Conformal Mapping Research Articles

A hybrid potential flow model for shedding flow around a circular cylinder

A Hybrid Potential Flow (HPF) model for flow around a circular cylinder in the subcritical Reynolds number range (300≤Re≤3×105\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$300 \\le Re \\le 3\ imes 10^5$$\\end{document}) is developed using a combination of elementary flow solutions and empirical data. By joining this developed near-body solution with von Karman’s model for the vortex wake, a complete solution for flow around a circular cylinder is calculated. Results for oscillatory forces, including the transverse lift force, due to vortex shedding as well as shedding frequencies are then calculated and presented. With the complete solution for flow around a cylinder calculated, the HPF model can be used as a step to calculate the flow around other bluff bodies using conformal mapping, an approach that has been developed and presented by the authors in a related paper.

Polynomial and rational convergence rates for Laplace problems on planar domains

Laplace problems on planar domains can be solved by means of least-squares expansions associated with polynomial or rational approximations. Here it is shown that, even in the context of an analytic domain with analytic boundary data, the difference in convergence rates may be huge when the domain is non-convex. Our proofs combine the theory of the Schwarz function for analytic continuation, potential theory for polynomial and rational approximation rates and the theory of crowding of conformal maps.

Effect of surfactants on enzymatic hydrolysis of PET

Enzyme is difficult to adsorb on the surface of high-crystallinity poly(ethylene terephthalate) fibers (PET，crystallinity of 48.56 %) due to the high hydrophobicity of PET fibers. Surfactants can effectively reduce the PET/water interface tension and improve the adsorption of enzyme on PET fibers, increasing the degradation of PET fibers promoted by enzyme. In this paper, different nonionic surfactants (Triton X-100, Triton X-114, Triton X-405, PEG 200, PEG 400, PEG 600, Tween 20, Tween 80) were selected to investigate the effect of surfactant on the degradation of PET fibers by enzyme. It was found that the addition of Triton X-405 can improve the concentration of degradation products. However, other surfactants selected in this paper, such as Tween-80 caused the decrease of the degradation yields. The results indicated that the structure of surfactant plays a very important roll on the degradation of PET catalyzed by enzyme. This is probably due to the transformation of enzyme conformation induced by surfactant, and affects on the activity of enzyme. The hydrophilic surface is beneficial to the adsorption of enzyme onto the surface of PET fibers, and further improve the degradation of PET fibers.

Mode-III fracture of four cracks emanating from an elliptical hole in one-dimensional orthorhombic quasicrystals with piezoelectric effect

In this study, a new generalized conformal mapping is introduced using the generalized complex variable method, focusing on the investigation of the anti-plane problem of an elliptical hole with four cracks in one-dimensional (1D) orthogonal piezoelectric quasicrystals. Under the consideration of electrically impermeable and electrically permeable boundary conditions, the governing equations of the anti-plane problem in 1D orthogonal piezoelectric quasicrystals are successfully simplified into an eighth-order partial differential equation and a fourth-order partial differential equation by introducing a potential function. Based on this, analytical solutions for the stress field and electric field are obtained, and the exact solution for the stress intensity factor (SIF) is derived. In addition, formulas for the electric displacement intensity factor (EDIF) and energy release rate are provided. Subsequently, the influence of geometric parameters and external loads on the field intensity factors and energy release rate is analyzed through numerical examples, providing a theoretical basis for engineering mechanics analysis.

