Numerical Conformal Mapping Research Articles

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.

Numerical Conformal Mappings From Multiply Connected Regions Onto Annular Domains With Slits

Numerical Conformal Mappings From Multiply Connected Regions Onto Annular Domains With Slits

Special finite elements with arbitrarily-shaped holes for heat conduction analysis of porous materials

Special finite elements are developed which adequately represent the state of heat conduction in the region around an arbitrarily-shaped hole. This is achieved with two assumed temperature fields. The frame field is defined along the element boundary while the intra-element field is independently approximated by using truncated T-complete solutions as trial functions for capturing hole effects. The geometrical irregularity of holes contribute to the difficulty in seeking T-complete solutions. By means of the charge simulation method and numerical conformal mapping technique, the systematic construction of T-complete solutions is shown which satisfy a priori the governing differential equation. Based on the modified variational principle, the two independent temperature fields are linked together at the element level, ensuring conformity across inter-element boundaries. The resultant element stiffness equation involves boundary integrals only, which provides a great flexibility in mesh generation with polygons. Numerical examples for the problems with a single, dual, and multiple holes are given to illustrate the efficiency and accuracy of the proposed method.

A Boundary Integral Equation Method for Computing Numerical Conformal Mappings onto the Disk with Rectilinear Slit and Spiral Slits Regions

This article proposes a boundary integral equation method for computing numerical conformal mappings of bounded multiply connected region Ω onto the disk with rectilinear slit and spiral slits regions, Ω1 and Ω2 Initially, the process involves calculating the boundary value of the canonical region. Cauchy’s integral formula can then be used to compute the mapping of the interior values. The effectiveness of the proposed method is demonstrated using several numerical examples.

Computation of plane potential flow around multi-element airfoils using the Schottky–Klein prime function

Potential flow is computed around multi-element airfoils in two dimensions using numerical conformal mapping combined with the calculus of the Schottky–Klein prime function. The corners at the trailing edges of the airfoils are successively removed by Kármán–Trefftz maps. The map from the domain exterior to disks to the domain exterior to the smooth images of the airfoils is computed using an extension of a Fourier series method for the disk due to Fornberg. The velocity potential for flow in the circle domain, with circulation calculated to satisfy the Kutta condition in the airfoil domain, is computed using the Schottky–Klein prime function. The computations are compared to examples from the aerospace engineering literature.

Preface

The 2022 International Conference on Mechanics and Applied Mathematics (MAM 2022) was successfully held in Guangzhou, China from November 18th-20th, 2022 (virtual conference). The conference is a leading international academic meeting and has the potential to bring new horizons and insights to the researchers and practitioners in the fields of mechanics and applied mathematics.The close relationship and interaction between mechanics and mathematics and the research activities on them has promoted the stable development of society and economy. MAM 2022 created an academic platform for scientific innovation, brought together experts, scholars, and scientists from all over the world to present their work and exchange information, promoted the industrial cooperation of academic achievements, and facilitated future business contacts among all the participants.The scope of the conference covered all areas of research in mechanics and applied mathematics, including Newtonian Mechanics, Explosion Mechanics, Elementary Algebraic Topology, Discrete Mathematics, Coupled Numerical Simulation, Mathematical Methods in Mechanics, etc. The conference brought together about 100 participants, primarily from China and India, but also from Malaysia, French, and other countries.A three-day program of the conference was divided into three sessions, including keynote speeches, oral presentations, and poster presentations. Each keynote speaker was given 30-40 minutes for keynote speeches and every presenter was given 10-15 minutes. Among them, Prof. Dr. Antti Rasila from Guangdong Technion-Israel Institute of Technology, China performed a keynote speech on the title: Conjugate Function Method and Numerical Conformal Mappings. He discussed the so-called conjugate function method, an algorithm based on properties of so-called conformal invariants, to obtain numerical representations of conformal mappings and its applications. Besides, he presented the basics of the algorithm, and discussed its potential applications related to various engineering problems, for example, stress analysis, crack propagation, inverse problems, and mesh generation. The insightful speeches of each keynote speakers triggered interesting discussions and a number of informal talks between all the participants.List of Committee member are available in this pdf.

