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Results for “conformal maps”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2020

Dissolution of a cylindrical disk in Hele-Shaw flow: a conformal-mapping approach

Here we apply conformal mapping to find the evolving shapes of a dissolving cylinder in a potential flow. Similar equations can be used to describe melting in a flowing liquid phase. Results are compared with microfluidic experiments and numerical simulations. Shapes predicted by conformal mapping agree almost perfectly with experimental observations, after a modest (20 %) rescaling of the time. Finite-volume simulations show that the differences with experiment are connected to the underlying assumptions of the analytical model: potential flow and diffusion-limited dissolution. Approximate solutions of the equations describing the evolution of the shape of the undissolved solid can be derived from a Laurent expansion of the mapping function from the unit circle. Asymptotic expressions for the evolution of the area of the disk and the shift in its centre of mass have been derived at low and high Péclet number. Analytic approximations to the leading-order Laurent coefficients provide additional insight into the mechanisms underlying pore-scale dissolution.

58 GEOSCIENCES↗
DOE OSTI2022

Noise effects on Padé approximants and conformal maps

Here, we analyze the properties of Padé and conformal map approximants for functions with branch points, in the situation where the expansion coefficients are only known with finite precision or are subject to noise. We prove that there is a universal scaling relation between the strength of the noise and the expansion order at which Padé or the conformal map breaks down. We illustrate this behavior with some physically relevant model test functions and with two non-trivial physical examples where the relevant Riemann surface has complicated structure.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

Conformal maps of viscous electron flow in the Gurzhi crossover

Herein we investigate the impact of geometric constriction on the viscous flow of electron liquid through quantum point contacts. We provide analysis on the electric potential distribution given the setup of a slit configuration and use the method of conformal mapping to obtain analytical results. The potential profile can be tested and contrasted experimentally with the scanning tunneling potentiometry technique. We discuss intricate physics that underlies the Gurzhi effect, i.e., the enhancement of conductivity in the viscous flow, and compare results for different boundary conditions. In addition, we calculate the temperature dependence of the momentum relaxation time as a result of impurity assisted quasiballistic interference effects and discuss various correlational corrections that lead to the violation of Matthiessen's rule in the hydrodynamic regime. We caution that spatially inhomogeneous profiles of current in the Gurzhi crossover between Ohmic and Stokes flows might also appear in the nonhydrodynamic limit where nonlocality plays an important role. This conclusion is corroborated by calculation of dispersive conductivity in the weakly impure limit.

2-dimensional systems↗
DOE OSTI2021

Superconducting diode effect via conformal-mapped nanoholes

A superconducting diode is an electronic device that conducts supercurrent and exhibits zero resistance primarily for one direction of applied current. Such a dissipationless diode is a desirable unit for constructing electronic circuits with ultralow power consumption. However, realizing a superconducting diode is fundamentally and technologically challenging, as it usually requires a material structure without a centre of inversion, which is scarce among superconducting materials. Here, we demonstrate a superconducting diode achieved in a conventional superconducting film patterned with a conformal array of nanoscale holes, which breaks the spatial inversion symmetry. We showcase the superconducting diode effect through switchable and reversible rectification signals, which can be three orders of magnitude larger than that from a flux-quantum diode. The introduction of conformal potential landscapes for creating a superconducting diode is thereby proven as a convenient, tunable, yet vastly advantageous tool for superconducting electronics. This could be readily applicable to any superconducting materials, including cuprates and iron-based superconductors that have higher transition temperatures and are desirable in device applications.

36 MATERIALS SCIENCE↗
DOE OSTI2024

CryoTRANS: predicting high-resolution maps of rare conformations from self-supervised trajectories in cryo-EM

