L-Stable Methods For Numerical Solution Of Structural Dynamics Equations And Multibody Dynamics Equations

Aerospace engineering contains massive structural and mechanism systems, such as satellite antennas and solar battery. When they are on working, they are the flexible structures; and when they are in the expansion process, they are the mechanism systems. The research on the dynamic behavior of the structure is usually based on the structural dynamics equation, and the research on mechanism systems is often based on multibody system dynamical equations. The semi-discrete structural dynamical equations belong to the initial value problem for differential equations, and dynamical equations of multi-body system belong to the initial value problem of the differential-algebraic equations. The main purpose of this dissertation is an attempt to study the numerical solution of these two class equations.Runge-Kutta methods are used in numerical solution of structural dynamical equations. When Runge-Kutta methods are employed to incremental formulations of structural dynamical equations, the iterative equations are derived, and two approaches to reduce the cost of computation are studied. We mention that some traditional direct integration methods belong to Runge-Kutta methods. Based on equilibrium equation governing motion of single degree of freedom system, the integration approximation operators of Runge-Kutta methods are derived. With the spectral radii of the integration approximation operators, we can analyze the stability of Runge-Kutta methods for second order ordinary differential equations. The influence of the numerical damping of various Runge-Kutta methods can be explained by spectral radii of the integration approximation operators. Numerical example demonstrates that the high order L-stable Runge-Kutta methods include ideal numerical damping, and can be used to get high precision solution of low frequency vibration, and, at the same time, very well to decay high frequency oscillatory . In exact solution the physical damping is a marvelous instrument to decay high frequency oscillatory , but in numerical solution it is little for the high frequency oscillatory. We conclude that in numerical solution of structural dynamical equations, it is not feasible to increase the physical damping for decaying the high frequency oscillatory, but it is feasible to employ the algorithm with numerical damping.Solvers which are suitable for high index differential-algebraic equation are used to solve the high index formulation of multibody system dynamics equations. Using the high index formulation of the multibody system dynamics equations, and employing the implicit differential equations, a few new solution variables are introduced, and the sparsity of original system matrix is kept. On the other hand, there are two disadvantages: (l)The acceleration variables and Lagrange multipliers are less precise than position variables and velocity variables. (2)The convergence of iterations is depend on consistent initial conditions. According to these two disadvantages, we present an amendment method which can improve the precision of acceleration variables and Lagrange multipliers, and convergence of iterations. Numerical examples demonstrate that the amendment method can reduce the cost of computation. Especially for small error tolerance, reduction of the cost of computation is considerable.Based on Taylor expansion, a new approach to construct block methods is derived; using the Pade approximation of the exponential function as the stability function of the block methods, we construct L-stability block methods. We point out that the block methods belong to Runge-Kutta methods. With the order conditions of Runge-Kutta methods, the order of L-stability block methods is presented. For implicit differential equations, parallel iterative formulae and local error estimates methods are derived. We develop a solver named PBIDE (Parallel Block solver for Implicit Differential Equations), which automatically adjusts the step size.For the order reduction while Runge-Kutta methods is applied to differential-algebraic equations, we put forward multistep block methods. The construction of multistep block methods is described, and order conditions and stability is studied. We prove that multi-step block methods are stiffly accurate, and the stage order is equal to the order. Numerical example demonstrates that there is no order reduction when multistep block methods is employed to index-2 differential-algebraic equations. A multistep block method of order and stage order 3 with 2 stage is constructed, and its stability function is a general Pade approximation of stability order 4. A multistep block method of order and stage order 4 with 3 stage is constructed, and its stability function is the rational approximation of exponential function as same as the stability function of Runge-Kutta methods. These two multistep block methods are L-stability. The Nordsieck vector expression of multistep block methods that is convenient to change the step size is derived. For implicit differential equations, we implement the multistep block method of order and stage order 3 with 2 stage and develop the solver named MBIDE(Multistep Block solver for Implicit Differential Equations).Finally, we test PBIDE and MBIDE with several problems, and compare these two solves with several well-known solvers on precision and efficiency. The problems chosen here for our tests are mainly taken from structure and multibody systems, and several addition test problems are take from another fields. The results show that the solves PBIDE and MBIDE are effective and reliable for stiff and differential-algebraic problems.