Set Of Linear Algebraic Equations Research Articles

Investigation of moving load-induced vibrations in layered functionally graded Terfenol-D beams: a differential quadrature method and analytical approach

The paper investigates the potential of the full-sinusoidal Fourier series as a solution form for the deflection of layered functionally graded beams associated with smart actuators, drawing on the fundamental principles of classical Fourier series theory. The current method simplifies the difficult beam problem to a set of linear algebraic equations by using the Duhamel integration technique and the orthogonality of the trial function. The study takes into account generalized boundary conditions and moving forces, which are seldom discussed in previous research. Under the action of a moving load, the boundary value problem for a functionally graded beam integrated with Terfenol-D is efficiently addressed using the approach of combined differential and integral quadrature. The generalized boundary conditions may be easily achieved by adjusting the stiffness of the restraining springs. The significant agreement between the differential quadrature solution and the Fourier series solution underlines the efficiency and accuracy of both methods. Furthermore, the influences of various crucial physical characteristics on the natural frequencies and the essential flow velocities are explored, including boundary stiffness, foundation parameters, and geometric parameters.

Thermal Spreading/Constriction From an Isothermal Source Into a Multilayer Orthotropic Semi-Infinite Flux Tube

Abstract Thermal spreading and constriction have been widely studied due to relevance in heat transfer across interfaces with imperfect contact and problems such as microelectronics thermal management. Much of the past work in this field addresses an isoflux source, with relatively lesser work on the isothermal source problem, which is of much relevance to heat transfer across rough interfaces. This work presents an analytical solution for thermal spreading/constriction resistance that governs heat flow from an isothermal source into a multilayer orthotropic semi-infinite flux tube. The mixed boundary condition due to the isothermal source is accounted for by writing a convective boundary condition with an appropriately chosen spatially-varying Biot number. A series solution for the temperature field is derived, along with a set of linear algebraic equations for the series coefficients. An expression for the nondimensional thermal spreading resistance is derived for Cartesian and cylindrical problems. It is shown that, depending on the values of various nondimensional parameters, heat transfer in either the thin film or the flux tube may dominate and govern the overall thermal spreading resistance. Results for a single-layered isotropic flux tube are derived as a special case of the general result, for which, good agreement with past work is demonstrated. An easy-to-use polynomial fit for this special case is presented. This work contributes a novel technique for solving mixed boundary problems involving an isothermal source, and may also help solve practical problems related to interfacial heat transfre and thermal management.

Galerkin finite block method in solid mechanics

In this paper, the Galerkin Finite Block Method (GFBM) is proposed for the first time to solve two-dimensional elasticity problems with functionally graded materials (FGMs). The physical variables are approximated with double layer Chebyshev polynomials with unknown coefficients for two-dimensional elasticity. The main idea is to establish a set of linear algebraic equations by the Galerkin method using Chebyshev polynomials in the weak form of the partial differential equations and the boundary conditions. By introducing the mapping technique, a block with irregular shape of the boundary is transformed from the Cartesian coordinates (x, y) to the normalized coordinates (ξ, η), |ξ|⩽1, |η|⩽1. The continuity conditions of the displacement and traction on the interface between two blocks are introduced both in the weak form and strong form respectively. Several numerical examples are presented and comparisons are made with the Finite Element Method (FEM) including a cracked FGM sheet, in order to demonstrate the accuracy and efficiency of the GFBM.