Anisotropic conformal change of conic pseudo-Finsler surfaces, I*

Abstract The present work is devoted to investigate anisotropic conformal transformation of conic pseudo-Finsler surfaces $(M,F)$, that is, $ F(x,y)\longmapsto \overline{F}(x,y)=e^{\phi(x,y)}F(x,y)$, where the function $\phi(x,y)$ depends on both position $x$ and direction $y$, contrary to the ordinary (isotropic) conformal transformation which depends on position only. If $F$ is a pseudo-Finsler metric, the above transformation does not yield necessarily a pseudo-Finsler metric. Consequently, we find out necessary and sufficient condition for a (conic) pseudo-Finsler surface $(M,F)$ to be transformed to a (conic) pseudo-Finsler surface $(M,\overline{F})$ under the transformation $\overline{F}=e^{\phi(x,y)}F$. In general dimension, it is extremely difficult to find the anisotropic conformal change of the inverse metric tensor in a tensorial form. However, by using the modified Berwald frame on a Finsler surface, we obtain the change of the components of the inverse metric tensor in a tensorial form. This progress enables us to study the transformation of the Finslerian geometric objects and the geometric properties associated with the transformed Finsler function $\overline{F}$. In contrast to isotropic conformal transformation, we have a non-homothetic conformal factor $\phi(x,y)$ that preserves the geodesic spray. Also, we find out some invariant geometric objects under the anisotropic conformal change. Furthermore, we investigate a sufficient condition for $\overline{F}$ to be dually flat or/and projectively flat. Finally, we study some special cases of the conformal factor $\phi(x,y)$. Various examples are provided whenever the situation needs.

Does the capacitor analogy model in fracture mechanics of elastic dielectrics constitute an appropriate approximation?

The analytical solution of an elliptic dielectric cavity in an infinite dielectric plate is taken as basis to investigate a Griffith crack problem which is obtained by letting the semi-minor axis tend towards zero. In the course of this, an erroneous conformal mapping, commonly employed in literature and correctly reproducing the electric field only in a part of the physical space, is rectified. Interpreting the elliptic interface as faces of a mechanically opened crack which is exposed to an oblique remote electric field, surface charges and electrostatic tractions are calculated. In contrast to the established capacitor analogy model, approximately yielding electric charge densities and Coulombic tractions from displacements and electric potentials in the undeformed crack configuration, the work at hand provides exact solutions accounting for different implications of the crack curvature and for the inclination of the electric field. Crack weight functions are finally used to calculate stress and electric displacement intensity factors. As turns out, the surface charges of the capacitor analogy represent an excellent substitute for the exact electric boundary conditions within a relevant range of parameters, whereas inaccurate Coulombic tractions in the vicinity of crack tips may lead to a significantly overestimated mode I stress intensity factor.

Fully developed flow through shrouded-fin arrays: exact and asymptotic solutions

The flow resistance, i.e. friction factor times Reynolds number ( $\,f\,{Re}$ ), of longitudinal-fin heat sinks with or without clearance between a shroud and the tips of the fins is an important parameter in thermal design. This is because it dictates the caloric resistance of the heat sink, i.e. change in bulk temperature of the fluid flowing through it. When there is no clearance and the common and oft-valid assumption of negligible fin thickness is invoked, $f\,{Re}$ corresponds to simply that of a rectangular duct. However, with clearance, only numerical results are available as per the well-known study by Sparrow, Baliga and Patankar (ASME J. Heat Transfer, vol. 100, 1978). We develop analytical formulae for $f\,{Re}$ for fully developed flow with clearance. The exact solution is provided by an integral formula derived via conformal mappings. Additionally, simple formulae are derived via asymptotic expansions in three cases: (1) the fin spacing is small compared to the fin height and clearance; (2) the clearance is small compared to the fin spacing, which is small compared to the fin height; (3) the same as case (2) but valid for larger clearances. The different asymptotic formulae are compared to the exact formula, and together cover almost the entire relevant parameter range (for fin spacing and clearance) with errors of less than 15 %. A formula for the limiting case of no clearance is shown to be more accurate, for any fin spacing, than a widely used correlation from the literature.

Sharp Coefficient Estimates for Analytic Functions Associated with Lemniscate of Bernoulli

The main purpose of this work is to study the third Hankel determinant for classes of Bernoulli lemniscate-related functions by introducing new subclasses of star-like functions represented by SLλ* and RLλ. In many geometric and physical applications of complex analysis, estimating sharp bounds for problems involving the coefficients of univalent functions is very important because these coefficients describe the fundamental properties of conformal maps. In the present study, we defined sharp bounds for function-coefficient problems belonging to the family of SLλ* and RLλ. Most of the computed bounds are sharp. This study will encourage further research on the sharp bounds of analytical functions related to new image domains.

Molecular Dynamics Simulation Reveal the Structure-Activity Relationships of Kainoid Synthases.

