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. ABSTRACTThe 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 © Copyright Policy - CC BY Related In: Results  -  Collection License getmorefigures.php?uid=PMC4487722&req=5 .flowplayer { width: px; height: px; } fig0035: Nodes placed outside the domain highlight an addition degree of freedom. Axis symmetry allows simplification of the problem to the first quadrant. Mentions: As an example of the complex variable boundary element method consider the twisting behavior of a homogeneous, isotropic shaft of an arbitrary, but uniform, cross section that is fixed at one end and subjected to a twisting couple at the other end. If the force and deformation behavior is of interest at some location somewhat removed from either end, then the stress and strain characteristics of the cross section as depicted in Fig. 7 are described by either of the following equations [3]:(38)∂2ψ(x,y)∂x2+∂2ψ(x,y)∂y2=0,(39)∂2ϕ(x,y)∂x2+∂2ϕ(x,y)∂y2=0.

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

Johnson AN, Hromadka TV - MethodsX (2015)

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fig0035: Nodes placed outside the domain highlight an addition degree of freedom. Axis symmetry allows simplification of the problem to the first quadrant.
Mentions: As an example of the complex variable boundary element method consider the twisting behavior of a homogeneous, isotropic shaft of an arbitrary, but uniform, cross section that is fixed at one end and subjected to a twisting couple at the other end. If the force and deformation behavior is of interest at some location somewhat removed from either end, then the stress and strain characteristics of the cross section as depicted in Fig. 7 are described by either of the following equations [3]:(38)∂2ψ(x,y)∂x2+∂2ψ(x,y)∂y2=0,(39)∂2ϕ(x,y)∂x2+∂2ϕ(x,y)∂y2=0.

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