Clifford algebra valued boundary integral equations for three-dimensional elasticity

Abstract

• Clifford analysis is applied to solve 3D problems of elasticity. • BIEs for Clifford algebra valued harmonic functions are derived. • The singularity-free BIEs are constructed without extra equations/variables/information. • The Clifford algebra valued singularity-free BEM is developed by a regular quadrature. • 3D problems of elasticity can be solved using the Clifford algebra valued BEM. Applications of Clifford analysis to three-dimensional elasticity are addressed in the present paper. The governing equation for the displacement field is formulated in terms of the Dirac operator and Clifford algebra valued functions so that a general solution is obtained analytically in terms of one monogenic function and one multiple-component spatial harmonic function together with its derivative. In order to solve numerically the three-dimensional problems of elasticity for an arbitrary domain with complicated boundary conditions, Clifford algebra valued boundary integral equations (BIEs) for multiple-component spatial harmonic functions at an observation point, either inside the domain, on the boundary, or outside the domain, are constructed. Both smooth and non-smooth boundaries are considered in the construction. Moreover, the singularities of the integrals are evaluated exactly so that in the end singularity-free BIEs for the observation point on the boundary taking values on Clifford numbers can be obtained. A Clifford algebra valued boundary element method (BEM) based on the singularity-free BIEs is then developed for solving three-dimensional problems of elasticity. The accuracy of the Clifford algebra valued BEM is demonstrated numerically.