Efficient and Accurate Computation of Arbitrary-Order Eigenpair Sensitivities Using Hypercomplex Automatic Differentiation

Juan C. Velasquez‐Gonzalez

doi:10.1002/nme.70245

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Efficient and Accurate Computation of Arbitrary-Order Eigenpair Sensitivities Using Hypercomplex Automatic Differentiation

2025-12-30

Juan C. Velasquez‐Gonzalez

DOI：10.1002/nme.70245

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Abstract

En 中文

Eigenvalue and eigenvector sensitivities with respect to design parameters are crucial for advancing design, optimization, and uncertainty quantification in structural systems. This paper introduces a novel, efficient, and general numerical method for computing arbitrary-order sensitivities of eigenpairs in self-adjoint undamped and underdamped systems. The proposed approach integrates Hypercomplex Automatic Differentiation (HYPAD) with a residual-based formulation to compute sensitivities with machine precision. Sensitivities are calculated in ascending order by solving a sequence of linear systems that share a common coefficient matrix. The method preserves the sparsity of the mass and stiffness matrices, allowing for efficient factorization and compatibility with current sparse direct solvers. The methodology is demonstrated through a numerical example under both undamped and underdamped conditions. Up to tenth-order sensitivities are computed with respect to multiple material and geometric parameters, showing excellent agreement with analytical solutions. Runtime analysis confirms that the computational cost per derivative remains constant, regardless of the order, underscoring the method's efficiency. Overall, the proposed approach offers a scalable and accurate framework for sensitivity analysis in large-scale eigenvalue problems.

Keywords:

EIGENVECTOR DERIVATIVES

EIGENVALUES

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