System Of Linear Algebraic Equations Research Articles

Numerical modeling of nonlinear deformation processes for shells of medium thickness

When modeling a nonlinear isotropic eight-node finite element, the main kinematic and physical relationships are determined. In particular, isoparametric approximations of the geometry and an unknown displacement increment vector, covariant and contravariant components of basis vectors, metric tensors, strain tensors (Cauchy - Green and Almansi) and true Cauchy stresses in the initial and current configuration are introduced. Next, a variational equation is introduced in the stress rates in the actual configuration without taking into account body forces and the Seth material is considered, where the Almansi strain tensor is used as the finite strain tensor. Linearization of this variational equation, discretization of the obtained relations (stiffness matrix, matrix of geometric stiffness) is carried out. The resulting expressions are written as a system of linear algebraic equations. Several test cases are considered. The problem of bending a strip into a ring is presented. This problem is solved analytically, based on kinematic and physical relationships. Examples of nonlinear deformation of cylindrical and spherical shells are also shown. The method proposed in this paper for constructing a three-dimensional finite element of the nonlinear theory of elasticity, using the Seth material, makes it possible to obtain a special finite element, with which it is quite realistic to calculate the stress state of shells of medium thickness using a single-layer approximation in thickness. The obtained results of test cases demonstrate the operability of the proposed technique.

Solving the Problem of Elasticity for a Layer with N Cylindrical Embedded Supports

The main goal of deformable solid mechanics is to determine the stress–strain state of parts, structural elements, and their connections. The most accurate results of calculations of this state allow us to optimize design objects. However, not all models can be solved using exact methods. One such model is the problem of a layer with cylindrical embedded supports that are parallel to each other and the layer boundaries. In this work, the supports are represented by cylindrical cavities with zero displacements set on them. The layer is considered in Cartesian coordinates, and the cavities are in cylindrical coordinates. To solve the problem, the Lamé equation is used, where the basic solutions between different coordinate systems are linked using the generalized Fourier method. By satisfying the boundary conditions and linking different coordinate systems, a system of infinite linear algebraic equations is created. For numerical realization, the method of reduction is used to find the unknowns. The numerical analysis has shown that the boundary conditions are fulfilled with high accuracy, and the physical pattern of the stress distribution and the comparison with results of similar studies indicate the accuracy of the obtained results. The proposed method for calculating the stress–strain state can be applied to the calculation of structures whose model is a layer with cylindrical embedded supports. The numerical results of the work make it possible to predetermine the geometric parameters of the model to be designed.

Straightforward Free Vibration Solutions of Open Cylindrical Shells by the Finite Integral Transform Method

Straightforward analytical free vibration solutions of non-Lévy-type open cylindrical shells are obtained in this study by the finite integral transform method, which was not achieved by conventional semi-inverse methods. Three double finite integral transforms are imposed on the high-order partial differential equations (PDEs) for the problems, with some boundary conditions input, by which the relations between the transformed quantities and specific unknowns are given. Substituting the corresponding inverse transforms into the other boundary conditions provides systems of linear algebraic equations for determining the final analytical solutions. Comprehensive benchmark results for representative open cylindrical shells under different boundary conditions are shown. The results are well verified by other solution methods. With the present solutions, the effects of geometric parameters and boundary conditions on the vibration behaviors are quantitatively revealed by a parametric study. Due to its rigorous and straightforward solution procedure, this study presents a solid easy-to-implement way to explore new analytical solutions of similar problems.

On the Uniqueness of Solution to Systems of Linear Algebraic Equations to Which the Inverse Problems of Gravimetry and Magnetometry Are Reduced: A Regional Variant

Conditions for the unique solvability of systems of linear algebraic equations to which many inverse problems of gravitational and magnetic exploration are reduced are considered. The mathematical statements of inverse problems take into account the sphericity of the Earth.

