MFS fading regularization method for the identification of boundary conditions from partial elastic displacement field data

Author(s):  
L. Caillé ◽  
J-L. Hanus ◽  
F. Delvare ◽  
N. Michaux-Leblond
Author(s):  
L. Caillé ◽  
J L. Hanus ◽  
F. Delvare ◽  
N. Michaux-Leblonda

A method is proposed to solve an inverse problem in twodimensional linear isotropic elasticity. The inverse problem consists of the determination of both the entire displacement field and the boundary conditions inaccessible to the measurement from the partial knowledge of the displacement field. The algorithm is based on a fading regularization method (FRM) and is numerically implemented using the method of fundamental solutions (MFS). The inverse technique is first validated with synthetic data and is then applied to the interpretation of experimental measurements obtained by digital image correlation (DIC).


2013 ◽  
Vol 586 ◽  
pp. 237-240 ◽  
Author(s):  
Lucie Šestáková

Most of fracture analyses often require an accurate knowledge of the stress/displacement field over the investigated body. However, this can be sometimes problematic when only one (singular) term of the Williams expansion is considered. Therefore, also other terms should be taken into account. Such an approach, referred to as multi-parameter fracture mechanics is used and investigated in this paper. Its importance for short/long cracks and the influence of different boundary conditions are studied. It has been found out that higher-order terms of the Williams expansion can contribute to more precise description of the stress distribution near the crack tip especially for long cracks. Unfortunately, the dependences obtained from the analyses presented are not unambiguous and it cannot be strictly derived how many of the higher-order terms are sufficient.


2021 ◽  
Author(s):  
Markus Even ◽  
Malte Westerhaus ◽  
Verena Simon

<p>The cavern field at Epe has been brined out of a salt deposit belonging to the lower Rhine salt flat, which extends under the surface of the North German lowlands and part of the Netherlands, and is used to store e.g. natural gas, brine and petroleum. Cavern convergence and operational pressure changes cause surface displacements that have been studied for this work with the help of SAR interferometry (InSAR) using distributed and persistent scatterers. Vertical and East-West movements have been determined based on Sentinel-1 data from ascending and descending orbit. Simple geophysical modeling is used to support InSAR processing and helps to interpret the observations. In particular, an approach is presented that allows to relate the deposit pressures with the observed surface displacements. Seasonal movements occurring over a fen situated over the western part of the storage site further complicate the analysis. Findings are validated with ground truth from levelling and groundwater level measurements.</p><p>For porous storage sites the geomechanic response can be described as elastic: displacement is almost proportional to reservoir pressure and displays the same pronounced seasonal behavior. At Epe the visco-elastic response of the salt layer has to be considered. The general appearance of the surface displacement is that of a strongly smoothed and shifted version of the cavern pressure curve. To deal with this situation a temporal model for displacement with pressure changes (pressure response) is derived that relates cavern pressure with observed displacement based on the theory for visco-elastic behavior of a Kelvin-Voigt body.</p><p>In order to deal successfully with the challenging displacement field at Epe several algorithmic improvements were implemented. To obtain a more complete picture of the displacement field DS pre-processing has been combined with StaMPS. Furthermore, StaMPS was modified in order to support unwrapping with a phase model composed of linear trend, pressure response and a seasonal component (caused by ground water level changes). Finally, refining the iterative estimation scheme of StaMPS helped avoiding leakage of the displacement signal to the spatially correlated noise term.</p><p>Determining vertical and east-west displacements from InSAR line-of-sight displacements is fundamental for interpretation and integration with levelling data. In this study, a basic method of orbit combination and another one supported by a simplistic geophysical model were applied in order to obtain 2D-displacements. For the basic method the north-south component was handled as if it were zero, while the geophysical model predicts the LOS effect of NS displacements. It assumes that caverns act as spherical pressure/volume sources embedded in an elastic half space (“Mogi” sources). To incorporate the visco-elastic component, each cavern is encompassed by a spherical salt shell that obeys the Kelvin-Voigt differential equations. The model is used here to describe either the parameters of the linear component of the displacement model or of the pressure response. A novelty of the orbit combinations implemented for this study is that the different components of the phase model are combined separately. This allows for a better understanding of the phenomena that contribute to the displacement field.</p>


2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Tuan H. Nguyen ◽  
Binh Thanh Tran

We consider the following Cauchy problem for the elliptic equation with inhomogeneous source in a rectangular domain with Dirichlet boundary conditions at x=0 and x=π. The problem is ill-posed. The main aim of this paper is to introduce a regularization method and use it to solve the problem. Some sharp error estimates between the exact solution and its regularization approximation are given and a numerical example shows that the method works effectively.


