A Hermite interpolation element-free Galerkin method for piezoelectric materials

Piezoelectric materials have played an important role in industry due to a number of beneficial properties. However, most numerical methods for the piezoelectric materials need mesh, in which the mesh generation and remeshing are prominent difficulties. This paper proposes a Hermite interpolation element-free Galerkin method (HIEFGM) for piezoelectric materials, where the Hermite approximate approach and interpolation element-free Galerkin method (IEFGM) are combined. Based on the constitutive equation, geometric equation, and Galerkin integral weak form, the HIEFGM formulation for piezoelectric materials is established. In the proposed method, the problem domain is discretized by many nodes rather than the meshes, so the pre-processing of numerical computation is simplified. Furthermore, a new approximation technique based on the moving least squares method and Hermite approximate approach is used to derive the approximation function of field quantities. The derived approximation function has the Kronecker delta property and considers the field quantity normal derivatives of boundary nodes, which avoids the problem of imposing the essential boundary conditions and improves the accuracy of meshless approximation. The effects of the scaling factor, node density, and node arrangement on the accuracy of the proposed method are investigated. Numerical examples are given for assessing the proposed method and the results uniformly demonstrate the proposed method has excellent performance in analyzing piezoelectric materials.

A Dimension Splitting-Interpolating Moving Least Squares (DS-IMLS) Method with Nonsingular Weight Functions

By introducing the dimension splitting method (DSM) into the improved interpolating moving least-squares (IMLS) method with nonsingular weight function, a dimension splitting–interpolating moving least squares (DS-IMLS) method is first proposed. Since the DSM can decompose the problem into a series of lower-dimensional problems, the DS-IMLS method can reduce the matrix dimension in calculating the shape function and reduce the computational complexity of the derivatives of the approximation function. The approximation function of the DS-IMLS method and its derivatives have high approximation accuracy. Then an improved interpolating element-free Galerkin (IEFG) method for the two-dimensional potential problems is established based on the DS-IMLS method. In the improved IEFG method, the DS-IMLS method and Galerkin weak form are used to obtain the discrete equations of the problem. Numerical examples show that the DS-IMLS and the improved IEFG methods have high accuracy.

A method for mobile device positioning using a sensor network of BLE beacons, approximation of the RSSI value and artificial neural networks

The paper considers the development of a method for positioning a mobile device using a sensor network of BLE-beacons, the approximation of RSSI values and artificial neural networks. The aim of the work is to develop a method for positioning small-scale industrial mechanization equipment for building unmanned systems for product movement tracking. The work is divided into four main parts: data synthesis, signal filtering, selection of BLE beacons, translation of the RSSI values into a distance, and multilateration. A simplified Kalman filter is proposed for filtering the input signal to suppress Gaussian noise. A description of two approaches to translating the RSSI value into a distance is given: an exponential approximation function with a coefficient of determination of 0.6994 and an artificial feedforward neural network. A comparison of the results of these approaches is carried out on several test samples: a training one, a test sample at a known distance (0–50 meters) and a test sample at an unknown distance (60–100 meters). The artificial neural network is shown to perform better in all experiments, except for the test sample at a known distance (0–50 meters), for which the r.m.s. error is higher by 0.02 m 2 than that for the approximation function, which can be neglected. An algorithm for positioning a mobile device based on the multilateration method is proposed. Experimental studies of the developed method have shown that the positioning error does not exceed 0.9 meters in a 5×5.5 m room under monitoring. The positioning accuracy of a mobile device using the proposed method in the experiment is 40.9 % higher. Experimental studies are also conducted in a 58.4×4.5 m room, showing more accurate results compared to similar studies.

A lower bound for the Hausdorff dimension of the set of weighted simultaneously approximable points over manifolds

Given a weight vector [Formula: see text] with each [Formula: see text] bounded by certain constraints, we obtain a lower bound for the Hausdorff dimension of the set [Formula: see text], where [Formula: see text] is a twice continuously differentiable manifold. From this we produce a lower bound for [Formula: see text] where [Formula: see text] is a general approximation function with certain limits. The proof is based on a technique developed by Beresnevich et al. in 2017, but we use an alternative mass transference style theorem proven by Wang, Wu and Xu (2015) to obtain our lower bound.

