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Modeling mixed boundary conditions in a Hilbert space with the complex variable boundary element method (CVBEM).

Johnson AN, Hromadka TV - MethodsX (2015)

Bottom Line: The Laplace equation that results from specifying either the normal or tangential force equilibrium equation in terms of the warping functions or its conjugate can be modeled as a complex variable boundary element method or CVBEM mixed boundary problem.The CVBEM is a well-known numerical technique that can provide solutions to potential value problems in two or more dimensions by the use of an approximation function that is derived from the Cauchy Integral in complex analysis.This paper highlights three customizations to the technique.•A least squares approach to modeling the complex-valued approximation function will be compared and analyzed to determine if modeling error on the boundary can be reduced without the need to find and evaluated additional linearly independent complex functions.•The nodal point locations will be moved outside the problem domain.•Contour and streamline plots representing the warping function and its complementary conjugate are generated simultaneously from the complex-valued approximating function.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Sciences, United States Military Academy, 601 Swift Road, West Point, NY 10996, USA.

ABSTRACT
The Laplace equation that results from specifying either the normal or tangential force equilibrium equation in terms of the warping functions or its conjugate can be modeled as a complex variable boundary element method or CVBEM mixed boundary problem. The CVBEM is a well-known numerical technique that can provide solutions to potential value problems in two or more dimensions by the use of an approximation function that is derived from the Cauchy Integral in complex analysis. This paper highlights three customizations to the technique.•A least squares approach to modeling the complex-valued approximation function will be compared and analyzed to determine if modeling error on the boundary can be reduced without the need to find and evaluated additional linearly independent complex functions.•The nodal point locations will be moved outside the problem domain.•Contour and streamline plots representing the warping function and its complementary conjugate are generated simultaneously from the complex-valued approximating function.

No MeSH data available.


Branch cut of the function ln(z − ζ), ζ ∈ Γ.
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fig0025: Branch cut of the function ln(z − ζ), ζ ∈ Γ.

Mentions: Letting node z1 be on the branch cut of the complex logarithm function ln(z − ζ) such that z ∈ Ω and ζ ∈ Γ (see Fig. 5), then (13) can be expanded as(14)ωˆk(z)=12πiRk−1(z)−12πi∑Pjk−1(z−zj)ln(z−zj)+Nmk(z),where is a polynomial of degree (k − 1) defined by(15)Pjk−1=(Njk(γj)−Nj−1k(γj−1))(z−zj)and ln(z − zj) is the principal value of the logarithm function. From the continuity of Gk(ζ), it is seen that at the nodal coordinate zj,(16)Njk(γj)−Nj−1k(γj−1)=0and that (z − zj) is a factor as shown in (15). In (14), the term appears due to the circuit around the branch point of the multiple-valued function ln(z − zj).


Modeling mixed boundary conditions in a Hilbert space with the complex variable boundary element method (CVBEM).

Johnson AN, Hromadka TV - MethodsX (2015)

Branch cut of the function ln(z − ζ), ζ ∈ Γ.
© Copyright Policy - CC BY
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4487722&req=5

fig0025: Branch cut of the function ln(z − ζ), ζ ∈ Γ.
Mentions: Letting node z1 be on the branch cut of the complex logarithm function ln(z − ζ) such that z ∈ Ω and ζ ∈ Γ (see Fig. 5), then (13) can be expanded as(14)ωˆk(z)=12πiRk−1(z)−12πi∑Pjk−1(z−zj)ln(z−zj)+Nmk(z),where is a polynomial of degree (k − 1) defined by(15)Pjk−1=(Njk(γj)−Nj−1k(γj−1))(z−zj)and ln(z − zj) is the principal value of the logarithm function. From the continuity of Gk(ζ), it is seen that at the nodal coordinate zj,(16)Njk(γj)−Nj−1k(γj−1)=0and that (z − zj) is a factor as shown in (15). In (14), the term appears due to the circuit around the branch point of the multiple-valued function ln(z − zj).

Bottom Line: The Laplace equation that results from specifying either the normal or tangential force equilibrium equation in terms of the warping functions or its conjugate can be modeled as a complex variable boundary element method or CVBEM mixed boundary problem.The CVBEM is a well-known numerical technique that can provide solutions to potential value problems in two or more dimensions by the use of an approximation function that is derived from the Cauchy Integral in complex analysis.This paper highlights three customizations to the technique.•A least squares approach to modeling the complex-valued approximation function will be compared and analyzed to determine if modeling error on the boundary can be reduced without the need to find and evaluated additional linearly independent complex functions.•The nodal point locations will be moved outside the problem domain.•Contour and streamline plots representing the warping function and its complementary conjugate are generated simultaneously from the complex-valued approximating function.

View Article: PubMed Central - PubMed

Affiliation: Department of Mathematical Sciences, United States Military Academy, 601 Swift Road, West Point, NY 10996, USA.

ABSTRACT
The Laplace equation that results from specifying either the normal or tangential force equilibrium equation in terms of the warping functions or its conjugate can be modeled as a complex variable boundary element method or CVBEM mixed boundary problem. The CVBEM is a well-known numerical technique that can provide solutions to potential value problems in two or more dimensions by the use of an approximation function that is derived from the Cauchy Integral in complex analysis. This paper highlights three customizations to the technique.•A least squares approach to modeling the complex-valued approximation function will be compared and analyzed to determine if modeling error on the boundary can be reduced without the need to find and evaluated additional linearly independent complex functions.•The nodal point locations will be moved outside the problem domain.•Contour and streamline plots representing the warping function and its complementary conjugate are generated simultaneously from the complex-valued approximating function.

No MeSH data available.