A HYBRID NUMERICAL/ANALYTICAL MODEL OF TRANSIENT SEEPAGE FOR VERTICAL FRACTURED WELL IN TIGHT GAS RESERVOIR BY USE OF FRACTAL THEORY AND CONFORMAL MAPPING METHOD

Insufficient consideration of the complex morphology of hydraulic fractures (HF) and heterogeneous physical properties of fractured reservoirs in seepage models can result in unreliable well testing analyses. The fractal porosity and permeability (FPP) model provides an effective method for characterizing reservoir heterogeneity in the near-wellbore zone. However, its application to scenarios involving irregularly-shaped hydraulic fracture networks and multiple fracture clusters is challenging due to the lack of spatial symmetry. To address this issue, this paper proposes a combined approach of FPP and conformal mapping (FPP-CM) to transform the region of fractured formation into the exterior of the unit disk domain using numerical conformal mapping. The transient seepage flow model of the vertical fracture well (VFW) is then established by coupling it with the FPP model. The typical curve of pressure transient behavior with the division of flow stages was plotted, and the model verification and sensitivity analysis of parameters were conducted. The results indicate that the fractal dimension primarily affects the formation linear flow stage and its subsequent flow stages; with a decrease in fractal dimension resulting in an increase in the position of the typical curve. For VFW with multiple HF wings, a decrease in the included angle of fracture wings causes an increase in the heterogeneity distribution of microfracture physical properties, resulting in an increase in the position of the pseudo-pressure derivative curve during the late flow stage.

Conformal capacity and polycircular domains

We study numerical conformal mapping of multiply connected planar domains with boundaries consisting of unions of finitely many circular arcs, so called polycircular domains. We compute the conformal capacities of condensers defined by polycircular domains. Experimental error estimates are provided for the computed capacity and, when possible, the rate of convergence under refinement of discretization is analyzed. The main ingredients of the computation are two computational methods, on one hand the boundary integral equation method combined with the fast multipole method and on the other hand the hp-FEM method. The results obtained with these two methods agree with high accuracy.

Polycircular domains, numerical conformal mappings, and moduli of quadrilaterals

We study numerical conformal mappings of planar Jordan domains with boundaries consisting of finitely many circular arcs, also called polycircular domains, and compute the moduli of quadrilaterals for these domains. Experimental error estimates are provided and, when possible, comparison to exact values or other methods are given. We also analyze the rate of convergence as a function of the number of degrees of freedom. The main ingredients of the computation are boundary integral equations combined with the fast multipole method.

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.

The GMRES(m) Method for Numerical Conformal Mapping of Bounded Multi-Connected Domains

It is difficult to solve conformal mapping functions of complex multi-connected domain. In order to overcome this difficulty, the problem of solving conformal mapping function was transformed into using the charge simulation method to solve a pair of conjugate harmonic functions on the problem domain. Using the proposed method, the linear systems were based on the boundary conditions. Then the charges can be computed by used GMRES(m)(The Generalized Minimal Residual) algorithm to solve the linear systems. The approximate conformal mapping functions are constructed accurately to map the bounded multi-connected domain onto three unbounded canonical slit domains. Numerical results show that the present method is effective.

Numerical computation of preimage domains for spiral slit regions and simulation of flow around bodies.

In this paper, we propose the iterative numerical methods to calculate the conformal preimage domains for the specified logarithmic spiral slit regions and develop the applications of conformal mappings in the simulations of the flow around bodies. Firstly, we postulate that the boundaries of the preimage domains mapped onto logarithmic spiral slits are ellipses. The lengths of the long axes of ellipses and the coordinates of the centers are calculated using our iterative methods. Secondly, each type of the presented iterative method calculates numerical conformal mappings via solving the boundary integral equation with the generalized Neumann kernel. Finally, numerical examples show the convergence and availability of our iterative methods and display the simulations of the flow around the bodies as an application.

Stress intensity factor analysis of kinked and hole crack in an infinite plate using numerical conformal mapping

The stress intensity factors (SIFs) of the kinked crack emanating from a hole in an infinite plate are calculated by Muskhelishvili’s formalism. A numerical Schwarz–Christoffel (S-C) conformal mapping method is proposed to calculate the mapping function which transforms the relatively complex configuration of the crack onto a unit circle. The application of this method to the cases of kinked crack and hole crack problems shows good accuracy of SIFs compared to the results of other researches. The performance of the application to the hole crack with multiple kinks shows the potential of this method in the analysis of crack with complex configuration.

Numerical Conformal Mapping with Rational Functions

New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the Dirichlet problem is solved by the “lightning Laplace solver” with poles exponentially clustered near each singularity. For polygons and circular polygons, further simplifications are possible.