Simulation of Radiation Charging of Microelectronic Equipment Cases for Space Applications

A model and a method for mathematical modeling of radiation charging of polymer microelectronic equipment housings with increased conductivity are developed, which are based on the application of the approximation function of the experimental dependence of the housing conductivity on the irradiation time obtained using parametric identification methods. The research results are aimed at developing composite polymer materials for microelectronic equipment housings with a conductivity that ensures the absence of electrostatic discharges and significantly increases the active life of spacecraft.

Rational Implementation of Fractional Calculus Operator Based on Quadratic Programming

When fractional calculus operators and models are implemented rationally, there exist some problems such as low approximation accuracy of rational approximation function, inability to specify arbitrary approximation frequency band, or poor robustness. Based on the error criterion of the least squares method, a quadratic programming method based on the frequency-domain error is proposed. In this method, the frequency-domain error minimization between the fractional operator s ± r and its rational approximation function is transformed into a quadratic programming problem. The results show that the construction method of the optimal rational approximation function of fractional calculus operator is effective, and the optimal rational approximation function constructed can effectively approximate the fractional calculus operator and model for the specified approximation frequency band.

The Component Weighted Median Absolute Deviations Problem

This paper considers the problem of robust modeling by using the well-known Least Absolute Deviation (LAD) regression. For that purpose, the approximation function is designed and analyzed, which is based on a certain component weight of the Weighted Median of Data. It is shown that the proposed approximation function is a piecewise constant function with finitely many pieces with respect to the model parameter. Thereby, an investigation of regions of constant values of the approximation function is conducted. It is established that the designed model based on the Component Weighted Median Absolute Deviations estimates a optimal model parameter on a finite set, which describes corresponding regions. Furthermore, the specified restriction of the approximation function is observed and analyzed, in order to examine the observed problem.

A Novel Robust Principal Component Analysis Algorithm of Nonconvex Rank Approximation

Noise exhibits low rank or no sparsity in the low-rank matrix recovery, and the nuclear norm is not an accurate rank approximation of low-rank matrix. In the present study, to solve the mentioned problem, a novel nonconvex approximation function of the low-rank matrix was proposed. Subsequently, based on the nonconvex rank approximation function, a novel model of robust principal component analysis was proposed. Such model was solved with the alternating direction method, and its convergence was verified theoretically. Subsequently, the background separation experiments were performed on the Wallflower and SBMnet datasets. Furthermore, the effectiveness of the novel model was verified by numerical experiments.

APROKSIMASI FUNGSI KONTINU TERBATAS DENGAN KONVOLUSI

Convolution is a mathematical operation on two functions that produces a new function that can be seen as a modified version of one of its original functions. The convolution operator has no identity element. However, it has an approximate identity. It can be found as a sequence of gk such that convolution of f and gk converges to f for k→∞. It implies that convolution can be used to approximate a function. In this article, we have proven basic theorems about approximation function by convolution for a bounded function in C(Rd). Konvolusi adalah suatu operasi pada dua fungsi dan menghasilkan suatu fungsi baru yang dapat dipandang sebagai versi modifikasi dari salah satu fungsi aslinya. Operasi konvolusi tidak memiliki unsur identitas. Namun, operasi konvolusi memiliki identitas hampiran, yakni dapat ditemukannya suatu barisan fungsi gk sehingga konvolusi dari f dan gk konvergen ke f untuk k→∞. Hal ini mengakibatkan konvolusi dapat digunakan untuk aproksimasi fungsi. Pada artikel ini dibuktikan teorema-teorema yang mendasari aproksimasi fungsi dengan konvolusi bagi fungsi terbatas di C(Rd) .