Time finite element based Moreau‐type integrators

With the postulation of the principle of virtual action, we propose, in this paper, a variational framework for describing the dynamics of finite dimensional mechanical systems, which contain frictional contact interactions. Together with the contact and impact laws formulated as normal cone inclusions, the principle of virtual action directly leads to the measure differential inclusions commonly used in the dynamics of nonsmooth mechanical systems. The discretization of the principle of virtual action in its strong and weak variational form by local finite elements in time provides a structured way to derive various time‐stepping schemes. The constitutive laws for the impulsive and nonimpulsive contact forces, ie, the contact and impact laws, are treated on velocity‐level by using a discrete contact law for the percussion increments in the sense of Moreau. Using linear shape functions and different quadrature rules, we obtain three different stepping schemes. Besides the well‐established Moreau time‐stepping scheme, we can present two alternative integrators referred to as symmetric and variational Moreau‐type stepping schemes. A suitable benchmark example shows the superiority of the newly proposed integrators in terms of energy conservation properties, accuracy, and convergence.     

Parallel asynchronous variational integrators

This paper presents a scalable parallel variational time integration algorithm for nonlinear elastodynamics with the distinguishing feature of allowing each element in the mesh to have a possibly different time step. Furthermore, the algorithm is obtained from a discrete variational principle, and hence it is termed parallel asynchronous variational integrator (PAVI). The underlying variational structure grants it outstanding conservation properties. Based on a domain decomposition strategy, PAVI combines a careful scheduling of computations with fully asynchronous communications to provide a very efficient methodology for finite element models with even mild distributions of time step sizes. Numerical tests are shown to illustrate PAVI's performance on both slow and fast networks, showing scalability properties similar to the best parallel explicit synchronous algorithms, with lower execution time. Finally, a numerical example in which PAVI needs ≈100 times less computing than an explicit synchronous algorithm is shown. Copyright © 2006 John Wiley & Sons, Ltd.     

Variational integrators – A continuous time approach

This article presents a family of variational integrators from a continuous time point of view. A general procedure for deriving symplectic integration schemes preserving an energy‐like quantity is shown, which is based on the principle of virtual work. The framework is extended to incorporate holonomic constraints without using additional regularization. In addition, it is related to well‐known partitioned Runge–Kutta methods and to other variational integration schemes. As an example, a concrete integration scheme is derived for the planar pendulum using both polar and Cartesian coordinates. Copyright © 2016 John Wiley & Sons, Ltd.     

Energy‐stepping integrators in Lagrangian mechanics

We present a class of integration schemes for Lagrangian mechanics, referred to as energy‐stepping integrators, that are momentum and energy conserving, symplectic and convergent. In order to achieve these properties we replace the original potential energy by a piecewise constant, or terraced approximation at steps of uniform height. By taking steps of diminishing height, an approximating sequence of energies is generated. The trajectories of the resulting approximating Lagrangians can be characterized explicitly and consist of intervals of piecewise rectilinear motion. We show that the energy‐stepping trajectories are symplectic, exactly conserve all the momentum maps of the original system and, subject to a transversality condition, converge to trajectories of the original system when the energy step is decreased to zero. These properties, the excellent long‐term behavior of energy‐stepping and its automatic time‐step selection property, are born out by selected examples of application, including the dynamics of a frozen Argon cluster, the spinning of an elastic cube and the collision of two elastic spheres. Copyright © 2009 John Wiley & Sons, Ltd.     

Force‐stepping integrators in Lagrangian mechanics

We formulate an integration scheme for Lagrangian mechanics, referred to as the force‐stepping scheme, which is symplectic, energy conserving, time‐reversible, and convergent with automatic selection of the time‐step size. The scheme also conserves approximately all the momentum maps associated with the symmetries of the system. The exact conservation of momentum maps may additionally be achieved by recourse to the Lagrangian reduction. The force‐stepping scheme is obtained by replacing the potential energy by a piecewise affine approximation over a simplicial grid or regular triangulation. By taking triangulations of diminishing size, an approximating sequence of energies is generated. The trajectories of the resulting approximate Lagrangians can be characterized explicitly and consist of piecewise parabolic motion, or free fall. Selected numerical tests demonstrate the excellent long‐term behavior of force‐stepping, its automatic time‐step selection property, and the ease with which it deals with constraints, including contact problems. Copyright © 2010 John Wiley & Sons, Ltd.     

