Ultra-Wideband Vertical Transition in Coplanar Stripline for Ultra-High-Speed Digital Interfaces

A design method for an ultra-wideband coplanar-stripline-based vertical transition that can be used for ultra-high-speed digital interfaces is proposed. A conventional via structure, based on a differential line (DL), inherently possesses performance limitations (<10 GHz) due to difficulties in maintaining constant line impedance and smooth electric field transformation, in addition to the effects of signal skews, FR4 fiber weave, and unbalanced EM interferences. DL-based digital interfaces may not meet the demands of ultra-high-speed digital data transmission required for the upcoming 6G communications. The use of a coplanar stripline (CPS), a type of planar balanced line (BL), for the vertical transition, along with the ultra-wideband DL-to-CPS transition, mostly removes the inherent and unfavorable issues of the DL and enables ultra-high-speed digital data transmission. The design process of the transition is simplified using the analytical design formulas, derived using the conformal mapping method, of the transition. The characteristic line impedances of the transition are calculated and found to be in close agreement with the results obtained from EM simulations. Utilizing these results, the CPS-based vertical transition, maintaining the characteristic line impedance of 100 Ω, is designed and fabricated. The measured results confirm its ultra-wideband characteristics, with a maximum of 1.6 dB insertion loss and more than 10 dB return loss in the frequency range of DC to 30 GHz. Therefore, the proposed CPS-based vertical transition offers a significantly wider frequency bandwidth, i.e., more than three times that of conventional DL-based via structures.

Clarification of Faber series and related applications to complex variable methods in two-dimensional elasticity

Faber series are used extensively in the application of complex variable methods to two-dimensional elasticity theory, for example, in the mechanical analysis of composite materials where Faber series representations of complex potentials lead to convenient expressions for the corresponding displacement and stress distributions. In many cases, the use of the Faber series is combined with conformal mapping techniques which "transfer" a boundary value problem defined in the elastic body (physical plane) to a simpler problem posed in an imaginary plane characterized by the conformal mapping. In several instances in the literature, however, little attention has been paid to the domain of definition of the Faber series in the imaginary plane leading often to misunderstandings and erroneous conclusions regarding the concept and feasibility of the use of the Faber series. In this paper, we present a thorough and rigorous examination of the representation of the Faber series in both the physical (occupied by the material) and imaginary (defined by the conformal mapping) plane. In addition, we show that replacing a truncated Faber series by a truncated Taylor series does not induce any additional errors in the numerical analysis of the corresponding boundary value problem. We anticipate that the discussion in this paper will help clarify any existing misinterpretations regarding the application of the Faber series and help further extend their use to a range of problems in composite mechanics.

Ultra-Wideband Differential Line-to-Balanced Line Transitions for Super-High-Speed Digital Transmission

A conventional differential line (DL), commonly used on typical digital circuit boards for transmitting high-speed digital data, has fundamental limitations on the maximum signal bandwidth (~10 GHz), mainly due to signal skew, multiple line coupling, and EM interference. Therefore, to support super-high-speed digital data transmission, especially for beyond 5G communications, a practical high-performance transmission structure for digital signals is required. Balanced lines (BLs) can transmit the differential signals with multiple advantages of ultra-wide bandwidth, common-mode rejection, reduced crosstalk, phase recovery, and skew reduction, which enable super-high-speed transmission. In order to utilize the BLs in the DL-based digital circuit, connecting structures between a DL and BLs are required, but the DL-to-BL transition structures dominate the operating bandwidth and signal properties. Therefore, in this paper, properties, and design methods for two ultra-wideband DL-to-BL transitions, i.e., DL-to-CPS (coplanar stripline) and DL-to-PSL (parallel stripline) transitions, are presented. Both implemented DL-to-CPS and DL-to-PSL transitions provide high-quality performance up to 40 GHz or higher, significantly enhancing the frequency bandwidth for the transmission of digital signals while providing compatibility with the DL-based PCBs. The fabricated DL-to-CPS transition performs well from DC to 40 GHz with an insertion loss of less than 0.86 dB and a return loss of more than 10 dB, and the fabricated DL-to-PSL transition also provides good performance from DC to 40 GHz, with an insertion loss of less than 1.34 dB and a return loss of more than 10 dB. Therefore, the proposed DL-to-BL transitions can be applied to achieve super-high-speed digital data transmission with over 40 GHz bandwidth, which is more than four times the bandwidth of the DL, supporting over 200 Gbps of digital data transmission on PCBs for the next generation of advanced communications.

