Conditioned Numerical Shape Functions on Unfitted Reduced Coarse Elements for Robust Analysis of Complex Solid Structures

The unfitted finite element (FE) methods offer significant advantages in engineering analysis by embedding the structure within a simple background mesh and eliminating the need for complex and labor-intensive mesh generation. A high solution accuracy can still be achieved via constructing, on each coarse element, standard polynomial shape functions, or numerical (piecewise) shape functions. However, the strategy incurs significant technical challenges due to the unavoidable occurrence of cut elements of arbitrarily small size that may greatly deteriorate the condition number of the stiffness matrix. To address the issue, we propose the concept of reduced coarse elements. By formulating high-order numerical shape functions as the product of a boundary interpolator and a boundary-interior mapping, a detailed condition number analysis reveals the dependence of the numerical shape function conditioning on the boundary interpolator. Based on these findings, we develop a new type of reduced coarse elements and their associated numerical shape functions to address the conditioning challenges. The stability, accuracy, convergence rate, and efficiency of the approach are tested through various numerical examples in comparison with other cutting-edge approaches. Its performance on a multi-material printed circuit board (PCB) example of 183 million fine mesh nodes is also tested.