Conformal Mapping Technique Research Articles

Theoretical development and validation of asphalt concrete density measurement using non-contact interdigital coplanar capacitive sensor

Asphalt concrete (AC) density serves as a critical criterion for quality control and quality assurance in asphalt pavement construction. Current non-destructive testing (NDT) approaches have shown potential for AC density measurement, but they present disadvantages such as inefficiency and unstable accuracy. This paper developed a novel NDT approach for AC density measurement based on non-contact interdigital coplanar capacitive sensor (ICCS). The feasibility of this approach was evaluated through both numerical and experimental validations. Firstly, the theoretical model of non-contact ICCS for AC density measurement was established based on conformal mapping technique, partial capacitance technique, and dielectric mixing model. Secondly, numerical simulations were performed to validate the developed theoretical model under two scenarios. Thirdly, laboratory experiments were conducted to measure capacitance data and predict density for three common types of ACs. Finally, the error of non-contact ICCS was discussed and compared with that of other NDT instruments. The results indicated that the developed theoretical model could establish the relationship between AC density and capacitance value, which provided the basis for AC density measurement using non-contact ICCS. It was also observed that the capacitance data obtained by the proposed model closely matched those computed by numerical simulations. The maximum errors between theoretical and numerical data were 4.10% and 1.08% for two cases respectively, which proved the feasibility of the theoretical model. Furthermore, experimental results indicated that the capacitance value of AC specimens increased by about 1 pF as the rolling time increases due to a decrease in the volume of air in AC, which reaffirms the validity of AC density measurement using non-contact ICCS. Additionally, the non-contact ICCS demonstrated excellent accuracy with a maximum error of 6.69%, which is comparable to the current mainstream NDT approaches. And it has the advantages of lower cost, simpler data processing and higher efficiency. The findings of this paper offer a new and reliable method and tool for non-destructive and large-scale AC density measurement.

Mechanism model for effect of eccentricity on capacitive control rod position indicators

The continuous identification technology for position signals based on the capacitive method has a good application potential in the field of the control rods position measurement in nuclear reactors. The main source for error of the Capacitive Control Rod Position Indicators (CCRPIs) comes from the effect of eccentricity. The conformal mapping technique was used in this paper, and the geometry of the CCRPIs could be transformed. A model for revealing the mechanism of the effect of eccentricity was established, and it was tested and verified by the method of numerical calculation. The characteristics of the distribution of the sensitive field was clarified by the model. The results indicate that the maximum error of the model is smaller than 13 %, and the eccentric effect of the indicator can be described by the model.

Directional pollutant extraction—the mathematics of the two-dimensional Aaberg exhaust hood

By deflecting unpolluted air away from the exhaust inlet by means of directional air jets, the Aaberg exhaust hood can focus the extraction flow it produces directly onto a source of contaminant and, thereby, achieve directional extraction of pollutant at high concentration. While significantly outperforming conventional hoods, such ‘jet-reinforced’ flows are notoriously challenging to control in practice and if operating incorrectly disperse, rather than extract, airborne pollutants. To provide a rapid predictive capability and insight into the role of device geometry, a mathematical model for the airflow created by the Aaberg exhaust hood is developed. Modelling the induced flow as potential flow, conformal mapping techniques are used to develop analytical solutions for the stream function and airspeed, from which we identify the region of pollutant capture. Investigating these flows as the directionality of the air jets and the ratio of jet-to-suction strengths are varied, our analytical solutions offer insights into the effective deployment of jet-reinforced exhaust hoods to improve indoor air quality in industrial workplaces.

