A two-loop procedure based on implicit Runge–Kutta method for index-3 DAE of constrained dynamic problems

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In this paper, a procedure is proposed to use implicit Runge–Kutta method as the integration method for the solution of index-3 differential–algebraic equation which comes from constrained dynamic problems. The position constraints of a multibody system are usually a set of nonlinear algebraic equations. The iteration of the position constraint equations is embedded into the iteration of the nonlinear algebraic equations that come from the implicit Runge–Kutta method. These two iterations construct a two-loop structure. By using the coordinate partitioning technique, the independent coordinates are picked out from the set of coordinates of the whole system. The basic unknowns of the proposed method are the independent velocities and positions. A symplectic implicit Runge–Kutta method is used as the integration method in the two-loop procedure so that the independent velocities and positions can be obtained. The iteration of the implicit Runge–Kutta method is the outer loop in the two-loop procedure. A fixed-point iteration is brought in the outer loop to avoid the numerical differentiation that is used in Newton’s method to get the Jacobian matrix of the right-hand function. Newton’s method is used in the inner loop to solve the nonlinear algebraic equations of the position constraints, and then the dependent position coordinates can be obtained. The dependent velocities may be determined by solving a set of linear algebraic equations. The variable step size strategy of the two-loop procedure, which is based on the estimation of the error and the convergence criterion, is proposed for the practical application of this method. Two numerical examples are presented to demonstrate the efficiency of the proposed method.