Configuration of 3D Indoor Positioning System Based on Bluetooth Beacons

2021 ◽  
Vol 27 (1) ◽  
pp. 32-40
Author(s):  
V. M. Grinyak ◽  
Keyword(s):  
Measurement Errors ◽  
Theoretical Calculations ◽  
Tracking System ◽  
Linear Equations ◽  
Estimation Procedure ◽  
Accuracy Evaluation ◽  
Algebraic Equations ◽  
Positioning Accuracy ◽  
Model Interpretation ◽  
Linear Algebraic Equations

This paper devoted to research of indoors navigation problems under poor or insufficient quality of satellite navigational data environment. The problem of object positioning in 3D space by Bluetooth devices located indoors forming a multi-position tracking system is considered in this research. Emphasized that in order to succeed for such system it is required to pre-estimate distinctive accuracy. The proposed model interpretation of the positioning problem as the system of linear equations. The classic model interpretation for method of least squares is used for resolution. General problem of linearization around reference resolution is the locality of its features. There are three concepts of problems solvability, such as fundamental solvability (observability), solvability in conditions of instrumental measurement errors and solvability under conditions of finite accuracy of computation on a computer. The first aspect of solvability is interpreted by the completeness of the rank of the corresponding system of linear algebraic equations, the second and third ones represents by the conditionality of the problem and the convergence of the iterative estimation procedure. The conducted experiments show that for the positioning problem the attributes of the linearized model are accurate enough to represent the original nonlinear problem. Such interpretation allows to build theoretical accuracy estimation priors for object coordinates evaluations and to identify the areas with insufficient positioning accuracy. In this paper there are results of expected accuracy evaluation for various system patterns with full-scale experiments proving the theoretical calculations. Experiments for problems with using SKYLAB Beacon VG01 Bluetooth transmitters and smartphone HUAWEI WAS-LX1 are presented and confirmed that math model with linear approximation defined by authors is usable for solving indoors navigation problems using Bluetooth signal. So, for good enough quantity and appropriate location of the tracks the achievable positioning accuracy could be as good as 0.2—0.3 meters for all three coordinates. Such accuracy allows to navigate small hovering objects such as drones. In general, it looks promising to use Bluetooth trackers for solving positioning problems for indoors environments.

A Diagonally Dominant Solution for the Cylinder End Problem

1981 ◽  
Vol 48 (4) ◽  
pp. 876-880 ◽  
Author(s):  
T. D. Gerhardt ◽  
Shun Cheng
Keyword(s):  
Infinite System ◽  
Linear Equations ◽  
Infinite Cylinder ◽  
Algebraic Equations ◽  
Precise Calculation ◽  
Linear Algebraic Equations ◽  
Present Solution ◽  
Diagonally Dominant ◽  
Expansion Coefficients

An improved elasticity solution for the cylinder problem with axisymmetric torsionless end loading is presented. Consideration is given to the specification of arbitrary stresses on the end of a semi-infinite cylinder with a stress-free lateral surface. As is known from the literature, the solution to this problem is obtained in the form of a nonorthogonal eigenfunction expansion. Previous solutions have utilized functions biorthogonal to the eigenfunctions to generate an infinite system of linear algebraic equations for determination of the unknown expansion coefficients. However, this system of linear equations has matrices which are not diagonally dominant. Consequently, numerical instability of the calculated eigenfunction coefficients is observed when the number of equations kept before truncation is varied. This instability has an adverse effect on the convergence of the calculated end stresses. In the current paper, a new Galerkin formulation is presented which makes this system of equations diagonally dominant. This results in the precise calculation of the eigenfunction coefficients, regardless of how many equations are kept before truncation. By consideration of a numerical example, the present solution is shown to yield an accurate calculation of cylinder stresses and displacements.

scholarly journals On the Solubility of Linear Algebraic Equations (Contd.)

1913 ◽  
Vol 12 ◽  
pp. 137-138
Author(s):  
John Dougall
Keyword(s):  
Linear Equations ◽  
Homogeneous System ◽  
Algebraic Equations ◽  
Similar Reasoning ◽  
Null Solution ◽  
Linear Algebraic Equations ◽  
The Given ◽  
Homogeneous Linear

A system of n non-homogeneous linear equations in n variables has one and only one solution if the homogeneous system obtained from the given system by putting all the constant terms equal to zero has no solution except the null solution.This may be proved independently by similar reasoning to that given for Theorem I., or it may be deduced from that theorem. We follow the latter method.

scholarly journals On the economical solution method for a system of linear algebraic equations

2004 ◽  
Vol 2004 (4) ◽  
pp. 377-410 ◽  
Author(s):  
Jan Awrejcewicz ◽  
Vadim A. Krysko ◽  
Anton V. Krysko
Keyword(s):  
Differential Equations ◽  
Boundary Value ◽  
Linear Equations ◽  
Three Dimensional ◽  
Boundary Element Methods ◽  
Minimal Amount ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Point Of Intersection ◽  
Stationary Heat Transfer

