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Baseline-free damage detection in bridges using acceleration records with the application of Laplacian
Published in Airong Chen, Xin Ruan, Dan M. Frangopol, Life-Cycle Civil Engineering: Innovation, Theory and Practice, 2021
The Laplacian or Laplace operator Δ is a differential operator given by the divergence of the gradient of a function on Euclidean space. It can be used to amplify anomalies in signals. (Ratcliffe 1997) used Laplacian of mode shapes to identify the location of stiffness damage as little as 10% in a uniform beam. A modified Laplacian was presented for lower damage values. The findings were supported by experiment on a steel beam with a cut. (Besio et al. 2006) adopted using Laplacian to detect differences on concentric electrode elements to measure the potentials of localized brain activities. Laplacian has various definitions depending on the studied case. For a multivariable domain, it is the sum of all the unmixed second partial derivatives of all variables. For example, inside a two-dimensional plan, it is defined by Equation 1. Δf=∂2f∂x2+∂2f∂y2
Solutions of typical three-dimensional consolidation problems
Published in Jian-Hua Yin, Guofu Zhu, Consolidation Analyses of Soils, 2020
where ∇2=∂2∂x2+∂2∂y2+∂2∂z2 is the Laplace operator in Cartesian co-ordinates.
Scalars and Vectors
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
The Laplace operator appears in many applications in physics and is often the generalisation of a second derivative in one dimension to similar problems in several dimensions.
Efficient Machine Learning-based Approach for Brain Tumor Detection Using the CAD System
Published in IETE Journal of Research, 2023
Mohamed Amine Guerroudji, Zineb Hadjadj, Mohamed Lichouri, Kahina Amara, Nadia Zenati
(a) Step (1): In the pre-processing phase, we correct the noise in the MRIs by the anisotropic filter proposed in [28], this step includes operations that allow the reduction of noise, intensity heterogeneity, and inter-slice intensity variation of images. The main objective of the author in [28] is to modify the diffusion equation (Equation (1)) to achieve anisotropic diffusion-based on: (1) Maximum homogenization of diffusion away from contours, and (2) Minimal diffusion on the level of contours. The heat diffusion equation was given by: Then, modified in [28] by: where is the Signal / Image; is the Divergence operator; is the Gradient Operator; is the Laplace operator; is the is a decreasing function with and.
Nonsmooth optimization by successive abs-linearization in function spaces
Published in Applicable Analysis, 2022
Andrea Walther, Olga Weiß, Andreas Griewank, Stephan Schmidt
For a bounded domain and a given desired state , consider the optimization problem where y and u represent the state and the control, respectively, some given function and , are parameters. Furthermore, is a linear elliptic differential operator of second order, e.g. the Laplace operator. Suppose, that the continuously differentiable operator is such that there exists a Lipschitz-continuously Fréchet differentiable solution operator , which gives the solution of the PDE in Equation (6) for any fixed control . As a very simple example one may consider as in [20]. Then, the reduced problem formulation is given by the optimization problem
An Intrinsic Geometrical Approach for Statistical Process Control of Surface and Manifold Data
Published in Technometrics, 2021
Xueqi Zhao, Enrique del Castillo
The Laplace operator of a twice differentiable function is minus the divergence of its gradient field: and measures the difference between f(x) and the average f(x) in a neighborhood around x. Given the second derivatives, it is a measure of curvature, and can be alternatively understood as minus the trace of the Hessian of f(x). The minus sign is for consistency with Equation (4). Note how the domain of the function here is n-dimensional Euclidean space. We next extend this definition to general manifolds, obtaining the main differential-geometric operator used in the sequence, the LB operator, widely used in computer graphics and machine learning (Belkin 2003; Kimmel 2004; Levy 2006; Reuter, Wolter, and Peinecke 2006; Patané 2014).