Cryogenic electron microscopy (cryo-EM) has revolutionized structural biology, enabling efficient determination of structures at near-atomic resolutions. However, a common challenge arises from the severe imbalance among various conformations of vitrified particles, leading to low-resolution reconstructions in rare conformations due to a lack of particle images in these quasi-stable states. We introduce CryoTRANS, a method that predicts high-resolution maps of rare conformations by constructing a self-supervised pseudo-trajectory between density maps of varying resolutions. This trajectory is represented by an ordinary differential equation parameterized by a deep neural network, ensuring retention of detailed structures from high-resolution density maps. By leveraging a single high-resolution density map, CryoTRANS significantly improves the reconstruction of rare conformations and has been validated on four real-world datasets: alpha-2-macroglobulin, actin-binding protein complexes, SARS-CoV-2 spike glycoprotein, and the 70S ribosome. CryoTRANS can also predict high-resolution structures in cryogenic electron tomography maps using a high-resolution cryo-EM map.Cryogenic electron microscopy (cryo-EM) has revolutionized structural biology, enabling efficient determination of structures at near-atomic resolutions. However, a common challenge arises from the severe imbalance among various conformations of vitrified particles, leading to low-resolution reconstructions in rare conformations due to a lack of particle images in these quasi-stable states. We introduce CryoTRANS, a method that predicts high-resolution maps of rare conformations by constructing a self-supervised pseudo-trajectory between density maps of varying resolutions. This trajectory is represented by an ordinary differential equation parameterized by a deep neural network, ensuring retention of detailed structures from high-resolution density maps. By leveraging a single high-resolution density map, CryoTRANS significantly improves the reconstruction of rare conformations and has been validated on four real-world datasets: alpha-2-macroglobulin, actin-binding protein complexes, SARS-CoV-2 spike glycoprotein, and the 70S ribosome. CryoTRANS can also predict high-resolution structures in cryogenic electron tomography maps using a high-resolution cryo-EM map.

47 OTHER INSTRUMENTATION↗
DOE OSTI2021

Quasi-periodic travelling gravity–capillary waves

We present a numerical study of spatially quasi-periodic travelling waves on the surface of an ideal fluid of infinite depth. This is a generalization of the classic Wilton ripple problem to the case when the ratio of wavenumbers satisfying the dispersion relation is irrational. We propose a conformal mapping formulation of the water wave equations that employs a quasi-periodic variant of the Hilbert transform to compute the normal velocity of the fluid from its velocity potential on the free surface. We develop a Fourier pseudo-spectral discretization of the travelling water wave equations in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on the torus. This leads to an overdetermined nonlinear least-squares problem that we solve using a variant of the Levenberg–Marquardt method. We investigate various properties of quasi-periodic travelling waves, including Fourier resonances, time evolution in conformal space on the torus, asymmetric wave crests, capillary wave patterns that change from one gravity wave trough to the next without repeating and the dependence of wave speed and surface tension on the amplitude parameters that describe a two-parameter family of waves.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2021

A multi-dimensional Child–Langmuir law for any diode geometry

While prior theoretical studies of multi-dimensional space-charge limited current (SCLC) assumed emission from a small patch on infinite electrodes, none have considered emission from an entire finite electrode. In this paper, we apply variational calculus (VC) and conformal mapping, which have previously been used to derive analytic solutions for SCLC density (SCLCD) for nonplanar one-dimensional geometries, to obtain mathematical relationships for any multi-dimensional macroscopic diode with finite cathode and anode. We first derive a universal mathematical relationship between space-charge limited potential and vacuum potential for any diode and apply this technique to determine SCLCD for an eccentric spherical diode. We then apply VC and the Schwartz–Christoffel transformation to derive an exact equation for SCLCD in a general two-dimensional planar geometry with emission from a finite emitter. Particle-in-cell simulations using VSim agreed within 4%–13% for a range of ratios of emitter width to gap distance using the thinnest electrodes practical for the memory constraints of our hardware, with the difference partially attributed to the theory's assumption of infinitesimally thin electrodes. After generalizing this approach to determine SCLCD for any orthogonal diode as a function of only the vacuum capacitance and vacuum potential, we derive an analytical formulation of the three-dimensional Child–Langmuir law for finite parallel rectangular and disk geometries. These results demonstrate the utility for calculating SCLCD for any diode geometry using vacuum capacitance and vacuum potential, which are readily obtainable for many diode geometries, to guide experiment and simulation development.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

BPS Wilson loop in $\mathcal{N}$ = 2 superconformal SU(N) “orientifold” gauge theory and weak-strong coupling interpolation

