Heterogeneous asynchronous time integrators built from the energy method for coupling newmark and α-schemes

The time integration procedure selected in computational structural dynamics must possess at least the stability and accuracy properties required for the convergence to the exact solution. Other desired properties are the unconditional stability for linear dynamics, second-order of accuracy, high frequency dissipation capabilities, self-starting, no overshoot, one step method and no inure than one set of implicit equations to be solved for each tinie step (single-step-single-solve format). In linear dynamics, the stability is classically assessed by a spectral study of the amplification matrix, whereas physical energy bounds are preferred in nonlinear dynamics. Popular a-schemes (HHT-α, WBZ-α, CH-α) are second-order accurate and provides numerical dissipation for spurious high frequencies due to the flute element discretization. To go beyond the staildard approach based on the same time integration scheme (homogeneous time integration scheme) and the same time step for all the finite elements of the mesh (synchronous time integration), the purpose of tins paper is to describe a general methodology for building Heterogeneous (different time integration schemes such as Newmark or a-schemes) Asynchronous (different time steps) Time Integrators (HATI) for computational dynamics. The key point for building the HATI methods is to cancel the interface pseudo-energy as introduced by Hughes in the so-called clergy method employed for proving the stability of implicit-explicit algorithms in its pioneer works on heterogeneous time integrators. By canceling the pseudo-energy at the interface between subdomains and assumillg a linear time variation of the Lagrange multipliers at the coarse time scale, the HATI method, called BCG-macro method, is derived. It can handle any dissipative cm-schemes (HHT-α, WBZ-cmα, CH-α), while preserving the secoud-order of accuracy when adopting different time steps. In addition to the energy argument (cancelation of the interface pseudo-energy), the stability and order of accuracy is proved by the spectral study of the amplification matrix

Low intrusive coupling of implicit and explicit integration schemes for structural dynamics: application to low energy impacts on composite structures

Simulation of low energy impacts on composite structures is a key feature in aeronautics. Unfortunately they are very expensive: on the one side, the structures of interest have large dimensions and need fine volumic meshes (at least locally) in order to capture damages. On the other side small time steps are required to ensure the explicit algorithms stability which are commonly used in these kind of simulations [4]. Implicit algorithms are in fact rarely used in this situation because of the roughness of the solutions that leads to prohibitive expensive time steps or even to non convergence of Newtonlike iterative processes. It is also observed that rough phenomenons are localized in space and time (near the impacted zone). It may therefore be advantageous to adopt a multiscale space/time approach by splitting the structure into several substructures owning there own space/time discretization and their own integration schemes. The purpose of this decomposition is to take advantage of the specificities of both algorithms families: explicit scheme focuses on rough areas while smoother (actually linear) parts of the solutions are computed with larger time steps with an implicit scheme. We propose here an implementation of the Gravouil-Combescure method (GC) [1] by the mean of low intrusive coupling between the implicit finite element analysis (FEA) code Z-set and the explicit FEA code Europlexus. Simulations of low energy impacts on composite stiffened panels are presented. It is shown on this application that time step ratios up to 5000 can be reached. However, computations related to the explicit domain still remain a bottleneck in terms of cpu time

A note on topological properties of volumes constructed from surfaces

Converting surfaces into a volume has a long interest in several communities, e.g. the computational mechanics community. This process involves having specific surfaces which can be converted into a solid, i.e., a volume. This paper presents in a clear and brief manner the topological properties conserved during surface to volume transformation. We state the limits of this approach if a specific volume structure is required. Volume structures can be a coarse volume organization or meshes. For that purpose, surface manifolds are mathematically turned into a volume manifold. Topological tools are presented to understand which properties are transmitted to the volume and which ones are unset. Developments are submitted both for continuous and discrete manifolds using CW-complexes

On dual Schur domain decomposition method for linear first-order transient problems

This paper addresses some numerical and theoretical aspects of dual Schur domain decomposition methods for linear first-order transient partial differential equations. In this work, we consider the trapezoidal family of schemes for integrating the ordinary differential equations (ODEs) for each subdomain and present four different coupling methods, corresponding to different algebraic constraints, for enforcing kinematic continuity on the interface between the subdomains. Method 1 (d-continuity) is based on the conventional approach using continuity of the primary variable and we show that this method is unstable for a lot of commonly used time integrators including the mid-point rule. To alleviate this difficulty, we propose a new Method 2 (Modified d-continuity) and prove its stability for coupling all time integrators in the trapezoidal family (except the forward Euler). Method 3 (v-continuity) is based on enforcing the continuity of the time derivative of the primary variable. However, this constraint introduces a drift in the primary variable on the interface. We present Method 4 (Baumgarte stabilized) which uses Baumgarte stabilization to limit this drift and we derive bounds for the stabilization parameter to ensure stability. Our stability analysis is based on the ``energy'' method, and one of the main contributions of this paper is the extension of the energy method (which was previously introduced in the context of numerical methods for ODEs) to assess the stability of numerical formulations for index-2 differential-algebraic equations (DAEs).Comment: 22 Figures, 49 pages (double spacing using amsart

A finite element method for level sets

Level set methods have recently gained much popularity to capture discontinuities, including their possible propagation. In this contribution we present a finite element approach for solving the governing equations of level set methods. After a review of the governing equations, the initialisation of the level sets, the discretisation on a finite domain and the stabilisation of the resulting finite element method will be discussed. Special attention will be given to the proper treatment of the internal boundary condition, which is achieved by exploiting the partition-of-unity property of finite element shape functions

Benchmark numérique pour intégrateurs explicites destinés à la dynamique d'impact

International audienceCe travail vise à établir un benchmark numérique pour les schémas d'intégration temporels explicites dédiés aux problèmes dynamiques non-linéaire avec impact. Ce benchmark est constitué de plusieurs cas tests, simples à implémenter et à analyser, dont quatre sont présentés ici. Chaque cas vise à tester une propriété numérique du schéma importante en dynamique non-linéaire avec impact : conservation de l'énergie à l'impact et en temps long, conservation du moment angulaire, capacité à surmonter des accumulations d'impacts... Certains schémas référents en dynamique d'impact seront testés à travers ce benchmark