Kainoid synthases are key enzymes in the biosynthesis of kainoids. Kainoids, as represented by DA and KA, are a class of naturally occurring non-protein amino acids with strong neurotransmitter activity in the mammalian central nervous system. Marine algae kainoid synthases include PnDabC from diatoms, which synthesizes domoic acid (DA), and DsKabC and GfKabC from red algae, which synthesize kainic acid (KA). Elucidation of the catalytic mechanism of kainoid synthases is of great significance for the rational design of better biocatalysts to promote the industrial production of kainoids for use in new drugs. Through modeling, molecular docking, and molecular dynamics simulations, we investigated the conformational dynamics of kainoid synthases. We found that the kainoid synthase complexes showed different stability in the simulation, and the binding and catalytic processes showed significant conformational transformations of kainoid synthase. The residues involved in specific interactions with the substrate contributed to the binding energy throughout the simulation process. Binding energy, the relaxed active pocket, electrostatic potential energy of the active pocket, the number and rotation of aromatic residues interacting with substrates during catalysis, and the number and frequency of hydrogen bonds between the individual functional groups revealed the structure-activity relationships and affected the degree of promiscuity of kainoid synthases. Our research enriches the understanding of the conformational dynamics of kainoid synthases and has potential guiding significance for their rational design.

Substructures of the Weyl group and their physical applications

We study substructures of the Weyl group of conformal transformations of the metric of (pseudo)Riemannian manifolds. These substructures are identified by differential constraints on the conformal factors of the transformations which are chosen such that their composition is associative. Mathematically, apart from rare exceptions, they are partial associative groupoids, not groups, so they do not have an algebra of infinitesimal transformations, but this limitation can be partially circumvented using some of their properties cleverly. We classify and discuss the substructures with two-derivatives differential constraints, the most famous of which being known as the harmonic or restricted Weyl group in the physics literature, but we also show the existence of a lightcone constraint which realizes a proper subgroup of the Weyl group. We then show the physical implications that come from invariance under the two most important substructures, concentrating on classical properties of the energy-momentum tensor and a generalization of the quantum trace anomaly. We also elaborate further on the harmonic substructure, which can be interpreted as partial gauge fixing of full Weyl invariance using BRST methods. Finally, we discuss how to construct differential constraints of arbitrary higher-derivative order and present, as examples, generalizations involving scalar constraints with four and six derivatives.

Conformally related vacuum gravitational waves and their symmetries

A special conformal transformation which carries a vacuum gravitational wave into another vacuum one is built by using Möbius-redefined time. It can either transform a globally defined vacuum wave into a vacuum sandwich wave, or carry the gravitational wave into itself. The first type, illustrated by linearly and circularly polarised vacuum plane gravitational waves, permutes the symmetries and the geodesics. Our second type is a pp wave with conformal O(1, 2) symmetry. An example inspired by molecular physics which seems to have escaped attention so far is an anisotropic generalisation of the familiar inverse-square profile and is reminiscent of Aichelburg-Sexl ultraboosts. The particle can escape, or perform circular periodic motion, or fall into the singularity.

Minimal mass optimization of tensegrity torsional structures

The advantage of high strength-to-mass ratio renders tensegrity structures suitable for application as lightweight structures. Therefore, the minimal mass optimization problem is one of the focal points in the research of tensegrity structures. Various minimal mass optimization configurations have been proposed for different types of loads. In this paper, we will continue this topic to deal with the minimal mass optimization problem of the tensegrity structure under torsional loads. The inspiration for this study stems from the constant radial angle feature of Michell structures, which is also present in the logarithmic spiral. By employing inverse stereographic projection with conformality, the logarithmic spiral can be mapped onto the sphere , the angle between the curve and the lines of latitude remains constant. On this basis, a type of tensegrity sphere structure combining the spherical logarithmic spiral and DHT (Double Helix Tensegrity) topology is proposed. Compared to the other two spheres with identical topology, the proposed tensegrity sphere structure significantly reduces the minimal mass under torsional loads. Furthermore, we have observed that when logarithmic spiral curves with opposite chirality are orthogonal, the optimal solution with minimal mass occurs. These findings demonstrate the potential of the combination of logarithmic spirals and conformal transformations, and positive effect of the orthogonal arrangement of members for mass optimization of tensegrity structures.

