Determining kernels in linear viscoelasticity

Barbara Kaltenbacher; Ustim Khristenko; Vanja Nikolić; Mabel Lizzy Rajendran; Barbara Wohlmuth

doi:10.1016/j.jcp.2022.111331

Determining kernels in linear viscoelasticity

Barbara Kaltenbacher, Ustim Khristenko, Vanja Nikolić^*, Mabel Lizzy Rajendran, Barbara Wohlmuth

^*Corresponding author for this work

Mathematics

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Abstract

In this work, we investigate the inverse problem of determining the kernel functions that best describe the mechanical behavior of a complex medium modeled by a general nonlocal viscoelastic wave equation. To this end, we minimize a tracking-type data misfit function under this PDE constraint. We perform the well-posedness analysis of the state and adjoint problems and, using these results, rigorously derive the first-order sensitivities. Numerical experiments in a three-dimensional setting illustrate the method.

Original language	English
Article number	111331
Number of pages	23
Journal	Journal of Computational Physics
Volume	464
Early online date	26 May 2022
DOIs	https://doi.org/10.1016/j.jcp.2022.111331
Publication status	Published - 1 Sept 2022

Bibliographical note

Acknowledgement:
The authors UK and BW gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG) within the project WO 671/11-1. The work of BK was supported by the Austrian Science Fund FWF under the grants P30054 and DOC 78. MLR acknowledges the support by the Laura Bassi Postdoctoral Fellowship (Technical University of Munich).

We would like to thank the reviewer for the careful reading of our manuscript and for the very helpful remarks.

Keywords

Viscoelasticity
Weakly singular kernels
Inverse problem

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title = "Determining kernels in linear viscoelasticity",

abstract = "In this work, we investigate the inverse problem of determining the kernel functions that best describe the mechanical behavior of a complex medium modeled by a general nonlocal viscoelastic wave equation. To this end, we minimize a tracking-type data misfit function under this PDE constraint. We perform the well-posedness analysis of the state and adjoint problems and, using these results, rigorously derive the first-order sensitivities. Numerical experiments in a three-dimensional setting illustrate the method.",

keywords = "Viscoelasticity, Weakly singular kernels, Inverse problem",

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AU - Kaltenbacher, Barbara

AU - Khristenko, Ustim

AU - Nikolić, Vanja

AU - Rajendran, Mabel Lizzy

AU - Wohlmuth, Barbara

N1 - Acknowledgement: The authors UK and BW gratefully acknowledge the support of the Deutsche Forschungsgemeinschaft (DFG) within the project WO 671/11-1. The work of BK was supported by the Austrian Science Fund FWF under the grants P30054 and DOC 78. MLR acknowledges the support by the Laura Bassi Postdoctoral Fellowship (Technical University of Munich). We would like to thank the reviewer for the careful reading of our manuscript and for the very helpful remarks.

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N2 - In this work, we investigate the inverse problem of determining the kernel functions that best describe the mechanical behavior of a complex medium modeled by a general nonlocal viscoelastic wave equation. To this end, we minimize a tracking-type data misfit function under this PDE constraint. We perform the well-posedness analysis of the state and adjoint problems and, using these results, rigorously derive the first-order sensitivities. Numerical experiments in a three-dimensional setting illustrate the method.

AB - In this work, we investigate the inverse problem of determining the kernel functions that best describe the mechanical behavior of a complex medium modeled by a general nonlocal viscoelastic wave equation. To this end, we minimize a tracking-type data misfit function under this PDE constraint. We perform the well-posedness analysis of the state and adjoint problems and, using these results, rigorously derive the first-order sensitivities. Numerical experiments in a three-dimensional setting illustrate the method.

KW - Viscoelasticity

KW - Weakly singular kernels

KW - Inverse problem

U2 - 10.1016/j.jcp.2022.111331

DO - 10.1016/j.jcp.2022.111331

M3 - Article

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VL - 464

JO - Journal of Computational Physics

JF - Journal of Computational Physics

M1 - 111331

ER -

Determining kernels in linear viscoelasticity

Abstract

Bibliographical note

Keywords

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