A mass-, kinetic energy- and helicity-conserving mimetic dual-field discretization for three-dimensional incompressible Navier-Stokes equations, part I: Periodic domains

Zhang, Yi; Palha, Artur; Gerritsma, Marc; Rebholz, Leo G.

doi:10.1016/j.jcp.2021.110868

Cited by 6 publications

(11 citation statements)

References 83 publications

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“…We now present the weak dual field formulation of the port-Hamiltonian model, inspired by the work of [29]. In this formulation, we combine together the primal and adjoint port-Hamiltonian systems described in the previous section.…”

Section: Weak Dual Field Formulationmentioning

confidence: 99%

“…By introducing the energy H T , it is possible to eliminate the need for a discrete Hodge star, at least in the linear case. However, the strength of the approach is not limited to the linear case: in [29], the construction of the adjoint system allows dealing with the non linear gyroscopic term in the Navier-Stokes equation in a linear fashion, by relying on the vorticity information from the dual system.…”

Section: Constitutive Equationsmentioning

confidence: 99%

“…One technique consists of strong imposition by direct assignment of the boundary degrees of freedom. Another interesting possibility consists of using a time staggering discretization, as proposed in [29], together with a weak imposition of the boundary conditions by means of the adjoint port-Hamiltonian systems. By time staggering discretization, we indicate the fact the solution of one system is computed at integer time instants, whereas for the second system, arising from the dual field formulation, the solution is computed at half-integer time instants.…”

Section: Boundary Conditions Enforcementmentioning

confidence: 99%

“…However, the discretization of constitutive equations remain problematic as those require a discrete Hodge star. Recently, a dual field formulation has been introduced to discretize the Navier-Stokes [29]. This methodology seeks variables in dual polynomial finite element spaces via a mixed finite element formulation and is capable of conserving mass, helicity and energy.…”

Section: Introductionmentioning

confidence: 99%

“…In this contribution, we show how the dual field formulation can be used for the systematic discretization of linear pH systems (but the framework is extendable to the state modulated case where one has a Poisson manifold, as already done in [29]). To show the generality of the approach, we detail the calculations in exterior calculus.…”

Section: Introductionmentioning

confidence: 99%

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Dual field structure-preserving discretization of port-Hamiltonian systems using finite element exterior calculus

Brugnoli¹,

Rashad²

2022

Preprint

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In this paper we propose a novel approach to discretize linear port-Hamiltonian systems while preserving the underlying structure. We present a finite element exterior calculus formulation that is able to mimetically represent conservation laws and cope with mixed open boundary conditions using a single computational mesh. The possibility of including open boundary conditions allows for modular composition of complex multi-physical systems whereas the exterior calculus formulation provides a coordinate-free treatment. Our approach relies on a dual-field representation of the physical system that is redundant at the continuous level but eliminates the need of mimicking the Hodge star operator at the discrete level. By considering the Stokes-Dirac structure representing the system together with its adjoint, which embeds the metric information directly in the codifferential, the need for an explicit discrete Hodge star is avoided altogether. By imposing the boundary conditions in a strong manner, the power balance characterizing the Stokes-Dirac structure is then retrieved at the discrete level via symplectic Runge-Kutta integrators based on Gauss-Legendre collocation points. Numerical experiments validate the convergence of the method and the conservation properties in terms of energy balance both for the wave and Maxwell equations in a three dimensional domain. For the latter example, the magnetic and electric fields preserve their divergence free nature at the discrete level.

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Section: Weak Dual Field Formulationmentioning

confidence: 99%

Section: Constitutive Equationsmentioning

confidence: 99%

Section: Boundary Conditions Enforcementmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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