Differential Forms Basis Functions for Better Conditioned Integral Equations

Fasenfest, B; White, D; Stowell, M; Rieben, R; Sharpe, R; Madsen, N; Rockway, J D; Champagne, N J; Jandhyala, V; Pingenot, J

doi:10.1109/APS.2005.1552646

Title: Differential Forms Basis Functions for Better Conditioned Integral Equations

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Abstract

Differential forms offer a convenient way to classify physical quantities and set up computational problems. By observing the dimensionality and type of derivatives (divergence,curl,gradient) applied to a quantity, an appropriate differential form can be chosen for that quantity. To use these differential forms in a simulation, the forms must be discretized using basis functions. The 0-form through 2-form basis functions are formed for surfaces. Twisted 1-form and 2-form bases will be presented in this paper. Twisted 1-form (1-forms) basis functions ({Lambda}) are divergence-conforming edge basis functions with units m{sup -1}. They are appropriate for representing vector quantities with continuous normal components, and they belong to the same function space as the commonly used RWG bases [1]. They are used here to formulate the frequency-domain EFIE with Galerkin testing. The 2-form basis functions (f) are scalar basis functions with units m{sup -2} and with no enforced continuity between elements. At lowest order, the 2-form basis functions are similar to pulse basis functions. They are used here to formulate an electrostatic integral equation. It should be noted that the derivative of an n-form differential form basis function is an (n+1)-form, i.e. the derivative of a 1-form basis function is a 2-form. Becausemore »« less

Authors:: Fasenfest, B; White, D; Stowell, M; Rieben, R; Sharpe, R; Madsen, N; Rockway, J D; Champagne, N J; Jandhyala, V; Pingenot, J

Publication Date:: Thu Jan 13 00:00:00 EST 2005

Research Org.:: Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 15017387

Report Number(s):: UCRL-CONF-209109
TRN: US200517%%418

DOE Contract Number:: W-7405-ENG-48

Resource Type:: Conference

Resource Relation:: Journal Volume: 4A; Conference: Presented at: 2005 IEEE AP-S International Symposium, Washington, DC, United States, Jul 03 - Jul 08, 2005

Country of Publication:: United States

Language:: English

Subject:: 42 ENGINEERING; DEGREES OF FREEDOM; ELECTROSTATICS; INTEGRAL EQUATIONS; PERFORMANCE; SCALARS; SIMULATION; TESTING; VECTORS

Citation Formats


                    Fasenfest, B, White, D, Stowell, M, Rieben, R, Sharpe, R, Madsen, N, Rockway, J D, Champagne, N J, Jandhyala, V, and Pingenot, J. Differential Forms Basis Functions for Better Conditioned Integral Equations.  United States: N. p., 2005. 
        Web.  doi:10.1109/APS.2005.1552646.

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                    Fasenfest, B, White, D, Stowell, M, Rieben, R, Sharpe, R, Madsen, N, Rockway, J D, Champagne, N J, Jandhyala, V, & Pingenot, J. Differential Forms Basis Functions for Better Conditioned Integral Equations.  United States.  https://doi.org/10.1109/APS.2005.1552646

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                    Fasenfest, B, White, D, Stowell, M, Rieben, R, Sharpe, R, Madsen, N, Rockway, J D, Champagne, N J, Jandhyala, V, and Pingenot, J. 2005.  
        "Differential Forms Basis Functions for Better Conditioned Integral Equations".  United States.  https://doi.org/10.1109/APS.2005.1552646.  https://www.osti.gov/servlets/purl/15017387.

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@article{osti_15017387,

  title        = {Differential Forms Basis Functions for Better Conditioned Integral Equations},

  author       = {Fasenfest, B and White, D and Stowell, M and Rieben, R and Sharpe, R and Madsen, N and Rockway, J D and Champagne, N J and Jandhyala, V and Pingenot, J},

  abstractNote = {Differential forms offer a convenient way to classify physical quantities and set up computational problems. By observing the dimensionality and type of derivatives (divergence,curl,gradient) applied to a quantity, an appropriate differential form can be chosen for that quantity. To use these differential forms in a simulation, the forms must be discretized using basis functions. The 0-form through 2-form basis functions are formed for surfaces. Twisted 1-form and 2-form bases will be presented in this paper. Twisted 1-form (1-forms) basis functions ({Lambda}) are divergence-conforming edge basis functions with units m{sup -1}. They are appropriate for representing vector quantities with continuous normal components, and they belong to the same function space as the commonly used RWG bases [1]. They are used here to formulate the frequency-domain EFIE with Galerkin testing. The 2-form basis functions (f) are scalar basis functions with units m{sup -2} and with no enforced continuity between elements. At lowest order, the 2-form basis functions are similar to pulse basis functions. They are used here to formulate an electrostatic integral equation. It should be noted that the derivative of an n-form differential form basis function is an (n+1)-form, i.e. the derivative of a 1-form basis function is a 2-form. Because the basis functions are constructed such that they have spatial units, the spatial units are removed from the degrees of freedom, leading to a better-conditioned system matrix. In this conference paper, we look at the performance of these differential forms and bases by examining the conditioning of matrix systems for electrostatics and the EFIE. The meshes used were refined across the object to consider the behavior of these basis transforms for elements of different sizes.},

  doi          = {10.1109/APS.2005.1552646},

  url          = {https://www.osti.gov/biblio/15017387},
  journal      = {},
number       = ,

  volume       = 4A,

  place        = {United States},

  year         = {Thu Jan 13 00:00:00 EST 2005},

  month        = {Thu Jan 13 00:00:00 EST 2005}

}

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