Computational Schemes for Flexible,Nonlinear Multi-Body Systems

This paper deals with the development of computational schemes for the dynamic analysis of flexible, nonlinear multi-body systems. The focus of the investigation is on the derivation of unconditionally stable time integration schemes for these types of problem. At first, schemes based on Galerkin and time discontinuous Galerkin approximations applied to the equations of motion written in the symmetric hyperbolic form are proposed. Though useful, these schemes require casting the equations of motion in the symmetric hyperbolic form, which is not always possible for multi-body applications. Next, unconditionally stable schemes are proposed that do not rely on the symmetric hyperbolic form. Both energy preserving and energy decaying schemes are derived that both provide unconditionally stable schemes for nonlinear multi-body systems. The formulation of beam and flexible joint elements, as well as of the kinematic constraints associated with universal and revolute joints. An automated time step selection procedure is also developed based on an energy related error measure that provides both local and global error levels. Several examples of simulation of realistic multi-body systems are presented which illustrate the efficiency and accuracy of the proposed schemes, and demonstrate the need for unconditional stability and high frequency numerical dissipation.     

An energy preserving/decaying scheme for nonlinearly constrained multibody systems

Several numerical time integration methods for multibody system dynamics are described: an energy preserving scheme and three energy decaying ones, which introduce high-frequency numerical dissipation in order to annihilate the nondesired high-frequency oscillations. An exhaustive analysis of these four schemes is done, including their formulation, and energy preserving and decaying properties by taking into account the presence of nonlinear algebraic constraints and the incrementation of finite rotations. A new energy preserving/decaying scheme is developed, which is well suited for either stiff or nonstiff nonlinearly constrained multibody systems. Examples on a series of test cases show the performance of the algorithms.     

Energy-momentum conserving integration of multibody dynamics

A rotationless formulation of multibody dynamics is presented, which is especially beneficial to the design of energy-momentum conserving integration schemes. The proposed approach facilitates the stable numerical integration of the differential algebraic equations governing the motion of both open-loop and closed-loop multibody systems. A coordinate augmentation technique for the incorporation of rotational degrees of freedom and associated torques is newly proposed. Subsequent to the discretization, size-reductions are performed to lower the computational costs and improve the numerical conditioning. In this connection, a new approach to the systematic design of discrete null space matrices for closed-loop systems is presented. Two numerical examples are given to evaluate the numerical properties of the proposed algorithms.     

Algorithms for improved fixed-time performance of Lyapunov-based economic model predictive control of nonlinear systems

This work presents algorithms for improved fixed-time performance of Lyapunov-based economic model predictive control (LEMPC) of nonlinear systems. Unlike conventional Lyapunov-based model predictive control (LMPC) schemes which typically utilize a quadratic cost function and regulate a process at a steady-state, LEMPC designs very often dictate time-varying operation to optimize an economic (typically non-quadratic) cost function. The LEMPC algorithms proposed here utilize a shrinking prediction horizon with respect to fixed (but potentially large) operation period to ensure improved performance, measured by the desired economic cost, over conventional LMPC by solving auxiliary LMPC problems and incorporating appropriate constraints, based on the LMPC solution, in their formulations at various sampling times. The proposed LEMPC schemes also take advantage of a predefined Lyapunov-based explicit feedback law to characterize their stability region while maintaining the closed-loop system state in an invariant set subject to bounded process disturbances. The LEMPC algorithms are demonstrated through a nonlinear chemical process example.     

Mixed perfectly-matched-layers for direct transient analysis in 2D elastic heterogeneous media

We are concerned with elastic waves arising in plane-strain problems in an elastic semi-infinite arbitrarily heterogeneous medium. Specifically, we discuss the development of a new mixed displacement–stress formulation for forward elastic wave simulations in perfectly-matched-layer (PML)-truncated heterogeneous media.To date, most PML formulations split the displacement and stress fields, resulting in non-physical components for each field. In this work, we favor unsplit schemes, primarily for the relative ease by which the resulting forms can be incorporated into existing codes, the ease by which the resulting semi-discrete forms can be integrated in time, and the ease by which they can be used in adjoint formulations arising in inverse problems, contrary to most past and current developments. We start by following classical lines, and apply complex-coordinate-stretching to the governing equations in the frequency domain, while retaining both displacements and stress quantities as unknowns. With the aid of auxiliary variables the resulting mixed form is rendered second-order in time, thereby allowing the use of standard time integration schemes. We report on numerical simulations demonstrating the stability and efficacy of the approach.     