Buckling analysis of rectangular plates with rotationally restrained edges subjected to linearly varying in-plane loading

This study addresses a comprehensive procedure for the buckling behavior of rectangular plates with rotationally restrained edges. They are subjected to uniaxial in-plane load that varies linearly from compression to bending. The load is applied to opposite edges with variable boundary conditions, ranging from simply supported to clamped. The other edges can also be free, guided, or between them. For the first time, the generalized integral transform technique (GITT) is applied to the buckling equation of such a plate. The integral kernel perfectly satisfies the plate boundary conditions and its constituent terms are limited to the approximate range of ±1 which prevents the numerical instability. The transformed equation is a set of linear algebraic equations which establish an eigenvalue problem. The plate buckling coefficient, and corresponding mode shape (contours of the buckled plate, the number of half-waves in each direction, etc.) are presented according to variations of the edge relative stiffness as well as the plate aspect ratio, the loading shape, and Poisson's ratio. The latter parameter has a significant effect on the buckling behavior when at least one of the edges is free, guided, or between them. This portion is developed for auxetic materials (negative Poisson's ratios), and it was concluded that the minimum buckling load mostly occurs as the Poisson's ratio tends to zero.

Time-dependent simplified [formula omitted] approximation for neutron transport equation

Time-dependent transport is still too computationally expensive for various practical purposes. On the other hand, the diffusion approximation is unable to provide accurate results for many real problems. To establish a middle ground between the accuracy of the transport and the simplicity of the diffusion approximations, the simplified spherical harmonics (SPN) method was introduced. The present paper aims to improve the accuracy of the time-dependent SP3 method by employing the simplified double-P1 (SDP1) approximation, which is based on the one-dimensional double-spherical harmonics DP1 equations. Using the finite element method to treat the spatial dependence of the SDP1 equations leads to a set of first-order ordinary differential equations on the time variable. These equations are then converted into a set of linear algebraic equations by discretizing the time variable with the implicit backward finite difference scheme. Our numerical results indicate that the SDP1 approximation could compute more precise values than the widely used SP3 method in transient problems.

Semi-analytical models for two-dimensional multispecies transport of sequentially degradation products influenced by rate-limited sorption subject to arbitrary time-dependent inlet boundary conditions

Most of the semi-analytical and analytical models employed to depict multidimensional, multispecies transport of sequentially degrading reaction products are built upon solving a set of coupled advection-dispersion equations (ADEs). Within these equations, sorption is considered as being equilibrium-controlled. However, it has been demonstrated that more realistic predictions of the transport of contaminants in the groundwater could be obtained by the use of a rate-limited sorption process instead of an equilibrium sorption assumption. This study is thus designed to devise semi-analytical models for the two-dimensional multispecies transport of a chemical mixture comprised of a parent compound and its degradation-daughter products which is influenced by rate-limited sorption subject to arbitrary time-dependent inlet boundary conditions. Three integral transforms are applied to generate a set of linear algebraic equations (AEs) resulting from the reductions of the ADEs. The contaminant concentration of each species is calculated by solving these AEs and then retransforming the solutions back to the original time-space domain. The simulation results obtained with our newly developed semi-analytical model are nearly identical to those generated using a numerical model based on the Laplace transform finite difference (LTFD) method. This confirms the validity of the new semi-analytical model. An investigation of the effect of rate-limited sorption on the migration of the contaminant plume is carried out using various time-dependent inlet boundary conditions. The sorption rate values vary from low to high, specifically, 0.05, 0.5, 5, and 50 years−1. The results show that the predicted concentrations of all contaminants within in the decay chain decrease as the sorption rate constant increases, for both constant and exponentially time-dependent boundary sources. However, under pulse loading boundary conditions, the concentrations of the later degradation products tend to increase with the sorption rate constant. The semi-analytical models created in this study, allow for different inlet boundary conditions, so can be utilized to simulate the transport of sequentially degrading reaction products. This greater accuracy makes them useful for many applications.

