Refined isogeometric analysis for generalized Hermitian eigenproblems
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We use refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems (Ku = λMu). rIGA conserves the desirable properties of maximum-continuity isogeometric analysis (IGA) while it reduces the solution cost by adding zero-continuity basis functions, which decrease the matrix connectivity. As a result, rIGA enriches the approximation space and reduces 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 [λ_s,λ_e] are of interest, we select several shifts σ_k ∈ [λ_s,λ_e] using a spectrum slicing technique. For each shift σ_k, the factorization cost of the spectral transformation matrix K − σ_k M controls the total computational cost of the eigensolution. Several multiplications of the operator matrix (K − σ_k M)^−1 M by vectors follow this factorization. Let p be the polynomial degree of the basis functions and assume that IGA has maximum continuity of p−1. When using rIGA, we introduce C^0 separators at certain element interfaces 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 moderate-size eigenproblems, the total observed computational cost reduction is O(p). In addition, rIGA improves the accuracy of every eigenpair of the first N_0 eigenvalues and eigenfunctions, where N_0 is the total number of modes of the original maximum-continuity IGA discretization.