Defect localization in fibre-reinforced composites by computing external volume forces from surface sensor measurements

Binder, Frank H; Schöpfer, Frank; Schuster, Thomas

doi:10.1088/0266-5611/31/2/025006

Cited by 14 publications

(23 citation statements)

References 35 publications

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“…The next lemma ensues directly from an application of theorem 30.4 in (cf. ).Lemma Suppose that f ∈ H 1 (0, T ; H ),

v_{0} \in H_{0}^{2} ((), normalΩ, {double-struckR}^{3}) \subset U

, and v 1 ∈ U . Then the unique solution v of satisfies

v \in (H^{1} (0, T; U) \cap H^{2} (0, T; H))

as well as

v \in L^{2} (0, T; H_{0}^{2} (Ω, R^{3})) .

…”

Section: Setting the Stagementioning

confidence: 97%

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On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

Seydel

Schuster

2016

Math Methods in App Sciences

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We compute a local linearization for the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as a solution of the nonlinear, dynamic, elastic wave equation, where the first Piola–Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense, the mathematical model of the direct problem is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article, we summarize some continuity results for this operator and deduce its Fréchet derivative as well as the adjoint of this derivative. Because the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton‐type methods, the knowledge of the Fréchet derivative and its adjoint is of utmost importance. Copyright © 2016 John Wiley & Sons, Ltd.

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“…The next lemma ensues directly from an application of theorem 30.4 in (cf. ).Lemma Suppose that f ∈ H 1 (0, T ; H ),

v_{0} \in H_{0}^{2} ((), normalΩ, {double-struckR}^{3}) \subset U

, and v 1 ∈ U . Then the unique solution v of satisfies

v \in (H^{1} (0, T; U) \cap H^{2} (0, T; H))

as well as

v \in L^{2} (0, T; H_{0}^{2} (Ω, R^{3})) .

…”

Section: Setting the Stagementioning

confidence: 97%

“…The next lemma ensues directly from an application of theorem 30.4 in [29] (cf. [30]). One principal technique to prove the results of this article takes advantage of Gronwall's lemma.…”

Section: Proofmentioning

confidence: 99%

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

Seydel

Schuster

2016

Math Methods in App Sciences

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“…Thus the excitation signal acts in x 3 -direction. We have taken −15 15 this approach for the excitation signal from [6]. The only difference is that their the wave is excited at four different locations.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Because of (2.32) it is easy to see that Q is linear in u ∈ L 2 (0, T ; U ). Furthermore it is proven in [6] that Q is continuous in u ∈ L 2 (0, T ; U ) and that the adjoint operator Q * has for all a ∈ L 2 (0, T ; R l ) the representation…”

Section: )mentioning

confidence: 99%

Identifying the stored energy of a hyperelastic structure by using an attenuated Landweber method

Seydel

Schuster

2017

Inverse Problems

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Abstract. We consider the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field as well as from surface sensor measurements. The displacement field is represented as a solution of Cauchy's equation of motion, which is a nonlinear, elastic wave equation. Hyperelasticity means that the first Piola-Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that a dictionary of suitable functions is available and the aim is to recover the stored energy with respect to this dictionary. The considered inverse problem is of vital interest for the development of structural health monitoring systems which are constructed to detect defects in elastic materials from boundary measurements of the displacement field, since the stored energy encodes the mechanical peroperties of the underlying structure. In this article we develope a numerical solver for both settings using the attenuated Landweber method. We show that the parameter-to-solution map satisfies the local tangential cone condition. This result can be used to prove local convergence of the attenuated Landweber method in case that the full displacement field is measured. In our numerical experiments we demonstrate how to construct an appropriate dictionary and show that our algorithm is well suited to localize damages in various situations.

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“…In parameter identification, one usually aims at the reconstruction of parameters that appear in the coefficients of a (partial) differential equation from the knowledge of the solution or the state of the system, or even boundary measurements thereof. Examples are electric impedance tomography [10], photoacoustic tomography, optical coherence tomography [14] or dynamic load monitoring [8]. The inverse problem we are concerned with in this article does not only aims at the reconstruction of parameters, but also requires the determination of the respective solution or state function.…”

mentioning

confidence: 99%

On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging

Nguyen¹,

Wald²

2022

IPI

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The Landau-Lifshitz-Gilbert equation yields a mathematical model to describe the evolution of the magnetization of a magnetic material, particularly in response to an external applied magnetic field. It allows one to take into account various physical effects, such as the exchange within the magnetic material itself. In particular, the Landau-Lifshitz-Gilbert equation encodes relaxation effects, i.e., it describes the time-delayed alignment of the magnetization field with an external magnetic field. These relaxation effects are an important aspect in magnetic particle imaging, particularly in the calibration process. In this article, we address the data-driven modeling of the system function in magnetic particle imaging, where the Landau-Lifshitz-Gilbert equation serves as the basic tool to include relaxation effects in the model. We formulate the respective parameter identification problem both in the all-at-once and the reduced setting, present reconstruction algorithms that yield a regularized solution and discuss numerical experiments. Apart from that, we propose a practical numerical solver to the nonlinear Landau-Lifshitz-Gilbert equation, not via the classical finite element method, but through solving only linear PDEs in an inverse problem framework.

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Defect localization in fibre-reinforced composites by computing external volume forces from surface sensor measurements

Cited by 14 publications

References 35 publications

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

Identifying the stored energy of a hyperelastic structure by using an attenuated Landweber method

On numerical aspects of parameter identification for the Landau-Lifshitz-Gilbert equation in Magnetic Particle Imaging

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