On the Uniqueness of Solution to Systems of Linear Algebraic Equations to Which the Inverse Problems of Gravimetry and Magnetometry Are Reduced: A Regional Variant

On the Uniqueness of Solution to Systems of Linear Algebraic Equations to Which the Inverse Problems of Gravimetry and Magnetometry Are Reduced: A Regional Variant

An Analytical Solution on Vibration Reduction and Shear Force Mitigation of Cantilever Mindlin Plates Using Orthogonal Ribs

Vibration of aircraft wings and the dynamic stress concentration at the clamped edge are important research topics due to concerns on the safety of aircrafts. To have a better understanding of the problem, the free and forced vibration response of a ribbed rectangular cantilever plate representing a section of an aircraft wing is investigated in this study. A new analytical solution is developed for the vibration analysis of rib stiffened cantilever plates using Mindlin plate and Timoshenko beam theories alongside the finite integral transform technique. The one- and two-dimensional integral transforms are applied to the governing equations of beams and plates, respectively, where the coupling force components at the interface between the base plate and the beam(s) can be automatically defined during the integral transform. Eventually, the partial differential equations are transformed into a system of linear algebraic equations in which its derivation is rigorous and easily implemented. Good agreements are found between the results of analytical solution, finite element analysis (FEA) and related literature. The solution is then employed to study the vibration suppression of cantilever plates and the shear force at the clamped edge. It is found that the insertion of a pair of orthogonal ribs in the plate can effectively reduce its vibration. An optimum orthogonal ribbing pattern is obtained using multi-objective particle swarm optimization (MOPSO) algorithm, taking into consideration both the vibration suppression of the plate and the maximum induced shear force at the corners of the clamped edge.

Numerical solution of stochastic Volterra integral equations based on uniform Haar wavelets by using direct method

Nowadays, wavelets have played a crucial role in approximating the solution of a wide range of problems arising in science and engineering. Wavelets have been used in numerous areas of applied mathematics as diverse as signal analysis, statistics, computer-aided geometric, image processing and numerical analysis. The uniform Haar wavelets are most widely used wavelets for solving stochastic integral equations, outperforming Legendre, Daubechies, etc. The uniform Haar wavelets are generated from pairs of piecewise constant functions and can be simply integrated. Furthermore, the uniform Haar functions are orthogonal and form a good transform basis. In this paper, we have proposed an efficient numerical method to solve a class of stochastic linear Volterra integral equations (SLVIEs) based on the uniform Haar wavelets. The aim of this paper is to apply numerical method to obtain approximate solution of SLVIEs. By using this method, the SLVIEs can be reduced to a linear system of algebraic equations. Also, the error analysis of this method is performed. Finally, some numerical examples are given to demonstrate pertinent features of this method. Furthermore, we compare obtained results from this method with the achieved results from relevant studies.

Explicit Chebyshev Petrov–Galerkin scheme for time-fractional fourth-order uniform Euler–Bernoulli pinned–pinned beam equation

Abstract In this research, a compact combination of Chebyshev polynomials is created and used as a spatial basis for the time fractional fourth-order Euler–Bernoulli pinned–pinned beam. The method is based on applying the Petrov–Galerkin procedure to discretize the differential problem into a system of linear algebraic equations with unknown expansion coefficients. Using the efficient Gaussian elimination procedure, we solve the obtained system of equations with matrices of a particular pattern. The L ∞ {L}_{\infty } and L 2 {L}_{2} norms estimate the error bound. Three numerical examples were exhibited to verify the theoretical analysis and efficiency of the newly developed algorithm.

An Approach to Solving Direct and Inverse Scattering Problems for Non-Selfadjoint Schrödinger Operators on a Half-Line

In this paper, an approach to solving direct and inverse scattering problems on the half-line for a one-dimensional Schrödinger equation with a complex-valued potential that is exponentially decreasing at infinity is developed. It is based on a power series representation of the Jost solution in a unit disk of a complex variable related to the spectral parameter by a Möbius transformation. This representation leads to an efficient method of solving the corresponding direct scattering problem for a given potential, while the solution to the inverse problem is reduced to the computation of the first coefficient of the power series from a system of linear algebraic equations. The approach to solving these direct and inverse scattering problems is illustrated by several explicit examples and numerical testing.