We consider the expectation value \( \left\langle \mathcal{W}\right\rangle \) of the circular BPS Wilson loop in \( \mathcal{N} \) = 2 superconformal SU( N ) gauge theory containing a vector multiplet coupled to two hypermultiplets in rank-2 symmetric and antisymmetric representations. This theory admits a regular large N expansion, is planar-equivalent to \( \mathcal{N} \) = 4 SYM theory and is expected to be dual to a certain orbifold/orientifold projection of AdS 5 × S 5 superstring theory. On the string theory side \( \left\langle \mathcal{W}\right\rangle \) is represented by the path integral expanded near the same AdS 2 minimal surface as in the maximally supersymmetric case. Following the string theory argument in [5], we suggest that as in the \( \mathcal{N} \) = 4 SYM case and in the \( \mathcal{N} \) = 2 SU( N ) × SU( N ) superconformal quiver theory discussed in [19], the coefficient of the leading non-planar 1/ N 2 correction in \( \left\langle \mathcal{W}\right\rangle \) should have the universal λ 3/2 scaling at large ’t Hooft coupling. We confirm this prediction by starting with the localization matrix model representation for \( \left\langle \mathcal{W}\right\rangle \) . We complement the analytic derivation of the λ 3/2 scaling by a numerical high-precision resummation and extrapolation of the weak-coupling expansion using conformal mapping improved Padé analysis.

1/N Expansion↗
DOE OSTI2021

Spatially Quasi-Periodic Water Waves of Infinite Depth

Abstract We formulate the two-dimensional gravity-capillary water wave equations in a spatially quasi-periodic setting and present a numerical study of solutions of the initial value problem. We propose a Fourier pseudo-spectral discretization of the equations of motion in which one-dimensional quasi-periodic functions are represented by two-dimensional periodic functions on a torus. We adopt a conformal mapping formulation and employ a quasi-periodic version of the Hilbert transform to determine the normal velocity of the free surface. Two methods of time-stepping the initial value problem are proposed, an explicit Runge–Kutta (ERK) method and an exponential time-differencing (ETD) scheme. The ETD approach makes use of the small-scale decomposition to eliminate stiffness due to surface tension. We perform a convergence study to compare the accuracy and efficiency of the methods on a traveling wave test problem. We also present an example of a periodic wave profile containing vertical tangent lines that is set in motion with a quasi-periodic velocity potential. As time evolves, each wave peak evolves differently, and only some of them overturn. Beyond water waves, we argue that spatial quasi-periodicity is a natural setting to study the dynamics of linear and nonlinear waves, offering a third option to the usual modeling assumption that solutions either evolve on a periodic domain or decay at infinity.

Wilkening, Jon (ORCID:0000000327827596)↗
DOE OSTI2023

Spatially quasi-periodic bifurcations from periodic traveling water waves and a method for detecting bifurcations using signed singular values

We present a method of detecting bifurcations by locating zeros of a signed version of the smallest singular value of the Jacobian. This enables the use of quadratically convergent root-bracketing techniques or Chebyshev interpolation to locate bifurcation points. Only positive singular values have to be computed, though the method relies on the existence of an analytic or smooth singular value decomposition (SVD). The sign of the determinant of the Jacobian, computed as part of the bidiagonal reduction in the SVD algorithm, eliminates slope discontinuities at the zeros of the smallest singular value. We use the method to search for spatially quasi-periodic traveling water waves that bifurcate from large-amplitude periodic waves. The water wave equations are formulated in a conformal mapping framework to facilitate the computation of the quasi-periodic Dirichlet-Neumann operator. We find examples of pure gravity waves with zero surface tension and overhanging gravity-capillary waves. In both cases, the waves have two spatial quasi-periods whose ratio is irrational. We follow the secondary branches via numerical continuation beyond the realm of linearization about solutions on the primary branch to obtain traveling water waves that extend over the real line with no two crests or troughs of exactly the same shape. The pure gravity wave problem is of relevance to ocean waves, where capillary effects can be neglected. Such waves can only exist through secondary bifurcation as they do not persist to zero amplitude. The gravity-capillary wave problem demonstrates the effectiveness of using the signed smallest singular value as a test function for multi-parameter bifurcation problems. This test function becomes mesh independent once the mesh is fine enough.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2022

Microgap breakdown with floating metal rod perturbations

Here, we report the characterization of microgap breakdown with perturbations from a metal rod floating between anode and cathode electrodes. The effects of the metal rod on the electric field distribution and the field enhancement factor are evaluated by numerical simulation and the conformal mapping method, and they indicate that the field emission regime is not reached. The breakdown voltages in the Townsend discharge regime are determined based on the voltage–current characteristics, which are obtained from two-dimensional fluid simulations. It is found that the breakdown characteristics can be significantly modulated by the floating metal rod, and the breakdown curve (breakdown voltage vs the net gap distance) is no longer U-shaped, which deviates from the conventional Paschen's law. The underlying physical mechanisms are related to the electric field enhancement, curved breakdown path, and nonuniform ion flux caused by the electric shielding effect. The results provide insights into breakdown characteristics in microscale discharges, which may promote conventional investigation of simplified clean gaps toward more complex conditions (e.g., with floating microparticles) in miniaturized plasma devices.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Extending the Mott–Gurney law to one-dimensional nonplanar diodes using point transformations