The analytical model of time-harmonic electric field and impedance spectrum in the multilayered interdigital electrode structure

Accurately determining the time-harmonic electric field and impedance spectrum in the multilayered interdigital electrode (IDE) transducers with lossy dielectrics is essential for the design of the structure, the property estimation of the dielectric, and performance optimization. This paper presents an analytical model of the electric field and impedance within a multilayered interdigitated electrode (IDE) when subjected to alternating excitation through the method of separation of variables (MSV). The equivalent circuit of IDE is established, where the impedance is the parallel of capacitive coupling and resistive coupling between electrodes. The general solutions for the electric field and impedance involving arbitrary numbers and complex permittivities of the dielectric layers are analytically derived. However, due to the approximation of boundary conditions, these physical quantities cannot be accurately obtained by the conventional analytical method using conformal mapping technique (CMT) and partial capacitance technique (PCT). The proposed model exhibits proficiency in generating precise electric field images and impedance values that closely correspond with simulated data, even when accounting for the frequency-dependent characteristics of permittivity and conductivity in lossy dielectrics. Moreover, the approaches to improve the sensitivity of a traditional three-layered IDE-based sensor are demonstrated. For a three-layered IDE-based capacitive sensor, improving the sensitivity can be accomplished through the incorporation of a ground plane at the substrate's bottom and increasing the metallization rate. Conversely, it is recommended to reduce the metallization rate, increase substrate thickness, and remove the ground plane in order to enhance the sensitivity of the IDE-based impedance sensor. The experimental results demonstrate that the proposed model generates outstanding impedance spectra (involving capacitance spectra and resistance spectra) results for various metallization ratios and top layers. This exemplifies the model's potential for predicting, designing, and enhancing a complex multilayered IDE-based transducer to achieve precise and sensitive detection.

Investigation of flower‐shaped microstrip antenna using conformal mapping technique

AbstractIn this paper, the flower‐shaped microstrip antenna (FSMA) is investigated theoretically and experimentally for the first time. A theoretical model is proposed to investigate the radiation characteristic of an FSMA using conformal mapping technique. The flower shape is mapped to a circular shape with the help of an appropriate mapping relation. The wave equation is solved in the transformed domain using Galerkin technique to obtain the approximate solution of eigenfunction and eigenvalue. Resonant frequency, internal field patterns, input impedance, and far‐field radiation patterns of the FSMA are investigated. Three antenna prototypes are fabricated for experimental verification. The acceptable agreement is found among the calculated, simulated, and measured data. Our theory shows an error of less than 1.45% in predicting the resonance frequency. Quality factor and bandwidth are also described. This theoretical approach not only quickly evaluates the performance of an FSMA but also provides the instruction to design an FSMA.

A numerical investigation on the vibration of a two-deck Euler–Bernoulli beam flooded by a potential flow

Abstract This work concerns the structural vibration of a bladeless wind turbine, modelled by a two-deck Euler–Bernoulli beam, due to a surrounding potential flow. The deflection is governed by the Euler–Bernoulli equation which is studied first by a linear theory and then computed numerically by a finite difference method in space with a collocation method over the arc length, and an implicit Euler method in time. The fluid motion in the presence of gravity is governed by the full Euler equations and solved by the time-dependent conformal mapping technique together with a pseudo-spectral method. Numerical experiments of excitation by a moving disturbance on the fluid surface with/without a stochastic noise are carried out. The random process involved in generating the noise on the water surface is driven by a Wiener Process. A Monte Carlo method is used for stochastic computations. The generated surface waves impinge on the beam causing structural vibration which is presented and discussed in detail. By elementary statistical analysis, the structural response subject to the stochastic hydrodynamic disturbance caused by white noise is found to be Gaussian.

Prediction of Resonant Frequency of an Equilateral Triangular Microstrip Antenna Using Conformal Mapping Technique

In this manuscript, Conformal Mapping Technique (CMT) is applied to predict the resonant frequency of an Equilateral Triangular Microstrip Antenna (ETMA). The ETMA is mapped into a simple circular geometry with the aid of CMT. The wave equation is solved in the transformed domain using Galerkin technique to obtain the approximate solution of eigenfunction and eigenvalue. Twenty-three measured data are collected from the open literature to validate our theory. Our theoretical results are also compared with several CAD models to show the effectiveness of the present theory. It is found that our proposed theory can predict the resonant frequency of the ETMA with an error of less than 1.5%, whereas those CAD models can produce a large error (sometimes, 10-19% error). Internal field patterns, input impedance, far-field radiation patterns, quality factor, bandwidth gain, and directivity of the ETMA for the dominant mode are also investigated using CMT.