Thermal spreading resistance from an isothermal source into a finite-thickness body

Thermal spreading resistance is encountered during two-dimensional thermal conduction from a hot surface into a body. Thermal spreading and the opposite problem of thermal constriction resistance are of much technological relevance in the design of heat sinks and thermal spreaders for thermal management of microelectronics and other heat-generating devices. Much of the past work in the theoretical analysis of thermal spreading is based on a source with given heat flux. In contrast, the isothermal source problem presents difficulties due to the resulting mixed nature of the boundary condition, due to which, only approximate solutions are available. This work derives the steady-state thermal spreading resistance from an isothermal source into a finite-thickness slab or cylinder. The mixed boundary condition is handled by posing it in the form of a spatially varying convective boundary condition, with a sufficiently large Biot number over the source to represent its isothermal nature. A series solution for the problem is derived, along a sufficient set of linear algebraic equations to determine the series coefficients. Results are shown to agree well with finite-element simulations. Results are compared with previously-reported approximate solutions within the parametric range of validity of the approximate solutions. The impact of key non-dimensional parameters on thermal spreading resistance is quantified. It is shown that thermal spreading resistance increases with decreasing size of the isothermal heat source, as expected. A third-order polynomial correlation with very good accuracy is proposed. This work advances the theoretical understanding of a problem for which only approximate solutions have been reported in the past. Results presented here offer a practical tool for thermal design and optimization of a variety of practical thermal management problems involving spreading or constriction.

Heat transfer in a vertically integrated semiconductor device with unequal layer widths

Vertically integrated semiconductor devices and interposer-based heterogeneous integration technologies are being commonly investigated for advanced microelectronic chips. Such multilayer structures often integrate layers of unequal lateral sizes. While most of the past literature on thermal modeling of multilayer semiconductor chips considers layers of uniform widths, heat transfer in multilayer systems with unequal widths orthogonal to the thickness direction has not been investigated much. This remains an important challenge towards effective thermal management of vertically integrated ICs and other semiconductor chips. This work presents an analytical model to determine the temperature distribution in a multilayer device with unequal layer widths. A series solution for temperature distribution in each layer is derived, with an independent set of eigenvalues for each layer. The coefficients of the series solutions are determined through the derivation of a set of linear algebraic equations in the coefficients. Results are found to be in good agreement with numerical simulations, and also agree well with past work for special cases. An analysis of the impact of unequal widths on heat transfer and temperature distribution is presented. It is shown that the lower the ratio of the layer widths, the greater is the peak temperature. On the other hand, layer thickness is found to have an interesting, non-monotonous impact on peak temperature, which is explained on the basis of the two-dimensional nature of heat transfer in this problem. It is found that for a representative SiGe-Si device, peak temperature increases by around 9% as the SiGe layer width is reduced to one-third compared to a baseline equal width case. The novelty of the present work is that it presents the thermal modeling of an important multilayer semiconductor architecture, which has not been addressed much in the past. Results presented here may help improve the thermal design of vertically integrated semiconductor devices and other similar multilayer devices.

Metasurface characterization based on eigenmode analysis and averaging of electromagnetic fields

A fully numerical homogenization technique for the retrieval of the effective surface susceptibilities of a periodic composite metasurface is developed in this work. We utilize the so-called dual field-flux finite element formulation scheme to accurately calculate the eigenmodes of a composite periodic metasurface, a scheme that possesses a crucial advantage: The capability of evaluating all field components and their derivatives accurately and with the same order of approximation is a requirement for our proposed technique. Next, we derive the generalized sheet transition condition equations for a general bianisotropic metasurface, which correlate the field components on either sides of the metasurface with its surface susceptibilities. At this point, we establish a new field averaging scheme for the acquisition of the average field components of the modes supported by the metasurface. Combining this computational information, we derive a set of linear algebraic equations based on the GSTCs and the average electromagnetic fields and numerically solve it, so as to obtain the effective surface susceptibilities of a bianisotropic metasurface. Comparison with the results of other techniques in the literature shows very good agreement, relatively to the resonance behavior of the returned values and their position at the frequency spectrum. The advantages that distinguish the proposed technique over other related methods are its foundation on the intrinsic modal information of the eigenmodes supported by the metasurface and its independence of any wave excitation schemes or involvement of analytical polarizability calculations.

