Dan Negrut

On an Implementation of the Hilber-Hughes-Taylor Method in the Context of Index 3 Differential-Algebraic Equations of Multibody Dynamics (DETC2005-85096)

The paper presents theoretical and implementation aspects related to a numerical integrator used for the simulation of large mechanical systems with flexible bodies and contact/impact. The proposed algorithm is based on the Hilber-Hughes-Taylor (HHT) implicit method and is tailored to answer the challenges posed by the numerical solution of index 3 differential-algebraic equations that govern the time evolution of a multibody system. One of the salient attributes of the algorithm is the good conditioning of the Jacobian matrix associated with the implicit integrator. Error estimation, integration step-size control, and nonlinear system stopping criteria are discussed in detail. Simulations using the proposed algorithm of an engine model, a model with contacts, and a model with flexible bodies indicate a 2 to 3 speedup factor when compared against benchmark MSC.ADAMS runs. The proposed HHT-based algorithm has been released in the 2005 version of the MSC.ADAMS/Solver.

Large-scale parallel multi-body dynamics with frictional contact on the graphical processing unit:

In the context of simulating the frictional contact dynamics of large systems of rigid bodies, this paper reviews a novel method for solving large cone complementarity problems by means of a fixed-point iteration algorithm. The method is an extension of the Gauss—Seidel and Gauss—Jacobi methods with over-relaxation for symmetric convex linear complementarity problems. Convergent under fairly standard assumptions, the method is implemented in a parallel framework by using a single instruction multiple data computation paradigm promoted by the Compute Unified Device Architecture library for graphical processing unit programming. The framework supports the simulation of problems with more than one million bodies in contact. Simulation thus becomes a viable tool for investigating the dynamics of complex systems such as ground vehicles running on sand, powder composites, and granular material flow.

An Implicit Runge–Kutta Method for Integration of Differential Algebraic Equations of Multibody Dynamics

When performing dynamic analysis of a constrained mechanical system, aset of index 3 Differential Algebraic Equations (DAE) describes the timeevolution of the model. This paper presents a state space DAE solutionframework that can embed an arbitrary implicit Ordinary DifferentialEquations (ODE) code for numerical integration of a reduced set of statespace ordinary differential equations. This solution framework isconstructed with the goal of leveraging with minimal effort establishedoff the shelf implicit ODE integrators for efficiently solving the DAEof multibody dynamics. This concept is demonstrated by embedding awell-known public domain singly diagonal implicit Runge–Kutta code inthe framework provided. The resulting L-stable, stiffly accurateimplicit algorithm is shown to be two orders of magnitude faster than astate of the art explicit algorithm when used to simulate a stiffvehicle model.

A Topology-Based Approach for Exploiting Sparsity in Multibody Dynamics in Cartesian Formulation*

In the present paper a new approach for solving for the accelerations and Lagrange multipliers when integrating multibody systems in descriptor form is given. When solving for these unknowns, the coef ficient matrix of the linear system to be solved has the particular form of an optimization matrix. Due to the conectivity among the bodies comprised within the system this matrix is likely to have a lar ge number of zero entries. Advantage is taken of both the special structure and the sparsity of the coef ficient matrix. Simple manipulations bring the

A Discussion of Low-Order Numerical Integration Formulas for Rigid and Flexible Multibody Dynamics

The premise of this work is that real-life mechanical systems limit the use of high order integration formulas due to the presence in the associated models of friction and contact/impact elements. In such cases producing a numerical solution necessarily relies on low order integration formulas. The resulting algorithms are generally robust and expeditious; their major drawback remains that they typically require small integration step-sizes in order to meet a user prescribed accuracy. This paper looks at three low order numerical integration formulas: Newmark, HHT, and BDF of order two. These formulas are used in two contexts. A first set of three methods is obtained by considering a direct index-3 discretization approach that solves for the equations of motion and imposes the position kinematic constraints. The second batch of three additional methods draws on the HHT and BDF integration formulas and considers in addition to the equations of motion both the position and velocity kinematic constraint equations. The first objective of this paper is to review the theoretical results available in the literature regarding the stability and convergence properties of these low order methods when applied in the context of multibody dynamics simulation. When no theoretical results are available, numerical experiments are carried out to gauge order behavior. The second objective is to perform a set of numerical experiments to compare these six methods in terms of several metrics: (a) efficiency, (b) velocity constraint drift, and (c) energy preservation. A set of simple mechanical systems is used for this purpose: a double pendulum, a slider crank with rigid bodies, and a slider crank with a flexible body represented in the floating frame of reference formulation.Copyright

