Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

Wöstehoff, A.; Schuster, Thomas

doi:10.1080/00036811.2014.940519

Cited by 8 publications

(16 citation statements)

References 20 publications

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“…All these constraints are necessary to prove the following uniqueness result for the solution of the IBVP and – for given

α \in scriptC ((), {((), κ^{[a]})}_{a = 1, 2}, {((), μ^{[b]})}_{b = 1, \dots, 7})

, which has been presented in . Theorem Let u and

ū

be two solutions to the IBVP and – corresponding to the parameters, initial values, and right‐hand sides ( α , u 0 , u 1 , f ) and

(\overset{α}{true}, ū_{0}, ū_{1}, \overset{f}{true})

, respectively.…”

Section: Setting the Stagementioning

confidence: 98%

“…We start by recapitulating an existing uniqueness result for the IBVP (8) and (4)-(6) from [25] as well as some important estimates that we need to prove our main results.…”

Section: Setting the Stagementioning

confidence: 99%

“…In this article, we provide prerequisites for solving numerically the inverse problem to be specified in the following. Inspired by Kaltenbacher and Lorenzi and following the authors of , we suppose to have a dictionary { C 1 , C 2 ,…, C N } consisting of functions C K = C K ( x , Y ), K = 1,…, N , given such that

C (x, Y) = \sum_{K = 1}^{N} α_{K} C_{K} (x, Y)

with positive constants α K >0, K = 1,…, N . In that way, the searched function C is projected onto a dictionary consisting of physically meaningful elements such as polyconvex functions .…”

Section: Introductionmentioning

confidence: 99%

“…All these constraints are necessary to prove the following uniqueness result for the solution of the IBVP (8) and (4) Á , which has been presented in [25].…”

mentioning

confidence: 99%

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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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“…All these constraints are necessary to prove the following uniqueness result for the solution of the IBVP and – for given

α \in scriptC ((), {((), κ^{[a]})}_{a = 1, 2}, {((), μ^{[b]})}_{b = 1, \dots, 7})

, which has been presented in . Theorem Let u and

ū

be two solutions to the IBVP and – corresponding to the parameters, initial values, and right‐hand sides ( α , u 0 , u 1 , f ) and

(\overset{α}{true}, ū_{0}, ū_{1}, \overset{f}{true})

, respectively.…”

Section: Setting the Stagementioning

confidence: 98%

“…We start by recapitulating an existing uniqueness result for the IBVP (8) and (4)-(6) from [25] as well as some important estimates that we need to prove our main results.…”

Section: Setting the Stagementioning

confidence: 99%

C (x, Y) = \sum_{K = 1}^{N} α_{K} C_{K} (x, Y)

with positive constants α K >0, K = 1,…, N . In that way, the searched function C is projected onto a dictionary consisting of physically meaningful elements such as polyconvex functions .…”

Section: Introductionmentioning

confidence: 99%

“…All these constraints are necessary to prove the following uniqueness result for the solution of the IBVP (8) and (4) Á , which has been presented in [25].…”

mentioning

confidence: 99%

See 2 more Smart Citations

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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“…For the investigations and proofs in this article we need at first some results the most of which are proven in [31] and [28]. The first one is a uniqueness result for the IBVP (1.6), (1.2)-(1.4) from [31]. In the following we assume that the conditions C K (x, 0) = 0 and ∇ Y C K (x, 0) = 0 are valid for the function C K : Ω × R 3×3 → R for all K = 1, ..., N and x ∈ Ω ⊂ R 3 .…”

Section: Denoting By T : D(t )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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Sequential Subspace Optimization for Recovering Stored Energy Functions in Hyperelastic Materials from Time-Dependent Data

Klein

Schuster

Wald

2021

Time-Dependent Problems in Imaging and Parameter Identification

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Uniqueness and stability result for Cauchy’s equation of motion for a certain class of hyperelastic materials

Cited by 8 publications

References 20 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

Sequential Subspace Optimization for Recovering Stored Energy Functions in Hyperelastic Materials from Time-Dependent Data

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