scholarly journals Null Field and Interior Field Methods for Laplace’s Equation in Actually Punctured Disks

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Hung-Tsai Huang ◽  
Ming-Gong Lee ◽  
Zi-Cai Li ◽  
John Y. Chiang
Keyword(s):  
Error Analysis ◽  
Numerical Solutions ◽  
Field Method ◽  
Boundary Integral ◽  
Laplace’S Equation ◽  
Algebraic Equations ◽  
Field Methods ◽  
Laplace's Equation ◽  
Null Field ◽  
Boundary Methods

For solving Laplace’s equation in circular domains with circular holes, the null field method (NFM) was developed by Chen and his research group (see Chen and Shen (2009)). In Li et al. (2012) the explicit algebraic equations of the NFM were provided, where some stability analysis was made. For the NFM, the conservative schemes were proposed in Lee et al. (2013), and the algorithm singularity was fully investigated in Lee et al., submitted to Engineering Analysis with Boundary Elements, (2013). To target the same problems, a new interior field method (IFM) is also proposed. Besides the NFM and the IFM, the collocation Trefftz method (CTM) and the boundary integral equation method (BIE) are two effective boundary methods. This paper is devoted to a further study on NFM and IFM for three goals. The first goal is to explore their intrinsic relations. Since there exists no error analysis for the NFM, the second goal is to drive error bounds of the numerical solutions. The third goal is to apply those methods to Laplace’s equation in the domains with extremely small holes, which are called actually punctured disks. By NFM, IFM, BIE, and CTM, numerical experiments are carried out, and comparisons are provided. This paper provides an in-depth overview of four methods, the error analysis of the NFM, and the intriguing computation, which are essential for the boundary methods.

scholarly journals Analysis of Dual Null Field Methods for Dirichlet Problems of Laplace’s Equation in Elliptic Domains with Elliptic Holes: Bypassing Degenerate Scales

2021 ◽  
Vol 37 ◽  
pp. 47-62
Author(s):  
Z.C. Li ◽  
◽  
H.T. Huang ◽  
L.P. Zhang ◽  
A.A. Lempert ◽  
...  
Keyword(s):  
Convergence Rates ◽  
Field Method ◽  
Laplace’S Equation ◽  
Algebraic Equations ◽  
Dirichlet Problems ◽  
Field Methods ◽  
Laplace's Equation ◽  
Null Field ◽  
Circular Holes ◽  
Circular Domains

Dual techniques have been used in many engineering papers to deal with singularity and ill-conditioning of the boundary element method (BEM). Our efforts are paid to explore theoretical analysis, including error and stability analysis, to fill up the gap between theory and computation. Our group provides the analysis for Laplace’s equation in circular domains with circular holes and in this paper for elliptic domains with elliptic holes. The explicit algebraic equations of the first kind and second kinds of the null field method (NFM) and the interior field method (IFM) have been studied extensively. Traditionally, the first and the second kinds of the NFM are used for the Dirichlet and Neumann problems, respectively. To bypass the degenerate scales of Dirichlet problems, the second and the first kinds of the NFM are used for the exterior and the interior boundaries, simultaneously, called the dual null field method (DNFM) in this paper. Optimal convergence rates and good stability for the DNFM can be achieved from our analysis. This paper is the first part of the study and mostly concerns theoretical aspects; the second part is expected to be devoted to numerical experiments.

scholarly journals AERODYNAMIC CHARACTERISTICS OF AIRFOIL USING PANEL METHOD

2017 ◽  
Vol 8 (1) ◽  
pp. 47
Author(s):  
Buyung Junaidin
Keyword(s):  
Potential Flow ◽  
Numerical Models ◽  
Linear Partial Differential Equation ◽  
Aerodynamic Characteristics ◽  
Specific Model ◽  
Boundary Integral ◽  
Panel Method ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Panel Methods

Potential flow over an airfoil plays an important historical role in the theory of flight. The governing equation for potential flow is Laplace’s equation, a widely studied linear partial differential equation. One of Green’s identities can be used to write a solution to Laplace’s equation as a boundary integral. Numerical models based on this approach are known as panel methods in the aerodynamics community. This paper introduces the availability of a computational tool for constructing numerical modelfor potential flow over an airfoil based on panel methods. Use of the software is illustrated by implementing a specific model using Hess and Smith panel method to compute the flow over a member of the NACA four-digit airfoils.

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