Arlequin framework for multi-model, multi-time scale and heterogeneous time integrators for structural transient dynamics

doi:10.1016/j.cma.2012.08.019

Computer Methods in Applied Mechanics and Engineering

Volume 254, February 2013, Pages 292-308

https://doi.org/10.1016/j.cma.2012.08.019 Get rights and content

Abstract

This paper presents a general framework for performing spatio-temporal multi-scale/multi-model coupling. Based on displacement continuity, this approach guarantees a global energy balance of the system in the context of the Arlequin method. It introduces specific interpolation of the Lagrange multipliers and parameters of time integration operators in the overlapping zone. A dual Schur implementation is then used. To illustrate the efficiency and robustness of the method, convergence analysis with numerical experiments is performed. Finally, 1D–1D and 2D–1D coupling applications are presented in which each subdomain is integrated in time, using different time steps and different schemes. This approach gives the user more ways of controlling computational behaviour via Newmark parameters, time steps and sizing the overlapping zones.

Highlights

► Arlequin coupling performed for beam/3D models for structural dynamic. ► Displacement continuity with multi-scale/heterogeneous time integrators strategies. ► No energy dissipation in the overlapping zones. ► Convergence order: 1–2, as a function of time steps and Newmark’s parameters. ► Efficiency/robustness of the method for low and high frequency loading.

Introduction

Performing numerical simulations of the dynamic response in rotating machinery is mostly based on beam structures represented by a coarse model. Taking into consideration local physical phenomena occurring at microscopic scales such as crack propagation and contact requires fine three dimensional modelling of the structure concerned. However, carrying out calculations on very finely meshed models can be very expensive in computational terms.

Many adaptive methods [19] proposing solutions for this kind of multiscale study have been developed, in order to enhance the accuracy of numerical approximations while reducing CPU time (Central Processing Unit), but in spite of their advantages, these multi-mesh methods are considered very expensive for implementation in practical computations.

Another approach is provided by multiscale methods. These consist in superimposing a very fine mesh including a zone of interest on a global domain with a coarse mesh. These methods are divided into two major classes: the first is based on coupling domains by using an overlapping zone [29], [5] in which the models concerned are superposed, and the second is the class of non overlapping methods in which coupling is performed through common interfaces [18].

The Arlequin method [8], [9] was developed in the context of multiscale overlapping methods. Using a partition of unity approach to modelling, this method provides a progressive passage between different models with enhanced flexibility by using multiple parameters [7]. In static and dynamic cases, the Arlequin method permits coupling dissimilar models with concurrent scales, such as 2D–1D and 3D–1D models. It is especially interesting for dynamic studies, as it can treat problems for which aspects like wave propagation are taken into consideration. It was shown in [2], that wave transition is guaranteed and problems of spurious reflections are avoided during transit between models provided the coupling parameters of this method are used correctly.

Moreover, being able to couple different models in the case where each one has its own time integration scheme is a very important aspect. This can be considered as the first step for introducing different time discretizations, as well as providing the key for coupling multiple finite element codes.

Extending the Arlequin method to such time multiscale applications can be particularly useful. In addition to the fact that high frequency waves transit correctly through an overlap area, using independent time integration schemes on each model permits controllable numerical dissipation in the high-frequency regime. On the other hand, the size of the time step must be a free parameter imposed by accuracy requirements. Therefore the scheme should be unconditionally stable.

Extending this approach to incompatible time steps (Fig. 1) provides the key for formulating them and proposing a general formalism making it possible to couple different models, each having its own space/time properties: finite element models, space discretization, time discretization, time scheme. Many incompatible time step approaches are currently being developed using the dual Schur-type approach and most methods are applied in a non overlapping context.

The Gravouil Combescure (GC) method [12] proposed the coupling of different Newmark schemes for both linear and nonlinear states. In this approach, Lagrange multipliers ( $λ$ ) at the interface were discretized on the finest time scale $(j)$ and continuity was imposed on velocities: $L_{1} {\dot{u}}_{j + 1}^{1} + L_{2} {\dot{u}}_{j + 1}^{2} = 0 .$

This method is stable for different time scales, but numerical dissipation was observed at the interface. In [11] the GC method was used in the Arlequin framework to couple discrete and finite element models. Energy dissipation at the interface was observed. In order to avoid this phenomenon, the authors suggested enriching the interpolation of the Lagrange multipliers at the interface when computing fine scale values $λ_{j}$ . The proposed high-order interpolation helped to decrease numerical dissipation, but did not eliminate it.

Based on the GC method, Pegon and Magonette [25] proposed an interfield parallel algorithm. In this approach the discretization at the interface between the coupled sub-domains was identical to the GC method. However, the main advantage was obtained through the parallelism of the processes involving the coarse and fine time steps.

In the same context, Prakash and Hjelmstad (PH) [26] developed an approach similar to the GC method. The main improvement concerned the time discretization on the interface. In this method coarse time scale discretization $(m)$ was used to ensure continuity at the interface by using Lagrange multipliers $λ_{m}$ and permitted significant computational time-saving. Velocity was also the kinematic quantity used to guarantee continuity at the interface while stability was ensured for different time scale ratios: $L_{1} {\dot{u}}_{m}^{1} + L_{2} {\dot{u}}_{m}^{2} = 0 .$

Later, the GC method was improved [20] and extended (GCbis method) so the discretization of the Lagrange multipliers could be performed on coarse time scales. Velocities were used to ensure continuity at the interface and stability was verified. One main difference between the two methods lies in the approximation of the Lagrange multipliers [20]. Using the Hughes energy method [15], it is possible to demonstrate that the PH and GCbis methods do not introduce or dissipate energy at the interface. On the other hand, the energy balance is not verified for either method. Numerical experiments performed in the Arlequin framework show that energy is introduced or dissipated in the overlapping domain.

