A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact,joints, and friction

We present a hard constraint, linear complementarity based, method for the simulation of stiff multibody dynamics with contact, joints and friction. The approach uses a linearization of the modified trapezoidal method, incorporates a Poisson restitution model at collision, and solves only one linear complementarity problem per time step when no collisions are encountered. We prove that, under certain assumptions, the method has order two, a fact that is also demonstrated by our numerical simulations. For the unconstrained (ODE) case, the method achieves second‐order convergence and absolute stability while solving only one linear system per step. When we use a special approximation of the Jacobian matrix for the case where the stiff forces originate in springs and dampers attached to two points in the system, the linear complementarity problem can be solved for any value of the time step and numerical simulation demonstrate that the method is stiffly stable. The method was implemented in UMBRA, an industrial‐grade virtual prototyping software. Copyright © 2005 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.     

A constraint‐stabilized time‐stepping approach for rigid multibody dynamics with joints,contact and friction

We present a method for achieving geometrical constraint stabilization for a linear‐complementarity‐based time‐stepping scheme for rigid multibody dynamics with joints, contact, and friction. The method requires the solution of only one linear complementarity problem per step. We prove that the velocity stays bounded and that the constraint infeasibility is uniformly bounded in terms of the size of the time step and the current value of the velocity. Several examples, including one for joint‐only systems, are used to demonstrate the constraint stabilization effect. Copyright © 2004 John Wiley & Sons, Ltd.     

Real‐time direct integration of reduced solid dynamics equations

A new method is developed here for the real‐time integration of the equations of solid dynamics based on the use of proper orthogonal decomposition (POD)–proper generalized decomposition (PGD) approaches and direct time integration. The method is based upon the formulation of solid dynamics equations as a parametric problem, depending on their initial conditions. A sort of black‐box integrator that takes the resulting displacement field of the current time step as input and (via POD) provides the result for the subsequent time step at feedback rates on the order of 1 kHz is obtained. To avoid the so‐called curse of dimensionality produced by the large amount of parameters in the formulation (one per degree of freedom of the full model), a combined POD–PGD strategy is implemented. Examples that show the promising results of this technique are included. Copyright © 2014 John Wiley & Sons, Ltd.     

Step size adjustment and extrapolation for time‐stepping schemes in non‐smooth dynamics

In this paper we use step size adjustment and extrapolation methods to improve Moreau's time‐stepping scheme for the numerical integration of non‐smooth mechanical systems, i.e. systems with impact and friction. The scheme yields a system of inclusions, which is transformed into a system of projective equations. These equations are solved iteratively. Switching points are time instants for which the structure of the mechanical system changes, for example, time instants for which a sticking friction element begins to slide. We show how switching points can be localized and how these points can be resolved by choosing a minimal step size. In order to improve the integration of non‐smooth systems in the smooth parts, we show how the time‐stepping method can be used as a base integration scheme for extrapolation methods, which allow for an increase in the integration order. Switching points are processed by a small time step, while time intervals during which the structure of the system does not change are computed with a larger step size and improved integration order. The overall algorithm, which consists of a time‐stepping module, an extrapolation module and a step size adjustment module, is discussed in detail and some examples are given. Copyright © 2008 John Wiley & Sons, Ltd.     

Resolving force indeterminacy in contact dynamics using compatibility conditions

Contact dynamics (CD) is a powerful method to solve the dynamics of large systems of colliding rigid bodies. CD can be computationally more efficient than classical penalty-based discrete element methods (DEM) for simulating contact between stiff materials such as rock, glass, or engineering metals. However, by idealizing bodies as perfectly rigid, contact forces computed by CD can be non-unique due to indeterminacy in the contact network, which is a common occurence in dense granular flows. We propose a frictionless CD method that is designed to identify only the unique set of contact forces that would be predicted by a soft particle method, such as DEM, in the limit of large stiffness. The method involves applying an elastic compatibility condition to the contact forces, which maintains no-penetration constraints but filters out force distributions that could not have arisen from stiff elastic contacts. The method can be used as a post-processing step that could be integrated into existing CD codes with minimal effort. We demonstrate its efficacy in a variety of indeterminate problems, including some involving multiple materials, non-spherical shapes, and nonlinear contact constitutive laws.     

