Inverse problems in elasticity
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Citations
A Biot model for the determination of material parameters of cancellous bone from acoustic measurements
Physical probing of cells
An iterative method for parameter identification and shape reconstruction
Inverse elastic scattering problems with phaseless far field data
An overview of inverse material identification within the frameworks of deterministic and stochastic parameter estimation
References
The Mathematical Theory of Finite Element Methods
Inverse Acoustic and Electromagnetic Scattering Theory
Related Papers (5)
Error Estimate Procedure in the Finite Element Method and Applications
Frequently Asked Questions (17)
Q2. What future works have the authors mentioned in the paper "Inverse problems in elasticity" ?
Some directions for further work directly related to topics presented in this article include ( i ) a more systematic development and testing of the reciprocity gap approach as a method for the identification of cracks, ( ii ) the study of convexity properties of functionals based on the error in constitutive equations, and develop similar functional in connection with the identification of nonlinear constitutive properties and ( iii ) the development of topological sensitivity techniques for time-domain formulations and in refined forms ( in particular based on higher-order expansions with respect to defect size ) and their integration in defect identification strategies.
Q3. What is the class of inverse problem of engineering interest?
The reconstruction of distributed parameters (such as the flexural stiffness or the mass density) of mechanical structures from vibrational data, i.e. measured values of eigenfrequencies and eigenmodal displacements, is a class of inverse problem of engineering interest, especially in connection with updating finite element (FE) models of mechanical structures, i.e. correcting FE models so that they agree best with measurements on the real structure.
Q4. What is the main drawback of the least squares functionals?
The major drawback of the least squares functionals are generally bad stability properties, in the sense that small data errors induce large errors in the solution.
Q5. What is the importance of the minimization of the shape of using such methods?
The minimization of J with respect to Γ using such methods needs in turn, for efficiency, the ability to evaluate the gradient of the functional J with respect to perturbations of the shape of Γ, in addition to J (Γ) itself.
Q6. How many cases of applied forces were generated?
Synthetic data was generated by solving the plate bending problem for the true distribution of D for nine cases of applied forces, in this case concentrated loads applied at nine different locations.
Q7. What is the topological derivative approach for inverse elastostatics?
In cases featuring only one characteristic length (one scatterer embedded in an unbounded medium an illuminated by a plane wave), the topological derivative approach essentially provides (up to a scaling factor) the lowest-order moment of the normalized scattering amplitude in the theory of low-frequency direct and inverse scattering [37, 53, 54].
Q8. What are some directions for further work related to the topics presented in this article?
Some directions for further work directly related to topics presented in this article include(i) a more systematic development and testing of the reciprocity gap approach as a method for the identification of cracks, (ii) the study of convexity properties of functionals based on the error in constitutive equations, and develop similar functional in connection with the identification of nonlinear constitutive properties and (iii) the development of topological sensitivity techniques for time-domain formulations and in refined forms (in particular based on higher-order expansions with respect to defect size) and their integration in defect identification strategies.
Q9. What is the fundamental balance equation of the dynamics of deformable bodies?
The fundamental balance equation of the dynamics of deformable bodies (an extension of Newton’s second law to a small material element) is then:div σ(x, t) + f(x, t) − ρ(x)ü(x, t) = 0 (7)where f(x, t) is a given distribution of body forces.
Q10. What is the need to keep the number of direct computations as low as possible?
The need to keep the number of direct computations as low as possible suggests instead to stick with classical gradient-based optimization algorithm.
Q11. What is the role of the virtual work principle in determining the quantity of displacement fields?
In such situations, the virtual work principle provides a direct link between theexperimental data and the unknown quantities, which are usually related to constitutive properties, defects or damage.
Q12. How many independent combinations of the elastic coefficients Cijkl(x) can be?
using ϕ[z] = ϕ(1)[z] in (41) for all y and all ϑ allows to find at most five independent linear combinations of the elastic coefficients δCijkℓ(x).
Q13. What are the types of inverse problems in the linear theory of elasticity?
In this article, several types of inverse problems arising in the linear theory of elasticity have been considered, revolving around the identification of material parameters, distributions of elastic moduli, and geometrical objects such as cracks and inclusions.
Q14. What is the topological derivative for acoustic scattering?
Numerical experiments based on the topological derivative in elastodynamics recently appeared in connnection with identification of cavities [26, 76] and of penetrable elastic inclusions [77].6.4.2. Topological derivative for acoustic scattering.
Q15. What is the corresponding integral in (134)?
Note that the last integral in (122), which is unaffected by whether time-harmonic or transient conditions are assumed, has been omitted in (136) for the sake of brevity.
Q16. What is the topological derivative for the case of a hard obstacle?
For the simplest case where B is the unit sphere, one hasDij(B, β, γ) = 3(1 − β)2 + β δijThe limiting situation β = 0 in (154) yields the expression of the topological derivative for the case of a hard (i.e. rigid) obstacle of vanishing size ε.6.4.3.
Q17. What are the techniques used to improve the effectiveness of the virtual field method?
To enhance the effectiveness and efficiency of this technique, a variety of techniques have been proposed for the construction of families of virtual fields w(x) tailored for specific classes of constitutive parameter identification problems.