Simulating Fuel Assembly Bowing: A Neutron-Diffusion Method Based on Arbitrary Quadrilateral Node and Conformal Mapping Technique

Fuel assembly bowing, widely observed in a pressurized water reactor (PWR), often results in an asymmetrical power distribution. This paper proposes a neutron-diffusion method that integrates the arbitrary quadrilateral node with the conformal mapping technique to characterize the impact of fuel assembly bowing on power distribution. The proposed method involves a nonlinear iteration process to solve the neutron-diffusion equation. The global coarse-mesh finite difference equation is established on the arbitrary quadrilateral nodes, which are redivided in response to fuel assembly bowing. The local two-node nodal expansion method equation is established on the rectangular nodes, which are mapped from the original arbitrary quadrilateral nodes using the conformal mapping technique. The proposed method has improved our self-developed core code, named SPARK, for PWRs. To verify this novel method, two distinct types of fuel assembly bowing are modeled based on the mini core. The reference results for these models were obtained using the Monte Carlo code NECP-MCX. The numerical results suggest a robust agreement between the biases of keff and power distributions and their corresponding reference results.

A matrix-form complex variable method for multiple non-circular tunnels in layered media

This paper presents a matrix-form complex variable method that can predict the stress and displacement field around multiple tunnels in layered media. The proposed matrix-form method avoids complicated coefficient determinations and iterative algorithm, which effectively improves the computation efficiency. Firstly, the expressions for the stress and displacement field are given under the generalized complex variable theory. The conformal mapping technique and fast Fourier transform algorithm are used to formulate the matrix-form expressions of boundary conditions. Considering the boundary conditions of the tunnels and soil interfaces, the global matrix solution for multiple tunnels in layered media are obtained by assembling the matrix solutions of all single layers. The present theory is verified by comparison with the existing literature, finite element analysis and field data in engineering practices, and several numerical examples are given to reveal the effects of elastic properties, tunnels' position, equivalent layer and the layer thickness surrounding tunnels.

Dynamics of multi-mode nonlinear internal waves in fluids of great depth

This paper is mainly concerned with the dynamics of strongly nonlinear internal waves in two-dimensional fluids of great depth. A fully nonlinear model for a three-layer fluid of great depth containing topography, a Miyata–Choi–Camassa (MCC)-type system, is developed based on the generalization of the Ablowitz–Fokas–Musslimani global-relation formulation for free-surface water waves. Mode-1 internal solitary waves are numerically found in the MCC-type equation using the Petviashvili iteration technique and in the full Euler equations based on the boundary integral equation method. The comparison between the results validates the broad applicability of the MCC-type system. Mode-2 internal solitary waves are found in the MCC-type equations and are shown to be a type of gap solitary wave. Using the multi-mode internal solitary-wave solutions in the MCC-type equations, we apply the conformal mapping technique to calculate streamlines and particle trajectories. For unsteady simulations, we propose a pseudo-spectral algorithm to handle the time-coupled equations and investigate the generation mechanism of multi-mode internal solitary waves due to the resonance effect of local protrusion on the rigid wall, as well as their collisions when the wall is flat. For flows past protrusion on the rigid wall, the flow speed can be divided into subcritical, transcritical, and supercritical. The shedding of multi-mode internal solitary waves can occur only in the transcritical region of flow speed. We implement the carving of bifurcation curves with inflection to achieve the regional delimitation and, based on this, the interpretation of the generation mechanism from a physical point of view.

