08:30
MS25: Discretizations of mixed-dimensional PDEs (Part 1)
Chair: Marie Rognes
08:30
25 mins
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A multi-layer reduced model for flow in porous media
Alessio Fumagalli, Anna Scotti
Abstract: In this work we present a new conceptual model to describe fluid flow in porous media in the presence of thin juxtaposed layers [1]. For instance,
geological faults are characterized by a high strain core surrounded on both sides by the so called damage zones, where a large number of smaller fractures enhance the permeability of the medium. Layered thin porous media are of interest also in the case of industrial applications. To perform efficient but accurate simulations in these domains we propose a hybrid dimensional model where objects of co-dimension 1 are coupled with the surrounding bulk medium and among themselves, accounting for fluid exchange with a generalized (mixed-dimensional) definition of the divergence operator [2].
The numerical discretization is based, in every dimension, on the classical Raviart-Thomas-Nedelec approximation for the flux and a constant piecewise representation of the pressure to obtain local mass conservation in the view of a coupling with a transport problem. The implementation is based on the library PorePy[3], a tool for fractured and deformable porous media The model is presented, analyzed, and tested in several configurations to prove its robustness in 2D and 3D, and in the presence of high contrast and heterogeneity of permeability.
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08:55
25 mins
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3d structure - 2d plate interaction model
Josip Tambaca, Matko Ljulj
Abstract: In this work we analyze a model of interaction of a linearized three-dimensional elastic structure with a thin elastic layer of possibly different material attached to it. By dimension reduction techniques we rigorously derive interaction model in five different regimes depending of the order of stiffness of the thin layer. Then we propose 3d structure - 2d plate model with the same asymptotics as the original 3d problem. In this way one does not need to identify a particular regime in advance. We then discuss possible discretizations of the 3d structure - 2d plate model. The work is motivated by the interaction of different blood vessel layers.
This work has been supported by the grant HRZZ 2735 of the Croatian Science Foundation.
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09:20
25 mins
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Mixed-dimensional coupled finite elements in FEniCS
Cécile Daversin-Catty, Marie Rognes
Abstract: Mixed-dimensional partial differential equations (PDEs) are equations coupling unknown fields defined over domains of differing topological dimension. Such mixed-dimensional PDEs naturally arise in a wide
range of fields including geology, bio-medicine, and fracture mechanics. Mixed-dimensional models can also be used to impose non-standard conditions over a subspace of lower dimension such as a part of the boundary or an interface between two domains, through a Lagrange multiplier.
Finite element discretizations of mixed-dimensional PDEs involve nested meshes of heterogeneous topological dimension. The assembly of such systems is non-standard and non-trivial especially with regard to the terms involved in the interactions between the different domains. In other words, automated solution of mixed-dimensional PDEs requires the design of both generic high level software abstractions
and lower level algorithms.
The FEniCS project aims at automating the numerical solution of mathematical models based on PDEs using finite element methods, and is organized as an open source collection of software components.
A core feature is a high-level domain-specific language for finite element spaces and variational forms close to mathematical syntax. While external FEniCS-based packages have been developed for simulating mixed-dimensional problems, there is an important need for embedding these features as an intrinsic part of the FEniCS library. We introduce an automated framework dedicated to mixed-dimensional problems
addressing this gap, which has already been used with diverse applications ranging from reservoir simulations to ion concentration dynamics models.
This talk gives an overview of the abstractions and algorithms involved. The introduced tools will be illustrated by concrete examples of applications highlighting their relevance for model in biomedicine in
general, with a focus on circulation, flow and exchange of tissue fluid in the brain.
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09:45
25 mins
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Assembly of multiscale linear PDE operators
Miroslav Kuchta
Abstract: In numerous applications the mathematical model consists of different
processes coupled across a lower dimensional manifold. Due to the multiscale cou-
pling, finite element discretization of such models presents a challenge. Assuming
that only singlescale finite element forms can be assembled we present here a sim-
ple algorithm for representing multiscale models as linear operators suitable for
Krylov methods. Flexibility of the approach is demonstrated by numerical examples
with coupling across dimensionality gap 1 and 2. Preconditioners for several of the
problems are discussed.
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