Schwarz-Christoffel Research Articles

Diffraction of P1, SV, or Rayleigh waves by an arbitrary-shaped canyon in an unsaturated half-plane

The diffraction of P1, SV, or Rayleigh waves by an arbitrary-shaped canyon in an unsaturated half-plane is investigated in this paper. The scattered wave potentials with unknown coefficients are solved by Fredlund’s theory and the Helmholtz’s decomposition. The general expressions of displacements, stresses, pore pressures, and seepage are obtained using complex variable method. The half-surface and the canyon surface are mapped onto semi-circles in the image plane with Möbius transformation and Schwarz–Christoffel mapping. The boundary-valued problems yield an infinite series of linear algebraic equations with unknown coefficients. The truncating numbers are selected to assure the convergence and accuracy of the solution. A parametric study is performed to investigate the site responses of the canyons by incident P1 waves in an unsaturated half-plane. Numerical results show that the aspect ratio, saturation degree, circular frequency, and angle of incident wave significantly affect the site response of the canyon. The effects of saturation degree on the ground motions are significant owing to the complex tri-interaction of the three phases in the unsaturated medium. The dynamic concentrations occur at the corner region of the canyon. The shielding effect of the canyon to obliquely incident P1 waves is obvious.

Boundary parameter matching for isogeometric analysis using Schwarz–Christoffel mapping

AbstractIsogeometric analysis has brought a paradigm shift in integrating computational simulations with geometric designs across engineering disciplines. This technique necessitates analysis-suitable parameterization of physical domains to fully harness the synergy between Computer-Aided Design and Computer-Aided Engineering analyses. Existing methods often fix boundary parameters, leading to challenges in elongated geometries such as fluid channels and tubular reactors. This paper presents an innovative solution for the boundary parameter matching problem, specifically designed for analysis-suitable parameterizations. We employ a sophisticated Schwarz–Christoffel mapping technique, which is instrumental in computing boundary correspondences. A refined boundary curve reparameterization process complements this. Our dual-strategy approach maintains the geometric exactness and continuity of input physical domains, overcoming limitations often encountered with the existing reparameterization techniques. By employing our proposed boundary parameter matching method, we show that even a simple linear interpolation approach can effectively construct a satisfactory analysis-suitable parameterization. Our methodology offers significant improvements over traditional practices, enabling the generation of analysis-suitable and geometrically precise models, which is crucial for ensuring accurate simulation results. Numerical experiments show the capacity of the proposed method to enhance the quality and reliability of isogeometric analysis workflows.

An Innovative Method for Wind Load Estimation in High-Rise Buildings Based on Green’s Function

High-rise buildings are inherently vulnerable to substantial wind-induced forces. The increasing complexity of building designs has posed challenges in calculating wind loads, while traditional methods involving physical models have proven to be intricate and time-consuming. In order to overcome these obstacles, this paper investigates a theoretical methodology aimed at streamlining the computation of wind loads. In the initial theoretical exploration, a simplified mathematical model based on Green’s function is introduced to take into account the interaction between wind loads and building geometry, while the model is not user-friendly and difficult to solve for complex polygonal buildings. To overcome this challenge, the study incorporates numerical simulations to extend the ideas and refine the methodology. To simplify the problem from a three-dimensional to a two-dimensional context, a bold tangential field assumption is made, assuming the wind pressure distribution remains similar across horizontal sections at different heights. The Schwarz–Christoffel formulation is then employed to facilitate the transformation. By integrating Green’s functions and conformal mapping to solve potential flow problems beyond the boundary layer, a comprehensive mathematical derivation is established. The above broadens the applicability of the mathematical theory and provides a new direction for estimations of high-speed wind load on buildings.

A Two-Dimensional Liquid Sloshing Analysis in a Partially Filled Complicated-Shape Tank by the Schwarz–Christoffel Transformation

A Two-Dimensional Liquid Sloshing Analysis in a Partially Filled Complicated-Shape Tank by the Schwarz–Christoffel Transformation

Harmonic-measure distribution functions of simply connected and doubly connected polygonal domains

The h-function encodes aspects of the behaviour of Brownian particles released from a specified fixed point inside the region. In turn, this behaviour is influenced by the shape of the region's boundary and the location of the fixed point. We compute the h-functions of numerous simply connected and doubly connected polygonal regions. The Schottky–Klein prime function plays a key role in computing h-functions of these regions. Also, in the computation, we use the Schwarz–Christoffel mappings, the Cayley-type mappings, and their generalisations, which are expressed in terms of the prime function. We also cross-check the validity of all the h-functions we find by rederiving them via numerical simulation of Brownian motion.