Bounds on geometric wakefields in collimators and step transitions of arbitrary cross sections

We present the wakefield conformal mapping technique that can be readily applied to the analysis of the radiation generated by an ultra-relativistic particle in the step transition and a collimator. We derive simple analytical expressions for the lower and upper bounds of both longitudinal and transverse wake potentials. We test the derived expressions against well-known formulas in several representative examples. The proposed method can greatly simplify the optimization of collimating sections and can become a useful tool in the shape optimization problems.

Hybrid technique of conformal mapping and Chebyshev collocation method for solving time–space fractional order wave equation

AbstractThis work presents a numerical approach for solving the time–space fractional‐order wave equation. The time‐fractional derivative is described in the conformal sense, whereas the space‐fractional derivative is given in the Caputo sense. The investigated technique is based on the third‐kind of shifted Chebyshev polynomials. The main problem is converted into a system of ordinary differential equations using conformal mapping, Caputo derivatives, and the properties of the third‐kind shifted Chebyshev polynomials. Then, the Chebyshev collocation method and the non‐standard finite difference method will be used to convert this system into a system of algebraic equations. Finally, this system can be solved numerically via Newton's iteration method. In the end, physics numerical examples and comparison results are provided to confirm the accuracy, applicability, and efficiency of the suggested approach.

Equivalent circuit model of the traveling wave electrode for lithium niobate thin film Mach-Zehnder modulators.

We propose an equivalent circuit model of the traveling wave electrode for lithium niobate thin film (TFLN) Mach-Zehnder modulators, in which the distributed capacitance and conformal mapping techniques are applied to calculate the microwave refractive index, microwave loss, and characteristic impedance. Their accuracies are verified by comparing with the results of the finite element method, and the relative errors are less than 3.282%, 1.776%, and 5.334%, respectively. The influence of the electrode's structural parameters on the modulation performances is analyzed, and a 3dB modulation bandwidth around 84GHz with an 8-mm-long traveling wave electrode is obtained.

Numerical conformal mapping for underground unlined tunnels with arbitrary shapes based on Symm’s method

Conformal mapping technique is important in theoretical analysis and numerical computation for the fields of stress and displacement. In general, a unlined tunnel with arbitrary shape has no analytical solution for conformal mapping. Therefore, the study of numerical method for conformal mapping has great significance. The basic functions of numerical conformal mapping are given based on Symm’s method in this paper. Furthermore, the inverse mapping functions were deduced according to the relationships between the boundary nodes in physical and mapped plane. Compared to the other numerical methods, the presented method has some advantages such that, it is simple in concept to be understood, and can give the mapping function without iteration process. The method can be used to the forward and inverse numerical conformal mappings for multiple underground unlined tunnels with arbitrary shapes in finite and infinite domains. With the help of method of fundamental solutions (MFS), the interpolation equations were proposed for multiple underground unlined tunnels with arbitrary shapes. Finally, several numerical examples for the groups of U-shaped and rectangle tunnels have been given to verify the effectiveness of this method. The numerical results can convergent to real cases, which show that the proposed method has the properties of good accuracy and strong adaptability.

Elastoplastic solutions for deep-buried twin tunnels with arbitrary shapes and various arrangements under biaxial in-situ stress field based on Mohr-Coulomb and generalized Hoek-Brown criteria

