Related papers: Variational Structures in Cochain Projection Based…
In this paper, we continue the construction of variational integrators adapted to contact geometry started in \cite{VBS}, in particular, we introduce a discrete Herglotz Principle and the corresponding discrete Herglotz Equations for a…
A canonical formalism and constraint analysis for discrete systems subject to a variational action principle are devised. The formalism is equivalent to the covariant formulation, encompasses global and local discrete time evolution moves…
We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method…
We introduce a new finite element (FE) discretization framework applicable for covariant split equations. The introduction of additional differential forms (DF) that form pairs with the original ones permits the splitting of the equations…
In this paper we develop, study, and test a Lie group multisymplectic integra- tor for geometrically exact beams based on the covariant Lagrangian formulation. We exploit the multisymplectic character of the integrator to analyze the energy…
We investigate discretization of $H(\mathrm{curl})$ and $H(\mathrm{div})$ in two and three space dimensions by partially discontinuous nodal finite elements, i.e., vector-valued Lagrange finite elements with discontinuity in certain…
Variational integrators are momentum-preserving and symplectic numerical methods used to propagate the evolution of Hamiltonian systems. In this paper, we introduce a new class of variational integrators that achieve fourth-order…
In this paper we explore the nonholonomic Lagrangian setting of mechanical systems in local coordinates on finite-dimensional configuration manifolds. We prove existence and uniqueness of solutions by reducing the basic equations of motion…
We establish a Lagrangian variational framework for general relativistic continuum theories that permits the development of the process of Lagrangian reduction by symmetry in the relativistic context. Starting with a continuum version of…
A low-order nonconforming finite element discretization of a smooth de Rham complex starting from the $H^2$ space in three dimensions is proposed, involving an $H^2$-nonconforming finite element space, a new tangentially continuous…
A mimetic spectral element discretization, utilizing a novel Galerkin projection Hodge star operator, of the macroscopic Maxwell equations in Hamiltonian form is presented. The idea of splitting purely topological and metric dependent…
A fractured poroelastic body is considered where the opening of the fractures is governed by a nonpenetration law while slip is described by a Coulomb-type friction law. This physical model results in a nonlinear variational inequality…
We propose and compare several projection methods applied to variational integrators for degenerate Lagrangian systems, whose Lagrangian is of the form $L = \vartheta(q) \cdot \dot{q} - H(q)$ and thus linear in velocities. While previous…
We investigate discretization strategies for a recently introduced class of energy-based models. The model class encompasses classical port-Hamiltonian systems, generalized gradient flows, and certain systems with algebraic constraints. Our…
In this paper we introduce discrete gradient methods to discretize irreversible port-Hamiltonian systems showing that the main qualitative properties of the continuous system are preserved using this kind discretizations methods.
We develop a geometric framework for the numerical integration of mechanical systems evolving on manifolds. After briefly reviewing classical numerical methods and highlighting their limitations and shortcomings in non-flat (non-Euclidean)…
Interconnected systems are an important class of mathematical models, as they allow for the construction of complex, hierarchical, multiphysics, and multiscale models by the interconnection of simpler subsystems. Lagrange--Dirac mechanical…
Following the discrete embedding formalism, we give a new derivation of the mid-point variational integrators as developed by J.M. Wendlandt and J.E. Marsden by defining an adapted order two discrete differential and integral calculus. This…
Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial…
The dynamical motion of mechanical systems possesses underlying geometric structures, and preserving these structures in numerical integration improves the qualitative accuracy and reduces the long-time error of the simulation. For a single…