A new tailored finite point method for strongly anisotropic diffusion equation on misaligned grids

Yang, Tinggan; Wang, Yihong

doi:10.1016/j.amc.2019.02.077

Cited by 5 publications

(2 citation statements)

References 31 publications

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“…Considerable literature has been devoted to solving anisotropic diffusion problems [36]. Diffusion over a surface is a particular case of strongly anisotropic diffusion with a zero-diffusion coefficient in the direction normal to the surface.…”

Section: Data Availability Statementmentioning

confidence: 99%

A self-consistent hybrid model of kinetic striations in low-current argon discharges

Kolobov¹,

Guzman²,

Arslanbekov³

2022

Plasma Sources Sci. Technol.

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A self-consistent hybrid model of standing and moving striations was developed for low-current DC discharges in noble gases. We introduced the concept of surface diffusion in phase space (r,u) (where u denotes the electron kinetic energy) described by a tensor diffusion in the nonlocal Fokker-Planck kinetic equation for electrons in the collisional plasma. Electrons diffuse along surfaces of constant total energy ε=u-eφ(r) between energy jumps in inelastic collisions with atoms. Numerical solutions of the 1d1u kinetic equation for electrons were obtained by two methods and coupled to ion transport and Poisson solver. We studied the dynamics of striation formation in Townsend and glow discharges in Argon gas at low discharge currents using a two-level excitation-ionization model and a “full-chemistry” model, which includes stepwise and Penning ionization. Standing striations appeared in Townsend and glow discharges at low currents, and moving striations were obtained for the discharge currents exceeding a critical value. These waves originate at the anode and propagate towards the cathode. We have seen two types of moving striations with the 2-level and full-chemistry models, which resemble the s and p striations previously observed in the experiments. Simulations indicate that processes in the anode region could control moving striations in the positive column plasma. The developed model helps clarify the nature of standing and moving striations in DC discharges of noble gases at low discharge currents and low gas pressures.

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Section: Data Availability Statementmentioning

confidence: 99%

A self-consistent hybrid model of kinetic striations in low-current argon discharges

Kolobov¹,

Guzman²,

Arslanbekov³

2022

Plasma Sources Sci. Technol.

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show abstract

“…2010)). It's very noticeable for extremely anisotropic cases such as in plasmas (Batischev et al (1999); Günter et al (2005)) and lots of methods have been proposed to address this issue in these specific cases (Soler et al (2020); van Es et al (2016); Tang and Wang (2017); Yang and Wang (2019)). Although it is rarely mentioned, such inaccuracy also impacts anisotropic Reaction/Diffusion systems, and the non-linear term describing the chemical reaction even amplifies this effect.…”

Section: Anisotropic Growthmentioning

confidence: 99%

Growth of oriented orthotropic structures with reaction/diffusion

Garnier

Schmidt

Rohmer

2022

Struct Multidisc Optim

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Lattice structures can present advantageous mechanical properties while remaining remarkably lightweight. Precise lattice design can however be tricky to set up with classical 3D modeling methods as it involves very fine details. Interestingly, natural porous structures can present such latticelike or membrane-like features which motivates to seek for more bio-inspired approaches to microstructure design. In this paper we present a novel method to grow lattice-like and membrane-like structures within an arbitrary shape and aligned along an oriented field. Our method relies on the use of a dedicated anisotropic Reaction-Diffusion system guided by an orthotropic diffusion tensor field. Assuming for instance the diffusion tensor to be related to the stress analysis of a given shape allows to generate emerging stripes patterns aligned along each one of the principal stress directions independently. A globally coherent mechanical model conforming to the initial shape boundary and infilled with oriented microstructures can therefore be synthesized. Further, we demonstrate the capability of this approach to handle other types of oriented fields such as obtained through optimization of material directions in scenarios with multiple load-cases. Our approach relies on spatially and temporally local operations allowing for efficient parallelization. This permits user-interaction and automated adaptation of the design, even for fine meshes over large volumes. For instance, a designer can locally erase or "draw" over the structure and let it regrow and adapt as well as enforce regions to be deliberately full or empty. The proposed approach yields smooth and conformal oriented anisotropic geometrical patterns. This is related to recent effort in the Structural Optimization community on the topic of optimized oriented infills and microstructure de-homogenization. One of the resulting designs is validated by means of a full scale general nonlinear analysis showcasing the advantageous properties of oriented microstructures for stability and robustness to buckling.

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Tailored finite point method for time fractional convection dominated diffusion problems with boundary layers

Wang

Cao

2020

Math Methods in App Sciences

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We propose a tailored finite point method (TFPM) for solving time fractional convection dominated diffusion equations in this paper. The main idea of TFPM is to firstly approximate the diffusion, convection coefficient near each grid by a constant, and then determine the weights of the finite difference scheme by using the exact solution of the convection diffusion equation with constant coefficients. This adaptation perfectly captures the rapid transition of the solutions which contain sharp boundary layers even with coarse meshes. The accuracy and stability of the scheme are rigorously analyzed. Numerical examples are shown to verify the accuracy and reliability of the proposed scheme.

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A new tailored finite point method for strongly anisotropic diffusion equation on misaligned grids

Cited by 5 publications

References 31 publications

A self-consistent hybrid model of kinetic striations in low-current argon discharges

A self-consistent hybrid model of kinetic striations in low-current argon discharges

Growth of oriented orthotropic structures with reaction/diffusion

Tailored finite point method for time fractional convection dominated diffusion problems with boundary layers

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