Modal Response Sensitivity Analysis With Arbitrary-Order Derivatives
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Solution Overview
Problem
Current methods for calculating sensitivities of eigenvalues and eigenvectors in structural systems are limited to first-order approximations, leading to numerical issues, complexity, and high computational costs, especially in complex systems, and lack generality.
Innovation Solution
A methodology using hypercomplex automatic differentiation (HYPAD) and semi-analytical expressions to compute arbitrary-order sensitivities of eigenvalues and eigenvectors, eliminating truncation and subtractive cancellation errors, and enabling efficient sensitivity analysis through a single general expression.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If first-order approximation methods are used for sensitivity analysis, then the computational process is simpler, but the accuracy is limited and numerical errors occur
Solution Approach 1:
The patent transforms the sensitivity analysis problem by changing the mathematical parameter representation from real numbers to dual numbers. This parameter transformation enables exact derivative computation through algebraic operations on dual numbers, where the imaginary unit satisfies ε²=0. The sensitivity derivatives are obtained as coefficients of the dual number components, eliminating the need for finite difference approximations and their associated truncation errors while maintaining computational tractability.
Solution Approach 2:
The patent replaces the traditional mechanical/computational approach of finite difference methods with an algebraic substitution method using dual numbers. Instead of numerically approximating derivatives through perturbation and division operations, the invention substitutes the real number system with the dual number system, where derivatives emerge naturally from algebraic expansion. This substitution eliminates subtractive cancellation errors and provides exact sensitivity information without requiring complex computational procedures.
2Measurement precision
If higher-order sensitivity analysis is performed, then the accuracy improves, but the computational cost increases significantly
Solution Approach 1:
The patent extends the dual number parameter transformation to higher-order derivatives by using higher-order dual numbers where the imaginary unit satisfies ε^(k+1)=0 for k-th order derivatives. This parameter transformation allows simultaneous computation of all derivatives up to order k through a single algebraic evaluation, avoiding the exponential growth of computational cost that would result from computing each derivative separately through multiple finite difference steps.
Solution Approach 2:
The patent performs preliminary transformation of the system parameters into dual number form before solving the eigenvalue problem. This preliminary action ensures that all derivative information is embedded in the parameter representation from the outset, allowing the eigenvalue solver to directly produce sensitivity information without requiring post-processing or additional computational steps for each derivative order.
3Ease of manufacture
If traditional finite difference methods are used, then the implementation is straightforward, but truncation and subtractive cancellation errors occur
Solution Approach 1:
The patent substitutes the finite difference computational mechanism with dual number algebra. In the dual number system, derivatives are obtained through exact algebraic expansion rather than numerical approximation. The sensitivity derivatives appear as coefficients in the dual number representation, eliminating subtractive cancellation errors that plague finite difference methods when small perturbation sizes are used. This substitution maintains implementation simplicity while dramatically improving numerical reliability.
Solution Approach 2:
The patent changes the fundamental parameter type from real numbers to dual numbers, which inherently encode derivative information. This parameter change transforms the computation from numerical approximation to exact algebraic evaluation, where the sensitivity derivatives are obtained as exact coefficients rather than approximations subject to truncation and round-off errors.
Data Source
AI summary
A method for determining the arbitrary-order sensitivities of eigenvalues and eigenvectors of a structural system includes the steps of receiving data related to the structural system including model parameters {α1, . . . , αj} of the structural system and a maximum order of the derivative m to be computed, computing system matrices [K] and [M] and their partial derivatives, solving real-value generalized eigenvalue problem, determining sensitivities for the eigenvalues and the eigenvectors, and presenting an output of the sensitivities for the eigenvalues and the eigenvectors.