A State-Space-Based Implicit Integration Algorithm for Differential-Algebraic Equations of Multibody Dynamics*

ABSTRACT An implicit numerical integration algorithm based on generalized coordinate partitioning is presented for the numerical solution of differential-algebraic equations of motion arising in multibody dynamics. The algorithm employs implicit numerical integration formulas to express independent generalized coordinates and their first time derivative as functions of independent accelerations at discrete integration times. The latter are determined as the solution of discretized equations obtained from state-space, second-order ordinary differential equations in the independent coordinates. All dependent variables in the formulation, including Lagrange multipliers, are determined by satisfying the full system of kinematic and kinetic equations of motion. The algorithm is illustrated using the implicit trapezoidal rule to integrate the constrained equations of motion for three stiff mechanical systems with different generalized coordinate dimensions. Results show that the algorithm is robust and has the ...

Using Nesterov's Method to Accelerate Multibody Dynamics with Friction and Contact

We present a solution method that, compared to the traditional Gauss-Seidel approach, reduces the time required to simulate the dynamics of large systems of rigid bodies interacting through frictional contact by one to two orders of magnitude. Unlike Gauss-Seidel, it can be easily parallelized, which allows for the physics-based simulation of systems with millions of bodies. The proposed accelerated projected gradient descent (APGD) method relies on an approach by Nesterov in which a quadratic optimization problem with conic constraints is solved at each simulation time step to recover the normal and friction forces present in the system. The APGD method is validated against experimental data, compared in terms of speed of convergence and solution time with the Gauss-Seidel and Jacobi methods, and demonstrated in conjunction with snow modeling, bulldozer dynamics, and several benchmark tests that highlight the interplay between the friction and cohesion forces.

GPU-based parallel computing for the simulation of complex multibody systems with unilateral and bilateral constraints: an overview

This work reports on advances in large-scale multibody dynamics simulation facilitated by the use of the Graphics Processing Unit (GPU). A description of the GPU execution model along with its memory spaces is provided to illustrate its potential parallel scientific computing. The equations of motion associated with the dynamics of large system of rigid bodies are introduced and a solution method is presented. The solution method is designed to map well on the parallel hardware, which is demonstrated by an order of magnitude reductions in simulation time for large systems that concern the dynamics of granular material. One of the salient attributes of the solution method is its linear scaling with the dimension of the problem. This is due to efficient algorithms that handle in linear time both the collision detection and the solution of the nonlinear complementarity problem associated with the proposed approach. The current implementation supports the simulation of systems with more than one million bodies on commodity desktops. Efforts are under way to extend this number to hundreds of millions of bodies on small affordable clusters.

A Topology-Based Approach to Exploiting Sparsity in Multibody Dynamics: Joint Formulation*

In this paper, advantage is taken of the problem structure in multibody dynamics simulation when the mechanical system is modeled using a minimal set of generalized coordinates. It is shown that the inertia matrix associated with any open- or closed-loop mechanism is positive definite by finding a simple mathematical expression for the quadratic form expressing the kinetic energy in an associated state space. Based on this result, an algorithm that efficiently solves for second time derivatives of the generalized coordinates is presented. Significant speed-ups accrue due to both the no fill-in factorization of the composite inertia matrix technique and the degree of parallelism attainable with the new algorithm.

Chrono: An Open Source Multi-physics Dynamics Engine

We provide an overview of a multi-physics dynamics engine called Chrono. Its forte is the handling of complex and large dynamic systems containing millions of rigid bodies that interact through frictional contact. Chrono has been recently augmented to support the modeling of fluid-solid interaction (FSI) problems and linear and nonlinear finite element analysis (FEA). We discuss Chrono’s software layout/design and outline some of the modeling and numerical solution techniques at the cornerstone of this dynamics engine. We briefly report on some validation studies that gauge the predictive attribute of the software solution. Chrono is released as open source under a permissive BSD3 license and available for download on GitHub.