Green’s Function Problem of Laplace Equation with Spherical and Prolate Spheroidal Boundaries by Using the Null-Field Boundary Integral Equation

2016 ◽  
Vol 13 (05) ◽  
pp. 1650020 ◽  
Author(s):  
Yu-Lung Chang ◽  
Ying-Te Lee ◽  
Li-Jie Jiang ◽  
Jeng-Tzong Chen
Keyword(s):  
Integral Equation ◽  
Laplace Equation ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Adaptive Observer ◽  
Degenerate Kernel ◽  
Prolate Spheroidal ◽  
Null Field ◽  
Prolate Spheroidal Coordinates ◽  
Spheroidal Coordinates

A systematic approach of using the null-field integral equation in conjunction with the degenerate kernel and eigenfunction expansion is employed to solve three-dimensional (3D) Green’s functions of Laplace equation. The purpose of using degenerate kernels for interior and exterior expansions is to avoid calculating the principal values. The adaptive observer system is addressed to employ the property of degenerate kernels in the spherical coordinates and in the prolate spheroidal coordinates. After introducing the collocation points on each boundary and matching boundary conditions, a linear algebraic system is obtained without boundary discretization. Unknown coefficients can be easily determined. Finally, several examples are given to demonstrate the validity of the present approach.

A Three-Dimensional Finite Element Method for Large Elastic Deformations of Ventricular Myocardium: II—Prolate Spheroidal Coordinates

1996 ◽  
Vol 118 (4) ◽  
pp. 464-472 ◽  
Author(s):  
K. D. Costa ◽  
P. J. Hunter ◽  
J. S. Wayne ◽  
L. K. Waldman ◽  
J. M. Guccione ◽  
...  
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Three Dimensional ◽  
Solution Time ◽  
Order Model ◽  
Prolate Spheroidal ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element ◽  
Prolate Spheroidal Coordinates ◽  
Spheroidal Coordinates

A three-dimensional finite element method for nonlinear finite elasticity is presented using prolate spheroidal coordinates. For a thick-walled ellipsoidal model of passive anisotropic left ventricle, a high-order (cubic Hermite) mesh with 3 elements gave accurate continuous stresses and strains, with a 69 percent savings in degrees of freedom (dof) versus a 70-element standard low-order model. A custom mixed-order model offered 55 percent savings in dof and 39 percent savings in solution time compared with the low-order model. A nonsymmetric 3D model of the passive canine LV was solved using 16 high-order elements. Continuous nonhomogeneous stresses and strains were obtained within 1 hour on a laboratory workstation, with an estimated solution time of less than 4 hours to model end-systole. This method represents the first practical opportunity to solve large-scale anatomically detailed models for cardiac stress analysis.

Convective and Microwave Drying of Prolate Spheroidal Solids: Modeling and Simulation

2019 ◽  
Vol 391 ◽  
pp. 233-238
Author(s):  
E. Gomes da Silva ◽  
E. Santana de Lima ◽  
W.M. Paiva Barbosa de Lima ◽  
A.G. Barbosa de Lima ◽  
J.J. Silva Nascimento ◽  
...  
Keyword(s):  
Mathematical Modeling ◽  
Heat Balance ◽  
Diffusion Theory ◽  
Microwave Drying ◽  
Drying Process ◽  
Balance Equations ◽  
Prolate Spheroidal ◽  
Moisture Migration ◽  
Prolate Spheroidal Coordinates ◽  
Spheroidal Coordinates

This paper focuses some fundamental aspects of combined convective and microwave drying of prolate spheroidal solids. A transient mathematical modeling based on the diffusion theory (mass and heat balance equations) written in prolate spheroidal coordinates was derived and the importance of this procedure on the analysis of the drying process of wet porous solid, is also presented. Results pointed to the behavior of the moisture migration and heating of the solid with different aspect ratio. Solids with higher area/volume relationships dry and heat faster.

scholarly journals A boundary integral method with volume-changing objects for ultrasound-triggered margination of microbubbles

2017 ◽  
Vol 836 ◽  
pp. 952-997 ◽  
Author(s):  
Achim Guckenberger ◽  
Stephan Gekle
Keyword(s):  
Drug Delivery ◽  
Integral Equation ◽  
Boundary Integral Equation ◽  
Integral Method ◽  
Three Dimensional ◽  
Formal Proof ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Drug Uptake ◽  
Periodic Domains

A variety of numerical methods exist for the study of deformable particles in dense suspensions. None of the standard tools, however, currently include volume-changing objects such as oscillating microbubbles in three-dimensional periodic domains. In the first part of this work, we develop a novel method to include such entities based on the boundary integral method. We show that the well-known boundary integral equation must be amended with two additional terms containing the volume flux through the bubble surface. We rigorously prove the existence and uniqueness of the solution. Our proof contains as a subset the simpler boundary integral equation without volume-changing objects (such as red blood cell or capsule suspensions) which is widely used but for which a formal proof in periodic domains has not been published to date. In the second part, we apply our method to study microbubbles for targeted drug delivery. The ideal drug delivery agent should stay away from the biochemically active vessel walls during circulation. However, upon reaching its target it should attain a near-wall position for efficient drug uptake. Though seemingly contradictory, we show that lipid-coated microbubbles in conjunction with a localized ultrasound pulse possess precisely these two properties. This ultrasound-triggered margination is due to hydrodynamic interactions between the red blood cells and the oscillating lipid-coated microbubbles which alternate between a soft and a stiff state. We find that the effect is very robust, existing even if the duration in the stiff state is more than three times lower than the opposing time in the soft state.

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