Theoretical solutions for elastoplastic analysis of deep-buried twin tunnels with arbitrary shapes and various arrangements are developed under biaxial in-situ stress field based on Mohr-coulomb (M–C) and generalized Hoek-Brown (G-H-B) criteria. The boundary stress transformation formulas adapted for arbitrary stress conditions on non-circular tunnel wall are provided in terms of conformal transformations. Regarding the plastic analysis, analytical expressions for stress characteristic relationships along slip lines under generalized Hoek-Brown criterion are derived. Subsequently, the semi-analytical solutions for plastic stresses of rock surrounding arbitrary-shaped tunnels are constructed based on Mohr-Coulomb and generalized Hoek-Brown criteria applying conformal mapping technique, slip-line method and finite difference-numerical iterative approach. On the basis of the discrete plastic stresses, interpolation functions are generated according to three-dimensional linear interpolation technique. And then, solutions for elastic stress function and plastic radius of twin tunnels with various arrangements are developed with improved mapping functions and newly defined fitness functions through differential evolution method. The accuracy of the semi-analytical solutions is verified by the numerical simulation. Meanwhile, the proposed solutions have the advantages of meshless characteristic, better convergence and higher computing efficiency comparing with the numerical simulation. From the point of engineering, the developed elastoplastic solutions can provide theoretical supports for the stress analysis, plastic radius estimation, safety clearances optimizations of deep-buried twin tunnels.

Reliability analysis of plate parts with multiple interacting hole-edge cracks using complex variable function

In this paper, a successive conformal mapping technique is used to propose an analytical method for solving the stress intensity factor (SIF) of asymmetrical hole-edge cracks on the basis of Muskhelishvili's complex variable function analysis method. Based on this analytical solution, the superposition method is used to get the SIF under the interaction of multiple cracks. Subsequently, according to the plastic collapse criterion, the residual strength under the interaction of two collinear asymmetric elliptical hole-edge cracks with different sizes on the side rail of a dumper's frame is analyzed through the method established. The failure probability of the plate part is obtained by the Kriging meta-model. This study provides a theoretical guide for the operation and maintenance plan of plate parts with multiple cracks.

A novel complex variable solution for the stress and displacement fields around a shallow non-circular tunnel

A novel complex variable solution is derived in this paper for the stress and displacement fields around a shallow non-circular tunnel. The proposed method allows explicit computations in matrix form and has advantages of considerable efficiency and high accuracy. Firstly, the stress and displacement boundary conditions for the shallow non-circular tunnel are obtained using the generalized complex variable theory and conformal mapping technique. Then, the fast Fourier transform algorithm is employed to effectively perform the Fourier series fitting for these boundary conditions, and matrix equations for the potential functions in the frequency domain are established. Later, a matrix solution for the potential functions is formulated to determine the stress and displacement fields around the shallow non-circular tunnel. Finally, select numerical analyses are conducted to verify the proposed method, discuss the key parameters in the algorithm, and reveal the effect of the acceleration strategy using FFT as well as the effect of non-circular geometries on the stress and displacement fields.

XFEM analysis for effectively modeling the singularity of the capacitor edge

In this paper, the edge effect of a parallel plate capacitor is numerically analyzed using the extended finite element method, which exhibits higher accuracy than the standard finite element method. The enrichment function, which plays an important role in this method, is numerically computed on the virtual plane in a simple manner using conformal mapping and complex number techniques. The complex velocity potential is used to derive the gradient of the enrichment function. In the implementation process, the inverse of the conformal mapping is derived numerically via triangular patches and the normalized coordinates within them because analytical derivation is challenging. Moreover, the coordinates are converged using the complex Newton–Raphson iterative method. The effectiveness of the proposed method is verified through numerical experiments. The L2 error norm of solution is used to evaluate the accuracy quantitatively. The proposed method exhibits smaller error norms in comparison with the standard finite element method.

