Abstractions and Automated Algorithms for Mixed Domain Finite Element Methods

Daversin-Catty, Cécile; Richardson, Chris N.; Ellingsrud, Ada J.; Rognes, Marie E.

doi:10.1145/3471138

Cited by 13 publications

(15 citation statements)

References 33 publications

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“…The reader will benefit from reading different approaches from ours in the work of several laboratories, including those of Mori (55,354,(394)(395)(396)(397)(398)(399)(400)(401)(402)(403)(404), Ellingsrud (405)(406)(407)(408)(409)(410), Sacco (411) and the many groups interested in glymphatics, sampled in (47)(48)(49)(50)(51)(52)(53)(54)(55)(56)(57). While few if any of these papers deal with potassium accumulation in tridomain systems, no doubt their methods could be usefully applied to those issues.…”

Section: Ion Transport We Define the Regionsmentioning

confidence: 99%

Structural Analysis of Fluid Flow in Complex Biological Systems

Eisenberg¹

2022

Preprint

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Biology is about structure. Structures within structures. Organs within animals, tissues within organs, cells within tissues, and molecules, often proteins within cells. The structures are so complex that they can only be described by numbers. No numbers are of more importance than those that describe proteins. The numbers that describe coordinates of its atoms, often determined by Patterson functions (which are inverse Fourier Transforms of intensities) of crystal diffraction. Without these numbers, structural biology would hardly exist. Without numbers, engineering would not exist. Numbers are surely needed by the engineers who produce the x-rays diffracting from atoms of protein crystals. Devices of engineering have function. They are built to implement equations. Engineering devices use structures to implement equations, when power is supplied at the right places, that produces appropriate flows. Flows are the essence of life. Equilibrium means death in most living systems. Flows in biological structures are hard to analyze because we do not know input output equations in advance. Sometimes we do not know their function. Flows, forces, and structures of life (like those of engineering) are related by field equations of conservation laws, partial differential equations, constrained by location and properties of structures. Constraints are boundary conditions located on the complicated domain of biological structure. Dealing with this complexity is simplified if one systematically analyzes structure using the most general field theory known, electricity described by the Maxwell equations, without significant known error. Currents are involved because flows of biology usually involve migration of charges, convection of water and solutes, diffusion of ions that form the plasma of life, and their interactions. Interactions can dominate function. Here I show how a few complex structures can be understood in engineering detail. This approach may be useful in dealing with biological and medical issues in many other cases. In one limited case—the clearance of a toxic waste (potassium ions) from the optic nerve—this approach seems to have succeeded.

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Section: Ion Transport We Define the Regionsmentioning

confidence: 99%

Structural Analysis of Fluid Flow in Complex Biological Systems

Eisenberg¹

2022

Preprint

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“…Using the time discretization given by Equations ( 36)- (37) for the potential and surface elevation time derivatives, we obtain the following fully discrete problem in the time domain: find…”

Section: Time Domainmentioning

confidence: 99%

A monolithic finite element formulation for the hydroelastic analysis of very large floating structures

Colomés

Verdugo

Akkerman

2022

Numerical Meth Engineering

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In this work we present a novel monolithic Finite Element method for the hydroelastic analysis of very large floating structures (VLFS) with arbitrary shapes that is stable, energy conserving, and overcomes the need of an iterative algorithm. The new formulation enables a fully monolithic solution of the linear free-surface flow, described by linear potential flow, coupled with floating thin structures, described by the Euler-Bernoulli beam or Poisson-Kirchhoff plate equations. The formulation presented in this work is general in the sense that solutions can be found in the frequency and time domains, it overcomes the need of using elements with C 1 continuity by employing a continuous/discontinuous Galerkin approach, and it is suitable for finite elements of arbitrary order. We show that the proposed approach can accurately describe the hydroelastic phenomena of VLFS with a variety of tests, including structures with elastic joints, variable bathymetry, and arbitrary structural shapes.

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“…In (10), ρ is the density of the fluid and μ its viscosity, A i = A i (s, t) denotes the cross-section area, while α i = α i (s, t) is a lumped flow parameter that depends on the domain geometry and the choice of axial velocity profile. We also define the (one-dimensional) normal stress induced by q and p as…”

Section: Perivascular Spaces As Topologically 1d-networkmentioning

confidence: 99%

“…A core Waterscales aim was to remedy this situation by designing numerical algorithms and software abstractions that allow for high-level specification and high-performance forward and reverse solution of models with multiscale features. We addressed this goal by designing and introducing mathematical software concepts together with lower-level algorithms for expressing, representing, and solving systems of PDEs coupled across interfaces or subdomains (Figure 8) [10]. These tools enable automated assembly and solution of a wide range of mixed finite element variational formulations, such as, e.g., the finite element spaces and formulations involved in the reduced perivascular flow models (10), interactions across the cell membrane in geometrically resolved models of excitable tissue [16,35] or fluid-structure interfaces [6].…”

Section: Computational Abstractions and Algorithmsmentioning

confidence: 99%

“…We addressed this goal by designing and introducing mathematical software concepts together with lower-level algorithms for expressing, representing, and solving systems of PDEs coupled across interfaces or subdomains (Figure 8) [10]. These tools enable automated assembly and solution of a wide range of mixed finite element variational formulations, such as, e.g., the finite element spaces and formulations involved in the reduced perivascular flow models (10), interactions across the cell membrane in geometrically resolved models of excitable tissue [16,35] or fluid-structure interfaces [6]. All algorithms are publicly and openly available via the FEniCS Project software [2,10].…”

Section: Computational Abstractions and Algorithmsmentioning

confidence: 99%

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