An incremental elastic-plastic Finite Element solver in a workstation cluster environment Part I. Formulations and parallel processing

We discuss solution schemes for the incremental elastic-plastic structural problem, discretized by means of the Finite Element method. Attention is focused on their formulation and implementation in a parallel computing environment defined by a cluster of workstations connected by means of a network. The availability of parallel computers allows one to consider possible formulations and solution strategies so far not considered competitive with the classical Newton-like schemes implying the definition of an elastic-plastic tangent stiffness matrix. The solution strategies herein considered are based on the explicit integration of the actual elastic-plastic rate problem. This, in turn, is phrased in terms of two different formulations, whose relative advantages—particularly with respect to their integration in parallel—are discussed. A − gl (displacemen plastic multiplier) formulation of the structural rate theory of plasticity [1], integrated by means of an explicit, element-by-element scheme, seems to be the most promising one.     

On the Modeling of Shells in Multibody Dynamics

Energy preserving/decaying schemes are presented for the simulation ofthe nonlinear multibody systems involving shell components. Theproposed schemes are designed to meet four specific requirements:unconditional nonlinear stability of the scheme, a rigorous treatmentof both geometric and material nonlinearities, exact satisfaction ofthe constraints, and the presence of high frequency numericaldissipation. The kinematic nonlinearities associated with arbitrarilylarge displacements and rotations of shells are treated in a rigorousmanner, and the material nonlinearities can be handled when theconstitutive laws stem from the existence of a strain energy densityfunction. The efficiency and robustness of the proposed approach isillustrated with specific numerical examples that also demonstrate theneed for integration schemes possessing high frequency numericaldissipation.     

Analysis of Joint Clearances in Multibody Systems

A methodology for the study of typical smooth joint clearances in multibody systems is presented. The proposed approach takes advantage of the analytical definition of the material surfaces defining the clearance, resulting in a formulation where the gap does not play a central role, as it happens in standard contact models. The contact forces are formulated in conserving form, such that the balance of total energy during the intermittent contact is exactly established in the discrete time integration scheme. Some numerical applications are presented, showing that the proposed methodology is very stable in long-term simulations with relatively large time step sizes. Therefore, it appears to be promising in terms of efficiency and robustness for the numerical analysis of real joints with clearances.     

A time-dependent Hamilton-Jacobi formulation of reachable sets for continuous dynamic games

We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equation's solution is continuous and defined throughout the state space, methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions. A numerical implementation of our formulation is described and has been released on the web. Its correctness is verified through a two vehicle, three dimensional collision avoidance example for which an analytic solution is available.     

Convergence of generalized-$\boldsymbol{\alpha}$ time integration for nonlinear systems with stiff potential forces

Generalized-\(\alpha\) time integration schemes, originally developed for application in structural dynamics, are increasingly popular throughout many branches of multibody system simulation. Their simple implementation and the opportunity to control the numerical dissipation make them highly appealing for use in broad fields of application.Initially introduced for the solution of linear ordinary differential equations, there have been several extensions to nonlinear structural dynamics and constrained multibody systems in various formulations.In the present paper, we consider the application to systems with very stiff potential forces (singular singularly perturbed systems) whose solution approaches in the limit case that of a constrained system (index-3 differential–algebraic equation). We give a convergence analysis comparing the highly oscillatory solutions of the stiff system to those of the associated constrained one and show that the classical second order convergence result holds for position coordinates as well as for appropriately projected errors on the velocity level.The theoretical results are verified by numerical experiments for a simple test example.     

Analysis of Flexible Multibody Systems with Intermittent Contacts

This paper is concerned with the dynamic analysis of flexible,nonlinear multibody systems undergoing intermittent contacts. Contact isassumed to be of finite duration, and the forces acting between thecontacting bodies which can be either rigid or deformable are explicitlycomputed during simulation. The modeling of contact consists of threeparts: a number of holonomic constraints that define the candidatecontact points on the bodies, a unilateral contact condition which istransformed into a holonomic constraint by the addition of a slackvariable, and a contact model which describes the relationship betweenthe contact force and the local deformation of the contacting bodies.This work is developed within the framework of energy preserving anddecaying time integration schemes that provide unconditional stabilityfor nonlinear, flexible multibody systems undergoing intermittentcontacts.     