An Efficient Technique for Algebraic System of Linear Equations Based on Neutrosophic Structured Element

Neutrosophic logic is frequently applied to the engineering technology, scientific administration, and financial matters, among other fields. In addition, neutrosophic linear systems can be used to illustrate various practical problems. Due to the complexity of neutrosophic operators, however, solving linear neutrosophic systems is challenging. This work proposes a new straightforward method for solving the neutrosophic system of linear equations based on the neutrosophic structured element (NSE). Here unknown and right-hand side vectors are considered as triangular neutrosophic numbers. Based on the NSE, analytical expressions of the solution to this equation and its degrees are also provided. Finally, several examples of the methodology are provided.

Qualitative Analysis for Solving a Fractional Integro-Differential Equation of Hyperbolic Type with Numerical Treatment Using the Lerch Matrix Collocation Method

In this research, we present a qualitative analysis for studying a new modification of a nonlinear hyperbolic fractional integro-differential equation (NHFIDEq) in dual Banach space CEE, J. Under some suitable conditions, the existence and uniqueness of a solution are demonstrated with the use of fixed-point theorems. The verification of the offered method has been conducted by applying the Lerch matrix collocation (LMC) method as a numerical treatment. The major motivation for selecting the LMC approach is that it reduces the solution of the given NHFIDEq to a matrix representation form corresponding to a linear system of algebraic equations; additionally, to demonstrate that the proposed strategy has better precision than alternative numerical methods, we study the error and the convergence analysis. Finally, we introduce numerical examples illustrating comparisons between the exact solutions and numerical solutions for different values of the Lerch parameters λ and time t as well as how the absolute error in each example is calculated.

On the Uniqueness of Solutions to Systems of Linear Algebraic Equations Resulting from the Reduction of Linear Inverse Problems of Gravimetry and Magnetometry: a Local Case

The paper considers issues of unique solvability of systems of linear algebraic equations to which many inverse problems of geophysics are reduced as a result of discretization. Examples of degenerate and nondegenerate systems of different dimensions arising from the interpretation of gravity and magnetometric data are given.

CONTROL OF BUNDLE OF TRAJECTORIES OF A LINEAR DISCRETE SYSTEM WITH FINITE SETS OF INITIAL AND FINAL STATES

The article proposes an analysis of the problem of control of trajectory bundle of a linear discrete system. The problem of trajectory bundle control arises in applications characterized by the deterministic uncertainty of the initial conditions of the system, control of a group of objects of a similar nature, etc. To such problems belongs, in particular, the control problem for a beam of charged particles. In the statement of the problem studied in the paper, the control function is scalar, the set of initial states and the set of final states contain a finite number of elements. At the same time, the system is completely controlled. The problem is to find a control that transfers the system from a set of initial conditions to a set of final states. The problem is to find a control that transfers the system from a set of initial conditions to a set of final states. In the article, the problem of controlling a bundle of trajectories is reduced to the problem of terminal control of a linear discrete system of higher dimension. At the same time, the resulting system has a block structure. This approach is new. Applying the formula that establishes the dependence between the initial and final states of a discrete system, the problem of terminal control is reduced to the problem of finding a solution to a system of linear algebraic equations. Using the structure of the matrix of the system, as well as the form of the general solution of the system of linear algebraic equations, we obtain a general solution of the terminal control problem. Applying the properties of the pseudoinverse matrix, we prove the theorem on conditions of the solution existence and establish the function, which gives the general solution of the trajectory bundle control problem. This result has an algorithmic orientation.

On the Uniqueness of Solutions to Systems of Linear Algebraic Equations Resulting from the Reduction of Linear Inverse Problems of Gravimetry and Magnetometry: a Local Case

On the Uniqueness of Solutions to Systems of Linear Algebraic Equations Resulting from the Reduction of Linear Inverse Problems of Gravimetry and Magnetometry: a Local Case

Reconstruction of a time-dependent wave source and the initial condition in a hyperbolic equation

This article is concerned with determination of a time-dependent wave source together with the initial condition in a hyperbolic equation. Based on the given data including the flux tensions of a string at end points and a nonlocal integral observation along with the wave displacement at two distinct instants of time, we prove the unique solvability of the inverse problem. Moreover, we estimate the modulus of continuity of the inverse problem under special conditions. The numerical solution of the problem is also presented by means of Bernstein spectral method, which converts the problem to a linear system of algebraic equations and then Tikhonov regularization technique is employed to derive reliable solutions. Regarding the perturbed boundary conditions, we use a regularized numerical technique to obtain stable derivatives. Numerical simulations are provided to show the effectiveness of the proposed scheme.

