3 DAE Exercises
The exercises are marked Fundamental, Computational, or Advanced. Hints indicate a starting point without reproducing a solution.
3.1 Fundamental
- For x'=z and 0=x^2+z^2-1, determine all consistent values of z(0) for a prescribed x(0). Which choices cease to be regular? Hint: inspect \partial g/\partial z.
- Starting from a holonomic constraint \Phi(\bm q)=0, derive the velocity and acceleration constraints and identify the terms that contain the multiplier. Hint: differentiate with the chain rule before substituting the equations of motion.
- Explain why enforcing only the acceleration constraint permits position drift. Hint: write the residual dynamics and count integrations.
3.2 Computational
- Implement the consistent projection of the course notes for the redundant-coordinate pendulum. Compare the correction obtained with two different diagonal scalings \bm W. Hint: report both scaled and physical residuals.
- Integrate the same pendulum without stabilization, with Baumgarte stabilization, and with projection. Compare position drift, velocity drift, and energy error. Hint: use identical time steps and nonlinear tolerances.
- Perform a three-level refinement study for one differential variable and one algebraic variable. Hint: estimate the two observed orders separately, using Order Reduction in DAE Integration of the course notes.
3.3 Advanced
- For an index-2 Hessenberg DAE, derive conditions under which the consistency Jacobian is nonsingular. Hint: use the implicit-function theorem on the differentiated algebraic constraint.
- Compare differential, perturbation, tractability, and strangeness indices for a constant-coefficient regular pencil. Hint: first transform the pencil to Weierstrass form and state the regularity assumptions.
- Design an event restart for a mechanical system that acquires a new constraint at impact. Hint: separate the impulsive velocity projection from the post-event consistent initialization.