Refined isogeometric analysis for generalized Hermitian eigenproblems

Abstract

We use the refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems $({Ku=\lambda Mu})$. The rIGA framework conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) discretizations while reducing the computation cost of the solution through partitioning the computational domain by adding zero-continuity basis functions. As a result, rIGA enriches the approximation space and decreases the interconnection between degrees of freedom. We compare computational costs of rIGA versus those of IGA when employing a Lanczos eigensolver with a shift-and-invert spectral transformation. When all eigenpairs within a given interval ${[\lambda_s,\lambda_e]}$ are of interest, we select several shifts ${\sigma_k\in[\lambda_s,\lambda_e]}$ using a spectrum slicing technique. For each shift $\sigma_k$, the cost of factorization of the spectral transformation matrix ${K-\sigma_k M}$ drives the total computational cost of the eigensolution. Several multiplications of the operator matrices ${(K-\sigma_k M)^{-1} M}$ by vectors follow this factorization. Let $p$ be the polynomial degree of basis functions and assume that IGA has maximum continuity of ${p-1}$, while rIGA introduces $C^0$ separators to minimize the factorization cost. For this setup, our theoretical estimates predict computational savings to compute a fixed number of eigenpairs of up to ${O(p^2)}$ in the asymptotic regime, that is, large problem sizes. Yet, our numerical tests show that for moderately-sized eigenproblems, the total computational cost reduction is $O(p)$. Nevertheless, rIGA improves the accuracy of every eigenpair of the first $N_0$ eigenvalues and eigenfunctions. Here, we allow $N_0$ to be as large as the total number of eigenmodes of the original maximum-continuity IGA discretization.