A nonsmooth frictional contact formulation for multibody system dynamics

We present a new node-to-face frictional contact element for the simulation of the nonsmooth dynamics of systems composed of rigid and flexible bodies connected by kinematic joints. The equations of motion are integrated using a nonsmooth generalized-α time integration scheme and the frictional contact problem is formulated using a mixed approach, based on an augmented Lagrangian technique and a Coulomb friction law. The numerical results are independent of any user-defined penalty parameter for the normal or tangential component of the forces and, the bilateral and the unilateral constraints are exactly fulfilled both at position and velocity levels. Finally, the robustness and the performance of the proposed algorithm are demonstrated by solving several numerical examples of nonsmooth mechanical systems involving frictional contact.     

Novel partitioned time integration methods for DAE systems based on L‐stable linearly implicit algorithms

Real‐time applications based on the principle of Dynamic Substructuring require integration methods that can deal with constraints without exceeding an a priori fixed number of steps. For these applications, first we introduce novel partitioned algorithms able to solve DAEs arising from transient structural dynamics. In particular, the spatial domain is partitioned into a set of disconnected subdomains and continuity conditions of acceleration at the interface are modeled using a dual Schur formulation. Interface equations along with subdomain equations lead to a system of DAEs for which both staggered and parallel procedures are developed. Moreover under the framework of projection methods, also a parallel partitioned method is conceived. The proposed partitioned algorithms enable a Rosenbrock‐based linearly implicit LSRT2 method, to be strongly coupled with different time steps in each subdomain. Thus, user‐defined algorithmic damping and subcycling strategies are allowed. Secondly, the paper presents the convergence analysis of the novel schemes for linear single‐Degree‐of‐Freedom (DoF) systems. The algorithms are generally A‐stable and preserve the accuracy order as the original monolithic method. Successively, these results are validated via simulations on single‐ and three‐DoFs systems. Finally, the insight gained from previous analyses is confirmed by means of numerical experiments on a coupled spring–pendulum system. Copyright © 2011 John Wiley & Sons, Ltd.     

State‐space time integration with energy control and fourth‐order accuracy for linear dynamic systems

A fourth‐order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented. It is derived by integrating the phase‐space representation and evaluating the resulting displacement and velocity integrals via integration by parts, substituting the time derivatives from the original differential equations. The resulting algorithm has an exact energy equation, in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix. This implies unconditional stability of the algorithm, and the relative phase error is of fourth‐order. An optional high‐frequency algorithmic damping is constructed by optimal combination of three different damping matrices, each proportional to either the mass or the stiffness matrix. This leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth‐order in the frequency. Thus, the low‐frequency response is virtually undamped, and the algorithm remains third‐order accurate even when algorithmic damping is included. The accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium, where the algorithmic damping is used to reduce dispersion due to the spatial discretization, leading to a smooth solution with a clearly defined wave front. Copyright © 2005 John Wiley & Sons, Ltd.     

Formulation of equations of motion of finite element form for vehicle–track–bridge interaction system with two types of vehicle model

Vehicle, track and bridge are considered as an entire system in this paper. Two types of vertical vehicle model are described. One is a one foot mass–spring–damper system having two‐degree‐of‐freedom, and the other is four‐wheelset mass–spring–damper system with two‐layer suspension systems possessing 10‐degree‐of‐freedom. For the latter vehicle model, the eccentric load of car body is taken into account. The rails and the bridge deck are modelled as an elastic Bernoulli–Euler upper beam with finite length and a simply supported Bernoulli–Euler lower beam, respectively, while the elasticity and damping properties of the rail bed are represented by continuous springs and dampers. The dynamic contact forces between the moving vehicle and rails are considered as internal forces, so it is not necessary to calculate the internal forces for setting up the equations of motion of the vehicle–track–bridge interaction system. The two types of equations of motion of finite element form for the entire system are derived by means of the principle of a stationary value of total potential energy of dynamic system. The proposed method can set up directly the equations of motion for sophisticated system, and these equations can be solved by step‐by‐step integration method, to obtain simultaneously the dynamic responses of vehicle, of track and of bridge. Illustration examples are given. Copyright 2004 © John Wiley & Sons, Ltd.     

Energy conserving and dissipative time finite element schemes for N‐body stiff problems

Petrov–Galerkin finite element method is adopted to develop a family of temporal integrators, which preserves the feature of energy conservation or numerical dissipation for non‐linear N‐body dynamical systems. This leads to an enhancement of numerical stability and the integrators may therefore offer some advantage for the numerical solution of stiff systems in long‐term simulations. Dynamically tuneable numerical integration is exploited to improve the accuracy of the time‐stepping schemes. Representative simulations for simple non‐linear systems show the performance of the schemes in controlling over or damping out unresolved high frequencies. Copyright © 2004 John Wiley & Sons, Ltd.     