Explicit calculation method for cell alignment in non-circular geometries

The alignment of spindle-shaped cells in two-dimensional geometries induces singular points called topological defects, at which the alignment angle of the cell cannot be defined. To control defects related to biological roles such as cell apoptosis, calculation methods for predicting the defect positions are required. This study proposes an explicit calculation method for predicting cell alignment and defect positions in non-circular geometries. First, a complex potential is introduced to describe the alignment angles of cells, which is used to derive an explicit formula for cell alignment in a unit disc. Then, the derived formula for the unit disc is extended to the case for non-circular geometries using a numerical conformal mapping. Finally, the complex potential allows a calculation of the Frank elastic energy, which can be minimized with respect to the defect positions to predict their equilibrium state in the geometry. The proposed calculation method is used to demonstrate a numerical prediction of multiple defects in circular and non-circular geometries, which are consistent with previous experimental results.

Conformal Geometry and Multimaterial Additive Manufacturing through Freeform Transformation of Building Layers

3D printing, formally known as additive manufacturing, creates complex geometries via layer-by-layer addition of materials. While 3D printing has been historically perceived as the static addition of build layers, 3D printing is now considered as a dynamic assembly process. In this context, here a new 3D printing process is reported that executes full degree-of-freedom (DOF) transformation (translating, rotating, and scaling) of each individual building layer while utilizing continuous fabrication techniques. Transforming individual building layers within the sequential layered manufacturing process enables dynamic transformation of the 3D printed parts on-the-fly, eliminating the time-consuming redesign steps. Preserving the locality of the transformation to each layer further enables the discrete conformal transformation, allowing objects such as vascular scaffolds to be optimally fabricated to properly fit within specific patient anatomy obtained from the magnetic resonance imaging (MRI) measurements. Finally, exploiting the freedom to control the orientation of each individual building layer, multimaterials, multiaxis 3D printing capability are further established for integrating functional modules made of dissimilar materials in 3D printed devices. This final capability is demonstrated through 3D printing a soft pneumatic gripper via heterogenous integration of rigid base and soft actuating limbs.

Novel cloaking lamellar structures for a screw dislocation dipole, a circular Eshelby inclusion and a concentrated couple

Using conformal mapping techniques, we design novel lamellar structures which cloak the influence of any one of a screw dislocation dipole, a circular Eshelby inclusion or a concentrated couple. The lamellar structure is composed of two half-planes bonded through a middle coating with a variable thickness within which is located either the dislocation dipole, the circular Eshelby inclusion or the concentrated couple. The Eshelby inclusion undergoes either uniform anti-plane eigenstrains or uniform in-plane volumetric eigenstrains. As a result, the influence of any one of the dislocation dipole, the circular Eshelby inclusion or the concentrated couple is cloaked in that their presence will not disturb the prescribed uniform stress fields in both surrounding half-planes.

Sound from aeroelastic vortex-fibre interactions

The motion of a line vortex moving past a one-dimensional flexible fibre is examined theoretically. A Schwarz-Christoffel conformal mapping enables the analytical solution of the potential flow field and its hydrodynamic moment on the flexible fibre, which is composed of a rigid segment constrained to angular motions on a wedge. The hydroelastic coupling of the vortex path and fibre motion affects the noise signature, which is evaluated for the special case of acoustically compact fibres embedded in a half plane. Results from this analysis attempt to address how the coupled interactions between vortical sources and flexible barbules on the upper surface of owl wings may contribute to their acoustic stealth. The analytical formulation is also amenable to application to vortex sound prediction from flexible trailing edges provided that an appropriate acoustic Green's function can be determined. This article is part of the theme issue 'Frontiers of aeroacoustics research: theory, computation and experiment'.

Analysis of Orthogonal Coupling Structure Based on Double Three-Contact Vertical Hall Device

A vertical Hall device is an important component of 3D Hall sensors, used for detecting magnetic fields parallel to the sensor surface. The Hall devices described in existing research still have problems, such as large offset voltage and low sensitivity. Aiming to solve these problems, this study proposes a double three-contact vertical Hall device with low offset voltage, and a conformal mapping analysis method to improve the sensitivity of the device. Secondly, an orthogonal coupling structure composed of two sets of double three-contact vertical Hall devices is proposed, which further reduces the offset voltage of the device. Finally, the TCAD simulation software was used to analyze the performance of the devices, and an existing vertical Hall device was compared to ours. The results show that the orthogonal coupling structure in this study exhibits better performance, reaching an average voltage sensitivity of 17.5222 mV/VT and an average offset voltage of about 0.075 mV. In addition, the structure has the same magnitude of offset voltage in the four phases of the rotating current method. This characteristic enables the back-end circuit to more accurately filter out the offset voltage and acquire the Hall signal.

Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI

We present a novel method to solve the accessory parameter problem arising in constructing conformal maps from a canonical simply connected planar region to the interior of a circular arc quadrilateral. The Schwarz–Christoffel accessory parameter problem, relevant when all sides have zero curvature, is also captured within our approach. The method exploits the isomonodromic tau function associated with the Painlevé VI equation. Recently, these tau functions have been shown to be related to certain correlation functions in conformal field theory and asymptotic expansions have been given in terms of tuples of the Young diagrams. After showing how to extract the monodromy data associated with the target domain, we show how a numerical approach based on the known asymptotic expansions can be used to solve the conformal mapping accessory parameter problem. The viability of this new method is demonstrated by explicit examples and we discuss its extension to circular arc polygons with more than four sides.

Capturing the flow beneath water waves

Recently, the authors presented two numerical studies for capturing the flow structure beneath water waves (Nachbin and Ribeiro-Junior 2014 Disc. Cont. Dyn. Syst. A34, 3135-3153 (doi:10.3934/dcds.2014.34.3135); Ribeiro-Junior et al. 2017 J. Fluid Mech.812, 792-814 (doi:10.1017/jfm.2016.820)). Closed orbits for irrotational waves with an opposing current and stagnation points for rotational waves were some of the issues addressed. This paper summarizes the numerical strategies adopted for capturing the flow beneath irrotational and rotational water waves. It also presents new preliminary results for particle trajectories, due to irrotational waves, in the presence of a bottom topography.This article is part of the theme issue 'Nonlinear water waves'.

Analyzing Single Giant Unilamellar Vesicles With a Slotline-Based RF Nanometer Sensor

Novel techniques that enable reagent free detection and analysis of single cells are of great interest for the development of biological and medical sciences as well as point-of-care health service technologies. Highly sensitive and broadband radio-frequency (RF) sensors are promising candidates for such a technique. In this work, we present a highly sensitive and tunable RF sensor, which is based on interference processes and built with a 100 nm slotline structure. The highly concentrated RF fields, up to ~1.76×107 V/m, enable strong interactions between Giant unilamellar vesicles (GUVs) and fields for high sensitivity operations. We also provide two modeling approaches to extract cell dielectric properties from measured scattering parameters. GUVs of different molecular compositions are synthesized and analyzed with the RF sensor at ~2 GHz, ~2.5 GHz, and ~2.8 GHz with an initial |S21 | min of ~-100 dB. Corresponding GUV dielectric properties are obtained. A one-dimensional scanning of single GUV is also demonstrated.

Asymmetric collapse by dissolution or melting in a uniform flow

An advection-diffusion-limited dissolution model of an object being eroded by a two-dimensional potential flow is presented. By taking advantage of the conformal invariance of the model, a numerical method is introduced that tracks the evolution of the object boundary in terms of a time-dependent Laurent series. Simulations of a variety of dissolving objects are shown, which shrink and collapse to a single point in finite time. The simulations reveal a surprising exact relationship, whereby the collapse point is the root of a non-analytic function given in terms of the flow velocity and the Laurent series coefficients describing the initial shape. This result is subsequently derived using residue calculus. The structure of the non-analytic function is examined for three different test cases, and a practical approach to determine the collapse point using a generalized Newton-Raphson root-finding algorithm is outlined. These examples also illustrate the possibility that the model breaks down in finite time prior to complete collapse, due to a topological singularity, as the dissolving boundary overlaps itself rather than breaking up into multiple domains (analogous to droplet pinch-off in fluid mechanics). The model raises fundamental mathematical questions about broken symmetries in finite-time singularities of both continuous and stochastic dynamical systems.