Numerical conformal mapping method vs. geometrical separation attitude: Combustion motor chamber internal flow simulation

AbstractThis paper describes the numerical solution of flow analysis in a two‐dimensional combustion chamber with two parallel inlets aided for propellants entrance, addressing a number of different problems that depend on each other: the flow affected by a discrete point in‐cell vortex, the velocity affected by the flow regime in the cell, and the combustion problem affected by the oxygen and fuel entries from the cell walls. The shape geometry of the combustion chamber is made from a square section connected with a semi‐circular one. The mathematical model of the problem consists of an inhomogeneous stream function Laplace equation coupled to partial differential mass conservation equation subjected to a wall non‐intrusion condition and known mass values in the wall openings. Algebraic conformal mapping transformation has been used for modeling the complex cell geometry. The solution was obtained by using the iterative Gauss–Seidel method for both the current function and the mass calculation. Comparison with another modeling method for calibration purpose was performed; considering each geometrical component separately whereas continuous conditions were applied between the two sections. The grid types are: (1) a uniform grid in Cartesian coordinates in the square part, and (2) a uniform circular grid in polar coordinates. It was found that different initial guesses have not effect on the final solution, but an informed initial guess affects the solution time convergence iterations number to solve the problem. Finally, the effect of the mixing vortex on the location of the flame front in the chamber was investigated.

Circular Slit Maps of Multiply Connected Regions with Application to Brain Image Processing

In this paper, we present a fast boundary integral equation method for the numerical conformal mapping and its inverse of bounded multiply connected regions onto a disk and annulus with circular slits regions. The method is based on two uniquely solvable boundary integral equations with Neumann-type and generalized Neumann kernels. The integral equations related to the mappings are solved numerically using combination of Nyström method, GMRES method, and fast multipole method. The complexity of this new algorithm is O((M + 1)n), where M+1 stands for the multiplicity of the multiply connected region and n refers to the number of nodes on each boundary component. Previous algorithms require O((M+1)^3 n^3) operations. The numerical results of some test calculations demonstrate that our method is capable of handling regions with complex geometry and very high connectivity. An application of the method on medical human brain image processing is also presented.

Bidirectional numerical conformal mapping based on the dipole simulation method

Many authors have studied the numerical computation of conformal mappings (numerical conformal mapping), and there are nowadays several efficient numerical schemes. Among them, Amano’s method offers a straightforward numerical procedure for computing conformal mappings based on the method of fundamental solutions, and it has been applied to several regions with great success. However, there are some difficulties in constructing a suitable conjugate harmonic function; therefore, it is required a little of craftmanship. In this paper, we construct another numerical scheme for computing conformal mappings based on the dipole simulation method and give mathematical theorems on the approximation error and the arrangements of the singular points. Several numerical results are also presented to ensure the effectiveness of our proposed method.

River reconstruction using a conformal mapping method

Accurate river bathymetry is required for applications including hydrodynamic flow modelling and understanding morphological processes. Bathymetric measurements are typically a set of depths at discrete points that must be reconstructed into a continuous surface. A number of algorithms exist for this reconstruction, including spline-based techniques and kriging methods. A novel and efficient method is introduced to produce a co-ordinate system fitted to the river path suitable for bathymetric reconstructions. The method is based on numerical conformal mapping and can handle topological features such as islands and branches in the river. Bathymetric surfaces generated using interpolation over a conformal map are compared to spline-based and kriging methods on a section of the Balonne River, Australia. The results show that the conformal mapping algorithm produces reconstructions comparable in quality to existing methods, preserves flow-wise features and is relatively insensitive to the number of sample points, enabling faster data collection in the field.

Numerical Conformal Mapping onto the Exterior Unit Disk with a Straight Slit and Logarithmic Spiral Slits

This paper presents a fast boundary integral equation method for numerical conformal mapping of unbounded multiply connected regions onto a disk with an infinite straight slit and finite logarithmic spiral slits. Some numerical examples are given to show the effectiveness of the proposed method.

Numerical Conformal Mapping onto the Entire Complex Plane Bounded with Finite Straight Slit and Logarithmic Spiral Slits

This paper presents a fast boundary integral equation method with for computing conformal mappings of multiply connected regions. We consider the canonical region consists of the entire complex plane bounded by a finite straight slit on the line Im ω = 0 and finite logarithmic spiral slits. Some numerical examples are given to show the effectiveness of the proposed method.