Research Papers

Analytical Formulae for Potential Integrals on Triangles

[+] Author and Article Information
Michael J. Carley

Department of Mechanical Engineering,
University of Bath,
Bath, BA2 7AY, UK
e-mail: m.j.carley@bath.ac.uk

Manuscript received January 27, 2012; final manuscript received October 15, 2012; accepted manuscript posted October 22, 2012; published online May 16, 2013. Assoc. Editor: Glaucio H. Paulino.

J. Appl. Mech 80(4), 041008 (May 16, 2013) (7 pages) Paper No: JAM-12-1035; doi: 10.1115/1.4007853 History: Received January 27, 2012; Revised October 15, 2012; Accepted October 22, 2012

The problem of evaluating potential integrals on planar triangular elements has been addressed using a polar coordinate decomposition, giving explicit formulae for the regular and for the principal value and finite part integrals used in hypersingular formulations. The resulting formulae are general, exact, easily implemented, and have only one special case, that of a field point lying in the plane of the element. Results are presented for the evaluation of the potential and its gradients, where the integrals must be treated as principal values or finite parts, for elements with constant and linearly varying source terms. These results are tested by application to a single triangular element, and to the evaluation of a potential gradient outside the unit cube. In both cases, the method is shown to be accurate and convergent.

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Topics: Errors
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Grahic Jump Location
Fig. 1

Reference triangle for quadrature

Grahic Jump Location
Fig. 3

Reference triangle for test of integration procedures. Points 1–5 are the projections onto the triangle plane of the field points. Points are 1: (-0.5,-0.1), 2: (0.1,0.1), 3: (0.1,-0.101), 4: (-0.1,-0.099), 5: (0.3,0.2).

Grahic Jump Location
Fig. 2

Integration over a general triangle (left) by subdivision into three triangles centered at the origin (right). The triangles shown dashed on the right have negative orientation and their contribution is subtracted from that of the other.

Grahic Jump Location
Fig. 4

Surface for testing of gradient calculation, cube with ℓ = 1/2

Grahic Jump Location
Fig. 5

Error versus discretization length for gradient of Laplace potential outside the unit cube. Symbols: computed error; solid line: ɛ = 2.72h2.12.



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