The present work proposes a novel optimal and exact method of solving large systems of linear algebraic equations. In the approach under consideration, the solution of a system of algebraic linear equations is found as a point of intersection of hyperplanes, which needs a minimal amount of computer operating storage. Two examples are given. In the first example, the boundary value problem for a three-dimensional stationary heat transfer equation in a parallelepiped inℝ3is considered, where boundary value problems of first, second, or third order, or their combinations, are taken into account. The governing differential equations are reduced to algebraic ones with the help of the finite element and boundary element methods for different meshes applied. The obtained results are compared with known analytical solutions. The second example concerns computation of a nonhomogeneous shallow physically and geometrically nonlinear shell subject to transversal uniformly distributed load. The partial differential equations are reduced to a system of nonlinear algebraic equations with the error ofO(hx12+hx22). The linearization process is realized through either Newton method or differentiation with respect to a parameter. In consequence, the relations of the boundary condition variations along the shell side and the conditions for the solution matching are reported.

Unsteady double-diffusive buoyancy-induced flow in a triangular solar collector shape enclosure with corrugated bottom wall

2017 ◽  
Vol 27 (6) ◽  
pp. 1282-1303 ◽  
Author(s):  
M.M. Rahman ◽  
Sourav Saha ◽  
Satyajit Mojumder ◽  
Khan Md. Rabbi ◽  
Hasnah Hasan ◽  
...  
Keyword(s):  
Mass Transfer ◽  
Solar Collector ◽  
Linear Equations ◽  
Bottom Wall ◽  
Algebraic Equations ◽  
Weighted Residual Method ◽  
Content Type ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Induced Flow

Purpose The purpose of this investigation is to determine the nature of the flow field, temperature distribution and heat and mass transfer in a triangular solar collector enclosure with a corrugated bottom wall in the unsteady condition numerically. Design/methodology/approach Non-linear governing partial differential equations (i.e. mass, momentum, energy and concentration equations) are transformed into a system of integral equations by applying the Galerkin weighted residual method. The integration involved in each of these terms is performed using Gauss’ quadrature method. The resulting non-linear algebraic equations are modified by the imposition of boundary conditions. Finally, Newton’s method is used to modify non-linear equations into the linear algebraic equations. Findings Both the buoyancy ratio and thermal Rayleigh number play an important role in controlling the mode of heat transfer and mass transfer. Originality/value Calculations are performed for various thermal Rayleigh numbers, buoyancy ratios and time periods. For each specific condition, streamline contours, isotherm contours and iso-concentration contours are obtained, and the variation in the overall Nusselt and Sherwood numbers is identified for different parameter combinations.

scholarly journals Development of a finite-element-based design sensitivity analysis for buckling and postbuckling of composite plates

1995 ◽  
Vol 1 (3) ◽  
pp. 255-274 ◽  
Author(s):  
Ruijiang Guo ◽  
Aditi Chattopadhyay
Keyword(s):  
Sensitivity Analysis ◽  
Finite Element ◽  
Linear Equations ◽  
Composite Plates ◽  
Material Model ◽  
Design Sensitivity ◽  
Algebraic Equations ◽  
Sensitivity Derivatives ◽  
Analysis Technique ◽  
Linear Algebraic Equations

A finite element based sensitivity analysis procedure is developed for buckling and postbuckling of composite plates. This procedure is based on the direct differentiation approach combined with the reference volume concept. Linear elastic material model and nonlinear geometric relations are used. The sensitivity analysis technique results in a set of linear algebraic equations which are easy to solve. The procedure developed provides the sensitivity derivatives directly from the current load and responses by solving the set of linear equations. Numerical results are presented and are compared with those obtained using finite difference technique. The results show good agreement except at points near critical buckling load where discontinuities occur. The procedure is very efficient computationally.

Bending of normally loaded simply supported rectangular plates in the large-deflection range

1969 ◽  
Vol 4 (3) ◽  
pp. 190-198 ◽  
Author(s):  
A Scholes ◽  
E L Bernstein
Keyword(s):  
Linear Equations ◽  
Rectangular Plates ◽  
Algebraic Equations ◽  
Simply Supported ◽  
Aspect Ratios ◽  
Linear Algebraic Equations ◽  
Non Linear ◽  
Out Of Plane ◽  
Linear Differential ◽  
Good Agreement

Means of solving the non-linear differential equations of plate bending are revieweed and a method based on minimizing the corresponding energy integral is selected as offering most advantages. The energy intergral can be approximated either by using finite-difference approximatons or by assuming a form of displacement variation. Two sets of non-linear algebraic equations (in the in-plane and out-of-plane deflections) are thus formed and, by substitution alternately in each set, the resulting linear equations are solved. Results for simply supported rectangular plates have been worked out in some detail; these are compared with tests made on plates of various aspect ratios. Good agreement on maximum values of stress and deflection was obtained.