Direct Method for Variational Problems Using Boubaker Wavelets

The wavelets have many applications in engineering and the sciences, especially mathematics. Recently, in 2021, the wavelet Boubaker (WB) polynomials were used for the first time to study their properties and applications in detail. They were also utilized for solving the Lane-Emden equation. The aim of this paper is to show the truncated Wavelet Boubaker polynomials for solving variation problems. In this research, the direct method using wavelets Boubaker was presented for solving variational problems. The method reduces the problem into a set of linear algebraic equations. The fundamental idea of this method for solving variation problems is to convert the problem of a function into one that involves a finite number of variables. Different numerical examples were given to demonstrate the applicability and validity of this method using the Matlab program. Also, the results of this technique were compared with the exact solution, and graphs were added to these examples to test the convergence of Wavelet Boubaker polynomials using this method.

Non-dyadic Haar Wavelet Algorithm for the Approximated Solution of Higher order Integro-Differential Equations

The objective of this study is to explore non-dyadic Haar wavelets for higher order integro-differential equations. In this research article, non-dyadic collocation method is introduced by using Haar wavelet for approximating the solution of higher order integrodifferential equations of Volterra and Fredholm type. The highest order derivatives in the integrodifferential equations are approximated by the finite series of non-dyadic Haar wavelet and then lower order derivatives are calculated by the process of integration. The integro-differential equations are reduced to a set of linear algebraic equations using the collocation approach. The Gauss - Jordan method is then used to solve the resulting system of equations. To demonstrate the efficiency and accuracy of the proposed method, numerous illustrative examples are given. Also, the approximated solution produced by the proposed wavelet technique have been compared with those of other approaches. The exact solution is also compared to the approximated solution and presented through tables and graphs. For various numbers of collocation points, different errors are calculated. The outcomes demonstrate the effectiveness of the Haar approach in resolving these equations.

Investigation of Higher Order Localized Approximations for a Fractional Pricing Model in Finance

In this work, by considering spatial uniform meshes and stencils having five adjacent discretization nodes, we furnish a numerical scheme to solve the time-fractional Black–Scholes (partial differential equation) PDE to price financial options under the generalized multiquadric radial basis function (RBF). The time-fractional derivative is estimated by an L1-scheme but the spatial variable is discretized using fourth-order RBF-FD methodology. As a matter of fact, the PDE problem is transformed in the form of a linear set of algebraic equations. To support analytical discussions, numerical tests are furnished and reveal the efficacy of the presented solver.

A Tau Approach for Solving Time-Fractional Heat Equation Based on the Shifted Sixth-Kind Chebyshev Polynomials

The time-fractional heat equation governed by nonlocal conditions is solved using a novel method developed in this study, which is based on the spectral tau method. There are two sets of basis functions used. The first set is the set of non-symmetric polynomials, namely, the shifted Chebyshev polynomials of the sixth-kind (CPs6), and the second set is a set of modified shifted CPs6. The approximation of the solution is written as a product of the two chosen basis function sets. For this method, the key concept is to transform the problem governed by the underlying conditions into a set of linear algebraic equations that can be solved by means of an appropriate numerical scheme. The error analysis of the proposed extension is also thoroughly investigated. Finally, a number of examples are shown to illustrate the reliability and accuracy of the suggested tau method.

