Abstract
In this work, we develop a specialized quadrature rule for trimmed domains, where the trimming curve is given implicitly by a real-valued function on the whole domain. We follow an error correction approach: In a first step, we obtain an adaptive subdivision of the domain in such a way that each cell falls in a predefined base case. We then extend the classical approach of linear approximation of the trimming curve by adding an error correction term based on a Taylor expansion of the blending between the linearized implicit trimming curve and the original one. This approach leads to an accurate method which improves the convergence of the quadrature error by one order compared to piecewise linear approximation of the trimming curve. It is at the same time efficient, since essentially the computation of one extra one-dimensional integral on each trimmed cell is required. Finally, the method is easy to implement, since it only involves one additional line integral and refrains from any point inversion or optimization operations. The convergence is analyzed theoretically and numerical experiments confirm that the accuracy is improved without compromising the computational complexity .
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The weights were chosen as \((\omega _{0,0},\omega _{1,0},\omega _{0,1},\omega _{1,1},\omega _{0,2},\omega _{1,2}) = \left (1,1,\frac {1}{\sqrt {2}},\frac {1}{\sqrt {2}},1,1\right )\) and the corresponding control points as (1, 0), (2, 0), (1, 1), (2, 2), (0, 1), (0, 2).
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Acknowledgements
The authors gratefully acknowledge the support provided by the Austrian Science Fund (FWF) through project NFN S11708 and by the European Research Council (ERC), project GA 694515.
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Scholz, F., Mantzaflaris, A., Jüttler, B. (2019). First Order Error Correction for Trimmed Quadrature in Isogeometric Analysis. In: Apel, T., Langer, U., Meyer, A., Steinbach, O. (eds) Advanced Finite Element Methods with Applications. FEM 2017. Lecture Notes in Computational Science and Engineering, vol 128. Springer, Cham. https://doi.org/10.1007/978-3-030-14244-5_15
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