Abstract
Variational integrators for dynamic systems with holonomic constraints are proposed based on Hamilton’s principle. The variational principle is discretized by approximating the generalized coordinates and Lagrange multipliers by Lagrange polynomials, by approximating the integrals by quadrature rules. Meanwhile, constraint points are defined in order to discrete the holonomic constraints. The functional of the variational principle is divided into two parts, i.e., the action of the unconstrained term and the constrained term and the actions of the unconstrained term and the constrained term are integrated separately using different numerical quadrature rules. The influence of interpolation points, quadrature rule and constraint points on the accuracy of the algorithms is analyzed exhaustively. Properties of the proposed algorithms are investigated using examples. Numerical results show that the proposed algorithms have arbitrary high order, satisfy the holonomic constraints with high precision and provide good performance for long-time integration.
Highlights
Numerical integrators for nonlinear dynamic mechanics should reflect the properties of the underlying mechanical systems [1,2]
In [5,39], the Galerkin variational integrators are based on the m points Gauss–Legendre quadrature rules, and choosing m + 1 equidistant points as the control points is shown to be equivalent to the collocation Gauss–Legendre method of order s = 2m
Combined with the construction process given it is clear that in Equations (30), (31) and (35), the approximation of the generalized coordinates and the Lagrange multipliers depends on the control points, the approximation of the integral depends on the quadrature rules, and the approximation of the holonomic constraint depends on the constraint points
Summary
Numerical integrators for nonlinear dynamic mechanics should reflect the properties of the underlying mechanical systems [1,2]. Galerkin variational integrators comprise a general method for constructing high-order variational integrators. They rely on the discretization of the action, which is approximated by the finite-dimensional function space and a numerical quadrature rule. For the constrained system, the Galerkin variational integrator with the highest order is constructed by taking the Lobatto quadrature rule [5]. The augmented Lagrangian is split into two parts, which makes it possible to use the Gauss–Legendre quadrature rule for constrained systems. This practice improves the accuracy of the algorithm and makes the construction of the algorithm more flexible.
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