Numerical Conformal Mappings Of Multiply Connected Regions Onto Slits Domains

In 1851,the famous Riemann theorem was proposed,which reveals the connection between any two simply connected regions on the extended complex plane.After years of development,we can explain this relationship as follows: any simply connected region whose boundary is not less than two points is conformally equivalent to the unit disk.Due to the above relationship,we can easily solve the issues defined in the complex domains by mapping the complex domains onto the unit disks.However,as a classic Dirichlet problem,solving the conformal mappings that satisfy special conditions is difficult.Even the proof of the Riemann theorem has puzzled the mathematical community for 61 years.Today,many issues with conformal mappings have been solved.Simultaneously,we can pay attention to conformal mappings onto slit domains,which are fundamental theories in analytic functions and potential theory and emerge in many methods of solving field theory problems.As is well known,the analytical expressions of conformal mapping functions can only be found in a few simple or specific regions due to the complexity of map types.Thus,the analytical formulas in the majority of practical situations are impossible to acquire,which has motivated numerous scholars to invest in research solving conformal mappings one after another,from which many efficient methods have been proposed.However,there are still a lot of flaws and gaps in the approaches for solving conformal mappings,even with the advancement of other scientific fields.For instance,on the one hand,there are still opportunities for development in accuracy,speed,and the theory of some methods.On the other hand,there aren’t many approaches to finding the preimage region of a specific domain with slits.Thus,given the shortcomings of the current methods for solving conformal mappings,the main work of this thesis is as follows:The relationship between analytic functions and conformal mappings,as well as two mainstream methods for calculating conformal mappings,are introduced,i.e.,the charge simulation method and the boundary integral equation method with the generalized Neumann kernel.The calculation method of conformal mappings based on the charge simulation method to map the bounded multiply connected regions onto the circular slits domain,the radial slits domain,and the parallel slits domain,respectively,is proposed.The generalized minimal residual is proposed to solve the ill-conditioned constraint equations and improve the accuracy of the conformal mapping functions.To calculate the conformal preimage domain of any given logarithmic spiral slit region,we offer an iterative method based on the boundary integral equation method with the generalized Neumann kernel,where the calculated preimage domain’s boundaries are ellipses.In the preimage domain,the irrotational plane flows,including the uniform flow,spiral point vortex,and point source,are simultaneously simulated.