390 research outputs found
Monotonicity-Based Regularization for Shape Reconstruction in Linear Elasticity
We deal with the shape reconstruction of inclusions in elastic bodies. For
solving this inverse problem in practice, data fitting functionals are used.
Those work better than the rigorous monotonicity methods from [5], but have no
rigorously proven convergence theory. Therefore we show how the monotonicity
methods can be converted into a regularization method for a data-fitting
functional without losing the convergence properties of the monotonicity
methods. This is a great advantage and a significant improvement over standard
regularization techniques. In more detail, we introduce constraints on the
minimization problem of the residual based on the monotonicity methods and
prove the existence and uniqueness of a minimizer as well as the convergence of
the method for noisy data. In addition, we compare numerical reconstructions of
inclusions based on the monotonicity-based regularization with a standard
approach (one-step linearization with Tikhonov-like regularization), which also
shows the robustness of our method regarding noise in practice.Comment: 26 pages, 15 figure
Iterative Methods for the Elasticity Imaging Inverse Problem
Cancers of the soft tissue reign among the deadliest diseases throughout the world and effective treatments for such cancers rely on early and accurate detection of tumors within the interior of the body. One such diagnostic tool, known as elasticity imaging or elastography, uses measurements of tissue displacement to reconstruct the variable elasticity between healthy and unhealthy tissue inside the body. This gives rise to a challenging parameter identification inverse problem, that of identifying the Lamé parameter μ in a system of partial differential equations in linear elasticity. Due to the near incompressibility of human tissue, however, common techniques for solving the direct and inverse problems are rendered ineffective due to a phenomenon known as the “locking effect”. Alternative methods, such as mixed finite element methods, must be applied to overcome this complication. Using these methods, this work reposes the problem as a generalized saddle point problem along with a presentation of several optimization formulations, including the modified output least squares (MOLS), energy output least squares (EOLS), and equation error (EE) frameworks, for solving the elasticity imaging inverse problem. Subsequently, numerous iterative optimization methods, including gradient, extragradient, and proximal point methods, are explored and applied to solve the related optimization problem. Implementations of all of the iterative techniques under consideration are applied to all of the developed optimization frameworks using a representative numerical example in elasticity imaging. A thorough analysis and comparison of the methods is subsequently presented
Uniqueness in inverse elastic scattering from unbounded rigid surfaces of rectangular type
Consider the two-dimensional inverse elastic scattering problem of
recovering a piecewise linear rigid rough or periodic surface of rectangular
type for which the neighboring line segments are always perpendicular.We
prove the global uniqueness with at most two incident elastic plane waves by
using near-field data. If the Lamé constants satisfy a certain condition,
then the data of a single plane wave is sufficient to imply the uniqueness.
Our proof is based on a transcendental equation for the Navier equation,
which is derived from the expansion of analytic solutions to the Helmholtz
equation. The uniqueness results apply also to an inverse scattering problem
for non-convex bounded rigid bodies of rectangular type
Local recovery of the compressional and shear speeds from the hyperbolic DN map
We study the isotropic elastic wave equation in a bounded domain with boundary. We show that local knowledge of the Dirichlet-to-Neumann map determines uniquely the speed of the p-wave locally if there is a strictly convex foliation with respect to it, and similarly for the s-wave speed.Peer reviewe
Direct and inverse elastic scattering problems for diffraction gratings
This paper is concerned with the direct and inverse scattering of time-harmonic plane elastic
waves by unbounded periodic structures (diffraction gratings). We present a variational approach
to the forward scattering problems with Lipschitz grating profiles and give a survey of recent
uniqueness and existence results. We also report on recent global uniqueness results within the
class of piecewise linear grating profiles for the corresponding inverse elastic scattering problems. Moreover, a discrete Galerkin method is presented to efficiently approximate solutions of direct
scattering problems via an integral equation approach. Finally, an optimization method for solving
the inverse problem of recovering a 2D periodic structure from scattered elastic waves measured
above the structure is discussed
Operator calculus approach to comparison of elasticity models for modelling of masonry structures
The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticity, micropolar elasticity and Cosserat elasticity. Evidently, a different macro-behaviour is expected depending on the specific theory used in the background. Although there have been several theoretical studies of different elasticity theories in recent years, there is still a lack of understanding of how modelling assumptions of different elasticity theories influence the modelling results of masonry structures. Therefore, a rigorous approach to comparison of different three-dimensional elasticity models based on quaternionic operator calculus is proposed in this paper. In this way, three elasticity models are described and spatial boundary value problems for these models are discussed. In particular, explicit representation formulae for their solutions are constructed. After that, by using these representation formulae, explicit estimates for the solutions obtained by different elasticity theories are obtained. Finally, several numerical examples are presented, which indicate a practical difference in the solutions
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