Convergence of asynchronous variational integrators in linear elastodynamics

Within the setting of linear elastodynamics of simple bodies, we prove that the discrete action functional obtained by following the scheme of asynchronous variational integrators converges in time. The convergence in space is assured by standard arguments when the finite element mesh is progressively refined. Our strategy exploits directly the action functional. In particular, we show that, if the asynchronicity of time steps and nodal initial data satisfy a boundedness condition, any sequence of stationary points of the discrete action functional is pre‐compact in the weak?* W^1,∞ topology and all its cluster points are stationary points for the continuous (in time) action. In this sense our proof is new with respect to existing ones. Copyright © 2007 John Wiley & Sons, Ltd.     

Replica time integrators

This paper is concerned with the classical problem of wave propagation in discrete models of nonuniform spatial resolution. We develop a new class of Replica Time Integrators (RTIs) that permit the two‐way transmission of thermal phonons across mesh interfaces. This two‐way transmissibility is accomplished by representing the state of the coarse regions by means of replica ensembles, consisting of collections of identical copies of the coarse regions. In dimension d, RTIs afford an O(n^d) speed‐up factor in sequential mode, and O(n^{d + 1}) in parallel, over regions that are coarsened n‐fold. In this work, we restrict ourselves to the solution of the 3d continuous wave equation, for both linear and non‐linear materials. By a combination of phase‐error analysis and numerical testing, we show that RTIs are convergent and result in exact two‐way transmissibility at the Courant–Friedrichs–Lewy limit for any angle of incidence. In this limit, RTIs allow step waves and high‐frequency harmonics to cross mesh interfaces in both directions without internal reflections or appreciable loss or addition of energy. The possible connections of RTIs with discrete‐to‐continuum approaches and, in particular, with the transition between molecular dynamics and continuum thermodynamics are also pointed to by way of future outlook. Copyright © 2011 John Wiley & Sons, Ltd.     

Midpoint rule for variational integrators on Lie groups

The midpoint rule provides a standard method to obtain symmetric, symplectic, and second‐order accurate variational integrators for mechanical systems whose configuration manifold is the vector space ?ⁿ. In this work, we discuss how to extend this rule to a generic finite‐dimensional Lie group G while retaining the same properties. We show that the function κ_G(g)=exp(½log(g)), g∈G plays a special role in the theory and, for G=SO(3), we give a compact formula to compute it. We also discuss sufficient conditions for the method to conserve momentum maps associated with left (or right) group actions. As an example, the variational integrator obtained from the midpoint rule is applied to simulating rigid body dynamics. The resulting integrator is compared with state‐of‐the‐art symmetric and second‐order accurate integrators for rigid body motion. Copyright © 2009 John Wiley & Sons, Ltd.     

Discontinuous variational time integrators for complex multibody collisions

The objective of the present work is to formulate a new class of discontinuous variational time integrators that allow the system to adopt two possibly different configurations at each sampling time t_k, representing predictor and corrector configurations of the system. The resulting sequence of configuration pairs then represents a discontinuous—or non‐classical—trajectory. Continuous or classical trajectories are recovered simply by enforcing a continuity constraint at all times. In particular, in systems subject to one‐sided contact constraints simulated via discontinuous variational time integrators, the predictor configuration is not required to satisfy the one‐sided constraints, whereas the corrector configuration is obtained by a closest‐point projection (CPP) onto the admissible set. The resulting trajectories are generally discontinuous, or non‐classical, but are expected to converge to classical or continuous solutions for decreasing time steps. We account for dissipation, including friction, by means of a discrete Lagrange–d'Alembert principle, and make extensive use of the spacetime formalism in order to ensure exact energy conservation in conservative systems, and the right rate of energy decay in dissipative systems. The structure, range and scope of the discontinuous variational time integrators, and their accuracy characteristics are illustrated by means of examples of application concerned with rigid multibody dynamics. Copyright © 2014 John Wiley & Sons, Ltd.     

A numerical test of long‐time stability for rigid body integrators

In the context of Hamiltonian ODEs, a necessary condition for an integrator to be symplectic or conjugate‐symplectic is that it nearly preserves the exact Hamiltonian. This paper introduces a numerical test of this necessity for rigid body methods. It turns out that several rigid body integrators proposed in literature fail this test. Hence, these integrators should be used with caution for long‐time simulation. Copyright © 2011 John Wiley & Sons, Ltd.     

