Conformal approach to physics simulations for thin curved 3D membranes

authored by: Igor Bogush, Vladimir M. Fomin, Oleksandr V. Dobrovolskiy
Abstract: Three-dimensional nanoarchitectures are widely used across various areas of physics, including spintronics, photonics, and superconductivity. In this regard, thin curved 3D membranes are especially interesting for applications in nano- and optoelectronics, sensorics, and information processing, making physics simulations in complex 3D geometries indispensable for unveiling new physical phenomena and the development of devices. Here, we present a general-purpose approach to physics simulations for thin curved 3D membranes, that allows for performing simulations using finite difference methods instead of meshless methods or methods with irregular meshes. The approach utilizes a numerical conformal mapping of the initial surface to a flat domain and is based on the uniformization theorem stating that any simply-connected Riemann surface is conformally equivalent to an open unit disk, a complex plane, or a Riemann sphere. We reveal that for many physical problems involving the Laplace operator and divergence, a flat-domain formulation of the initial problem only requires a modification of the equations of motion and the boundary conditions by including a conformal factor and the mean/Gaussian curvatures. We demonstrate the method's capabilities for case studies of the Schrödinger equation for a charged particle in static electric and magnetic fields for 3D geometries, including C-shaped and ring-shaped structures, as well as for the time-dependent Ginzburg-Landau equation.
External Organisation(s): Technische Universität Braunschweig
Leibniz Institute for Solid State and Materials Research Dresden (IFW)
Moldova State University
Type: Article
Journal: Computer physics communications
Volume: 315
ISSN: 0010-4655
Publication date: 10.2025
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Hardware and Architecture, General Physics and Astronomy
Electronic version(s): https://doi.org/10.1016/j.cpc.2025.109736 (Access: Open)

BibTeX

@article{f3cd273919584183999edd3c8909fda5,
title = "Conformal approach to physics simulations for thin curved 3D membranes",
abstract = "Three-dimensional nanoarchitectures are widely used across various areas of physics, including spintronics, photonics, and superconductivity. In this regard, thin curved 3D membranes are especially interesting for applications in nano- and optoelectronics, sensorics, and information processing, making physics simulations in complex 3D geometries indispensable for unveiling new physical phenomena and the development of devices. Here, we present a general-purpose approach to physics simulations for thin curved 3D membranes, that allows for performing simulations using finite difference methods instead of meshless methods or methods with irregular meshes. The approach utilizes a numerical conformal mapping of the initial surface to a flat domain and is based on the uniformization theorem stating that any simply-connected Riemann surface is conformally equivalent to an open unit disk, a complex plane, or a Riemann sphere. We reveal that for many physical problems involving the Laplace operator and divergence, a flat-domain formulation of the initial problem only requires a modification of the equations of motion and the boundary conditions by including a conformal factor and the mean/Gaussian curvatures. We demonstrate the method's capabilities for case studies of the Schr{\"o}dinger equation for a charged particle in static electric and magnetic fields for 3D geometries, including C-shaped and ring-shaped structures, as well as for the time-dependent Ginzburg-Landau equation.",
keywords = "3D membranes, Conformal mapping, Quantum mechanics, Superconductivity, Time-dependent Ginzburg-Landau",
author = "Igor Bogush and Fomin, {Vladimir M.} and Dobrovolskiy, {Oleksandr V.}",
note = "Publisher Copyright: {\textcopyright} 2025 The Author(s)",
year = "2025",
month = oct,
doi = "10.1016/j.cpc.2025.109736",
language = "English",
volume = "315",
journal = "Computer physics communications",
issn = "0010-4655",
publisher = "Elsevier BV",
}