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High-Order Shape Functions in the Scaled Boundary Finite Element Method Revisited

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Abstract

The scaled boundary finite element method (SBFEM) is a semi-analytical approach to solving partial differential equations, in which a finite element approximation is deployed for the domain’s boundary, while analytical solutions are sought to describe the behavior in the interior of the domain. Since the inception of SBFEM, a number of different shape functions have been applied to interpolate the solution on the boundary. The overarching goal of this communication is to review the respective advantages and disadvantages of the available interpolants in the context of the SBFEM and develop recommendations regarding their application. In addition, we discuss in detail the discretization employed in the so-called diagonal SBFEM.

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Notes

  1. Exponential convergence can be achieved by using a p-refinement for sufficiently smooth problems or an hp-refinement if singularities are present in the solution.

  2. We assume here for ease of notation that the solution u(x, y) is a scalar field. The extension to vector fields is straightforward and analogous to other finite element approaches.

  3. While it would generally be possible to omit one of the sine or cosine terms to create an odd-numbered basis of shape functions, this would lead to ill-conditioning in the steps that follow.

  4. While other authors refer to the combination of IGA and SBFEM as SBIGA [41], SIGA [42, 43], or IGA-SBFEM [61] we will stick to the term SBFEM since in the context of this discussion the approaches differ only in the choice of shape functions.

  5. It can be noted that—compared to Fourier shape functions—the basis functions \(\varvec{\uppsi }(s)\) are relatively expensive to compute, which may be a drawback when constructing a nodal basis of high order.

  6. This is because \(\tilde{\mathbf {E}}_\mathbf {1}\) involves second derivatives of the Cartesian coordinates w.r.t. \(\eta\) (see Eqs. (36), (20), (11) in [34]), and the Coordinates are interpolated using standard finite element shape functions. All other terms in \(\tilde{\mathbf {E}}_\mathbf {1}\) involve derivatives of the diagonal shape functions and thus vanish when nodal quadrature is applied.

  7. This argument, of course, assumes that both approaches use the same nodal positions and quadrature scheme. In Ref. [33], GLL-points are used to construct the shape functions just like in SEM. In most other publications on the diagonal SBFEM, GLC points are employed which would be a possible yet highly uncommon choice in other approaches.

  8. The interested reader is referred to Ref. [14], in particular Sect. 4.1, which contains a similar argument to demonstrate the equivalence of different mass lumping techniques.

  9. Physically, this case corresponds to a steady-state heat conduction problem with unit thermal diffusivity in the absence of body loads.

  10. Again, it should be noted that in this example, where the geometry consists of straight edges only, the NURBS shape functions reduce to simple B-splines.

  11. Again assuming the somewhat loose definition of p-refinement as discussed in Sect. 4.1.1.

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Authors and Affiliations

  1. University of Duisburg-Essen, Universitätsstraße 15, 45141, Essen, Germany

    Hauke Gravenkamp

  2. University of New South Wales Sydney, Sydney, 2052, Australia

    Albert A. Saputra & Sascha Duczek

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  1. Hauke Gravenkamp
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Gravenkamp, H., Saputra, A.A. & Duczek, S. High-Order Shape Functions in the Scaled Boundary Finite Element Method Revisited. Arch Computat Methods Eng 28, 473–494 (2021). https://doi.org/10.1007/s11831-019-09385-1

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