Non-singular boundary integral formulations for plane interior potential problems

2002 ◽  
Vol 53 (7) ◽  
pp. 1751-1762 ◽  
Author(s):  
W. S. Hwang ◽  
L. P. Hung ◽  
C. H. Ko
Keyword(s):  
Boundary Integral ◽  
Singular Boundary ◽  
Potential Problems

Virtual Boundary Integral Method for Anisotropic Potential Problems

2012 ◽  
Vol 155-156 ◽  
pp. 370-374
Author(s):  
Xin Rong Jiang ◽  
Xin Fu
Keyword(s):  
Integral Method ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Singular Boundary ◽  
Boundary Integrals ◽  
Science And Engineering ◽  
Numerical Examples ◽  
Potential Problems ◽  
Virtual Boundary ◽  
Virtual Boundary Element

Heat conduction in anisotropic materials has important applications in science and engineering. In this paper the virtual boundary element method (VBEM) is applied to solve these problems. Due to the fact of a virtual boundary outside the real boundary, the VBEM does not need to treat the singular boundary integrals, and thus, is more accurate and convenient than the traditional one. Numerical examples are presented, to demonstrate the efficiency and accuracy of this method.

Shape Sensitivity Formulation for Axisymmetric Thermal Conducting Solids

1993 ◽  
Vol 207 (3) ◽  
pp. 209-216 ◽  
Author(s):  
Boo Youn Lee
Keyword(s):  
Shape Change ◽  
Direct Method ◽  
Design Sensitivity ◽  
Boundary Integral ◽  
Singular Boundary ◽  
Concept Design ◽  
Shape Sensitivity ◽  
Material Derivative ◽  
Thermal Conducting ◽  
Thermal Diffuser

A direct differentiation method is presented for the shape design sensitivity analysis of axisymmetric thermal conducting solids. Based purely on the standard boundary integral equation (BIE) formulation, a new BIE is derived using the material derivative concept. Design derivatives in terms of shape change are directly calculated by solving the derived BIE. The present direct method has a computational advantage over the adjoint variable method, in the sense that it avoids the problem of solving for the adjoint system with the singular boundary condition. Numerical accuracy of the method is studied through three examples. The sensitivities by the present method are compared with analytic sensitivities for two problems of a hollow cylinder and a hollow sphere, and are then compared with those by finite differences for a thermal diffuser problem. As a practical application to numerical optimization, an optimal shape of the thermal diffuser to minimize the weight under a prescribed constraint is found by use of an optimization routine.

scholarly journals Forward

2007 ◽  
Vol 1 (1) ◽  
Author(s):  
Thomas J. Rudolphi
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Equation Method ◽  
Equation Approach ◽  
Equation Formulation ◽  
Integral Equation Formulation ◽  
Potential Problems ◽  
Integral Equation Approach

