Improved Numerical Robustness of the X-FFT Solver via Internal Scaling
The recently introduced X-FFT solver improves the spatial accuracy of FFT-based homogenization methods for two-dimensional thermal homogenization problems without compromising their numerical efficiency. Through the use of an X-FEM discretization, optimal error convergence rates of the discretization error are achieved, and the developed X-FFT preconditioner guarantees a mesh-independent upper bound on the condition number. In the work at hand, we introduce an internal scaling of the enriched shape functions that prevents numerical instabilities and simplifies the X-FFT preconditioner. Our numerical studies demonstrate that the accuracy and the efficiency of the X-FFT solver remain unaffected by the internal scaling.
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