Elastic fields of double branched and Kalthoff–Winkler cracks in a half-plane

We demonstrate in this paper a combination of the Schwarz–Christoffel mapping and Muskhelishvili’s approach with fractional function series in solving the elastic fields of a cracked half-plane, and zoom in on two typical problems, a double branched crack with two rays emanating from one point on the edge and two edge cracks spaced by a certain distance. Typical loading conditions are considered, including far-field uniform tensile stress and concentrated loads along either the tangential or the normal direction of the free surface. We supply a semi-analytic solution to those boundary-value problems in the cracked half-plane, and validate the theory by comparing the theoretical results in terms of stress fields, stress intensity factors (SIFs) and crack opening displacement (COD) with those from finite-element simulations. The theoretical approach shows how two edge cracks may shield the stress intensity factors of each other in a quantitative manner. For the typical Kalthoff–Winkler cracks of length a and being spaced by a distance d, their SIFs KI decay with decreasing d, and KI=KI0−KI1[1−exp(−a/d)]. It converges to KI0—the SIF of a single edge crack when d approaches to infinity. Those observations and the theory approach itself provide a general way to analyze the mechanical consequence of edge cracks in engineering practice.

Schwarz–Christoffel Mapping from a Rectangle

Schwarz–Christoffel Mapping from a Rectangle

Design of an L-Shaped Array Vernier Permanent Magnet Machine for Unmanned Aerial Vehicle Propulsion Using a Schwarz–Christoffel Mapping-Based Equivalent Magnetic Network Model

The Vernier permanent magnet (VPM) machines supersede conventional permanent magnet synchronous machine (PMSM) topologies in terms of torque-density and cogging torque. This paper presents a fractional-slot L-shaped magnet VPM machine, including peculiar designs for permanent magnet (PM) housing and rotor core construction for small-scale EVs such as unmanned aerial vehicles (UAVs). The idea for selecting the PM arrangement is based on combining the V-shaped and spoke-array PM topologies for achieving a higher torque-density than V-shaped PM machines and a lower cogging torque than spoke-array PM machines. The rotor core is created from non-integrated segments to form the flux-barriers in end-portions of PM housings. In this way, the leakage flux lines in the end-portion of the PMs reduce, resulting in enhanced flux linkage and power factor. A 630 W, 12-slot/8-pole motor is designed and prototyped for model validation purposes. An innovative equivalent magnetic network (EMN) model is established for analytical prediction of the performance of the machine. The Schwarz-Christoffel mapping is used to create the rotor core permeance network due to its special shaping. Two innovative pentagonal-shaped mesh cells are used in the air-gap region for more accurate capturing of flux behavior. The model is validated by comparing the results with the finite element results and experimental measurements.

Conformal Mapping of a Z-Shaped Domain

For the problem of conformal mapping of a half-plane onto a Z-shaped domain with arbitrary geometry, an efficient method is developed for finding parameters of the Schwarz–Christoffel integral, i.e., the preimages of the vertices (prevertices) and the pre-integral multiplier. Special attention is given to the situation of crowding prevertices, in which case conventional integration methods face significant difficulties. For this purpose, the concept of a cluster is introduced, its center is determined, and all integrand binomials with prevertices from this cluster are expanded into a fast-convergent series by applying a unified scheme. Next, the arising integrals are reduced to single or double series in terms of Gauss hypergeometric functions F(a, b, c, q). The fast convergence of the resulting expansions is ensured by applying formulas for analytic continuation of F(a, b, c, q) to a neighborhood of the point q = 1 and using numerically stable recurrence relations. The constructed expansions are also fairly efficient for choosing initial approximations for prevertices in Newton’s iteration method. By using the leading terms of these expansions, the approximations for the prevertices are expressed in explicit form in terms of elementary functions, and the subsequent iterations ensure the fast convergence of the algorithm. After finding the parameters in the integral, the desired mapping is constructed as a combination of power series expansions at prevertices, regular expansions at the preimage of the center of symmetry, a Laurent series in a semi-annulus, and special series near the preimages of the vertical segments. Numerical results demonstrate the high efficiency of the developed method, especially in the case of strong crowding of prevertices.

