Refined isogeometric analysis for generalized Hermitian eigenproblems
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We use the refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems (Ku=λ 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 [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 cost of factorization of the spectral transformation matrix K-σ_k M drives the total computational cost of the eigensolution. Several multiplications of the operator matrices (K-σ_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.
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