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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.


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Simultaneous contour plots of the streamline and potential functions. The two real-valued functions,  and , relate stress and strain within the domain and are the real and complex components of the complex-valued approximating function .
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fig0050: Simultaneous contour plots of the streamline and potential functions. The two real-valued functions, and , relate stress and strain within the domain and are the real and complex components of the complex-valued approximating function .

Mentions: Fig. 8 displays the approximate boundary for 6 node (a) collocation and (b) least squares models taken along a very small portion of the elliptical boundary. The deviation of Fig. 8a is due to error. Fig. 8b shows some error, but nearly as much. Fig. 9 shows the relative error for the 6-node model, respectively. The CVBEM relative error along the quarter elliptical section boundary is computed as where is the CVBEM approximate solution and ϕ(x, y) is the exact solution. The quarter model of Fig. 7 was chosen to take advantage of the problem symmetry and to demonstrate the imposition of ϕ boundary conditions along the exterior curved edge and ψ along the interior straight edge which is the side extending from the origin to the point (a, 0) and the line extending from the origin to the point (0, b) where a and b are 6.25 and 3.75 respectively. Table 1 summarizes the exact and computed warping function and shear–stress values at points in Ω using the collocation method. Table 2 demonstrates the exact and warping function calculation using least squares for the same number of basis functions, which are the nodes. The graphical depiction of the CVBEM in Fig. 10 uses computer programs Matlab, Mathematica, and MATLink to model the mixed boundary problem. Fig. 11 is a time analysis of the efficiency of collocation compared to least squares.


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

Johnson AN, Hromadka TV - MethodsX (2015)

Simultaneous contour plots of the streamline and potential functions. The two real-valued functions,  and , relate stress and strain within the domain and are the real and complex components of the complex-valued approximating function .
© Copyright Policy
Related In: Results  -  Collection

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getmorefigures.php?uid=PMC4487722&req=5

fig0050: Simultaneous contour plots of the streamline and potential functions. The two real-valued functions, and , relate stress and strain within the domain and are the real and complex components of the complex-valued approximating function .
Mentions: Fig. 8 displays the approximate boundary for 6 node (a) collocation and (b) least squares models taken along a very small portion of the elliptical boundary. The deviation of Fig. 8a is due to error. Fig. 8b shows some error, but nearly as much. Fig. 9 shows the relative error for the 6-node model, respectively. The CVBEM relative error along the quarter elliptical section boundary is computed as where is the CVBEM approximate solution and ϕ(x, y) is the exact solution. The quarter model of Fig. 7 was chosen to take advantage of the problem symmetry and to demonstrate the imposition of ϕ boundary conditions along the exterior curved edge and ψ along the interior straight edge which is the side extending from the origin to the point (a, 0) and the line extending from the origin to the point (0, b) where a and b are 6.25 and 3.75 respectively. Table 1 summarizes the exact and computed warping function and shear–stress values at points in Ω using the collocation method. Table 2 demonstrates the exact and warping function calculation using least squares for the same number of basis functions, which are the nodes. The graphical depiction of the CVBEM in Fig. 10 uses computer programs Matlab, Mathematica, and MATLink to model the mixed boundary problem. Fig. 11 is a time analysis of the efficiency of collocation compared to least squares.

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.


Related in: MedlinePlus