In this article, a brief description of Discrete Mechanics and Variational Integrators which preserve the symplectic structure of the flow will be provided and a Newton-Raphson algorithm that can be used to solve implicit equations on the SO(3) manifold will be developed. These techniques will be used to simulate the rotational dynamics of a rigid body evolving on the Lie Group SO(3).
Shanghai Jiao Tong University（上海交通大学）；Central Scientific and Research Institute “Elektropribor”（中央科研研究所“Elektropribor”）
This paper compares two basic approaches to solving ordinary differential equations, which form the basis for attitude computation in strapdown inertial navigation systems, namely, the Taylor series expansion approach that was used in its low-order form for deriving all mainstream algorithms and the functional iterative integration approach developed recently. They are respectively applied to solve the kinematic equations of major attitude parameters, including the quaternion, the Rodrigues vector and the rotation vector. Specifically, the mainstream algorithms, which have relied on the simpli…
Discrete variational methods have shown an excellent performance in numerical simulations of different mechanical systems. In this paper, we introduce an iterative method for discrete variational methods appropriate for boundary value problems. More concretely, we explore a parallelization strategy that leverages the power of multicore CPUs and GPUs (graphics cards). We study this parallel method for first-order and second-order Lagrangians and we illustrate its excellent behavior in some interesting applications, namely Zermelo's navigation problem, a fuel-optimal navigation problem, and an i…
Discrete variational methods show excellent performance in numerical simulations of different mechanical systems. In this paper, we introduce an iterative procedure for the solution of discrete variational equations for boundary value problems. More concretely, we explore a parallelization strategy that leverages the capabilities of multicore CPUs and GPUs (graphics cards). We study this parallel method for higher-order Lagrangian systems, which appear in fully-actuated problems and beyond. The most important part of the paper is devoted to a precise study of different convergence conditions f…
In this paper, we propose a variationally consistent technique for decreasing the maximum eigenfrequencies of structural dynamics related finite element formulations. Our approach is based on adding a symmetric positive-definite term to the mass matrix that follows from the integral of the traction jump across element boundaries. The added term is weighted by a small factor, for which we derive a suitable, and simple, element-local parameter choice. For linear problems, we show that our mass-scaling method produces no adverse effects in terms of spatial accuracy and orders of convergence. We i…
A novel implicit integration scheme for the Discrete Element Method (DEM) based on the variational integrator approach is presented. The numerical solver provides a fully dynamical description that, notably, reduces to an energy minimisation scheme in the quasi-static limit. A detailed derivation of the numerical method is presented for the Hookean contact model and tested against an established open source DEM package that uses the velocity-Verlet integration scheme. These tests compare results for a single collision, long-term stability and statistical quantities of ensembles of particles. N…
We consider variational time integration using continuous Galerkin Petrov methods applied to evolutionary systems of changing type. We prove optimal-order convergence of the error in a cGP-like norm and conclude the paper with some numerical examples and conclusions.
A variational framework for bond-based peridynamics with spatially varying horizons and its asynchronous time integration
Friedrich-Alexander-Universität Erlangen-Nürnberg（埃尔朗根-纽伦堡大学）；Universität Siegen（锡根大学）；California Institute of Technology（加州理工学院）；Queensland University of Technology（昆士兰科技大学）
It is well known that symplectic integrators lose their near energy preservation properties when variable step sizes are used. The most common approach to combine adaptive step sizes and symplectic integrators involves the Poincaré transformation of the original Hamiltonian. In this article, we provide a framework for the construction of variational integrators using the Poincaré transformation. Since the transformed Hamiltonian is typically degenerate, the use of Hamiltonian variational integrators based on Type II or Type III generating functions is required instead of the more traditional L…
We propose a new discrete element method supporting general polyhedral meshes. The method can be understood as a lowest-order discontinuous Galerkin method parametrized by the continuous mechanical parameters (Young's modulus and Poisson's ratio). We consider quasi-static and dynamic elasto-plasticity, and in the latter situation, a pseudo-energy conserving time-integration method is employed. The computational cost of the time-stepping method is moderate since it is explicit and used with a naturally diagonal mass matrix. Numerical examples are presented to illustrate the robustness and versa…
We develop an explicit, second-order, variational time integrator for full body dynamics that preserves the momenta of the continuous dynamics, such as linear and angular momenta, and exhibits near-conservation of total energy over exponentially long times. In order to achieve these properties, we parametrize the space of rotations using exponential local coordinates represented by a rescaled form of the Rodrigues rotation vector and we systematically derive the time integrator from a discrete Lagrangian function that yields discrete Euler-Lagrange equations amenable to explicit, closed-form s…
Slimplectic Integrators: Variational Integrators for General Nonconservative Systems
McGill University（麦吉尔大学）；Walter Burke Institute for Theoretical Physics, California Institute of Technology（加州理工学院沃尔特·伯克理论物理研究所）；Cornell University（康奈尔大学）