Asynchronous collision integrators: Explicit treatment of unilateral contact with friction and nodal restraints

This article presents asynchronous collision integrators and a simple asynchronous method treating nodal restraints. Asynchronous discretizations allow individual time step sizes for each spatial region, improving the efficiency of explicit time stepping for finite element meshes with heterogeneous element sizes. The article first introduces asynchronous variational integration being expressed by drift and kick operators. Linear nodal restraint conditions are solved by a simple projection of the forces that is shown to be equivalent to RATTLE. Unilateral contact is solved by an asynchronous variant of decomposition contact response. Therein, velocities are modified avoiding penetrations. Although decomposition contact response is solving a large system of linear equations (being critical for the numerical efficiency of explicit time stepping schemes) and is needing special treatment regarding overconstraint and linear dependency of the contact constraints (for example from double‐sided node‐to‐surface contact or self‐contact), the asynchronous strategy handles these situations efficiently and robust. Only a single constraint involving a very small number of degrees of freedom is considered at once leading to a very efficient solution. The treatment of friction is exemplified for the Coulomb model. Special care needs the contact of nodes that are subject to restraints. Together with the aforementioned projection for restraints, a novel efficient solution scheme can be presented. The collision integrator does not influence the critical time step. Hence, the time step can be chosen independently from the underlying time‐stepping scheme. The time step may be fixed or time‐adaptive. New demands on global collision detection are discussed exemplified by position codes and node‐to‐segment integration. Numerical examples illustrate convergence and efficiency of the new contact algorithm. Copyright © 2013 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons, Ltd.     

Lagrangian mechanics and variational integrators on two‐spheres

Euler–Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two‐spheres. The geometric structure of a product of two‐spheres is carefully considered in order to obtain global equations of motion. Both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints. We derive global expressions for the Euler–Lagrange equations on two‐spheres, which are more compact than existing equations written in terms of angles. Since the variational integrators are derived from Hamilton's principle, they preserve the geometric features of the dynamics such as symplecticity, momentum maps, or total energy, as well as the structure of the configuration manifold. Computational properties of the variational integrators are illustrated for several mechanical systems. In addition, Lie group variational integrators can be used to integrate Lagrangian flows on more general homogeneous spaces. This is achieved by lifting the discrete Hamilton's principle on homogeneous spaces to a discrete variational principle on the Lie group that is constrained by a discrete connection. Copyright © 2009 John Wiley & Sons, Ltd.     

A variationally consistent fractional time‐step integration method for incompressible and nearly incompressible Lagrangian dynamics

In this paper we present a fractional time‐step method for Lagrangian formulations of solid dynamics problems. The method can be interpreted as belonging to the class of variational integrators which are designed to conserve linear and angular momentum of the entire mechanical system exactly. Energy fluctuations are found to be minimal and stay bounded for long durations. In order to handle incompressibility, a mixed formulation in which the pressure appears explicitly is adopted. The velocity update over a time step is split into deviatoric and volumetric components. The deviatoric component is advanced using explicit time marching, whereas the pressure correction for each time step is computed implicitly by solving a Poisson‐like equation. Once the pressure is known, the volumetric component of the velocity update is calculated. In contrast with standard explicit schemes, where the time‐step size is determined by the speed of the pressure waves, the allowable time step for the proposed scheme is found to depend only on the shear wave speed. This leads to a significant advantage in the case of nearly incompressible materials and permits the solution of truly incompressible problems. Copyright © 2005 John Wiley & Sons, Ltd.     

A simple and effective single‐step time marching technique based on adaptive time integrators

In this work, a novel time marching procedure is presented in which adaptive time integration parameters are locally computed, allowing different spatial and temporal distributions. As a consequence, a more versatile technique arises, enabling an enhanced performance. The main features of the proposed technique are: (1) it is simple; (2) it has guaranteed stability; (3) it is an efficient non‐iterative adaptive single‐step procedure; (4) it provides enhanced accuracy; (5) it enables advanced controllable algorithmic dissipation in the higher modes; (6) it is truly self‐starting; and (7) it is entirely automatic, requiring no input parameter from the user. The proposed technique is very complete, providing the main positive attributes that are requested from an effective time marching procedure. Numerical results are presented along the paper, illustrating the excellent performance of the method. Copyright © 2016 John Wiley & Sons, Ltd.     

Asynchronous multi‐domain variational integrators for non‐linear problems

We develop an asynchronous time integration and coupling method with domain decomposition for linear and non‐linear problems in mechanics. To ensure stability in the time integration and in coupling between domains, we use variational integrators with local Lagrange multipliers to enforce continuity at the domain interfaces. The asynchronous integrator lets each domain step with its own time step, using a smaller time step where required by stability and accuracy constraints and a larger time step where allowed. We show that in practice the time step is limited by accuracy requirements rather than by stability requirements. Copyright © 2008 John Wiley & Sons, Ltd.     