Fourier‐based numerical approximation of the Weertman equation for moving dislocations

This work addresses the numerical approximation of solutions to a dimensionless form of the Weertman equation, which models a steadily moving dislocation and is an important extension (with advection term) of the celebrated Peierls‐Nabarro equation for a static dislocation. It belongs to the class of nonlinear reaction‐advection‐diffusion integro‐differential equations with Cauchy‐type kernel, thus involving an integration over an unbounded domain. In the Weertman problem, the unknowns are the shape of the core of the dislocation and the dislocation velocity. The proposed numerical method rests on a time‐dependent formulation that admits the Weertman equation as its long‐time limit. Key features are (1) time iterations are conducted by means of a new, robust, and inexpensive Preconditioned Collocation Scheme in the Fourier domain, which allows for explicit time evolution but amounts to implicit time integration, thus allowing for large time steps; (2) as the integration over the unbounded domain induces a solution with slowly decaying tails of important influence on the overall dislocation shape, the action of the operators at play is evaluated with exact asymptotic estimates of the tails, combined with discrete Fourier transform operations on a finite computational box of size L; (3) a specific device is developed to compute the moving solution in a comoving frame, to minimize the effects of the finite‐box approximation. Applications illustrate the efficiency of the approach for different types of nonlinearities, with systematic assessment of numerical errors. Converged numerical results are found insensitive to the time step, and scaling laws for the combined dependence of the numerical error with respect to L and to the spatial step size are obtained. The method proves fast and accurate and could be applied to a wide variety of equations with moving fronts as solutions; notably, Weertman‐type equations with the Cauchy‐type kernel replaced by a fractional Laplacian.     

A stabilized numerical solution for the dynamic contact of the bodies having very stiff constraint on the contact point

For the numerical analysis of dynamic contact problem where the contact constraint is imposed by a very stiff massless spring between the bodies, it is shown that a stabilized time integration solution can be obtained without spurious oscillations by imposing the velocity and acceleration constraints as well as the displacement constraint on the contact point. For the velocity and acceleration contact constraints which are crucial for the numerical stability, the time derivatives of the spring deformation are computed by using the Newmark time integration rule of structural dynamics. With the numerical experiments the necessity of the velocity and acceleration contact constraints and the necessity of unconditionally stable time integration rule for the very stiff spring are demonstrated.     

Hybrid state‐space time integration in a rotating frame of reference

A time integration algorithm is developed for the equations of motion of a flexible body in a rotating frame of reference. The equations are formulated in a hybrid state‐space, formed by the local displacement components and the global velocity components. In the spatial discretization the local displacements and the global velocities are represented by the same shape functions. This leads to a simple generalization of the corresponding equations of motion in a stationary frame in which all inertial effects are represented via the classic global mass matrix. The formulation introduces two gyroscopic terms, while the centrifugal forces are represented implicitly via the hybrid state‐space format. An angular momentum and energy conserving algorithm is developed, in which the angular velocity of the frame is represented by its mean value. A consistent algorithmic damping scheme is identified by applying the conservative algorithm to a decaying response, which is rendered stationary by an increasing exponential factor that compensates the decay. The algorithmic damping is implemented by introducing forward weighting of the mean values appearing in the algorithm. Numerical examples illustrate the simplicity and accuracy of the algorithm. Copyright © 2011 John Wiley & Sons, Ltd.     

Dynamic analysis of structures with unilateral constraints: Numerical integration and reduction of structural equations

Structural dynamic equations with unilateral constraints upon the displacements, velocities and accelerations are employed in order to represent vibrating elastic structures with normal, oblique impact and friction interaction points. For obtaining a numerical integration scheme the Lagrange multipliers and a minimum work approach are employed at each time step. The algorithm is presented as an extension of the generalized Newmark scheme. It seems to retain the asymptotic features of the original one. The reduction of the number of dynamic degrees of freedom of the unilaterally constrained structures is carried out by representing the equations of motion in modal co-ordinates of the unconstrained structure and truncating the dynamic contributions of higher modes. The presented techniques have been verified by investigating free longitudinal vibroimpact motion laws of an elastic vibroconverter and free longitudinal and bending vibration of a vibroconverter interacting with a moving rigid body by oblique impact and friction forces.     

