This paper presents a new methodology demonstrating the feasibility and advantages of a state-time formulation for dynamic simulation of complex multibody systems which shows potential advantages for exploiting massively parallel computing resources. This formulation allows time to be discretized and parameterized so that it can be treated as a variable in a manner similar to the system state variables. As a consequence of such a state-time discretization scheme, the system of governing equations yields to a set of loosely coupled linear-quadratic algebraic equations that is well-suited in structure for some families of nonlinear algebraic equations solvers. The goal of this work is to develop efficient multibody dynamics algorithm that is extremely scalable and better able to fully exploit anticipated immensely parallel computing machines (tera flop, pecta flop and beyond) made available to it. KeywordMultibody dynamics, State-time formulation, Scalable algorithm, Parallel computing Nomenclature B: Typical body B with its mass center B * F kA : k th applied force acting on the body B f mC : m th unknown constraint force acting on the body B r kA : k th position vector from B * to application point of F kA r B * jm : m th position vector from B * to application point (joint) of f mC
A novel state-time (ST) formulation for the simulation and analysis of the dynamic behavior of complex multibody systems is presented. The method proposes a computationally fast algorithm which is better able to fully exploit anticipated future immensely parallel computing resources (e.g. pecta flop machines and beyond) than existing multibody algorithms. The intent of the algorithm is to yield significantly reduced simulation turnaround time in situations where massively parallel (>10 6 processors) computing resources are available to it. It is shown that as a consequence of such a ST discretization scheme, the system of governing equations yields a set of loosely coupled nonlinear algebraic equations which is at most quadratic in the ST variables, with significant linear components. As such, it is well-suited in structure for nonlinear algebraic equations solvers. The linear-quadratic (LQ) structure of these equations further permits the use of a special solution scheme, which is expected to yield superior performance relative to more traditional Newton-Raphson type schemes when applied to large general systems. Nomenclature BTypical body B with its mass center B * F k A k th applied force acting on the body B f mC m th unknown constraint force acting on the body B r k A k th position vector from B * to application point of F k A r B * jm m th position vector from B * to application point (joint) of f mC T l l th applied concentrated moment acting on the body B N Newtonian reference frame I B/B * inertia dyadic of body B w.r.t. B * N ω B absolute angular velocity of body B N α B absolute angular acceleration of body B ε i jk the standard indicial cyclic permutation operator x B absolute displacement vector of B * N ω B × angular velocity cross product matrix C = N C Bdirection cosine matrix from the local frame B to N ψ i (t) and ϕ i (t) i th member of local family of C 1 and C 0 continuous shape functions A, B, D, E coefficient matrices in the partioned form of the discretized Newton's second Law R 1 , R 2 right-hand side column matrices in the partioned form of the discretized Newton's second Law and geometric constraint equations
When performing the dynamic simulation of stiff mechanical systems, implicit type integration schemes are usually required to preserve stability. This article presents a new implicit time integrator, which is a particular application of a novel state-time formulation recently developed by the authors in a more general scope. The proposed scheme is constructed with the intent of benefitting from the accuracy and apparent robustness thus far achieved with this algorithm in an integration context. This is realized by first setting up the weighted residual form of the governing equations of the system in a form associated with the application of a time marching integration scheme. The resulting algebraic equations are then solved, minimizing the error of integration time step in a generalized energy sense, allowing one to capture the stiff behavior of solution in an efficient manner. Examples are provided to show the proposed method performance when dealing with a stiff system.
This paper outlines the parallel implementation of a newly developed multibody system dynamics formulation. The methodology provides the means for the dynamic simulation to be parallelized temporally as well as spatially which will allow better exploitation of anticipated massively parallel computing resources. This will have three advantages: First, the system of equations may now be coarse grain parallelized to a far greater degree allowing an increased number of processors to be effectively utilized. Secondly, this will significantly reduce the fraction of serial operations and thus should increase speedup (reduced turn-around). Finally, the method allows temporal scale of each variable to be adjusted independently and as such offer considerable advantage for the efficient and accurate modeling and simulation of multiscale behaviors. These gains can be accomplished by discretizing a special form of the equations of motion in both temporal and spatial domains. Examples are provided to clarify the application of this scheme with particular attention on time domain parallelization.
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