<br /><br /> <table width="530" border="0" cellspacing="0" cellpadding="0"> <tr> <td align="left" valign="top"> <a name="abstract"></a> <span class="subtitle" style="font-weight:bold">Abstract</span><br /> <p><img src="http://ejbe.libraries.rutgers.edu/files/rizzo.gif" align=left HSPACE=20>This is the first of two special issues of the Electronic Journal of Boundary Elements dedicated to Frank Rizzo. To say that Frank Rizzo played an important role in the development of what he referred to as “boundary integral equations� would not give much credit to where much credit is due. While it could be argued that the use of integral equations to formulate and form a computational basis of many of the problems of applied mathematics and engineering would probably have been inevitably developed, it was Frank’s seminal work on using the integral equation approach to classical elastostatics that set a whole new research area into motion. His dissertation (which we thought would be of interest to include in this issue) topic, as suggested by his mentor Marvin Stippes at the University of Illinois, and subsequently so well documented in the oft-cited paper “An Integral Equation Approach to Boundary Value Problems of Classical Elastostatics�, Quarterly of Applied Mechanics, 1967, represented the quantum step in the use of integral equations for classical scalar potential problems to the vector potential problems of practical engineering significance. The theoretical basis for this development was Betti’s reciprocal work theorem with the fundamental (response to a point force) solution of the equations of elastostatics, but it was Frank Rizzo who actually breathed the new life into this classical mathematics. A nontrivial contribution of Frank’s original work was to not only to achieve the singular integral equation formulation, but also the systematic methodology of reducing the elegant integral equation formulation to well conditioned, linear algebraic equations by proper analytical integration of the singular terms. Those combined theoretical and practical developments by Frank set into motion a whole new and modern approach to numerically solving partial differential equations, at least of the elliptic type. With Frank’s hard work and the recognition of its elegance and potential by several of his early disciples, the integral equation method blossomed into a powerful and practical computational methodology that would eventually be called “boundary elements�. Amongst the early disciples of the integral equation method, several of which contributed significantly to advancing the methodology to a sophisticated and now mature state, are the authors of this issue and its sequel dedicated to Frank. It is undoubtedly fair to say that most of these authors were, at one time or even continuously, colleagues and personal friends of Frank Rizzo. Frank’s contributions to the boundary integral equation method spanned nearly four decades, from roughly 1964 to 2001. I, too, have been very privileged to become involved with this field in the 1970’s and later to work side by side with Frank, especially in that part of the development of the methodology for what is now referred to as “hypersingular� integral equations. I’m sure that all the present authors can recall numerous occasions and conversations with Frank on a technical point or issue regarding the application of “his� boundary integral method to their own problem of interest. Throughout his productive career, his easy going, collegial, engaging, yet rigorous style earned him respect and admiration that surely befits the “father� of modern boundary integral methods. This commemorative sequence of two issues represents only a small token of tribute and recognition that Frank Rizzo so much deserves for his “singular� contributions to the field that he virtually invented, developed, promoted and nurtured to maturity. Thomas J. Rudolphi Iowa State University <br /><br /><br /> </td> </tr> </table>

Precorrected-FFT Accelerated Singular Boundary Method for Large-Scale Three-Dimensional Potential Problems

2017 ◽  
Vol 22 (2) ◽  
pp. 460-472 ◽  
Author(s):  
Weiwei Li ◽  
Wen Chen ◽  
Zhuojia Fu
Keyword(s):  
Computational Complexity ◽  
Large Scale ◽  
Three Dimensional ◽  
Iteration Step ◽  
Singular Boundary ◽  
Matrix Vector Multiplication ◽  
Boundary Method ◽  
Potential Problems ◽  
Speed Up ◽  
Singular Boundary Method

AbstractThis study makes the first attempt to accelerate the singular boundary method (SBM) by the precorrected-FFT (PFFT) for large-scale three-dimensional potential problems. The SBM with the GMRES solver requires computational complexity, where N is the number of the unknowns. To speed up the SBM, the PFFT is employed to accelerate the SBM matrix-vector multiplication at each iteration step of the GMRES. Consequently, the computational complexity can be reduced to . Several numerical examples are presented to validate the developed PFFT accelerated SBM (PFFT-SBM) scheme, and the results are compared with those of the SBM without the PFFT and the analytical solutions. It is clearly found that the present PFFT-SBM is very efficient and suitable for 3D large-scale potential problems.

Alternative Boundary-Integral Representations of Ship Waves

2002 ◽  
Author(s):  
Francis Noblesse ◽  
Chi Yang ◽  
Dane Hendrix ◽  
Rainald Lo¨hner
Keyword(s):  
Integral Representation ◽  
Fundamental Problem ◽  
Boundary Integral ◽  
Ship Hull ◽  
Integral Representations ◽  
Singular Boundary ◽  
Ship Waves ◽  
Classical Boundary ◽  
The Green Function ◽  
Boundary Integral Representation

The fundamental problem of determining the free-surface potential flow that corresponds to a given flow at a ship hull surface is reconsidered. Stokes’ theorem is used to transform the dipole distribution over the ship hull surface in the classical boundary-integral representation of the velocity potential. This Stokes’ transformation yields a weakly-singular boundary-integral representation that defines the potential in terms of the Green function G and related functions that are no more singular than G. Accordingly, the velocity representation only involves functions that are no more singular than ∇G.

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