Structure Evolution of 2D Covalent Organic Frameworks Unveiled by Single-Crystal X-ray Diffraction.

We report 9 crystal structures of a two-dimensional (2D) covalent organic framework (COF), including the parent Py-1P, 5 derivatives formed by chemical reactions, and 3 dynamic states by solvent exchange/loss. Structure details of these porous crystals, including stacking mode, interlayer distance, pore aperture, and incline angle, before, during, and after conversion processes in solution, were unveiled by single-crystal X-ray diffraction with resolutions up to 0.85 Å. The structure evolution is triggered by stepwise conformational transformation of the molecular building blocks in 2D COF, while their long-range ordering remained unsacrificed.

Free-form and multi-physical metamaterials with forward conformality-assisted tracing.

Transformation theory, active control and inverse design have been mainstream in creating free-form metamaterials. However, existing frameworks cannot simultaneously satisfy the requirements of isotropic, passive and forward design. Here we propose a forward conformality-assisted tracing method to address the geometric and single-physical-field constraints of conformal transformation. Using a conformal mesh composed of orthogonal streamlines and isotherms (or isothermal surfaces), this method quasi-analytically produces free-form metamaterials using only isotropic media. The geometric nature of this approach allows for universal regulation of both dissipative thermal fields and non-dissipative electromagnetic fields. We experimentally demonstrate free-form thermal cloaking in both two and three dimensions. Additionally, the multi-physical functionalities of our method, including optical cloaking, bending and thermo-electric transparency, confirm its broad applicability. Our method features improvements in efficiency, accuracy and adaptability over previous approaches. This study provides an effective method for designing complex metamaterials with arbitrary shapes across various physical domains.

QCD critical point, Lee-Yang edge singularities, and Padé resummations

I analyze the trajectory of the Lee-Yang edge singularities of the QCD equation of state in the complex baryon chemical potential (μB) plane for different values of the temperature by using the recent lattice results for the Taylor expansion coefficients up to eighth order in μB and various resummation techniques that blend in Padé expansions and conformal maps. By extrapolating from this information, I estimate for the location of the QCD critical point: Tc≈100 MeV, μc≈580 MeV. I also estimate the crossover slope at the critical point to be α1≈9∘ and further constrain the nonuniversal mapping parameters between the three-dimensional Ising model and QCD equations of state. Published by the American Physical Society 2024

Mammalian cell display with automated oligo design and library assembly allows for rapid residue level conformational epitope mapping

Precise epitope determination of therapeutic antibodies is of great value as it allows for further comprehension of mechanism of action, therapeutic responsiveness prediction, avoidance of unwanted cross reactivity, and vaccine design. The golden standard for discontinuous epitope determination is the laborious X-ray crystallography method. Here, we present a combinatorial method for rapid mapping of discontinuous epitopes by mammalian antigen display, eliminating the need for protein expression and purification. The method is facilitated by automated workflows and tailored software for antigen analysis and oligonucleotide design. These oligos are used in automated mutagenesis to generate an antigen receptor library displayed on mammalian cells for direct binding analysis by flow cytometry. Through automated analysis of 33930 primers an optimized single condition cloning reaction was defined allowing for mutation of all surface-exposed residues of the receptor binding domain of SARS-CoV-2. All variants were functionally expressed, and two reference binders validated the method. Furthermore, epitopes of three novel therapeutic antibodies were successfully determined followed by evaluation of binding also towards SARS-CoV-2 Omicron BA.2. We find the method to be highly relevant for rapid construction of antigen libraries and determination of antibody epitopes, especially for the development of therapeutic interventions against novel pathogens.

