scholarly journals Recovery of a Lamé parameter from displacement fields in nonlinear elasticity models

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hugo Carrillo ◽  
Alden Waters
Keyword(s):  
Stability Result ◽  
Elliptic Systems ◽  
Stability Estimate ◽  
Two Dimensions ◽  
Small Displacement ◽  
Displacement Fields ◽  
Linear Type ◽  
Basic Model ◽  
First Case ◽  
Elasticity Problems

Abstract We study some inverse problems involving elasticity models by assuming the knowledge of measurements of a function of the displaced field. In the first case, we have a linear model of elasticity with a semi-linear type forcing term in the solution. Under the hypothesis the fluid is incompressible, we recover the displaced field and the second Lamé parameter from power density measurements in two dimensions. A stability estimate is shown to hold for small displacement fields, under some natural hypotheses on the direction of the displacement, with the background pressure fixed. On the other hand, we prove in dimensions two and three a stability result for the second Lamé parameter when the displacement field follows the (nonlinear) Saint-Venant model when we add the knowledge of displaced field solution measurements. The Saint-Venant model is the most basic model of a hyperelastic material. The use of over-determined elliptic systems is new in the analysis of linearization of nonlinear inverse elasticity problems.

scholarly journals An inverse problem of radiative potentials and initial temperatures in parabolic equations with dynamic boundary conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
El Mustapha Ait Ben Hassi ◽  
Salah-Eddine Chorfi ◽  
Lahcen Maniar
Keyword(s):  
Inverse Problem ◽  
Boundary Conditions ◽  
Parabolic Equations ◽  
Stability Result ◽  
Stability Estimate ◽  
Lipschitz Stability ◽  
Dynamic Boundary Conditions ◽  
Logarithmic Convexity ◽  
Dynamic Boundary ◽  
Logarithmic Stability

Abstract We study an inverse problem involving the restoration of two radiative potentials, not necessarily smooth, simultaneously with initial temperatures in parabolic equations with dynamic boundary conditions. We prove a Lipschitz stability estimate for the relevant potentials using a recent Carleman estimate, and a logarithmic stability result for the initial temperatures by a logarithmic convexity method, based on observations in an arbitrary subdomain.

scholarly journals Demonstration of Three-Dimensional Indoor Visible Light Positioning with Multiple Photodiodes and Reinforcement Learning

Sensors ◽  
2020 ◽  
Vol 20 (22) ◽  
pp. 6470
Author(s):  
Zhuo Zhang ◽  
Huayang Chen ◽  
Weikang Zeng ◽  
Xinlong Cao ◽  
Xuezhi Hong ◽  
...  
Keyword(s):  
Reinforcement Learning ◽  
Visible Light ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Location Based Services ◽  
Positioning Accuracy ◽  
The Third ◽  
Basic Model ◽  
3D Space ◽  
3D Positioning

To provide high-quality location-based services in the era of the Internet of Things, visible light positioning (VLP) is considered a promising technology for indoor positioning. In this paper, we study a multi-photodiodes (multi-PDs) three-dimensional (3D) indoor VLP system enhanced by reinforcement learning (RL), which can realize accurate positioning in the 3D space without any off-line training. The basic 3D positioning model is introduced, where without height information of the receiver, the initial height value is first estimated by exploring its relationship with the received signal strength (RSS), and then, the coordinates of the other two dimensions (i.e., X and Y in the horizontal plane) are calculated via trilateration based on the RSS. Two different RL processes, namely RL1 and RL2, are devised to form two methods that further improve horizontal and vertical positioning accuracy, respectively. A combination of RL1 and RL2 as the third proposed method enhances the overall 3D positioning accuracy. The positioning performance of the four presented 3D positioning methods, including the basic model without RL (i.e., Benchmark) and three RL based methods that run on top of the basic model, is evaluated experimentally. Experimental results verify that obviously higher 3D positioning accuracy is achieved by implementing any proposed RL based methods compared with the benchmark. The best performance is obtained when using the third RL based method that runs RL2 and RL1 sequentially. For the testbed that emulates a typical office environment with a height difference between the receiver and the transmitter ranging from 140 cm to 200 cm, an average 3D positioning error of 2.6 cm is reached by the best RL method, demonstrating at least 20% improvement compared to the basic model without performing RL.

scholarly journals A Hölder-logarithmic stability estimate for an inverse problem in two dimensions

2015 ◽  
Vol 23 (1) ◽  
Author(s):  
Matteo Santacesaria
Keyword(s):  
Inverse Problem ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Stability Estimate ◽  
Two Dimensions ◽  
Positive Energy ◽  
Two Dimensional ◽  
Ill Posed ◽  
Dirichlet To Neumann Map ◽  
Logarithmic Stability

