Improve the ill-conditioning for small physical covers by area normalized method in numerical manifold method (NMM)

With the weight functions on mathematical covers and the degrees of freedom (DOFs) on physical covers, the numerical manifold method (NMM) achieves a unified and natural simulation for continuous-discontinuous analysis. In the NMM, a small physical cover is often unavoidable when a mathematical node approaches a physical line. Such small physical cover may degrade the conditioning of the discretized system equations, which is termed as the small-physical-cover trouble (ST). In this paper, a modified NMM is proposed to fix the ST, where the area normalized method is proposed to precondition the local cover functions. The preconditioning is completed prior to the assembly of the discretized system equations, so the complicated matrix operations such as matrix inversion for large matrix can be avoided. With the local pixel representation of the finite cover systems, the area can be directly evaluated according to the number of pixels for each pixelated physical cover. The whole process does not need to integrate the polygonal domain, but only involves simple algebraic operations. The proposed approach features a simple principle and convenient numerical implementation. Several numerical examples demonstrate that the proposed approach can effectively improve the ill-conditioning induced by small physical covers.