Conformal approach to physics simulations for thin curved 3D membranes

TL;DR

This paper introduces a conformal mapping-based numerical method for physics simulations on thin curved 3D membranes, enabling finite difference approaches in complex geometries relevant to nano- and optoelectronics.

Contribution

It presents a novel conformal mapping technique that simplifies physics simulations on curved membranes by transforming them into flat domains, improving computational efficiency and accuracy.

Findings

Successfully applied to Schrödinger and Ginzburg-Landau equations

Handled complex geometries like C-shaped and ring-shaped structures

Enhanced simulation accuracy with conformal mapping approach

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.