Friction and Wetting Transitions of Magnetic Droplets on Micropillared Superhydrophobic Surfaces

Reliable characterization of wetting properties is essential for the development and optimization of superhydrophobic surfaces. Here, the dynamics of superhydrophobicity is studied including droplet friction and wetting transitions by using droplet oscillations on micropillared surfaces. Analyzing droplet oscillations by high‐speed camera makes it possible to obtain energy dissipation parameters such as contact angle hysteresis force and viscous damping coefficients, which indicate pinning and viscous losses, respectively. It is shown that the dissipative forces increase with increasing solid fraction and magnetic force. For 10 µm diameter pillars, the solid fraction range within which droplet oscillations are possible is between 0.97% and 2.18%. Beyond the upper limit, the oscillations become heavily damped due to high friction force. Below the lower limit, the droplet is no longer supported by the pillar tops and undergoes a Cassie–Wenzel transition. This transition is found to occur at lower pressure for a moving droplet than for a static droplet. The findings can help to optimize micropillared surfaces for low‐friction droplet transport.     

A family of noniterative integration methods with desired numerical dissipation

A new family of unconditionally stable integration methods for structural dynamics has been developed, which possesses the favorable numerical dissipation properties that can be continuously controlled. In particular, it can have zero damping. This numerical damping is helpful to suppress or even eliminate the spurious participation of high frequency modes, whereas the low frequency modes are almost unaffected. The most important improvement of this family method is that it involves no nonlinear iterations for each time step, and thus it is very computationally efficient when compared with a general second‐order accurate integration method, such as the constant average acceleration method. Copyright © 2014 John Wiley & Sons, Ltd.     

Interval uncertain method for multibody mechanical systems using Chebyshev inclusion functions

This study proposes a new uncertain analysis method for multibody dynamics of mechanical systems based on Chebyshev inclusion functions The interval model accounts for the uncertainties in multibody mechanical systems comprising uncertain‐but‐bounded parameters, which only requires lower and upper bounds of uncertain parameters, without having to know probability distributions. A Chebyshev inclusion function based on the truncated Chebyshev series, rather than the Taylor inclusion function, is proposed to achieve sharper and tighter bounds for meaningful solutions of interval functions, to effectively handle the overestimation caused by the wrapping effect, intrinsic to interval computations. The Mehler integral is used to evaluate the coefficients of Chebyshev polynomials in the numerical implementation. The multibody dynamics of mechanical systems are governed by index‐3 differential algebraic equations (DAEs), including a combination of differential equations and algebraic equations, responsible for the dynamics of the system subject to certain constraints. The proposed interval method with Chebyshev inclusion functions is applied to solve the DAEs in association with appropriate numerical solvers. This study employs HHT‐I3 as the numerical solver to transform the DAEs into a series of nonlinear algebraic equations at each integration time step, which are solved further by using the Newton–Raphson iterative method at the current time step. Two typical multibody dynamic systems with interval parameters, the slider crank and double pendulum mechanisms, are employed to demonstrate the effectiveness of the proposed methodology. The results show that the proposed methodology can supply sufficient numerical accuracy with a reasonable computational cost and is able to effectively handle the wrapping effect, as cosine functions are incorporated to sharpen the range of non‐monotonic interval functions. Copyright © 2013 John Wiley & Sons, Ltd.     

The particle finite element method: a powerful tool to solve incompressible flows with free‐surfaces and breaking waves

Particle Methods are those in which the problem is represented by a discrete number of particles. Each particle moves accordingly with its own mass and the external/internal forces applied to it. Particle Methods may be used for both, discrete and continuous problems. In this paper, a Particle Method is used to solve the continuous fluid mechanics equations. To evaluate the external applied forces on each particle, the incompressible Navier–Stokes equations using a Lagrangian formulation are solved at each time step. The interpolation functions are those used in the Meshless Finite Element Method and the time integration is introduced by an implicit fractional‐step method. In this manner classical stabilization terms used in the momentum equations are unnecessary due to lack of convective terms in the Lagrangian formulation. Once the forces are evaluated, the particles move independently of the mesh. All the information is transmitted by the particles. Fluid–structure interaction problems including free‐fluid‐surfaces, breaking waves and fluid particle separation may be easily solved with this methodology. Copyright © 2004 John Wiley & Sons, Ltd.