Recent studies have applied variational calculus, conformal mapping, and point transformations to generalize the one-dimensional (1D) space-charge limited current density (SCLCD) and electron emission mechanisms to nonplanar geometries; however, these assessments have focused on extending the Child–Langmuir law (CLL) for SCLCD in vacuum. Since the charge in the diode is independent of the coordinate system (i.e., covariant), we apply bijective point transformations to extend the Mott–Gurney law (MGL) for the SCLCD in a collisional or semiconductor gap to nonplanar 1D geometries. This yields a modified MGL that replaces the Cartesian gap distance with a canonical gap distance that may be written generally in terms of geometric scale factors that are known for multiple geometries. We tabulate results for common geometries. Such an approach may be applied to any current density, including non-space-charge limited gaps and SCLCD that may fall between the CLL and MGL.

Garner, Allen L. [Purdue Univ., West Lafayette, IN↗
DOE OSTI2023

Spatially quasi-periodic water waves of finite depth

We present a numerical study of spatially quasi-periodic gravity-capillary waves of finite depth in both the initial value problem and travelling wave settings. We adopt a quasi-periodic conformal mapping formulation of the Euler equations, where one-dimensional quasi-periodic functions are represented by periodic functions on a higher-dimensional torus. We compute the time evolution of free surface waves in the presence of a background flow and a quasi-periodic bottom boundary and observe the formation of quasi-periodic patterns on the free surface. Two types of quasi-periodic travelling waves are computed: small-amplitude waves bifurcating from the zero-amplitude solution and larger-amplitude waves bifurcating from finite-amplitude periodic travelling waves. We derive weakly nonlinear approximations of the first type and investigate the associated small-divisor problem. We find that waves of the second type exhibit striking nonlinear behaviour, e.g. the peaks and troughs are shifted non-periodically from the corresponding periodic waves due to the activation of quasi-periodic modes.

Science & Technology - Other Topics↗
DOE OSTI2025

Invariant Forms of Dissolution Fingers

Dissolution of fractured and porous media introduces a positive feedback between fluid flow and reactant transport, leading to the emergence of pronounced, fingerlike channels. We investigate the formation of these structures using a microfluidic Hele-Shaw cell with a soluble bottom. Our experiments show that the shape of dissolution fingers is invariant and reveals itself over time as the fingers extend into the system. Here, by combining reactive-transport theory and conformal mapping techniques we derive these invariant forms. We relate these results to natural dissolution fingers in karst landscapes, and illustrate how to determine the groundwater flow rate responsible for their formation based on the finger shape.

Flows in porous media↗
DOE OSTI2022

Narrow Slot Gasket Grooves And Step Changes In Width

We provide corrections to the slot capacitance and inverse inductance per unit length for slot gasket groove geometries using an approximate conformal mapping approach. We also provide corrections for abrupt step changes in slot width along with boundary discontinuity conditions for implementation in the various slot models.

42 ENGINEERING↗
DOE OSTI2023

Multipole-based Cable Braid Magnetic Penetration Model for Conducting Wires

In this report, we investigate the effects of conductor losses in a multipole-based cable braid magnetic penetration model. Our multipole model uses a mesh of the actual cable geometry, which enables us to model more complicated structures. After summarizing the first principles model formulation, we consider a one-dimensional array of wires, for which an analytical solution is known in the lossless case. We extend this solution to the lossy case by using a complex-valued radius. We also model this structure analytically using a conformal-mapping solution. We then compare both the self-impedance and the transfer impedance results from our first principles cable braid electromagnetic penetration model to those obtained using the analytical solutions. An analysis for various frequencies (and skin depths) usually encountered in cable modeling is reported. These results are found in good agreement up to a radius to half spacing ratio of about 0.7, demonstrating a robustness needed for many commercial and non-commercial cables.

36 MATERIALS SCIENCE↗