Refined isogeometric analysis for generalized Hermitian eigenproblems.

We use refined isogeometric analysis (rIGA) to solve generalized Hermitian eigenproblems (K u = λ M u). 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. • The article proposes to use rIGA discretizations to solve generalized Hermitian eigenproblems. • The shift-and-invert spectral transformation is implemented for eigenpair calculation. • Computational cost of rIGA is compared to that of high-continuity IGA discretization. • Total computational cost of eigenanalysis is reduced by up to O (p) when using rIGA instead of IGA. • A better accuracy of first N0 eigenvalues and eigenfunctions is achieved by using rIGA instead of IGA. [ABSTRACT FROM AUTHOR]