Time discontinuous Galerkin methods with energy decaying correction for non‐linear elastodynamics

In this paper a new time discontinuous Galerkin (TDG) formulation for non‐linear elastodynamics is presented. The new formulation embeds an energy correction which ensures truly energy decaying, thus allowing to achieve unconditional stability that, as shown in the paper, is not guaranteed by the classical TDG formulation. The resulting method is simple and easily implementable into existing finite element codes. Moreover, it inherits the desirable higher‐order accuracy and high‐frequency dissipation properties of the classical formulation. Numerical results illustrate the very good performance of the proposed formulation. Copyright © 2010 John Wiley & Sons, Ltd.     

The effects of element shape on the critical time step in explicit time integrators for elasto‐dynamics

In this paper, the effects of element shape on the critical time step are investigated. The common rule‐of‐thumb, used in practice, is that the critical time step is set by the shortest distance within an element divided by the dilatational (compressive) wave speed, with a modest safety factor. For regularly shaped elements, many analytical solutions for the critical time step are available, but this paper focusses on distorted element shapes. The main purpose is to verify whether element distortion adversely affects the critical time step or not. Two types of element distortion will be considered, namely aspect ratio distortion and angular distortion, and two particular elements will be studied: four‐noded bilinear quadrilaterals and three‐noded linear triangles. The maximum eigenfrequencies of the distorted elements are determined and compared to those of the corresponding undistorted elements. The critical time steps obtained from single element calculations are also compared to those from calculations based on finite element patches with multiple elements. Copyright © 2014 John Wiley & Sons, Ltd.     

线性振动系统初值问题的变分原理和时域离散解法

研究线性振动系统初值问题的时域求解方法。首先，定义了形式较为一般的卷积，给出了有关的一些性质。然后导出了线性振动初值问题泛函为卷积形式的一类变量变分原理。应用这一变分原理，建立了初值问题时域离散求解的方法，包括分单元离散、展开离散和分段递推等求解方法     

Two implementations of IRK integrators for real‐time multibody dynamics

Recently, several authors have proposed the use of implicit Runge–Kutta (IRK) integrators for the dynamics of multibody systems. On the other hand, Newmark‐type or structural integrators have shown to be appropriate when real‐time performance is demanded in that field. Therefore, the following question arises: might the IRK integrators be suitable for real‐time purposes? And, provided the answer is positive: might they be preferable to the Newmark‐type family? This paper reports an investigation which has been conducted by the authors in order to get insight into the two questions formulated above. Since, based on previous experiences, it can be suspected that the performance of the integrators may be dependent on the type of dynamic formulation applied, the following three formulations have been considered for the study: a global penalty formulation in dependent natural co‐ordinates (many constraints), a topological semi‐recursive penalty formulation in dependent relative co‐ordinates (few constraints), and a topological semi‐recursive formulation in independent relative co‐ordinates (no constraints). As representative of the IRK family, a two‐stage SDIRK integrator has been selected due to its low associated computational burden, while, on the side of the structural integrators, the trapezoidal rule has been chosen. Two alternative implementations have been proposed to combine the dynamic formulations and the SDIRK integrator. A very demanding maneuver of the whole model of a vehicle has been simulated through all the possible combinations dynamic‐formulation/integrator, for different time‐steps. Conclusions have been drawn based on the obtained results, which provide some practical criteria for those interested in achieving real‐time performance for large and complex multibody systems. Copyright © 2005 John Wiley & Sons, Ltd.     

Time‐parallel implicit integrators for the near‐real‐time prediction of linear structural dynamic responses

The time‐parallel framework for constructing parallel implicit time‐integration algorithms (PITA) is revisited in the specific context of linear structural dynamics and near‐real‐time computing. The concepts of decomposing the time‐domain in time‐slices whose boundaries define a coarse time‐grid, generating iteratively seed values of the solution on this coarse time‐grid, and using them to time‐advance the solution in each time‐slice with embarrassingly parallel time‐integrations are maintained. However, the Newton‐based corrections of the seed values, which so far have been computed in PITA and related approaches on the coarse time‐grid, are eliminated to avoid artificial resonance and numerical instability. Instead, the jumps of the solution on the coarse time‐grid are addressed by a projector which makes their propagation on the fine time‐grid computationally feasible while avoiding artificial resonance and numerical instability. The new PITA framework is demonstrated for a complex structural dynamics problem from the aircraft industry. Its potential for near‐real‐time computing is also highlighted with the solution of a relatively small‐scale problem on a Linux cluster system. Copyright © 2006 John Wiley & Sons, Ltd.