AbstractThe problem of the recovery of a real-valued potential in the two-dimensional Schrödinger equation at positive energy from the Dirichlet-to-Neumann map is considered. It is know that this problem is severely ill-posed and the reconstruction of the potential is only logarithmic stable in general. In this paper a new stability estimate is proved, which is explicitly dependent on the regularity of the potentials and on the energy. Its main feature is an efficient

A Parallel Adaptive Treecode Algorithm for Evolution of Elastically Stressed Solids

2014 ◽  
Vol 15 (2) ◽  
pp. 365-387 ◽  
Author(s):  
Hualong Feng ◽  
Amlan Barua ◽  
Shuwang Li ◽  
Xiaofan Li
Keyword(s):  
Arithmetic Operation ◽  
Summation Method ◽  
Two Dimensions ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Point Distribution ◽  
Long Time ◽  
Elasticity Problems ◽  
Matrix Vector ◽  
Speedup Factor

AbstractThe evolution of precipitates in stressed solids is modeled by coupling a quasi-steady diffusion equation and a linear elasticity equation with dynamic boundary conditions. The governing equations are solved numerically using a boundary integral method (BIM). A critical step in applying BIM is to develop fast algorithms to reduce the arithmetic operation count of matrix-vector multiplications. In this paper, we develop a fast adaptive treecode algorithm for the diffusion and elasticity problems in two dimensions (2D). We present a novel source dividing strategy to parallelize the treecode. Numerical results show that the speedup factor is nearly perfect up to a moderate number of processors. This approach of parallelization can be readily implemented in other treecodes using either uniform or non-uniform point distribution. We demonstrate the effectiveness of the treecode by computing the long-time evolution of a complicated microstructure in elastic media, which would be extremely difficult with a direct summation method due to CPU time constraint. The treecode speeds up computations dramatically while fulfilling the stringent precision requirement dictated by the spectrally accurate BIM.

22.—Analytic, Generalized, Hyperanalytic Function Theory and an Application to Elasticity

1975 ◽  
Vol 73 ◽  
pp. 317-331 ◽  
Author(s):  
Robert P. Gilbert ◽  
Wolfgang L. Wendland
Keyword(s):  
Integral Equations ◽  
Open Problem ◽  
Function Theory ◽  
Elliptic Systems ◽  
Representation Formula ◽  
Volterra Integral Equations ◽  
First Order ◽  
Elasticity Problems ◽  
Hypercomplex Algebra

SynopsisThough it is still an open problem for which class of first-order elliptic systems Carleman's theorem holds, this is proven here for a certain class of systems (with analytic coefficients) for which Douglis introduced the hypercomplex algebra and hyperanalytic functions. The proof is based on a representation formula generalising Vekua's approach with Volterra integral equations in C2 to more than two unknowns. The representation formula is of its own interest because it provides the generation of complete families of solutions. The equations of plane inhomogeneous elasticity problems lead to a system of the desired class.

scholarly journals Semilinear Elliptic Systems With Exponential Nonlinearities in Two Dimensions

2006 ◽  
Vol 6 (2) ◽  
Author(s):  
Djairo G. de Figueiredo ◽  
João Marcos do Ó ◽  
Bernhard Ruf
Keyword(s):  
Positive Solutions ◽  
A Priori ◽  
Elliptic Systems ◽  
Two Dimensions ◽  
Smooth Domain ◽  
A Priori Bounds ◽  
Semilinear Elliptic Systems ◽  
Semilinear Elliptic

AbstractWe establish a priori bounds for positive solutions of semilinear elliptic systems of the formwhere Ω is a bounded and smooth domain in ℝ

Calculation of 3D internal displacement fields from 3D X-ray computer tomographic images

1995 ◽  
Vol 449 (1937) ◽  
pp. 537-554 ◽  
Keyword(s):  
Displacement Field ◽  
Permanent Deformation ◽  
Gradient Algorithm ◽  
Internal Displacement ◽  
Small Displacement ◽  
Lagrange Equations ◽  
Displacement Fields ◽  
Plane Shear ◽  
Tomographic Images ◽  
X Ray

We present a new method for computing the internal displacement fields associated with permanent deformations of 3D composite objects with complex internal structure for fields satisfying the small displacement gradient approximation of continuum mechanics. We compute the displacement fields from a sequence of 3D X-ray computed tomography (CT) images. By assuming that the intensity of the tomographic images represents a conserved property which is incompressible, we develop a constrained nonlinear regression model for estimation of the displacement field. Successive linear approximation is then employed and each linear subsidiary problem is solved using variational calculus. We approximate the resulting Euler-Lagrange equations using a finite set of linear equations using finite differencing methods. We solve these equations using a conjugate gradient algorithm in a multiresolution framework. We validate our method using pairs of synthetic images of plane shear flow. Finally, we determine the 3D displacement field in the interior of a cylindrical asphalt/aggregate core loaded to a state of permanent deformation.

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