An interpolatory directional splitting-local discontinuous Galerkin method with application to pattern formation in 2D/3D

Castillo, Paul; Gómez, Sergio

doi:10.1016/j.amc.2021.125984

Cited by 4 publications

(3 citation statements)

References 33 publications

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“…where τ = t n+1 − t n and U n h is the vector approximation of u h (•, t n ). The efficiency of combining some discontinuous Galerkin methods with an interpolatory approximation of the nonlinear term as spatial discretization on classical meshes and the SS-OS time marching scheme was assessed by Castillo and Gómez in [12,13].…”

Section: Fully Discrete Schemementioning

confidence: 99%

High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems

Gómez¹

2021

Preprint

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We propose an efficient method for the numerical solution of a general class of two dimensional semilinear parabolic problems on polygonal meshes. The proposed approach takes advantage of the properties of the serendipity version of the Virtual Element Method (VEM), which not only significantly reduces the number of degrees of freedom compared to the classical VEM but also, under certain conditions on the mesh allows to approximate the nonlinear term with an interpolant in the Serendipity VEM space; which substantially improves the efficiency of the method. An error analysis for the semi discrete formulation is carried out, and an optimal estimate for the error in the L 2 -norm is obtained. The accuracy and efficiency of the proposed method when combined with a second order Strang operator splitting time discretization is illustrated in our numerical experiments, with high order approximations up to order 6.

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Section: Fully Discrete Schemementioning

confidence: 99%

High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems

Gómez¹

2021

Preprint

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“…Nature is replete of porous structures with unique curvature topologies-from the simplest example of soap films with constant mean curvatures (Plateau's law [1,2] ) to the morphogenesisdriven Turing patterns [3,4] in animal skin pigmentation; from biological systems such as trabecular bone [5][6][7] and vascular networks [8,9] to non-biological systems like porous ceramics, nanoporous gold, [10] and block copolymers [11] (see Figure 1). Driven by complex relaxation dynamics and nonequilibrium phenomena such as self-assembly, [12][13][14] pattern formation, [15][16][17] and phase ordering kinetics, [18,19] both understanding and tuning such physics behind the natural emergence of complex curvature topologies opens up new avenues for advances in materials engineering.…”

Section: Introductionmentioning

confidence: 99%

Inverse Designing Surface Curvatures by Deep Learning

Guo,

Sharma,

Kumar

2024

Advanced Intelligent Systems

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Smooth and curved microstructural topologies found in nature—from soap films to trabecular bone—have inspired several mimetic design spaces for architected metamaterials and bio‐scaffolds. However, the design approaches so far are ad hoc, raising the challenge: how to systematically and efficiently inverse design such artificial microstructures with targeted topological features? Herein, surface curvature is explored as a design modality and a deep learning framework is presented to produce topologies with as‐desired curvature profiles. The inverse design framework can generalize to diverse topological features such as tubular, membranous, and particulate features. Moreover, successful generalization beyond both the design and data space is demonstrated by inverse designing topologies that mimic the curvature profile of trabecular bone, spinodoid topologies, and periodic nodal surfaces for application in bio‐scaffolds and implants. Lastly, curvature and mechanics are bridged by showing how topological curvature can be designed to promote mechanically beneficial stretching‐dominated deformation over bending‐dominated deformation.

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“…Oruç [20] developed an efficient wavelet collocation method for the nonlinear 2D Fisher-KPP equation and extended Fisher-Kolmogorov equation. Castillo and Gómez [21] used the local discontinuous Galerkin method with the symmetric Strang operator splitting scheme to solve Fisher-KPP. Kabir [22] considered a model with travelling wave solutions with a qualitatively similar structure to that observed in the Fisher-KPP equation and performed numerical simulations.…”

Section: Introductionmentioning

confidence: 99%

An Unconditionally Stable Positivity-Preserving Scheme for the One-Dimensional Fisher–Kolmogorov–Petrovsky–Piskunov Equation

Kim

Lee

et al. 2021

Discrete Dynamics in Nature and Society

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In this study, we present an unconditionally stable positivity-preserving numerical method for the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation in the one-dimensional space. The Fisher–KPP equation is a reaction-diffusion system that can be used to model population growth and wave propagation. The proposed method is based on the operator splitting method and an interpolation method. We perform several characteristic numerical experiments. The computational results demonstrate the unconditional stability, boundedness, and positivity-preserving properties of the proposed scheme.

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An interpolatory directional splitting-local discontinuous Galerkin method with application to pattern formation in 2D/3D

Cited by 4 publications

References 33 publications

High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems

High-order interpolatory Serendipity Virtual Element Method for semilinear parabolic problems

Inverse Designing Surface Curvatures by Deep Learning

An Unconditionally Stable Positivity-Preserving Scheme for the One-Dimensional Fisher–Kolmogorov–Petrovsky–Piskunov Equation

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