A recent method was proposed in [21] in which different time integrators (Newmark, HHT, Simo, Krenk, Verlet) were introduced in a unified framework. As with all the methods mentioned above, global stability is ensured through the stability of each scheme while continuity at the interface is controlled through velocities. A significant difference with the abovementioned methods lies in the energy balance. In this approach, zero numerical dissipation is guaranteed and verified by the energy balance of the Newmark scheme. Mention should be made that this energy conservation is established through a strong condition of the energy balance. This means that the link at the interface is imposed through a constant Lagrange multiplier during the time increment.

It can be seen that energy balance aspects in a time multi-scale context are not always satisfied by existing methods unless a strong condition is imposed. Here, we mainly seek to highlight this problem and propose an acceptable solution without imposing constraints on the Lagrange multipliers $λ$ .

This paper is structured as follows. After the introduction, we describe the energy conservation aspects for continuous Arlequin formulations. In the third section, we consider the Arlequin framework in which various subdomains are integrated in time using different time schemes with homogeneous time steps. Thus distributed Newmark parameters are derived by using unity partition functions and by assuming displacement continuity on the overlapping zones. This distribution of parameters ensures the Newmark scheme of the energy balance. In the fourth section, we consider the case where subdomains can be integrated in time with different time steps and different Newmark schemes. In this case, a new Newmark parameter distribution is derived. This distribution is based on an interpolation of the Lagrange multipliers on the fine time steps via a geometric-arithmetic progression and the displacement continuity assumption. This new distribution allows energy conservation and better control of numerical dissipation. Numerical experiments in which a classical convergence order of Newmark schemes is obtained are described in the fifth section. In the sixth section, 1D–1D and 2D–1D coupled models with different time steps and/or different Newmark schemes are presented and commented.

Section snippets

Reference problem

Firstly, we focused on an elastodynamics problem over the time interval $[0, T]$ , in order to define the reference problem. An isotropic elastic body is considered to be represented by a regular domain $Ω$ . Let $u, \dot{u}$ and $\ddot{u}$ denote the displacement, velocity and acceleration fields, let $σ$ denote the Cauchy stress tensor and let $f$ be the prescribed body force per unit volume. The infinitesimal strain tensor $ε$ is defined by the symmetric part of the displacement gradient. The stress tensor is defined in

The Arlequin method with heterogeneous time integrators and a homogeneous time step

In this section, we consider the time space discretized formulation of the previous continuous one where each subdomain has its own finite element model and mesh. The implicit Newmark integration algorithm is used with different parameters ( $γ_{i}, β_{i}$ ) for each subdomain $Ω_{i}$ , with $γ_{i} ⩾ \frac{1}{2}$ and $β_{i} ⩾ \frac{γ_{i}}{2}$ .

Discretized problem

We consider the same problem as that used in the above sections. The initial domain $Ω$ is split into two subdomains $Ω_{1}$ and $Ω_{2}$ with overlapping zone $Ω_{0}$ . Let $Δ T$ and $Δ t$ be the time steps associated with $Ω_{1}$ and $Ω_{2}$ , respectively. $Δ T$ denotes the coarse time step, $Δ t$ denotes the fine time step and $Δ T = m Δ t$ . The Newmark schemes used have two different sets of parameters: $(γ_{1}, β_{1})$ and $(γ_{2}, β_{2})$ . For the sake of simplicity, we illustrate the coupling method by considering one step $Δ T$ from $t_{0}$ to $t_{m} = t_{0} + Δ T$ as shown

Convergence analysis

In this section we study the convergence of the methods presented previously. Two cases are taken into consideration: a single step and a heterogeneous time step coupling. For each example, Newmark average acceleration and Newmark dissipative schemes are applied. A four-DoF model problem [4] is used in order to perform this first analysis. The reference system is composed of four masses $m_{i} = 20000 kg$ and four springs $k_{i} = 10^{6} N / m$ . We assume that this system is undamped and unforced. The compact

Applications

In this section, we present two applications and discuss several numerical experiments. The first example is concerned with 1D–1D coupled bar models under the action of traction-compression loading. The second example is a 1D (Beam)–2D (Plane stresses) coupled model under the action of traction-compression loading.

Conclusion

In this article we proposed a general formalism leading to space–time multi-model-multiscale/multi-scheme coupling. This formalism permits multiscale/multi-model coupling since it is based on the Arlequin method. We showed that using an interpolated Newmark parameter in the recovering area ensures the continuity of the kinematic fields in a homogeneous time step/multi-scheme framework. Furthermore, when heterogeneous time scales are applied, we showed that the continuity of displacements in the

Acknowledgement

The authors thank the EDF Research and Development Division (AMA/T63) of Clamart (France) for its financial support.

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Arlequin framework for multi-model, multi-time scale and heterogeneous time integrators for structural transient dynamics

Abstract

Highlights

Introduction

Section snippets

Reference problem

The Arlequin method with heterogeneous time integrators and a homogeneous time step

Discretized problem

Convergence analysis

Applications

Conclusion

Acknowledgement

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