26 research outputs found
On the port-Hamiltonian structure of the Navier-Stokes equations for reactive flows
We consider the problem of finding an energy-based formulation of the Navier-Stokes equations for reactive flows. These equations occur in various applications, e. g., in combustion engines or chemical reactors. After modeling, discretization, and model reduction, important system properties as the energy conservation are usually lost which may lead to unphysical simulation results. In this paper we introduce a port-Hamiltonian formulation of the one-dimensional Navier-Stokes equations for reactive flows. The port-Hamiltonian structure is directly associated with an energy balance, which ensures that a temporal change of the total energy is only due to energy flows through the boundary.DFG, SFB 1029, Substantial efficiency increase in gas turbines through direct use of coupled unsteady combustion and flow dynamic
Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems
When computing the eigenstructure of matrix pencils associated with the
passivity analysis of perturbed port-Hamiltonian descriptor system using a
structured generalized eigenvalue method, one should make sure that the
computed spectrum satisfies the symmetries that corresponds to this structure
and the underlying physical system. We perform a backward error analysis and
show that for matrix pencils associated with port-Hamiltonian descriptor
systems and a given computed eigenstructure with the correct symmetry structure
there always exists a nearby port-Hamiltonian descriptor system with exactly
that eigenstructure. We also derive bounds for how near this system is and show
that the stability radius of the system plays a role in that bound
Structure-Preserving Model-Reduction of Dissipative Hamiltonian Systems
Reduced basis methods are popular for approximately solving large and complex
systems of differential equations. However, conventional reduced basis methods
do not generally preserve conservation laws and symmetries of the full order
model. Here, we present an approach for reduced model construction, that
preserves the symplectic symmetry of dissipative Hamiltonian systems. The
method constructs a closed reduced Hamiltonian system by coupling the full
model with a canonical heat bath. This allows the reduced system to be
integrated with a symplectic integrator, resulting in a correct dissipation of
energy, preservation of the total energy and, ultimately, in the stability of
the solution. Accuracy and stability of the method are illustrated through the
numerical simulation of the dissipative wave equation and a port-Hamiltonian
model of an electric circuit
Symplectic Model Reduction of Hamiltonian Systems
In this paper, a symplectic model reduction technique, proper symplectic
decomposition (PSD) with symplectic Galerkin projection, is proposed to save
the computational cost for the simplification of large-scale Hamiltonian
systems while preserving the symplectic structure. As an analogy to the
classical proper orthogonal decomposition (POD)-Galerkin approach, PSD is
designed to build a symplectic subspace to fit empirical data, while the
symplectic Galerkin projection constructs a reduced Hamiltonian system on the
symplectic subspace. For practical use, we introduce three algorithms for PSD,
which are based upon: the cotangent lift, complex singular value decomposition,
and nonlinear programming. The proposed technique has been proven to preserve
system energy and stability. Moreover, PSD can be combined with the discrete
empirical interpolation method to reduce the computational cost for nonlinear
Hamiltonian systems. Owing to these properties, the proposed technique is
better suited than the classical POD-Galerkin approach for model reduction of
Hamiltonian systems, especially when long-time integration is required. The
stability, accuracy, and efficiency of the proposed technique are illustrated
through numerical simulations of linear and nonlinear wave equations.Comment: 25 pages, 13 figure