Overhauser elements in boundary element analysis.

*(English)*Zbl 0726.65129Authors’ summary: “The accuracy and the merit of the Overhauser cubic spline as an isoparametric representation in solving two-dimensional potential problems by the boundary element method is investigated. These functions are used to form a curvilinear boundary element which is intrinsically \(C^ 1\)-continuous between the elements. The resulting Overhauser element avoids the computational inefficiencies suffered by general cubic splines which require an additional variable to enforce derivative continuity between elements. Several numerical examples of phenomena governed by both the Poisson and biharmonic equations are presented and compared with existing numerical results or exact solutions.”

{Reviewer’s remark: The fundamental solution of the biharmonic equation is stated incorrectly in equation (6).}

{Reviewer’s remark: The fundamental solution of the biharmonic equation is stated incorrectly in equation (6).}

Reviewer: D.Kershaw (Lancaster)

##### MSC:

65N38 | Boundary element methods for boundary value problems involving PDEs |

65D10 | Numerical smoothing, curve fitting |

65R20 | Numerical methods for integral equations |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J40 | Boundary value problems for higher-order elliptic equations |

35C15 | Integral representations of solutions to PDEs |

31A30 | Biharmonic, polyharmonic functions and equations, Poisson’s equation in two dimensions |

45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |

##### Keywords:

Poisson equation; Overhauser cubic spline; boundary element method; numerical examples; biharmonic equations
PDF
BibTeX
XML
Cite

\textit{C. V. Camp} and \textit{G. S. Gipson}, Math. Comput. Modelling 15, No. 3--5, 59--69 (1991; Zbl 0726.65129)

Full Text:
DOI

##### References:

[1] | Ligget, J.A.; Salmon, J.R., Cubic spline boundary elements, International journal of numerical methods in engineering, 17, 543-556, (1981) · Zbl 0466.76093 |

[2] | Ortiz, J.C., An improved boundary element analysis system for the solution of Poisson’s equation, () |

[3] | Walters, H.G., Techniques for boundary element analysis in elastostatics influenced by geometric modelling, () |

[4] | Camp, C.V.; Gipson, G.S., A boundary element method for viscous flows at low Reynolds number, Engineering analysis, (1988), accepted for publication by · Zbl 0729.73255 |

[5] | Camp, C.V.; Gipson, G.S., Boundary element analysis of nonhomogeneous biharmonic phenomena, Computational mechanics, (1988), accepted for publication by · Zbl 0810.73001 |

[6] | () |

[7] | Zienkiewicz, O.C., The finite element method, (1977), McGraw-Hill Maidenhead, UK · Zbl 0435.73072 |

[8] | Kreyszig, E., Advanced engineering mathematics, (1983), Wiley & Sons · Zbl 0589.00002 |

[9] | Overhauser, A.W., Analytic definition of curves and surfaces by parabolic blending, Ford motor company technical report, SL68-40, (1968) |

[10] | Brewer, J.A., Three dimensional design by graphical man-computer communication, () |

[11] | Riccardella, P.C., An implementation of the boundary integral technique for planar problems in elasticity and elastoplasticity, () |

[12] | Gipson, G.S., Boundary element fundamentalsâ€”basic concepts and recent developments in the Poisson equation, (1987), Computational Mechanics Publications Boston · Zbl 0708.73088 |

[13] | Timoshenko, S.T.; Woinowsky-Krieger, S., Theory of plates and shells, (1959), McGraw-Hill New York · Zbl 0114.40801 |

[14] | Lebedev, N.N.; Skalskaya, I.P.; Ulfiyans, Y.S., Worked problems in applied mathematics, (1965), Dover Publications New York, (Translated by R.A. Silverman) |

[15] | Brewer, J.A.; Anderson, D.C., Visual interaction with overhauser curves and surfaces, Computer graphics, 11, 132-137, (1977) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.