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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.


CVBEM [5].
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fig0010: CVBEM [5].

Mentions: Along the boundary Γ, or exterior to the problem domain union boundary, there are defined n nodal points. For development purposes, the n nodes are assumed defined on Γ [8]. Later, we will move the nodes outward away from the boundary to demonstrate an addition degree of freedom. The simple closed contour, Γ, in Fig. 2 is divided into n boundary elements, Γj−1, Γj,…,Γn. For each boundary element, an interpolating polynomial will be used to create a piecewise continuous global interpolation function. In Fig. 2, the boundary, Γ, is “severed” at s = 0 and in the positive direction spans until s = L, the arc length of Γ. In Fig. 3, the boundary is “flattened” and the piecewise function presented. Here, k = 1 is chosen, and the complex polynomials are uniquely defined as first order linear functions.


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

Johnson AN, Hromadka TV - MethodsX (2015)

CVBEM [5].
© Copyright Policy
Related In: Results  -  Collection

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

fig0010: CVBEM [5].
Mentions: Along the boundary Γ, or exterior to the problem domain union boundary, there are defined n nodal points. For development purposes, the n nodes are assumed defined on Γ [8]. Later, we will move the nodes outward away from the boundary to demonstrate an addition degree of freedom. The simple closed contour, Γ, in Fig. 2 is divided into n boundary elements, Γj−1, Γj,…,Γn. For each boundary element, an interpolating polynomial will be used to create a piecewise continuous global interpolation function. In Fig. 2, the boundary, Γ, is “severed” at s = 0 and in the positive direction spans until s = L, the arc length of Γ. In Fig. 3, the boundary is “flattened” and the piecewise function presented. Here, k = 1 is chosen, and the complex polynomials are uniquely defined as first order linear functions.

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.