LDG2: A Variant of the LDG Flux Formulation for the Spectral Volume Method

The local discontinuous Galerkin (LDG) viscous flux formulation was originally developed by Cockburn and Shu for the discontinuous Galerkin setting and later extended to the spectral volume setting by Wang and his collaborators. Unlike the penalty formulations like the interior penalty and the BR2 schemes, the LDG formulation requires no length based penalizing terms and is compact. However, computational results using LDG are dependant of the orientation of the faces especially for unstructured and non uniform grids. This results in lower solution accuracy and stiffer stability constraints as shown by Kannan and Wang. In this paper, we develop a variant of the LDG, which not only retains its attractive features, but also vastly reduces its unsymmetrical nature. This variant (aptly named LDG2), displayed higher accuracy than the LDG approach and has a milder stability constraint than the original LDG formulation. In general, the 1D and the 2D numerical results are very promising and indicate that the approach has a great potential for 3D flow problems.     

Energy consistent algorithms for dynamic finite deformation plasticity

This paper addresses the theoretical development and numerical implementation of energy consistent algorithms for dynamic elastoplasticity, emphasizing finite strain constitutive formulations so that unconditional stability of the algorithms is assured even in the fully nonlinear regime. The key concept behind energy consistency is the requirement that the discretized system obey an a priori stability estimate, which may be derived in general using the first and second laws of thermodynamics. This approach to computational dynamic plasticity differs from typical application of traditional algorithms (such as Newmark or Hilber–Hughes–Taylor-α methods), where local time integration schemes for plasticity laws are developed somewhat independently from the global time integration scheme for the equations of motion, without explicit consideration of thermodynamical restrictions. Two algorithms based on both additive and multiplicative finite deformation plasticity model are formulated within the energy consistent framework. Both algorithms possess the desirable feature of nonlinear stability of previous energy–momentum algorithms for elastodynamics.     

Simulation of Wheels in Nonlinear,Flexible Multibody Systems

This paper is concerned with the modeling of wheels within the framework of finite element-based dynamic analysis of nonlinear, flexible multibody systems. The overall approach to the modeling of wheels is broken into four distinct parts: a purely kinematic part describing the configuration of the wheel and contacting plane, a unilateral contact condition giving rise to a contact force, the friction forces associated with rolling and/or sliding, and a model of the deformations in the wheel tire. The formulation of these various aspects of the problem involves a combination of holonomic and non-holonomic constraints enforced via the Lagrange multiplier technique. This work is developed within the framework of energy-preserving and decaying time integration schemes that provide unconditional stability for nonlinear, flexible multibody systems involving wheels. Strategies for dealing with the transitions from rolling to sliding and vice-versa are discussed and are found to be more efficient than the use of a continuous friction law. Numerical examples are presented that demonstrate the efficiency and accuracy of the proposed approach.     

Nonlinear transient finite element analysis of rotor-bearing-stator systems

This paper extends the finite element scheme to handle the highly nonlinear interfacial fields generated in the fluid filled annulli of squeeze film and journal bearings so as to model the transient response of rotor-bearingstator systems. Since such simulations are highly nonlinear, direct numerical integration schemes are employed to generate the overall response. In this context, the paper gives consideration to such items as (i) numerical efficiency/stability, (ii) comparison of implicit and explicit schemes, (iii) determines extent of response nonlinearity as well as (iv) extensively benchmarks the overall concept/methodologies.     

Asynchronous Schwarz methods applied to constrained mechanical structures in grid environment

This paper deals with the parallel solution of the stationary obstacle problem with convection–diffusion operator. The obstacle problem can be formulated by various ways and in the present study it is formulated like a multivalued problem. Another formulation by complementary problem is also considered. Appropriate discretization schemes are considered for the numerical solution on decentralised memory machines by using parallel synchronous and asynchronous Schwarz alternating algorithms. The considered discretization schemes ensure the convergence of the parallel synchronous or asynchronous Schwarz alternating methods on one hand for the solution of the multivalued problem and on the other hand for the solution of the complementary problem. Finally the implementation of the algorithms is described and the results of parallel simulations are presented.     