A problem for a material surface attached to the boundary of an elastic semi-plane

A problem for a nano-sized material surface attached to the boundary of an elastic isotropic semi-plane is considered. The semi-plane and the material surface are assumed to be in a plane strain state under the action of a normal external tractions applied to the material surface. The material surface is modeled using the Steigmann–Ogden form of surface energy. The problem is solved using integral representations of stresses and displacements. With the help of these representations, the problem can be reduced to either a system of two singular integral equations or a single singular integral equation. Two types of material surface tip conditions are considered: tip conditions with an uncompensated surface prestress (free tip conditions) and tip conditions with a compensated surface prestress term. The numerical solution of the system of singular integral equations is obtained by expanding each unknown function into a series based on the Chebyshev polynomials. Then, the approximations of the unknown functions can be obtained from a system of linear algebraic equations. Accuracy of the numerical procedure is studied. Various numerical examples for different values of the surface energy parameters are considered. It is shown that both the surface parameters and the type of tip conditions have significant influence on the behavior of the material system.

A collocation method based on roots of Chebyshev polynomial for solving Volterra integral equations of the second kind

This paper mainly studies numerical solution to the Volterra integral equation of the second kind. By using the roots of Chebyshev polynomial as collocation points, a new collocation method is proposed to solve the Volterra integral equation of the second kind. The proposed method firstly interpolates the product of the kernel function and the unknown solution at the roots of Chebyshev polynomial. Then, the Volterra integral equation is transformed into a system of linear algebra equations by properties of Chebyshev polynomials. Finally, the numerical solution of the Volterra integral equation is obtained by the Chebyshev polynomial interpolation. In addition, the error estimates of the proposed method are provided in a semi-posteriori sense; and numerical examples are given to show effectiveness of the proposed method.

A New Approach to Increase the Accuracy of a Numerical Scheme Using Modified Partial Differential Equation : Application to Meshless Methods

Meshless methods are gaining popularity for numerically solving the governing equations of Fluid Dynamics primarily because they do not require the generation of grid around the given geometry in the computational domain. They just require a point distribution with a proper connectivity, which is extremely simple to generate as compared to the grid around complex geometries. However, obtaining a higher order accurate numerical solution is difficult in case of meshless methods as it requires either a solution of a larger system of linear algebraic equations, or defect correction method coupled with inner iterations. This method becomes too much complicated if still further higher order accuracy is desired. Here an alternate method based on Modified Partial Differential Equations has been used to obtain higher order accurate solution using the meshless methods. The basic idea behind the new approach is to explicitly remove the truncation error from the Modified Partial Differential Equation (formed from the governing equations) and numerically solve the resulting equation with the meshless technique. This technique has been successfully applied for the simulation of subsonic, weak transonic, strong transonic and supersonic flow past NACA0012 aerofoil. A good match of the numerical results with the standard test results of AGARD/GAMM workshop has been.

Sound Wave Radiation from Partially Lined Duct

The radiation of sound waves from partially lined duct is treated by using the mode-matching method in conjunction with the Wiener-Hopf technique. The solution is obtained by modification of the Wiener-Hopf technique and involves an infinite series of unknowns which are determined from an infinite system of linear algebraic equations. Numerical solution of this system is obtained for various values of the problem parameters, whereby the effects of these parameters on the sound diffraction are studied. A perfect agreement is observed when the results of radiated field are compared numerically with a similar work existing in the literature.

Scattering of Sound Waves in Two Stepped Non-uniformly Lined Duct

Scattering of sound waves in two stepped cylindrical duct which walls are coated with different acoustically absorbent materials is investigated by using Wiener-Hopf technique directly and by determining scattering matrices. First, by using Fourier transform technique we obtain a couple of modified Wiener-Hopf equations whose solutions involve four sets of infinitely many unknown expansion coefficients providing systems of linear algebraic equations. Then we determine scattering matrices of the problem and we state the total transmitted field by using generalized scattering matrix method. Numerical results are compared for different parameters.