Novel solver for dynamic surfaces

Physics-based modeling integrates dynamics and geometry. The standard methods to solve the Lagrangian equations use a direct approach in the spatial domain. Though extremely powerful, it requires time consuming discrete-time integration. In this paper, we propose to use an indirect approach using the Transformation Theory. In particular, we use z-transform from the digital signal processing theory, and formulate a general, novel, unified solver that is applicable for various models and behavior. The convergence and accuracy of the solver are guaranteed if the temporal sampling period is less than the critical sampling period, which is a function of the physical properties of the model. Our solver can seamlessly handle curves, surfaces and solids, and supports a wide range of dynamic behavior. The solver does not depend on the topology of the model, and hence supports non-manifold and arbitrary topology. Our numerical techniques are simple, easy to use, stable, and efficient. We develop an algorithm and a prototype software simulating various models and behavior. Our solver preserves physical properties such as energy, linear momentum, and angular momentum. This approach will serve as a foundation for many applications in many fields.