Accuracy and stability of a set of free-surface time-domain boundary element models based on B-splines

Büchmann, Bjarne

doi:10.1002/(sici)1097-0363(20000515)33:1<125::aid-fld5>3.0.co;2-q

Cited by 14 publications

(13 citation statements)

References 8 publications

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“…Most such methods are similar to spectral element methods for volume problems [Patera 1984] in that the surface is decomposed in to elements or patches, and a high order polynomial representation is used to represent the unknowns on each patch. The spectral element theory suggests that such methods should converge algebraically with respect to increasing numbers of patches, and spectrally with respect to increasing order of the polynomials on each patch, but those convergence rates are only achieved if patch surfaces are also represented to high order [Büchmann 2000;Manier 1995]. This last point is a significant complicating factor, because enforcing the Galerkin or collocation condition requires the time-consuming computation of integrals of products of polynomials and singular kernels over curved surfaces [Wang et al 2000;Newman 1986].…”

Section: Higher-order Methodsmentioning

confidence: 99%

A meshless, spectrally accurate, integral equation solver for molecular surface electrostatics

Kuo

Tidor

White

2008

J. Emerg. Technol. Comput. Syst.

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The need to determine electrostatic fields in domains bounded by molecular surfaces arises in a number of nanotechnology applications including: biomolecule design, carbon nanotube simulation, and molecular electron transport analysis. Molecular surfaces are typically smooth, without the corners common in electrical interconnect problems, but are often so geometrically complicated that numerical evaluation of the associated electrostatic fields is extremely time-consuming. In this paper we describe and demonstrate a meshless spectrally-accurate integral equation method that only requires a description of the molecular surface in the form of a collection of surface points. Our meshless method is a synthesis of techniques, suitably adapted, including: spherical harmonic surface interpolation, spectral-element-like integral equation discretization, integral desingularization via variable transformation, and matrix-implicit iterative matrix solution. The spectral accuracy of this combined method is verified using analytically solvable sphere and ellipsoid problems, and then its accuracy and efficiency is demonstrated numerically by solving capacitance and coupled Poisson/linearized Poisson-Boltzmann problems associated with a commonly used model of a molecule in solution. The results demonstrate that for a tolerance of 10 −3 this new approach reduces the number of unknowns by as much as two orders of magnitude over the more commonly used flat panel methods.

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Section: Higher-order Methodsmentioning

confidence: 99%

A meshless, spectrally accurate, integral equation solver for molecular surface electrostatics

Kuo

Tidor

White

2008

J. Emerg. Technol. Comput. Syst.

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“…For an evaluation point at r(x, y, z): (13) where r f is such that P( r f ) = r (see (9)), |J| is the Jacobian of the mapping function P. Note that the basis function, originally defined on the local patch, is also being used to represent the solution in the global surface through the mapping function:…”

Section: Integration Over Curved Surfacesmentioning

confidence: 99%

“…Therefore, there is much interest in developing higher order methods [8,19,26] that can achieve faster convergence and reduce problem size. In [9,26], the use of a higher order basis based on B-splines resulted in faster algebraic convergence, while in [8,19], the aim was to attain spectral convergence. In this paper, we propose a new kind of higher order basis and demonstrate spectral convergence (error decays exponentially with number of unknowns).…”

Section: Introductionmentioning

confidence: 99%

A spectrally accurate integral equation solver for molecular surface electrostatics

Kuo

White

2006

Proceedings of the 2006 IEEE/ACM International Conference on Computer-Aided Design - ICCAD '06

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Electrostatic analysis of complicated molecular surfaces arises in a number of nanotechnology applications including: biomolecule design, carbon nanotube simulation, and molecular electron transport. Molecular surfaces are typically smooth, without the corners common in electrical interconnect problems, and are candidates for methods with higher order convergence than that of the commonly used flat panel methods. In this paper we describe and demonstrate a spectrally accurate approach for analyzing molecular surfaces described by a collection of surface points. The method is a synthesis of several techniques, and starts by using least squares to fit a high order spherical harmonic surface representation to the given points. Then this analytic representation is used to construct a differentiable map from the molecular suface to a cube, an orthogonal basis is generated on the rectangular cube surfaces, and a change of variables is used to desingularize the required integrals of products of basis functions and Green's function. Finally, an efficient method for solving the discretized system using a matrix-implicit scheme is described. The combined method is demonstrated on an analytically solvable sphere problem, capacitance calculation of complicated molecular surface, and a coupled Poisson/Poisson-Boltzmann problem associated with a biomolecule. The results demonstrate that for a tolerance of 10 −3 this new approach requires one to two orders of magnitude fewer unknowns than a flat panel method.

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Section: Introductionmentioning

confidence: 99%

A Spectrally Accurate Integral Equation Solver for Molecular Surface Electrostatics

Kuo¹,

White²

2006

2006 IEEE/ACM International Conference on Computer Aided Design

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Electrostatic analysis of complicated molecular surfaces arises in a number of nanotechnology applications including: biomolecule design, carbon nanotube simulation, and molecular electron transport. Molecular surfaces are typically smooth, without the corners common in electrical interconnect problems, and are candidates for methods with higher order convergence than that of the commonly used flat panel methods. In this paper we describe and demonstrate a spectrally accurate approach for analyzing molecular surfaces described by a collection of surface points. The method is a synthesis of several techniques, and starts by using least squares to fit a high order spherical harmonic surface representation to the given points. Then this analytic representation is used to construct a differentiable map from the molecular suface to a cube, an orthogonal basis is generated on the rectangular cube surfaces, and a change of variables is used to desingularize the required integrals of products of basis functions and Green's function. Finally, an efficient method for solving the discretized system using a matrix-implicit scheme is described. The combined method is demonstrated on an analytically solvable sphere problem, capacitance calculation of complicated molecular surface, and a coupled Poisson/Poisson-Boltzmann problem associated with a biomolecule. The results demonstrate that for a tolerance of 10-3 this new approach requires one to two orders of magnitude fewer unknowns than a flat panel method.

show abstract

Accuracy and stability of a set of free-surface time-domain boundary element models based on B-splines

Cited by 14 publications

References 8 publications

A meshless, spectrally accurate, integral equation solver for molecular surface electrostatics

A meshless, spectrally accurate, integral equation solver for molecular surface electrostatics

A spectrally accurate integral equation solver for molecular surface electrostatics

A Spectrally Accurate Integral Equation Solver for Molecular Surface Electrostatics

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