3D-1D coupling on non conforming meshes via a three-field optimization based domain decomposition

Berrone, Stefano; Grappein, Denise; Scialò, Stefano

doi:10.1016/j.jcp.2021.110738

Cited by 11 publications

(14 citation statements)

References 29 publications

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“…The behaviour here observed is quite different from the one observed in Ref. [1], where a larger effect of the mesh parameters on the conditioning was instead observed. Further, system (39) is known to be even better conditioned than the corresponding system (37), [18].…”

Section: Test Problem 1 (Tp1)contrasting

confidence: 99%

“…Different coupling conditions could be considered, for example adding a pressure continuity constraint and consequently not linking the flux definition to the pressure jump, as done in Ref. [1]. The choice of the interface condition depends of course on the properties of the interface, and thus on the kind of application.…”

Section: Notation and Formulation Of The Fully 3d Coupled Problemmentioning

confidence: 99%

“…Here an extension of the approach described in Ref. [1] for 3D-1D coupled elliptic problems is proposed, allowing to deal with discontinuous solution at the interfaces. This is relevant for a large variety of practical applications, as e.g.…”

Section: Introductionmentioning

confidence: 99%

“…The original work in Ref. [1] deals with continuous solution at the boundary between the 3D and the 1D domains. Here, while keeping the same structure of this original method, different interface variables are introduced for the domain decomposition process, which result in a novel setting for the PDE constrained optimization problem described.…”

Section: Introductionmentioning

confidence: 99%

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A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

Berrone¹,

Grappein²,

Scialò³

2022

Preprint

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A numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined function spaces a well posed 3D-1D problem is obtained from the original fully 3D problem and the solution is then found by a PDE-constrained optimization reformulation. This is a domain decomposition strategy in which unknown interface variables are introduced and a suitably defined cost functional, expressing the error in fulfilling interface conditions, is minimized constrained by the constitutive equations on the subdomains. The resulting discrete problem is robust with respect to geometrical complexity thanks to the use of independent discretizations on the various subdomains. Meshes of different sizes can be used without affecting the conditioning of the discrete linear system, and this is a peculiar aspect of the considered formulation. An efficient resolution strategy is further proposed, based on the use of a gradient based solver and yielding a method ready for parallel implementation. A numerical experiment on a problem with known analytical solution shows the accuracy of the method, and two examples on more complex configurations are proposed to address the applicability of the approach to practical problems.

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Section: Test Problem 1 (Tp1)contrasting

confidence: 99%

Section: Notation and Formulation Of The Fully 3d Coupled Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

Berrone¹,

Grappein²,

Scialò³

2022

Preprint

Self Cite

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show abstract

“…Coupled partial differential equation problems on 3D and 1D domains arise from the application of dimensional reduction models to equi-dimensional problems where cylindrical or nearlycylindrical inclusions with small cross sectional size are embedded in a larger 3D domain, [11,13,1]. Indeed, the treatment of such narrow and elongated regions as one-dimensional manifolds reduces the overhead in simulations related to the generation of a computational mesh inside the inclusions.…”

Section: Introductionmentioning

confidence: 99%

Extended Finite Elements for 3D-1D coupled problems via a PDE-constrained optimization approach

Grappein¹,

Scialò²,

Vicini³

2022

Preprint

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In this work we propose the application of the eXtended Finite Element Method (XFEM) to the simulation of coupled elliptic problems on 3D and 1D domains, arising from the application of geometrical reduction models to fully three dimensional problems with cylindrical, or nearly cylindrical, inclusions with small radius. In this context, the use of non conforming meshes is widely adopted, even if, due to the presence of singular source terms for the 3D problems, mesh adaptation near the embedded 1D domains may be necessary to improve solution accuracy and to recover optimal convergence rates. The XFEM are used here as an alternative to mesh adaptation. The choice of a suitable enrichment function is presented and an effective quadrature strategy is proposed. Numerical tests show the effectiveness of the approach.

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A mixed‐dimensional model for direct current simulations in the presence of a thin high‐resistivity liner

Fumagalli,

Panzeri,

Formaggia

et al. 2023

Numerical Meth Engineering

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SummaryIn this work, we present a mixed‐dimensional mathematical model to obtain the electric potential and current density in direct current simulations when a thin liner is included in the modelled domain. The liner is used in landfill management to prevent leakage of leachate from the waste body into the underground and is made of a highly‐impermeable high‐resistivity plastic material. The electrodes and the liner have diameters and thickness, respectively, that are much smaller than their other dimensions, thus their numerical simulation might be too costly in an equi‐dimensional setting. Our approach is to approximate them as objects of lower dimension and derive the corresponding equations. The obtained mixed‐dimensional model is validated against laboratory experiments of increasing complexity to show the reliability of the proposed mathematical model. Our tests also show that configurations with current and voltage electrodes on either sides of the liner confining the landfill may be effective in detecting damages of the membrane.

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3D-1D coupling on non conforming meshes via a three-field optimization based domain decomposition

Cited by 11 publications

References 29 publications

A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

Extended Finite Elements for 3D-1D coupled problems via a PDE-constrained optimization approach

A mixed‐dimensional model for direct current simulations in the presence of a thin high‐resistivity liner

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