On multi-time-step monolithic coupling algorithms for elastodynamics
Section snippets
Introduction and motivation
Coupled problems (such as fluid–structure interaction, structure–structure interaction and thermal–structure interaction) have been the subject of intense research in recent years in both computational mechanics and applied mathematics. The report compiled by the Blue Ribbon Panel on Simulation-Based Engineering Science emphasizes that the ability to solve coupled problems will be vital to accelerate the advances in engineering and science through simulation [1]. Developing stable and accurate
Newmark family of time-stepping schemes
Consider a system of second-order ordinary differential equations of the following form: where t denotes time, T denotes the time interval of interest, M is a symmetric positive definite matrix, K is a symmetric positive semidefinite matrix, and a superposed dot denotes derivative with respect to the time. The above system of equations can arise from a semi-discrete finite element discretization of the governing equations in linear elastodynamics [25]. In this case, M is
Governing equations for multiple subdomains
We now write governing equations for multiple subdomains. We will also outline various ways to write subdomain interface conditions, and discuss their pros and cons. To this end, let us divide the domain Ω into S non-overlapping subdomains, which will be denoted by . That is, We shall assume that the meshes in the subdomains are conforming along the subdomain interface, as shown in Fig. 1. There are several ways to enforce the continuity along the interface, and
Proposed multi-time-step coupling method
The aim of this paper is to solve Eqs. (15a), (15b) numerically by allowing each subdomain to have its own time-step and its own time integrator from the Newmark family of time stepping schemes. We first introduce notation that will help in presenting the proposed multi-time-step coupling method in a concise manner.
Stability analysis using the energy method
We shall employ the energy method to show the stability of the proposed multi-time-step coupling method. The energy method is a popular strategy employed in Mathematical Analysis to derive estimates and to perform stability analysis. The method is widely employed in the theory of partial differential equations [39], and numerical analysis [25], [40]. The basic idea behind the energy method is to choose an appropriate norm (which is referred to as the energy norm) and show that the solution is
Split degree-of-freedom lumped parameter system
Consider a split agree of freedom whose motion can be described by the following system of ordinary differential/algebraic equations: The following parameters are used: , , and the stiffness of springs are and . The subdomain time-steps are taken as and . The system time-step is taken as . The values of the external forces are taken to be zero, that is and . The
On energy conserving vs. energy preserving coupling
In this section we address the energy preserving and energy conserving properties of the proposed multi-time-step coupling method. Two different notions of energy preserving will be considered. In particular, the following questions will be answered:
- (a)
Does the coupling method add or extract energy from the system of subdomains in comparison with the case of no coupling?
- (b)
Do the interface forces perform net work?
- (c)
Under what conditions does the coupling method conserve the total energy of the system
On the performance of backward difference formulae and implicit Runge–Kutta schemes
In the numerical analysis literature, backward difference formulae (BDF) and implicit Runge–Kutta (IRK) schemes have been the schemes of choice for solving DAEs [18], [41]. The following quote by Petzold has been a popular catch-phrase for promoting BDF schemes: “BDF is so beautiful that it is hard to imagine something else could be better” [41, p. 481]. This statement may be true for first-order DAEs that arise from modeling of physical systems involving dissipation. But these two families of
Representative numerical results
Using several canonical problems, we illustrate that the proposed multi-time-step coupling method possesses the following desirable properties:
- (I)
All subdomains can subcycle simultaneously. That is, .
- (II)
The method can handle multiple subdomains.
- (III)
Drift in displacements along the subdomain interface is not significant.
- (IV)
Under fixed subdomain time-steps, the accuracy of numerical solutions can be improved by decreasing the system time-step.
- (V)
For a fixed system time-step, accuracy of the
Concluding remarks
We have developed a multi-time-step coupling method that can handle multiple subdomains with different time-steps in different subdomains. The coupling method can couple implicit and explicit time-stepping schemes under the Newmark family even with disparate time-steps of more than two orders of magnitude in different subdomains. A systematic study on the energy preservation and energy properties of the proposed coupling method is presented, and the corresponding sufficient conditions are also
Acknowledgements
The authors acknowledge the support of the National Science Foundation under Grant No. CMMI 1068181. The opinions expressed in this paper are those of the authors and do not necessarily reflect that of the sponsor(s).
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Saeid Karimi, Graduate Student.