Formulas for Computing Euler-Type Integrals and Their Application to the Problem of Constructing a Conformal Mapping of Polygons

This paper deals with Euler-type integrals and the closely related Lauricella function , which is a hypergeometric function of many complex variables. For new analytic continuation formulas are found that represent it in the form of Horn hypergeometric series exponentially converging in corresponding subdomains of, including near hyperplanes of the form, . The continuation formulas and identities for found in this paper make up an effective apparatus for computing this function and Euler-type integrals expressed in terms of it in the entire complex space , including complicated cases when the variables form one or several groups of closely spaced neighbors. The results are used to compute parameters of the Schwarz–Christoffel integral in the case of crowding and to construct conformal mappings of polygons.

Steady-State Flow Through a Subsurface Reservoir with a Displaced Fault and its Poro-elastic Effects on Fault Stresses

We consider steady-state single-phase confined flow through a subsurface porous layer containing a displaced, fully conductive fault causing a sudden jump in the flow path, and we employ (semi-)analytical techniques to compute the corresponding pressures and fault stresses. In particular, we obtain a new solution for the pressure field with the aid of conformal mapping and a Schwarz–Christoffel transformation. Moreover, we use an existing technique to compute the poro-elastic stress field with the aid of inclusion theory. The additional resistance to fluid flow provided by a displaced fault, relative to the resistance in a layer without a fault, is a function of dip angle, fault throw divided by reservoir height, and reservoir width divided by reservoir height. Fluid flow has a larger effect on fault stresses in case of injection than in case of depletion, where injection with up-dip flow results in increased zones of fault slip near the bottom of the reservoir. Opposedly, injection with down-dip flow results in increased slip near the top of the reservoir. An order-of-magnitude estimate of the effect of steady-state flow across displaced faults in the Groningen natural gas reservoir shows that the effect on fault stresses is probably negligible. A similar estimate of the effect in low-enthalpy geothermal doublets indicates that steady-state flow may possibly play a small role, in particular close to the injector, but site-specific assessments will be necessary to quantify the effect.

Sequential method conformal mappings

The well-known, very important Schwarz–Christoffel integral does not yet completely solve the problem of mapping a half-plane onto a predetermined polygon. This integral includes parameters (inverse images of the polygon), the relationship of which with the lengths of the edges of the polygon is not known in advance. The main difficulty in using the Schwarz–Christoffel formula lies in determining these parameters. If this difficulty can be overcome in some effective way, then the Schwarz–Christoffel formula expands the range of conformally mapped regions so much that it can be considered “universal”, given that the curvilinear boundary of the mapped region can be approximated by a broken line. Thus, together with the Schwarz–Christoffel formula, a new direction in the theory of functions arises - numerical methods of conformal mapping. However, these approximate numerical methods were developed independently of the Schwarz–Christoffel formula.

An analytical model of the detecting structure of electrostatic inductive electric field sensor

To analyze and design the detecting structure of electrostatic inductive electric field sensors, it is crucial to determine the capacitance. This study proposes a model for the theoretical analysis of the capacitance of the detecting structure of the electrostatic inductive electric field sensor. The aim is to achieve high calculation accuracy and deepen the fundamental understanding of high performance. Using Schwarz–Christoffel mapping, a reasonably simplified and approximative solution program is developed. The precision is comparable to that of a finite element solver. The calculation results exhibit good agreement with the simulation results from finite element analysis when various parameter ratios are used. Notably, the calculation results of the analytical model are verified by the measured data of a realistic electrostatic sensor. Using the analytical model, a parametric study of the detecting structure is conducted, leading to the determination of the final optimized, technically achievable device structures.

Analytical Calculation of Hybrid Excitation Transverse Flux Permanent Magnet Synchronous Linear Motor Based on Schwarz–Christoffel Mapping

An analytical calculation method of thrust and suspension force based on Schwarz–Christoffel (SC) mapping is presented for hybrid excitation transverse flux permanent magnet synchronous linear motor (HETFPMSLM) applied to a maglev train. The effectiveness of the analytical calculation is verified by three-dimensional finite element analysis. Firstly, the structure and principle of HETFPMSLM are analyzed, and its longitudinal two-dimensional SC mapping model is proposed. The magnetic field of HETFPMSLM is calculated by MATLAB’s SC Toolbox. Then the suspension force and thrust are calculated by the Maxwell stress tensor method, which is compared with the 3D finite element to verify the accuracy of the analytical calculation.

