{U }}^{T}\mathbf{M }\dot{\mathbf{U }}\Big ]^{n+1}_{n} = \varDelta t \langle \dot{\mathbf{U }}_{n} \rangle ^{T} \langle \mathbf F _{n} \rangle + \langle \dot{\mathbf{U }}_{n} \rangle ^{T} \langle \mathbf I _{n} \rangle \end{aligned}$$

(82)

From (81b) we obtain:

$$\begin{aligned} \Big [\frac{1}{2}\dot{\mathbf{U }}^{T}{} \mathbf M \dot{\mathbf{U }}\Big ]^{n+1}_{n}= & {} \displaystyle [\mathbf U _{n}]^{T} \langle \mathbf F _{n} \rangle + \frac{\varDelta t}{4} [\mathbf W _n]^{T} \langle \mathbf F _{n} \rangle + \frac{[\mathbf U _{n}]}{\varDelta t}^{T} \langle \mathbf I _{n} \rangle + \frac{1}{4} [\mathbf W _n]^{T} \langle \mathbf I _{n} \rangle \nonumber \\= & {} \displaystyle [\mathbf U _{n}]^{T} \langle \mathbf F _{n} \rangle + \frac{[\mathbf U _{n}]}{\varDelta t}^{T} \langle \mathbf I _{n} \rangle + \frac{1}{4} [\mathbf W _n]^{T}\Big (\varDelta t\langle \mathbf F _{n} \rangle + \langle \mathbf I _{n} \rangle \Big )\nonumber \\= & {} \displaystyle [\mathbf U _{n}]^{T} \langle \mathbf F _{n} \rangle + \frac{[\mathbf U _{n}]}{\varDelta t}^{T} \langle \mathbf I _{n} \rangle + \frac{1}{4} [\mathbf W _n]^{T} M \langle \mathbf W _{n} \rangle \end{aligned}$$

(83)

Then, the discretized energy balance can be written as follows:

$$\begin{aligned} \Big [\frac{1}{2}\dot{\mathbf{U }}^{T}{} \mathbf M \dot{\mathbf{U }} - \frac{1}{8} \mathbf W ^{T} \mathbf M \mathbf W \Big ]^{n+1}_{n} = [\mathbf U _{n}]^{T} \langle \mathbf F _{n} \rangle + \frac{[\mathbf U _{n}]}{\varDelta t}^{T} \langle \mathbf I _{n} \rangle \end{aligned}$$

(84)

which can also be denoted:

$$\begin{aligned} \varDelta W_{{ kin},n+1} + \varDelta W_{{ comp},n+1} +\varDelta W_{{ int},n+1} = \varDelta W_{{ ext},n+1} + \varDelta W_{IC,n+1} \end{aligned}$$

(85)

The expression of energies involved in the above balance energy are:

$$\begin{aligned}&\varDelta W_{{ kin},n+1} = \Big [\frac{1}{2}\dot{\mathbf{U }}^{T}{} \mathbf M \dot{\mathbf{U }} \Big ]^{n+1}_{n} \end{aligned}$$

(86)

$$\begin{aligned}&\varDelta W_{{ comp},n+1} = \Big [- \frac{1}{8} \mathbf W ^{T} \mathbf M \mathbf W \Big ]^{n+1}_{n} \end{aligned}$$

(87)

$$\begin{aligned}&\varDelta W_{{ ext},n+1} = [\mathbf U _{n}]^{T} \langle \mathbf F _{{ ext},n} \rangle \end{aligned}$$

(88)

$$\begin{aligned}&\varDelta W_{{ int},n+1} = [\mathbf U _{n}]^{T} \langle \mathbf F _{{ int},n} \rangle \end{aligned}$$

(89)

$$\begin{aligned}&\varDelta W_{IC,n+1} = \frac{[\mathbf U _{n}]}{\varDelta t}^{T} \langle \mathbf I _{n} \rangle \end{aligned}$$

(90)

and $\varDelta W_{{ kin}}$, $\varDelta W_{{ int}}$, $\varDelta W_{{ comp}}$, $\varDelta W_{{ ext}}$, $\varDelta W_{IC}$ are, respectively, the increments over the time step of the kinetic, internal, complementary, external and impact/contact energies with explicit time integrator.

It can be noticed that the quantities inside bracket on the left hand side are path independent quantities (conservative quantities) (see [50]). The balance energy (85) corresponds to a generalization of the Newmark energy balance given in [57] to the case of contact/impact dynamics.

1.2 Appendix 2.2: Explicit–implicit HATI for contact/impact problems

In order to check the energy properties of the proposed explicit–implicit HATI, we recall the discrete energy balance equation between $t_{0}$ and $t_{m}$:

$$\begin{aligned}&\displaystyle \varDelta W^I_{{ kin},m} + \varDelta W^I_{{ int},m} \nonumber \\&\qquad +\,\sum _{j=1}^{m} \left( \varDelta W^E_{{ kin},j} + \varDelta W^E_{{ int},j} + \varDelta W^E_{{ comp},j}\right) = \varDelta W^I_{{ ext},m} \\&\qquad +\,\varDelta W^I_{D,m} + \sum _{j=1}^{m} (\varDelta W^E_{ext,j} + \varDelta W^E_{IC,j} + \varDelta W^E_{D,j}) \nonumber \\&\qquad +\, \varDelta W_{interface}\nonumber \end{aligned}$$

(91)

The expressions of the increments over a time step of the kinetic $\varDelta W_{{ kin}}$, internal $\varDelta W_{{ int}}$, complementary $\varDelta W_{{ comp}}$, external $\varDelta W_{{ ext}}$ and impact/contact $\varDelta W_{IC}$ energies are given, respectively, in Eqs. (86), (89), (87), (88) and (90). In addition, the definitions of the additional energy terms are given below:

$$\begin{aligned}&\varDelta W^k_{D,e} = [\mathbf U ^{k}_{e}]^{T} \langle \mathbf F ^{k}_{D,e} \rangle \ ; \quad e= 0, (j-1); \ k=I, E \end{aligned}$$

(92)

$$\begin{aligned}&\varDelta W_{{ interface}} = [\mathbf U ^{I}_0]^T (L^{I}_{G})^{T} [\varvec{\varLambda }_{G,0}] + \sum _{j=0}^{m-1} [\mathbf U ^{E}_{j}]^T (L^{E}_{G})^T [\varvec{\varLambda }_{G,j}] \end{aligned}$$

(93)

Appendix 3: Time error indicator

In contact dynamics, due to velocity discontinuities, the convergence cannot be observed using uniform norm as demonstrated in [2]. For this reason, Moreau [69] introduced the convergence in the sense of filled-in-graph using the Hausdorff distance to measure the error with respect to a reference solution. It is shown in [2] that an equivalent absolute $l_1$-norm gives the same order of convergence as the Hausdorff norm and can be used thanks to its easy implementation. For this purpose, we introduce the relative error indicator as follows:

$$\begin{aligned} e_f = \frac{\displaystyle \varDelta t \sum _{i=1}^{N} |f_i - f(t_i)|}{\displaystyle \varDelta t\sum _{i=1}^{N}|f(t_i)|} = \frac{\displaystyle \sum _{i=1}^{N} |f_i - f(t_i)|}{\displaystyle \sum _{i=1}^{N}|f(t_i)|} \end{aligned}$$

(94)

where N is the number of time steps in the time interval [0, T], $f_i$ the numerical results and $f(t_i)$ is the reference results at time $t_i$. f indicates generalized coordinate, displacement or velocity.