1995 ◽  
Vol 408 ◽  
Author(s):  
Vijay Shastry ◽  
Diana Farkas

AbstractThe elastic displacement field solution of a semi-infinite crack in an anisotropic body, calculated using a complex variable approach due to Sih and Liebowitz, is usually used by atomistic simulations of fracture. The corresponding expression for the displacement field of a finite crack is numerically cumbersome since it involves multiple square roots of complex numbers. In this study, displacement field of the crack is calculated by superposing the displacements of dislocations in an equivalent double pileup, equilibrated under mode I conditions. An advantage of this method is its extensibility to atomistic studies of more complex systems containing multiple cracks or interfaces. The pileup representation of the finite crack is demonstrated as being equivalent to its corresponding continuum description using the example of a double ended crack in α-Fe, loaded in mode I. In these examples, the interatomic interaction in α-Fe is described by an empirical embedded atom (EAM) potential.


1991 ◽  
Vol 238 ◽  
Author(s):  
R. Bonnet ◽  
M. Loubradou

ABSTRACTIn many materials, crystalline interfaces are facetted. The experimental evidence is that on each side of an interfacial ledge, or along the facets meeting along a common line, low energy atomic structural units are preserved which accommodate elastically angular or/and length misfit(s). Each facet can be considered as a Somigliana dislocation (SD) whose core is extended on the facet. The elastic displacement field of a SD is derived in an anisotropic continuum, for any orientation of the facet relative to a given Cartesian frame. From an atomic point of view, the translation state of the two crystals on each side of the facet is defined. The dislocation content attached to a ledge or a dihedral angle formed by two joining facets along a common side is also analyzed. The local elastic field related to these cases are derived and applications are presented for depicting the positions of the atomic columns in theoretical plots. Comparisons are made with some other theoretical works and HRTEM images. Examples illustrate the application of the Somigliana model to grain boundaries in hexagonal crystals (Mg, WC), and an interphase interface Ni3AI/Ni3Nb.


2011 ◽  
Vol 702-703 ◽  
pp. 11-17 ◽  
Author(s):  
Brent L. Adams ◽  
David T. Fullwood ◽  
J.A. Basinger ◽  
T. Hardin

Significant advances are reported in the application of HR-EBSD to the imaging of the dislocation structure of polycrystalline materials. The central assumption of the method is the compatibility of the total displacement field, which relates the (Nye) dislocation tensor to the (partially measurable) curl of the elastic displacement field. Two key challenges must be addressed, including: a) the fundamental limitation imposed by the electron-opacity of typical materials, which limits the measurement of gradients in the displacement field in the direction normal to the sample surface; and b) the inability of HR-EBSD to recover the spherical (elastic) distortions of the lattice. This second challenge can be overcome if a traction free boundary condition is applied. It is illustrated that consideration of the familiar stress equilibrium relations gives additional information, which may enable estimates of the missing components of the Nye tensor. An example of application of HR-EBSD to a Mg-Ce sample is presented.


2016 ◽  
Vol 31 (06) ◽  
pp. 1650012
Author(s):  
Guglielmo Fucci

In this work, we analyze the Casimir energy and force for a thick piston configuration. This study is performed by utilizing the spectral zeta function regularization method. The results we obtain for the Casimir energy and force depend explicitly on the parameters that describe the general self-adjoint boundary conditions imposed. Numerical results for the Casimir force are provided for specific types of boundary conditions and are also compared to the corresponding force on an infinitely thin piston.


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