Extension of the generalized-α method for constrained systems in mechanics: analysis and applications

Differential-algebraic equations are mathematical building blocks used to model real-world systems. One example is a swinging pendulum, which can be modeled by a differential algebraic equation. Since most differential-algebraic equations do not have explicit analytical solutions, except for simple systems, numerical methods must be used to approximate solutions that cannot be obtained otherwise. An important characteristic of numerical methods is its accuracy. Accuracy tells us how close the numerical approximation remains to the exact solution. We are also concerned with the concept of stiffness. Stiffness is a property that can disrupt the accuracy of a numerical method. Stiff differential-algebraic equations may require very small time-steps to obtain satisfactory results, which provides an issue with efficiency. Further, stiff differential-algebraic equations can result in a reduction of accuracy. A major challenge is how to deal effectively and efficiently with stiff differential-algebraic equations.

In this work, I have developed an extension to the generalized-α method, a type of numerical method used to solve differential equations. This extension allows us to solve differential-algebraic equations with mixed constraints, while retaining the accuracy of original numerical method. For a differential-algebraic equation, a constraint is a condition of the problem which the solution must satisfy. For the pendulum example, the constraint is given by the length of the string, which must remain the same. The structure of this numerical method allows it to be very efficient when implemented. This extension satisfies each constraint directly and is able to deal with problems that may have a nonconstant mass. In this thesis, I developed an accuracy theory for the extension of the generalized-α method, and investigated its suitability towards stiff differential-algebraic equations. I also provide numerical examples to illustrate the convergence results.