Comparison of several formulations and integration methods for the resolution of DAEs formulations in event-driven simulation of nonsmooth frictionless multibody dynamics

This article is devoted to the comparison of numerical integration methods for nonsmooth multibody dynamics with joints, unilateral contacts and impacts in an industrial context. With an event-driven strategy, the smooth dynamics, which is integrated between two events, can be equivalently formulated as a Differential Algebraic Equation (DAE) of index 1, 2 or 3. It is well known that these reformulations are no longer equivalent when a numerical time-integration technique is used. The drift-off effect and the stability of the numerical scheme strongly depend on the index of the formulation. But, besides the standard properties of accuracy and stability of the DAE solvers, the event-driven context imposes some further requirements that are crucial for a robust and efficient event-driven strategy. In this article, several state-of-the-art numerical time integration methods for each formulation are compared: the generalized-\(\alpha\) scheme for index-3 formulation and stabilized index-2 formulation, (Partitioned) Runge–Kutta Half-Explicit Method of order 5 (HEM5 and PHEM56) for index-2 DAEs with projection techniques, and Runge–Kutta explicit scheme of order 5, the Dormand–Prince scheme (DOPRI5), for index-1 DAEs with projection techniques (MDOP5). We compare these schemes in terms of efficiency, violation of the constraints and the way they handle stiff dynamics on numerous industrial benchmarks, where a CAD software is in this loop. One of the major conclusions is that the index-2 DAEs solvers prove to be better than other schemes to maintain low violations at position and acceleration levels. The best compromise allows us to design efficient event-driven solvers. When the dynamics is stiff, implicit schemes outperform explicit and half-explicit methods which are sometimes unable to compute the dynamics when the system’s frequency range is wide. Furthermore, in industrial context, some solvers fail to reproduce the properties that they enjoy in theory. This is particularly true for half-explicit schemes when the Jacobian of the constraints has not full rank.     

Seismic analysis of base isolated buildings

Summary  This paper presents a survey of the numerical simulation of base isolation systems for the vibration control of buildings and their equipment, primarilly against earthquakes. Base isolation has received much attention in the recent twenty years and many buildings have been protected using this technology. The article focusses mainly on the different numerical methods used in the analysis of base isolated buildings. The conventional form of solving the equations of motion governing the seismic response of building structures with nonlinear base isolation consists of using monolithic step by step integration methods. As an efficient alternative static condensation and block iterative schemes can be applied. The particularities of the equations of motion of buildings equiped with various base isolation systems are described. The linear theory of base isolated buildings is then presented. After this, numerical solution techniques for the analysis of the seismic response of buildings with isolation systems are developed in detail in the paper. Finally, numerical results for elastic and inelastic structures are described. A complete set of references coverning a wide range of studies is included.     

Modeling Spatial Motion of 3D Deformable Multibody Systems with Nonlinearities

A computational strategy for modeling spatial motion of systems of flexible spatial bodies is presented. A new integral formulation of constraints is used in the context of the floating frame of reference approach. We discuss techniques to linearize the equations of motion both with respect to the kinematical coupling between the deformation and rigid body degrees of freedom and with respect to the geometrical nonlinearities (inclusion of stiffening terms). The plastic behavior of bodies is treated by means of plastic multipliers found as the result of fixed-point type iterations within a time step. The time integration is based on implicit Runge Kutta schemes with arbitrary order and of the RadauIIA type. The numerical results show efficiency of the developed techniques.     

Energy preserving schemes for nonlinear Hamiltonian systems of wave equations: Application to the vibrating piano string

This paper considers a general class of nonlinear systems, “nonlinear Hamiltonian systems of wave equations”. The first part of our work focuses on the mathematical study of these systems, showing central properties (energy preservation, stability, hyperbolicity, finite propagation velocity, etc.). Space discretization is made in a classical way (variational formulation) and time discretization aims at numerical stability using an energy technique. A definition of “preserving schemes” is introduced, and we show that explicit schemes or partially implicit schemes which are preserving according to this definition cannot be built unless the model is trivial. A general energy preserving second order accurate fully implicit scheme is built for any continuous system that fits the nonlinear Hamiltonian systems of wave equations class. The problem of the vibration of a piano string is taken as an example. Nonlinear coupling between longitudinal and transversal modes is modeled in the “geometrically exact model”, or approximations of this model. Numerical results are presented.