A well-posed multilayer model for granular avalanches with μ(I) rheology

The problem of finding a relaxation time spectrum best fitting dynamic moduli data in the least-squares sense is shown to be well-posed and to yield a discrete spectrum, provided the data cannot be fitted exactly, i.e., without any deviation of data and calculated values. Properties of the resulting spectrum are discussed. Examples of discrete spectra obtained from simulated literature data and experimental literature data on polymers are given. The problem of smoothing discrete spectra when continuous ones are expected is discussed. A detailed study of an integral transform inversion under the non-negativity constraint is given in Appendix.

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Convergence and stability analysis of the half thresholding based few-view CT reconstruction

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0003 ◽

2020 ◽

Vol 28 (6) ◽

pp. 829-847

Author(s):

Hua Huang ◽

Chengwu Lu ◽

Lingli Zhang ◽

Weiwei Wang

Keyword(s):

Noise Suppression ◽

Reconstructed Image ◽

Reference Image ◽

Projection Data ◽

Ct Reconstruction ◽

Reconstruction Algorithms ◽

Ill Posed ◽

Thresholding Algorithm ◽

The Stability ◽

Well Posed

AbstractThe projection data obtained using the computed tomography (CT) technique are often incomplete and inconsistent owing to the radiation exposure and practical environment of the CT process, which may lead to a few-view reconstruction problem. Reconstructing an object from few projection views is often an ill-posed inverse problem. To solve such problems, regularization is an effective technique, in which the ill-posed problem is approximated considering a family of neighboring well-posed problems. In this study, we considered the {\ell_{1/2}} regularization to solve such ill-posed problems. Subsequently, the half thresholding algorithm was employed to solve the {\ell_{1/2}} regularization-based problem. The convergence analysis of the proposed method was performed, and the error bound between the reference image and reconstructed image was clarified. Finally, the stability of the proposed method was analyzed. The result of numerical experiments demonstrated that the proposed method can outperform the classical reconstruction algorithms in terms of noise suppression and preserving the details of the reconstructed image.

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Polyanalytic boundary value problems for planar domains with harmonic Green function

Analysis and Mathematical Physics ◽

10.1007/s13324-021-00569-2 ◽

2021 ◽

Vol 11 (3) ◽

Author(s):

Heinrich Begehr ◽

Bibinur Shupeyeva

Keyword(s):

Green Function ◽

Boundary Value Problems ◽

Arbitrary Order ◽

Boundary Value ◽

Planar Domains ◽

Schwarz Problem ◽

Bianalytic Functions ◽

Neumann Type ◽

The Neumann Problem ◽

Well Posed

AbstractThere are three basic boundary value problems for the inhomogeneous polyanalytic equation in planar domains, the well-posed iterated Schwarz problem, and further two over-determined iterated problems of Dirichlet and Neumann type. These problems are investigated in planar domains having a harmonic Green function. For the Schwarz problem, treated earlier [Ü. Aksoy, H. Begehr, A.O. Çelebi, AV Bitsadze’s observation on bianalytic functions and the Schwarz problem. Complex Var Elliptic Equ 64(8): 1257–1274 (2019)], just a modification is mentioned. While the Dirichlet problem is completely discussed for arbitrary order, the Neumann problem is just handled for order up to three. But a generalization to arbitrary order is likely.

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Elastostatics of Bernoulli–Euler Beams Resting on Displacement-Driven Nonlocal Foundation

Nanomaterials ◽

10.3390/nano11030573 ◽

2021 ◽

Vol 11 (3) ◽

pp. 573

Author(s):

Marzia Sara Vaccaro ◽

Francesco Paolo Pinnola ◽

Francesco Marotti de Sciarra ◽

Raffaele Barretta

Keyword(s):

Boundary Conditions ◽

Structural Engineering ◽

Beam Deflection ◽

Soil Reaction ◽

Differential Problem ◽

Reaction Field ◽

Ill Posed ◽

Elastic Responses ◽

Benchmark Solutions ◽

Well Posed

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler’s model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt’s original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.

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