Landmark-free geometric methods in biological shape analysis

In this paper, we propose a new approach for computing a distance between two shapes embedded in three-dimensional space. We take as input a pair of triangulated genus zero surfaces that are topologically equivalent to spheres with no holes or handles, and construct a discrete conformal map f between the surfaces. The conformal map is chosen to minimize a symmetric deformation energy Esd(f) which we introduce. This measures the distance of f from an isometry, i.e. a non-distorting correspondence. We show that the energy of the minimizing map gives a well-behaved metric on the space of genus zero surfaces. In contrast to most methods in this field, our approach does not rely on any assignment of landmarks on the two surfaces. We illustrate applications of our approach to geometric morphometrics using three datasets representing the bones and teeth of primates. Experiments on these datasets show that our approach performs remarkably well both in shape recognition and in identifying evolutionary patterns, with success rates similar to, and in some cases better than, those obtained by expert observers.

Uniform stress fields inside multiple inclusions in an elastic infinite plane under plane deformation

Multiple elastic inclusions with uniform internal stress fields in an infinite elastic matrix are constructed under given uniform remote in-plane loadings. The method is based on the sufficient and necessary condition imposed on the boundary value of a holomorphic function that guarantees the existence of the holomorphic function in a multiply connected region. The unknown shape of each of the multiple inclusions is characterized by a conformal mapping. This work focuses on a major large class of multiple inclusions characterized by a simple condition that covers and is much beyond the known related results reported in previous works. Extensive examples of multiple inclusions with or without geometrical symmetry are shown. Our results showed that the inclusion shapes obtained for the uniformity of internal stress fields are independent of the remote loading only when all of the multiple inclusions have the same shear modulus as that of the matrix. Moreover, specific conditions are derived on remote loading, elastic constants of the inclusions and uniform internal stress fields, which guarantee the existence of multiple symmetric inclusions or multiple rotationally symmetrical inclusions with uniform internal stress fields.

How round is a protein? Exploring protein structures for globularity using conformal mapping

We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map f between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy E_S(f) that measures the distance of f from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that E_S(f) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy E_S(f) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that E_S(f) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry.

Statistical analyses of brain surfaces using Gaussian random fields on 2-D manifolds

Interest in the morphometric analysis of the brain and its subregions has recently intensified because growth or degeneration of the brain in health or illness affects not only the volume but also the shape of cortical and subcortical brain regions, and new image processing techniques permit detection of small and highly localized perturbations in shape or localized volume, with remarkable precision. An appropriate statistical representation of the shape of a brain region is essential, however, for detecting, localizing, and interpreting variability in its surface contour and for identifying differences in volume of the underlying tissue that produce that variability across individuals and groups of individuals. Our statistical representation of the shape of a brain region is defined by a reference region for that region and by a Gaussian random field (GRF) that is defined across the entire surface of the region. We first select a reference region from a set of segmented brain images of healthy individuals. The GRF is then estimated as the signed Euclidean distances between points on the surface of the reference region and the corresponding points on the corresponding region in images of brains that have been coregistered to the reference. Correspondences between points on these surfaces are defined through deformations of each region of a brain into the coordinate space of the reference region using the principles of fluid dynamics. The warped, coregistered region of each subject is then unwarped into its native space, simultaneously bringing into that space the map of corresponding points that was established when the surfaces of the subject and reference regions were tightly coregistered. The proposed statistical description of the shape of surface contours makes no assumptions, other than smoothness, about the shape of the region or its GRF. The description also allows for the detection and localization of statistically significant differences in the shapes of the surfaces across groups of subjects at both a fine and coarse scale. We demonstrate the effectiveness of these statistical methods by applying them to study differences in shape of the amygdala and hippocampus in a large sample of normal subjects and in subjects with attention deficit/hyperactivity disorder (ADHD).

Flattening maps for the visualization of multibranched vessels

In this paper, we present two novel algorithms which produce flattened visualizations of branched physiological surfaces, such as vessels. The first approach is a conformal mapping algorithm based on the minimization of two Dirichlet functionals. From a triangulated representation of vessel surfaces, we show how the algorithm can be implemented using a finite element technique. The second method is an algorithm which adjusts the conformal mapping to produce a flattened representation of the original surface while preserving areas. This approach employs the theory of optimal mass transport. Furthermore, a new way of extracting center lines for vessel fly-throughs is provided.

Scattering Parameters Representing Imperfections in Precision Coaxial Air Lines

Scattering parameter expressions are developed for the principal mode of a coaxial air line. The model allows for skin-effect loss and dimensional variations in the inner and outer conductors. Small deviations from conductor circular cross sections are conformally mapped by the Bergman kernel technique. Numerical results are illustrated for a 7 mm air line. An error analysis reveals that the accuracy of the scattering parameters is limited primarily by the conductor radii measurement precision.