[Paper Review] Hamiltonian inclusions with convex dissipation with a view towards applications

Marius Buliga|arXiv (Cornell University)|Oct 8, 2008

Nonlinear Partial Differential Equations23 references7 citations

TL;DR

This paper introduces a generalized Hamiltonian formalism for dynamical systems with convex dissipation, extending classical mechanics by incorporating subdifferential inclusions to model energy dissipation. The key contribution is a unified framework that recovers Mielke's quasistatic rate-independent processes in the limit and enables a new class of dynamical brittle damage models using Ambrosio-Tortorelli-type energy and 1-homogeneous dissipation, yielding a differential inclusion with a maximal damage propagation speed of order $\gamma\sqrt{1+c^2}$.

ABSTRACT

We propose a generalization of hamiltonian mechanics, as a hamiltonian inclusion with convex dissipation function. We obtain a dynamical version of the approach of Mielke to quasistatic rate-independent processes. Then we show that a class of models of dynamical brittle damage can be formulated in this setting.

Motivation & Objective

• To generalize Hamiltonian mechanics by incorporating convex dissipation via subdifferential inclusions, enabling energy-consistent modeling of irreversible processes.
• To reformulate Mielke's quasistatic rate-independent processes in a dynamical context, bridging static and dynamic damage mechanics.
• To develop a new class of dynamical brittle damage models using energy functionals of Ambrosio-Tortorelli type and 1-homogeneous dissipation.
• To derive a differential inclusion for damage evolution that captures non-smooth, rate-independent damage propagation with a finite maximal speed.

Proposed method

• Formulates a generalized Hamiltonian system as a subdifferential inclusion: $ J\dot{z} - D_z H(t,z) \in \partial_{\dot{z}} \mathcal{R}(z,\dot{z}) $, where $ \mathcal{R} $ is a convex dissipation function.
• Uses a Hamiltonian $ H(t,q,p) = T(p) + \mathcal{E}(t,q) $, with $ T $ as the Fenchel conjugate of kinetic energy, and a 1-homogeneous dissipation function to model rate-independent behavior.
• Applies the formalism to a continuum mechanics setting with displacement $ \mathbf{u} $, momentum $ \mathbf{p} $, damage $ d $, and damage driving variable $ y $, governed by coupled evolution equations.
• Derives the momentum balance equation $ \text{div}(\phi(d)\mathbf{S}) + \mathbf{f}(t) = \dot{\mathbf{p}} $ and the damage evolution via $ \dot{d} = b y $, with $ y $ governed by a subdifferential inequality.
• Replaces the subdifferential inequality with an equivalent differential inclusion: $ -\left(\ddot{d} + \gamma^2 d + \gamma c \phi'(d) w(\nabla\mathbf{u}) - \gamma^2 c^2 \Delta d\right) \in \gamma c \, S(\dot{d}) $, where $ S $ is the subdifferential of a convex function.
• Establishes that the system admits a solution satisfying energy balance and kinematic admissibility, with boundary conditions on $ \partial\Omega $ for $ y $ and $ d $.

Experimental results

Research questions

• RQ1 Can a Hamiltonian formalism be extended to include convex dissipation functions to model irreversible, rate-independent processes?
• RQ2 How can Mielke’s quasistatic rate-independent processes be recovered as a limiting case of a dynamical Hamiltonian system with dissipation?
• RQ3 Can a dynamical brittle damage model be formulated using an Ambrosio-Tortorelli-type energy functional and a 1-homogeneous dissipation function?
• RQ4 What is the propagation speed of damage in such a model, and does it remain finite?
• RQ5 Can symplectic integrators be applied to this generalized Hamiltonian system for stable, energy-conserving numerical discretization?

Key findings

• The proposed Hamiltonian inclusion with convex dissipation generalizes classical Hamiltonian mechanics and recovers Mielke’s quasistatic rate-independent processes in the limit of vanishing inertia.
• The model for brittle damage is derived from an Ambrosio-Tortorelli-type energy functional and a 1-homogeneous dissipation function, leading to a subdifferential inclusion for damage evolution.
• The damage evolution is governed by $ \dot{d} = b y $, with $ y $ satisfying a subdifferential inequality that leads to the differential inclusion $ -\left(\ddot{d} + \gamma^2 d + \gamma c \phi'(d) w(\nabla\mathbf{u}) - \gamma^2 c^2 \Delta d\right) \in \gamma c \, S(\dot{d}) $.
• The system exhibits a finite maximal damage propagation speed of order $ \gamma\sqrt{1 + c^2} $, derived from the structure of the differential inclusion and the choice of parameters.
• The momentum balance equation $ \text{div}(\phi(d)\mathbf{S}) + \mathbf{f}(t) = \dot{\mathbf{p}} $ is recovered, with stress $ \mathbf{S} = Dw(\nabla\mathbf{u}) $, and boundary conditions are naturally enforced.
• The formalism allows for symplectic discretization, suggesting potential for long-term energy stability in numerical simulations.