Energy-Consistent Integration of Mechanical Systems with Singular or Configuration-Dependent Mass Matrices using Livens Principle

Core Concepts

A novel energy-consistent time-stepping scheme is proposed for the numerical integration of mechanical systems with singular or configuration-dependent mass matrices, based on the Livens principle.

Abstract

The content presents a novel approach for the numerical integration of mechanical systems with singular or configuration-dependent mass matrices, based on the Livens principle. The key highlights are: Livens principle is introduced as a unifying framework that combines Lagrangian and Hamiltonian viewpoints, avoiding the need to invert the mass matrix. The Livens principle is extended to handle holonomically constrained mechanical systems, where the mass matrix can be singular. A structure-preserving time-stepping scheme is developed by directly discretizing the Euler-Lagrange equations from the Livens principle, using Gonzalez discrete gradients. The proposed integrator discretely conserves a generalized energy function, which differs from the total energy if the mass matrix is configuration-dependent. The method is demonstrated on two representative examples: a mass-spring system with a singular mass matrix, and a nonlinear spring pendulum in spherical coordinates. The numerical results show that the proposed scheme exactly satisfies the holonomic constraints and preserves the generalized energy function up to machine precision.

Stats

The mass matrix M(q) can be singular, and may depend on the configuration q. The total energy of the system is given by Etot(q,v) = T(q,v) + V(q).

Quotes

"Livens principle allows for an advantageous universal description due to its mixed character." "The novel scheme algorithmically conserves a general energy function and aims at the preservation of momentum maps corresponding to symmetries of the system."

Key Insights Distilled From

Energy-consistent integration of mechanical systems based on Livens principle

by Philipp L. K... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2312.02825.pdf

Deeper Inquiries

How can the proposed framework be extended to handle nonholonomic constraints or more general types of constraints?

The proposed framework based on Livens principle can be extended to handle nonholonomic constraints by incorporating the concept of nonintegrable constraints. Nonholonomic constraints are constraints that cannot be derived from a potential function, unlike holonomic constraints. These constraints involve velocity-dependent restrictions on the system's motion, making them more complex to handle. To address nonholonomic constraints within the Livens principle framework, one approach is to introduce Lagrange multipliers associated with the nonholonomic constraints. By incorporating these multipliers into the augmented action integral, the resulting Euler-Lagrange equations will include terms related to the nonholonomic constraints. This extension allows for the inclusion of nonholonomic constraints in the system's dynamics and ensures that the constraints are satisfied throughout the simulation. Furthermore, for more general types of constraints beyond holonomic and nonholonomic constraints, the framework can be adapted to accommodate them by formulating appropriate constraint equations and incorporating them into the variational principle. The key lies in defining the constraints effectively within the action integral and deriving the corresponding Euler-Lagrange equations to ensure that the constraints are enforced in the system's dynamics.

What are the potential limitations or drawbacks of the Livens principle-based approach compared to other structure-preserving integration methods?

While the Livens principle-based approach offers advantages such as avoiding the inversion of mass matrices and providing a unified framework for Lagrangian and Hamiltonian mechanics, it also has some limitations and drawbacks compared to other structure-preserving integration methods. One limitation is the complexity of handling nonholonomic constraints within the Livens principle framework. Nonholonomic constraints introduce additional challenges in formulating the augmented action integral and deriving the corresponding equations of motion. This complexity can make the implementation and numerical integration of systems with nonholonomic constraints more challenging compared to other methods specifically designed for such constraints. Another drawback is the potential computational cost associated with the implicit nature of the equations derived from the Livens principle. The need for iterative methods like Newton's method to solve the implicit equations in each time step can lead to increased computational overhead, especially for large-scale systems or systems with intricate constraints. Additionally, the requirement for a consistent and accurate discretization scheme to preserve the energy and momentum properties of the system can pose challenges in practice. Ensuring that the discrete integrator accurately captures the conservation laws of the system while maintaining numerical stability can be a non-trivial task, especially for complex systems.

Can the ideas presented in this work be applied to the development of higher-order variational integrators for mechanical systems with singular or configuration-dependent mass matrices?

Yes, the ideas presented in this work can be applied to the development of higher-order variational integrators for mechanical systems with singular or configuration-dependent mass matrices. By building upon the foundation of the Livens principle-based approach and the structure-preserving integration scheme proposed in the study, it is possible to extend these concepts to higher-order variational integrators. One approach to developing higher-order variational integrators for systems with singular or configuration-dependent mass matrices is to enhance the discretization scheme used in the proposed framework. By incorporating higher-order discrete derivatives or integrating more sophisticated numerical techniques, such as implicit-explicit methods or symplectic integrators, the accuracy and stability of the integrator can be improved. Furthermore, the principles of energy conservation and momentum preservation inherent in the Livens principle-based approach can be leveraged in the design of higher-order variational integrators. By ensuring that the discrete integrator accurately captures the system's energy and momentum properties at a higher order of accuracy, it is possible to develop advanced numerical methods that maintain the structure-preserving properties of the system while accounting for singular or configuration-dependent mass matrices. In conclusion, the ideas presented in this work provide a solid foundation for the development of higher-order variational integrators tailored to handle the complexities of mechanical systems with singular or configuration-dependent mass matrices.