### On The Relaxed Continuity Approach For The Selfregular Traction-Bie.

#### Abstract

The ‘relaxed continuity’ hypothesis adopted on the self-regular traction-BIE is

investigated for bidimensional problems. The self-regular traction-BIE, a fully regular

equation, is derived from Somigliana stress identity, which contains hypersingular integrals.

Due to the presence of hypersingular integrals the displacement field is required to achieve

C1, Hölder continuity. This condition is not met by the use of standard conforming elements,

based on C0 interpolation functions, which only provide a piecewise C1, continuity. Thus, a

relaxed continuity hypothesis is adopted, allowing the displacement field to be C1, piecewise

continuous at the vicinity of the source point. The self-regular traction-BIE makes use of the

displacement tangential derivatives, which are not part of the original BIE. The tangential

derivatives are obtained from the derivative of the element interpolation functions. Therefore,

two possible sources of error, which are the discontinuity of the displacement gradients at

inter-element nodes and the approximation of the displacement tangential derivatives, are

introduced. In order to establish the dominant error, non-conforming elements are

implemented since they satisfy the continuity requirement at each collocation point. Standard

Gaussian integration scheme is applied in the evaluation of all integrals involved. Quadratic,

cubic and quartic isoparametric boundary elements are employed. Some numerical results

are presented comparing the accuracy of conforming and non-conforming elements on the

self-regular traction-BIE and highlighting the dominant error.

investigated for bidimensional problems. The self-regular traction-BIE, a fully regular

equation, is derived from Somigliana stress identity, which contains hypersingular integrals.

Due to the presence of hypersingular integrals the displacement field is required to achieve

C1, Hölder continuity. This condition is not met by the use of standard conforming elements,

based on C0 interpolation functions, which only provide a piecewise C1, continuity. Thus, a

relaxed continuity hypothesis is adopted, allowing the displacement field to be C1, piecewise

continuous at the vicinity of the source point. The self-regular traction-BIE makes use of the

displacement tangential derivatives, which are not part of the original BIE. The tangential

derivatives are obtained from the derivative of the element interpolation functions. Therefore,

two possible sources of error, which are the discontinuity of the displacement gradients at

inter-element nodes and the approximation of the displacement tangential derivatives, are

introduced. In order to establish the dominant error, non-conforming elements are

implemented since they satisfy the continuity requirement at each collocation point. Standard

Gaussian integration scheme is applied in the evaluation of all integrals involved. Quadratic,

cubic and quartic isoparametric boundary elements are employed. Some numerical results

are presented comparing the accuracy of conforming and non-conforming elements on the

self-regular traction-BIE and highlighting the dominant error.

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Güemes 3450

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Phone: 54-342-4511594 / 4511595 Int. 1006

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**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**