Abstract
ABSTRACT We investigate an energy-based formulation of the two-field poroelasticity model and the related multiple-network model as they appear in geosciences or medical applications. We propose a port-Hamiltonian formulation of the system equations, which is beneficial for preserving important system properties after discretization or model-order reduction. For this, we include the commonly omitted second-order term and consider the corresponding first-order formulation. The port-Hamiltonian formulation of the quasi-static case is then obtained by (formally) setting the second-order term zero. Further, we interpret the poroelastic equations as an interconnection of a network of submodels with internal energies, adding a control-theoretic understanding of the poroelastic equations.
Highlights
The port-Hamiltonian framework constitutes an energy-based model paradigm that offers a systematic approach for the interactions of systems with each other and with the environment and extends Hamiltonian systems to open physical systems
Afterwards, we show that the poroelastic equations can be formulated in a corresponding way, leading to a portHamiltonian partial differential-algebraic equation
We start with the definition of a pH-DAE in the finite-dimensional framework, which we will mimic in the pH-partial differential-algebraic equation (PDAE) setting
Summary
The port-Hamiltonian (pH) framework constitutes an energy-based model paradigm that offers a systematic approach for the interactions of (physical) systems with each other and with the environment and extends Hamiltonian systems to open physical systems. The definition of pH systems was extended to cover implicit systems [13,14,15,16], resulting in so-called port-Hamiltonian differential-algebraic equations (pHDAEs) This system class covers a wide range of equations and simplifies the mathematical theory in many aspects, almost all results are obtained for finite-dimensional DAEs. Extensions of the pH framework to (constrained) infinite-dimensional systems typi cally do not exist in a general form, but rather consider particular applications or special model classes [4, 17,18,19,20,21,22,23]. The full strength of the pH approach, becomes visible in the network case, including multiple pressure variables Such models typically appear in medical applica tions, such as cerebral infusion tests [36,37] or the investigation of cerebral oedema [38].
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