A time‐parallel implicit method for accelerating the solution of non‐linear structural dynamics problems

The parallel implicit time‐integration algorithm (PITA) is among a very limited number of time‐integrators that have been successfully applied to the time‐parallel solution of linear second‐order hyperbolic problems such as those encountered in structural dynamics. Time‐parallelism can be of paramount importance to fast computations, for example, when space‐parallelism is unfeasible as in problems with a relatively small number of degrees of freedom in general, and reduced‐order model applications in particular, or when reaching the fastest possible CPU time is desired and requires the exploitation of both space‐ and time‐parallelisms. This paper extends the previously developed PITA to the non‐linear case. It also demonstrates its application to the reduction of the time‐to‐solution on a Linux cluster of sample non‐linear structural dynamics problems. Copyright © 2008 John Wiley & Sons, Ltd.     

Energy‐conserving and decaying Algorithms in non‐linear structural dynamics

A generalized formulation of the Energy‐Momentum Methodwill be developed within the framework of the Generalized‐α Methodwhich allows at the same time guaranteed conservation or decay of total energy and controllable numerical dissipation of unwanted high frequency response. Furthermore, the latter algorithm will be extended by the consistently integrated constraints of energy and momentum conservation originally derived for the Constraint Energy‐Momentum Algorithm. The goal of this general approach of implicit energy‐conserving and decaying time integration schemes is, to compare these algorithms on the basis of an equivalent notation by the means of an overall algorithmic design and hence to investigate their numerical properties. Numerical stability and controllable numerical dissipation of high frequencies will be studied in application to non‐linear structural dynamics. Among the methods considered will be the Newmark Method, the classical α‐methods, the Energy‐Momentum Methodwith and without numerical dissipation, the Constraint Energy‐Momentum Algorithm and the Constraint Energy Method. Copyright © 1999 John Wiley & Sons, Ltd.     

An explicit discontinuous Galerkin method for non‐linear solid dynamics: Formulation,parallel implementation and scalability properties

An explicit‐dynamics spatially discontinuous Galerkin (DG) formulation for non‐linear solid dynamics is proposed and implemented for parallel computation. DG methods have particular appeal in problems involving complex material response, e.g. non‐local behavior and failure, as, even in the presence of discontinuities, they provide a rigorous means of ensuring both consistency and stability. In the proposed method, these are guaranteed: the former by the use of average numerical fluxes and the latter by the introduction of appropriate quadratic terms in the weak formulation. The semi‐discrete system of ordinary differential equations is integrated in time using a conventional second‐order central‐difference explicit scheme. A stability criterion for the time integration algorithm, accounting for the influence of the DG discretization stability, is derived for the equivalent linearized system. This approach naturally lends itself to efficient parallel implementation. The resulting DG computational framework is implemented in three dimensions via specialized interface elements. The versatility, robustness and scalability of the overall computational approach are all demonstrated in problems involving stress‐wave propagation and large plastic deformations. Copyright © 2007 John Wiley & Sons, Ltd.     

Multi‐time‐step explicit–implicit method for non‐linear structural dynamics

We present a method with domain decomposition to solve time‐dependent non‐linear problems. This method enables arbitrary numeric schemes of the Newmark family to be coupled with different time steps in each subdomain: this coupling is achieved by prescribing continuity of velocities at the interface. We are more specifically interested in the coupling of implicit/explicit numeric schemes taking into account material and geometric non‐linearities. The interfaces are modelled using a dual Schur formulation where the Lagrange multipliers represent the interfacial forces. Unlike the continuous formulation, the discretized formulation of the dynamic problem is unable to verify simultaneously the continuity of displacements, velocities and accelerations at the interfaces. We show that, within the framework of the Newmark family of numeric schemes, continuity of velocities at the interfaces enables the definition of an algorithm which is stable for all cases envisaged. To prove this stability, we use an energy method, i.e. a global method over the whole time interval, in order to verify the algorithms properties. Then, we propose to extend this to non‐linear situations in the following cases: implicit linear/explicit non‐linear, explicit non‐linear/explicit non‐linear and implicit non‐linear/explicit non‐linear. Finally, we present some examples showing the feasibility of the method. Copyright © 2001 John Wiley & Sons, Ltd.     

Flow‐induced vibrations of non‐linear cables. Part 2: Simulations

In this paper, we present simulations of flow interacting with non‐linear cables. We first consider the case of a pre‐stretched straight cable subject to uniform inflow, which eventually assumes a catenary‐like equilibrium position. We then simulate the flow induced by a riser of an S shape at equilibrium, subject to time‐periodic forcing at one of its ends. We demonstrate that the models and algorithms developed in Part 1 of this work can be used effectively in simulating flow‐structure interactions in non‐linear systems of industrial complexity. Copyright © 2002 John Wiley & Sons, Ltd.     

Accuracy of one‐step integration schemes for damped/forced linear structural dynamics

This paper proposes an energy‐based measure for the evaluation of the local truncation error of two‐level one‐step integration schemes. The measure applies to multiple degree of freedom systems and does not necessarily require modal reduction to a scalar model; it naturally handles the structural damping and external forcing terms that are generally and mistakenly neglected in error analyses, and it segregates the error associated with the free and forced response components of the problem. To illustrate the approach, two examples associated with the application of the trapezoidal scheme and of a high‐order scheme proposed in the literature are analyzed. The latter reveals the shortcomings of the standard approach that is based on the undamped/unforced linear oscillator and therefore highlights the need for the proposed framework. Indeed, the scheme order of accuracy is below expectation when structural damping or external forcing is considered, in the numerically dissipative setting. Developments on the basis of the time discontinuous Galerkin (TDG) method are then proposed to recover the scheme high‐order accuracy. Additionally, they show the similarity that exists between schemes related to the TDG method and the ones obtained by integration by parts of the equation of motion. Copyright © 2014 John Wiley & Sons, Ltd.     

Flow‐induced vibrations of non‐linear cables. Part 1: Models and algorithms

In this paper, we develop governing equations for non‐linear cables as well as a formulation for the coupled flow‐structure problem. The structure is discretized with second‐order accuracy while the flow is discretized using spectral/hp elements in the context of the arbitrary Lagrangian–Eulerian formulation (ALE). Several benchmark problems are considered and the computational implementation is detailed. In the second part of this work large‐scale simulation examples are presented. Copyright © 2002 John Wiley & Sons, Ltd.     

A novel non‐linearly explicit second‐order accurate L‐stable methodology for finite deformation: hypoelastic/hypoelasto‐plastic structural dynamics problems

A novel non‐linearly explicit second‐order accurate L‐stable computational methodology for integrating the non‐linear equations of motion without non‐linear iterations during each time step, and the underlying implementation procedure is described. Emphasis is placed on illustrative non‐linear structural dynamics problems employing both total/updated Lagrangian formulations to handle finite deformation hypoelasticity/hypoelasto‐plasticity models in conjunction with a new explicit exact integration procedure for a particular rate form constitutive equation. Illustrative numerical examples are shown to demonstrate the robustness of the overall developments for non‐linear structural dynamics applications. Copyright © 2004 John Wiley & Sons, Ltd.