Enhanced and efficient scaled boundary finite elements for mechanical problems under the action of body loads

Abstract

The Scaled Boundary Finite Element Method (SBFEM) can be viewed as a semi-analytical approach, which has been successfully applied to singular mechanical problems with vanishing force terms. Interpreted in the context of composite Duffy approximations based on star-shape polytopes (S-elements), and using an operator-adapted principle, the shape functions are determined by imposing an energy-orthogonality constraint. In the presence of non-vanishing forcing terms, the SBFEM approximation converges at sub-optimal rates. In the current paper, a strategy is proposed to overcome this drawback by the enrichment of the classic SBFEM spaces with energy-orthogonal bubble Duffy functions. The orthogonality property leads to a decoupling of the approximate solutions into components of boundary trace and interior (bubble) functions. The bubble functions can be solved separately, thus reducing the computational cost. A priori energy-error estimates are provided demonstrating optimal rates of convergence. The convergence rates are confirmed by two and three-dimensional numerical verification tests.