Analytical prediction and analysis of the detent force with the feedback harmonics for the direct drive motion system

AbstractIn the direct drive motion system, there are obvious feedback harmonics caused by the various disturbances, which have important influence on the thrust characteristics of the permanent magnet synchronous linear motor (PMSLM). This paper presents the influence of the feedback harmonics on the detent force for the direct drive motion system. Firstly, the Maxwell's equation and the Schwarz–Christoffel (SC) mapping‐based method are used to accurately establish the magnetic field model in the air‐gap considering the slotting effect and end effect. Then, based on the control model of the motion system, the drive current caused by the feedback harmonics are calculated considering the non‐linearity of the drive circuit. The detent force of PMSLM is analytically represented and analyzed. Finally, the validity of the theoretical analysis is verified by the finite‐element method (FEM) and experiments. The results shown that the feedback harmonics will be coupled with the detent force and intensify the couplings among different sources of the thrust harmonics, which will lead to new coupled thrust harmonics, deteriorating the dynamic precision of the direct drive motion system.

A semi-analytical solution to the stress intensity factors of branched cracks

Crack branching is ubiquitous in engineering practice, and it often takes place when a crack is subjected to dynamic stress fields, or runs into heterogeneous regions. The mechanical analysis of branched cracks is of great significance in safety analysis and crack-path engineering. In this work we developed a theoretical method to calculate the stress intensity factors (SIFs) of branched cracks. By employing both Schwarz–Christoffel mapping and Muskhelishvili approach, we present an asymptotic approximation for the conformal mapping and SIFs of arbitrary branched cracks are then readily derived. We further demonstrate the convenience of this analytical approach to obtain the SIFs of forked crack as well as four-branched cracks. The theoretical solutions are validated by using finite-element simulations. It is shown that the semi-analytical approach agrees well with the FEM calculations on SIFs. The analytical methods supply a general way to solve the SIFs and therefore the energy release rate of branched cracks. It can then be adopted to understand crack splitting and crack network engineering.

Numerical Computation on the Hydrodynamic Characteristics of an AUV near Uneven Seabed Based on Schwarz-Christoffel Mapping

Numerical Computation on the Hydrodynamic Characteristics of an AUV near Uneven Seabed Based on Schwarz-Christoffel Mapping

Moduli of quadrilaterals and quasiconformal reflection

We study the interior and exterior moduli of polygonal quadrilaterals. The main result is a formula for a conformal mapping of the upper half plane onto the exterior of a convex polygonal quadrilateral. We prove this by a careful analysis of the Schwarz-Christoffel transformation and obtain the so-called accessory parameters and then the result in terms of the Lauricella hypergeometric function. This result enables us to understand the dissimilarities of the exterior and interior of a convex polygonal quadrilateral. We also give a Mathematica algorithm for the computation. Then we study the special case of an isosceles trapezoidal polygon L and obtain two-sided estimates for the coefficient of quasiconformal reflection over L in terms of special functions and geometric parameters of L.

Water wave models using conformal coordinates

A review of nonlinear water wave problems in complex domains is presented. These include non-smooth highly variable bottom topographies and boundaries as for example forked channel regions. The physical and mathematical relevance of each case is discussed. The Schwarz–Christoffel mapping is the common modeling tool for these different problems. The mapping produces conformal coordinates which allow the problems’ dimension to be reduced and the weakly nonlinear partial differential system of equations to be simplified. Usually conformal mapping is adopted while preserving the Laplace operator. The main focus in the present review is to highlight some water wave problems where the underlying differential operators are not invariant under the conformal change of coordinates. The Schwarz–Christoffel mapping is used to produce a boundary fitted, curvilinear conformal coordinate system. The models reported include solitary waves on branching channels, and its reduction to nonlinear waves on graphs, as well as three dimensional surface-waves propagating over highly variable ridges. Of particular interest is nonlinear water waves on graphs, which has important applications and a scarce mathematical literature. Conformal mapping plays a crucial role regarding a systematic incorporation of forking angles into the equations, as not done before, and deducing a generalized compatibility condition at the graph nodes.

Solution of the parameter problem of the Schwarz–Christoffel conformal mapping of the interior (exterior) of a circle onto the interior (exterior) of a polygon

We propose an analytical solution of the parameter problem of the Schwarz–Christoffel conformal mapping of the interior (exterior) of a circle onto the interior (exte-rior) of a polygon by use of the behavior of the Newtonian simple layer and logarithmic potentials equal to a constant inside of a simply connected domain.