scholarly journals SECTIONING OF HIGH DIMENSIONAL BANDED MATRICES

2014 ◽  
pp. 14-21
Author(s):  
Dmytro Fedasyuk ◽  
Pavlo Serdyuk ◽  
Yuriy Semchyshyn
Keyword(s):  
Linear Equations ◽  
Computing Time ◽  
Mathematical Physics ◽  
High Dimensional ◽  
Systems Of Equations ◽  
Algebraic Equations ◽  
Banded Matrices ◽  
Linear Algebraic Equations ◽  
Systems Of Linear Equations ◽  
The Subject

Solving high dimensional systems of linear algebraic equations is of use to many problems of mathematical physics, in particular, it is one of the main subgoals at solving systems of equations in partial derivatives. Distributed solving of high dimensional systems of linear equations allows to reduce computing time, especially in cases when these matrices can not be kept in one computer's RAM. The subject of this study is the search of optimal high dimensional matrices sectioning algorithms for distributed solving systems of linear algebraic equations.

An Algorithm for Systematic Generation of Subscales With Application to a Two-Disk Friction System

1997 ◽  
Author(s):  
Yasir S. Zargar ◽  
Kambiz Farhang
Keyword(s):  
Scaling Law ◽  
Linear Equations ◽  
Constrained Systems ◽  
Algebraic Equations ◽  
Dimensionless Parameters ◽  
Scale Factors ◽  
Linear Algebraic Equations ◽  
Scale Models ◽  
Disk Friction ◽  
Integer Solutions

Abstract In this paper we propound an algorithm for development of scale models of complex physical systems. The algorithm systematically treats the three sub-tasks of scaling, which includes creation of dimensionless parameters, application of scaling law and similitude requirements, and determination of scale factors. Optimum set of dimensionless parameters are obtained in which the pertinent variables with only integer powers appear. This is achieved by seeking integer solutions to appropriate set of linear algebraic equations in the powers, using a simple iterative technique. Application of the scaling law and similitude requirements along with a logarithmic transformation leads to a set of linear algebraic equations in the scale factors. In many applications, over-constrained systems are resulted for which true scaling can not be achieved. Two techniques are introduced in the proposed algorithm which circumvent the difficulty in treating over-constrained systems. The first method is based on a relaxation technique, whereas the second method introduces the concept of auxiliary variables to obtain the least squares solutions to the new set of linear equations. The effectiveness of the algorithm is demonstrated through its application in finding scale models of a two disk friction system.

scholarly journals Comparison of methods for solving systems of linear equations occurring in tasks with a changing configuration of the contact surface

2021 ◽  
pp. 1-28
Author(s):  
Pavel Sergeevich Aronov ◽  
Mikhail Pavlovich Galanin ◽  
Aleksandr Sergeevich Rodin
Keyword(s):  
Contact Surface ◽  
Linear Equations ◽  
Iterative Algorithms ◽  
Relaxation Method ◽  
Iterative Refinement ◽  
Jacobi Method ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Gauss Method ◽  
Thermomechanical Processes

The article discusses iterative algorithms for solving systems of linear algebraic equations arising when solving problems of contact interaction of thermoelastic bodies with a changing configuration of the contact surface. The implementations of the modified symmetric successive upper relaxation method (MSSOR), modified successive upper relaxation method (MSOR), modified Jacobi method (MJOR) and Uzawa method as well as algorithms based on the Gauss method with two options for iterative refinement, allowing to consider the exit from the contact of the contact border sections, are presented. The results of the application of these algorithms to the demo problem that simulates thermomechanical processes in the section of the fuel element, which includes two tablets and a fragment of the shell, are presented.

Combined method for correction interval systems of linear algebraic equations

Metrologiya ◽  
2021 ◽  
pp. 17-39
Author(s):  
A. N. Bazhenov ◽  
A. Yu. Telnova
Keyword(s):  
Linear Equations ◽  
Interval Uncertainty ◽  
Algebraic Equations ◽  
System Matrix ◽  
Combined Method ◽  
Interval System ◽  
Linear Algebraic Equations ◽  
Linear Interval ◽  
Systems Of Linear Equations ◽  
Interval Systems

The possibility of application of the interval analysis for data processing in the field of spectral analysis is considered. It is assumed that the data have interval uncertainty; therefore the problem of finding unknown concentrations is posed as a linear interval tolerance problem. The incompatibility of the interval system of linear algebraic equations is shown for the initial data using the apparatus of the recognizing functional. The relevance of the topic is due to the need for regularization of inconsistent interval systems of linear equations. The idea of S. P. Shary of a combined method for correcting a linear tolerance problem has been implemented. A new method for managing the solution by changing the linear algebraic equations interval system matrix elements radii has been developed. The research results can be used for example, to calculate the substance’s concentrations by measurement of the characteristic X-ray radiation.

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