Modeling Electrochemical Migration and Growth of Isolated Metal Particles

Electrokinetic phenomena within complex structures are relevant in microfluidics. For example, ion concentration polarization is used for electrokinetic trapping for enhanced biosensing using molecular probes1. Concentration polarization near ion-selective membranes also plays an important role in separation systems for desalination2. Aside from microfluidics, electrochemical growth-dissolution phenomenon has been reported in lithium ion battery systems where lithium plating and subsequent growth of dendrites can exacerbate the loss of cyclable lithium through the formation of isolated Lithium (i-Li) islands3. Initially thought to be “dead”, these islands were shown to migrate from one electrode to the other through a deposition-dissolution mechanism3.We present a mathematical solution for the growth and migration of an electrochemically active metal particle in a background current. A broad range of phenomena such as viscous fingering4, diffusion-limited aggregation4 and electrochemical deposition5 follow Laplacian growth and have been traditionally described using conformal map-dynamics in two dimensions. Some non-Laplacian phenomena like electrochemical transport6,7 and advection-diffusion-limited aggregation6 fall into the conformally invariant category8 and can still be simplified using conformal-mapping techniques. Our solution applies conformal mapping to the non-Laplacian growth of the metal particle to evaluate the role of particle morphology in the evolution of the phase boundary. In addition to migration, dissolution-deposition was found to lead to formation of cusps on the phase boundary under certain conditions. The solution is applicable for a general class of problems with a reactive post or particle in an applied background flux. Analytical solutions such as the one presented here are expected to augment numerical simulations and lead to expressions that capture conditions for the onset of morphological instabilities. References S. Park, B. Sabbagh, R. Abu-Rjal, and G. Yossifon, Lab Chip, 22, 814–825 (2022) https://pubs.rsc.org/en/content/articlehtml/2022/lc/d1lc00864a.D. Deng et al., Desalination, 357, 77–83 (2015).F. Liu et al., Nature 2021 600:7890, 600, 659–663 (2021) https://www.nature.com/articles/s41586-021-04168-w.J. Mathiesen, I. Procaccia, H. L. Swinney, and M. Thrasher, Europhys Lett, 76, 257 (2006) https://iopscience.iop.org/article/10.1209/epl/i2006-10246-x.D. A. Kessler, J. Koplik, and H. Levine, http://dx.doi.org/10.1080/00018738800101379, 37, 255–339 (2006) https://www.tandfonline.com/doi/abs/10.1080/00018738800101379.M. Z. Bazant, J. Choi, and B. Davidovitch, Phys Rev Lett, 91, 045503 (2003) https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.91.045503.Z. Gu et al., Phys Rev Fluids, 7, 033701 (2022) https://journals.aps.org/prfluids/abstract/10.1103/PhysRevFluids.7.033701.M. Z. Bazant, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 460, 1433–1452 (2004) https://royalsocietypublishing.org/doi/10.1098/rspa.2003.1218.

Dynamic response of a shallow lined circular tunnel by incident P1 and SV waves in an unsaturated half-plane

The dynamic response of a shallow lined circular tunnel in the unsaturated half-plane during earthquakes is investigated in this paper using the complex function method. The wave fields of the unsaturated medium with unknown coefficients are solved by Fredlund’s theory and Helmholtz’s decomposition based on the momentum and mass balance equations. General expressions for the displacements and total stresses of the solid skeleton, pore pressures, and seepage are derived using the complex function method and conformal mapping technique. The half-plane surface and the tunnel in the physical domain are mapped onto two bonded ring regions in the image plane using two conformal mapping functions. The boundary value problem yields a series of algebraic equations by the boundary conditions and the continuity conditions along the liner-medium interface. The unknown coefficients in the infinite series of algebraic equations can be solved numerically by truncating the series number. A parametric study is performed to investigate the dynamic responses of the medium and the liner by the incident P1 waves. Numerical results reveal that the embedded depth of the tunnel, the saturation degree of the medium, the liner-medium rigidity ratio, the frequency and angle of the incident P1 waves significantly affect the dynamic responses of the medium and the liner. The shielding effect on the tunnel to the incident P1 waves is obvious. The response of the liner-medium system to the incident waves is independent of the saturation degree of the unsaturated medium when its value is high. The effect of the liner-medium rigidity ratio is great on the dynamic stress concentrations of the liner-medium system and weak on the displacements along the half surface. The frequency of the incident excitation has a great influence on the displacements of the half surface.