Multidual Sensitivity Method in Ray-Tracing Transport Simulations

Balcer, Matthew; Millwater, Harry; Favorite, Jeffrey A.

doi:10.1080/00295639.2021.1883949

Cited by 2 publications

(2 citation statements)

References 21 publications

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“…Using this approach, the error in the derivative approximation can be reduced to machine precision because no subtraction cancellation error is present. In addition, the method has no truncation error; therefore, there is no dependence on the parameter h [69]. In general, a step size of h = 1 is used for simplicity.…”

Section: Arbitrary-order Differentiation Using Hypadmentioning

confidence: 99%

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Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD)

et al. 2023

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The calculation of accurate arbitrary-order sensitivities of eigenvalues and eigenvectors is crucial for structural analysis applications, including topology optimization, system identification, finite element model updating, damage detection, and fault diagnosis. Current approaches to obtaining sensitivities for eigenvalues and eigenvectors lack generality, are complicated to implement, prone to numerical errors, and are computationally expensive. In this work, a novel methodology is introduced that uses hypercomplex automatic differentiation (HYPAD) and semi-analytical expressions to obtain arbitrary-order sensitivities for eigenfrequency problems. The new methodology exhibits no sign of truncation nor subtractive cancellation errors regardless of the order of the sensitivity, it is general, and can obtain any high-order sensitivities with the simplicity of first-order computations. A numerical example is presented to verify the accuracy of the method, where the free vibration of a homogeneous cantilever beam is studied. For this problem, up to third-order sensitivities of the eigenvalues and eigenvectors with respect to the material and geometrical parameters were obtained, considering the cases of close and distinct eigenvalues. The results were verified using analytical equations, showing excellent agreement for the eigenvalues and the eigenvectors. The new method promises to facilitate the computation of sensitivities for eigenfrequency problems into routine practice and commercial software.

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Section: Arbitrary-order Differentiation Using Hypadmentioning

confidence: 99%

“…For more background about HYPAD and multidual numbers, the interested reader is directed to [48,49,69].…”

Section: Arbitrary-order Differentiation Using Hypadmentioning

confidence: 99%

Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD)

et al. 2023

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HYPAD-UQ: A Derivative-Based Uncertainty Quantification Method Using a Hypercomplex Finite Element Method

Balcer

Aristizábal

Tabares

et al. 2023

Journal of Verification, Validation and Uncertainty Quantification

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A derivative-based Uncertainty Quantification (UQ) method called HYPAD-UQ that utilizes sensitivities from a computational model was developed to approximate the statistical moments and Sobol' indices of the model output. HYPercomplex Automatic Differentiation (HYPAD) was used as a means to obtain accurate high-order partial derivatives from computational models such as finite element analyses. These sensitivities are used to construct a surrogate model of the output using a Taylor series expansion and subsequently used to estimate statistical moments (mean, variance, skewness, and kurtosis) and Sobol' indices using algebraic expansions. The uncertainty in a transient linear heat transfer analysis was quantified with HYPAD-UQ using first-order through seventh-order partial derivatives with respect to seven random variables encompassing material properties, geometry, and boundary conditions. Random sampling of the analytical solution and the regression-based stochastic perturbation finite element method were also conducted to compare accuracy and computational cost. The results indicate that HYPAD-UQ has superior accuracy for the same computational effort compared to the regression-based stochastic perturbation finite element method. Sensitivities calculated with HYPAD can allow higher-order Taylor series expansions to be an effective and practical UQ method.

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Multidual Sensitivity Method in Ray-Tracing Transport Simulations

Cited by 2 publications

References 21 publications

Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD)

Arbitrary-Order Sensitivity Analysis of Eigenfrequency Problems with Hypercomplex Automatic Differentiation (HYPAD)

HYPAD-UQ: A Derivative-Based Uncertainty Quantification Method Using a Hypercomplex Finite Element Method

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