Unraveling and in-situ observation on the interfacial mechanical behavior of the NEPE propellant reinforced by NPBA via molecular dynamics simulation

Excellent mechanical properties are the lifeline for the application of solid propellants. Understanding the strengthening mechanism of bonding agent is of great significance for design and application of high-performance bonding agents. In this study, the strengthening mechanism of neutral polymer bonding agent (NPBA) for the NEPE propellant and dynamic mechanical response were explored through MD simulation. Significant difference in bonding role for RDX and HMX surface is first revealed with a more effective improvement in HMX surface. NPBA requires higher peak pulling force to achieve conformational transformation compared with binder throughout the peeling process. The chain orientation and movement undergo significant changes in nitroglycerin environment and an interface peeling model is proposed to reveal the potential mechanism. Finally, the effect of NPBA insertion on interfacial micro-peeling process indicates that no significant detachment of NPBA until the binder was severely deformed, resulting in a higher pulling force required.

Improved energy method and agglomeration influence of carbon nanotubes on polymer composites

In this paper, the geometric analysis of carbon nanotubes (CNTs) without external loading is carried out by energy method. Based on the theory of molecular mechanics, an improved mechanical model is proposed to predict the energy of armchair carbon nanotubes under stress-free conditions, and the diameter of CNTs is estimated according to the principle of minimum energy. The results show that the diameter obtained by the improved model is larger, but basically consistent with that obtained by conformal mapping. The inversion energy term is added to the modified model, and the inversion energy term related to atomic curvature is characterized by the conization angle. It can be seen from the error that the inversion energy of carbon nanotubes can not be neglected in the stress-free state, especially in the case of small diameter. The agglomeration of nanotubes is one of the important factors, which affects the effective elastic modulus of nanocomposites. Here, a new micro-mechanics model consisting of both agglomeration of CNTs and pure matrix is also presented to analyze its effect on the effective elastic modulus. It is noted from the results that the stiffness of nanocomposites is very sensitive to the CNTs agglomeration.

Permittivity measurement of high-frequency substrate based on the double-sided parallel-strip line resonator method

With the development of 5G technology, the accurate measurement of the complex permittivity of a printed circuit board (PCB) in the wide frequency range is crucial for the design of high-frequency circuits. In this paper, a microwave measurement device and method based on the double-sided parallel-strip line (DSPSL) resonator have been developed to measure the complex permittivity of typical PCBs in the vertical direction. The device includes the DSPSL resonator, the DSPSL coupling probe, a pressure monitor, a Farran C4209 vector network analyzer (100 K to 9 GHz), and a FEV-10-PR-0006 frequency multiplier (75–110 GHz). Based on transmission line theory, the physical model of the DSPSL resonator was established, and the relative permittivity and loss angle tangent value of the dielectric substrate were calculated using conformal transformation. To excite the resonator, the DSPSL coupling probe with a good transmission effect was designed, which consists of DSPSL microstrip line (MSL) transition structure and an MSL-WR10 rectangular waveguide converter. To reduce the air gap between the sample and the metal guide band and dielectric support block, and to improve test accuracy, a mechanical pressure device is added to the top of the DSPSL resonator. Based on the DSPSL resonator, we have used the device to test four typical PCBs, namely, polytetrafluoroethylene, Rogers RT/duroid®5880, Rogers RO3006®, and Rogers RO3010®. The results show that the maximum error of the relative permittivity is less than 3.05%, and the maximum error of the loss angle tangent is less than 1.27 × 10−4.

The cosmological collider in R 2 inflation

Starobinsky's R 2 inflation manifests a best-fit scenario for the power spectrum of primordial density fluctuations. Observables derived from the slow-roll picture of the R 2 model in the Einstein frame relies on the conformal transformation of the metric, which inevitably induces a unique exponential-type couplings of the rolling scalaron with all matter fields during inflation. The “large-field” nature of the R 2 model further invokes non-negligible time and scale dependence to the matter sector through such an exponential coupling, modifying not only the dynamics of matter perturbations on superhorizon scales but also their decay rates. In this work, we identify the simplest observable of the cosmological collider physics built in the background of R 2 inflation, focusing on the so-called “quantum primordial clock” signals created by the non-local propagation of massive scalar perturbations. Our numerical formalism based on the unique conformal coupling can have extended applications to (quasi-)single-field inflationary models with non-trivial couplings to gravity or models that originated from the f(R) modification of gravity.