Algebraic Methods for Achieving Super-Resolution by Digital Antenna Arrays

The actual modern problem of developing and improving measurement and observation systems (including robotic ones) is to increase the volume and quality of the information received. Increasing the angle resolution to values significantly exceeding the Rayleigh criterion, i.e. achieving super-resolution is one of important ways to solve the problem. Angular super-resolution which makes it possible to detail images of research objects and their individual fragments, improves the quality of solutions to detection, recognition and identification problems, increases the range of such systems. In many papers methods developed by authors to achieve a super-resolution based on approximate solutions of inverse problems in the form of Fredholm integral equation of the first kind of convolution type called algebraic are presented. The methods used, as well as their varieties, make it possible to reduce solutions of inverse problems posed to solving sets of linear algebraic equations (SLAE). This paper presents results of further improvement of algebraic methods based on intelligent analysis of received signals. It is shown that their use in systems based on digital antenna arrays makes it possible to increase the achieved degree of exceeding the Rayleigh criterion. In the course of numerical experiments with a mathematical model, the stability of the solutions obtained and their adequacy were confirmed. The numerical results obtained open the following possibilities: (1) obtaining images of studied objects with a resolution exceeding the Rayleigh criterion by 4 to 10 times, (2) determining the angular coordinates of individual small-sized objects as part of multi-element complex objects (group targets), (3) clarifying boundaries of extended objects and their individual elements, (4) localizing individual bright objects on a smoothly inhomogeneous reflective background. Applying presented new methods does not require a significant computing power, what allows you to work in a real time mode using relatively simple and inexpensive computing devices. The ways of further improvement of presented algebraic methods for solving applied inverse problems are described.

Moment-Fourier approach to ion parallel fluid closures and transport for a toroidally confined plasma

A general method of solving the drift kinetic equation is developed for an axisymmetric magnetic field. Expanding a distribution function in general moments, a set of ordinary differential equations is obtained. Successively expanding the moments and magnetic-field involved quantities in Fourier series, a set of linear algebraic equations is obtained. The set of full (Maxwellian and non-Maxwellian) moment equations is solved to express the first-order density, temperature, and flow velocity in terms of radial gradients of the zeroth-order pressure and temperature. Closure relations that connect parallel heat flux density and viscosity to the radial gradients and parallel gradients of temperature and flow velocity are also obtained by solving the non-Maxwellian moment equations. The closure relations combined with the linearized fluid equations reproduce the same solution obtained directly from the full moment equations. The method can be generalized to derive closures and transport for an electron-ion plasma and a multi-ion plasma in a general magnetic field.

Modified harmonic balance method for forced nonlinear vibration of beam on an elastic foundation

A modified harmonic balance method has been presented and adopted for the nonlinear vibration problem of beams on an elastic foundation. By this method, a nonlinear algebraic equation along with a set of linear algebraic equations are required to be solved, which reduces computation effort. The results determined by the present method are compared with the corresponding results obtained by the existing method. Besides, the effect of various parameters on nonlinear vibrational behavior is examined.

Numerical Solution of High-Order Linear Fredholm Integro-Differential Equations by Lucas Collocation Method

In this paper, a useful matrix approach for high-order linear Fredholm integro-differential equations with initial boundary conditions expressed as Lucas polynomials is proposed. Using a matrix equationwhich is equivalent to a set of linear algebraic equations the method transforms to integro-differential equation. When compared to other methods that have been proposed in the literature, the numerical results from the suggested technique reveal that it is effective and promising. And also, error estimation of the scheme was derived. These results were compared with the exact solutions and the other numerical methods to the tested problems.

Separation of ion component from solid hydrocarbon materials by laser ablation

Plasma usually consists of multiple ion component. Ion-component separation occurs in various conditions, and profoundly affects the plasma dynamic evolution. In this work, ion-component separation in two-ion-component plasma is investigated in the hydrodynamic condition. Starting from the Landau-Fokker-Planck equations of two-ion-component plasma, the ion transport equations are reduced through the Chapman-Enskog approach. The transport equations are then transformed into a set of linear algebraic equations and solved by expanding the perturbed ion distribution functions into the series of Sonine polynomials. The diffusive ion mass flows with inclusion of baro-diffusion, thermo-diffusion and electro-diffusion are thus obtained. With these efforts, the complete ion fluid equations are presented, which can be used to describe the processes of ion-component separation. We evaluate ion-component separation in the case of a solid CH plate target ablated with a laser pulse, by solving the ion diffusion equation with the hydro states output from the one-dimensional radiative hydro code Multi-1D. The simulation results show that ion-component separation mainly occurs around ablation front and under-dense region, and that the effect of ion-species separation on plasma hydrodynamic evolution is minor and can be neglected. For those physical processes sensitive to ion concentration such as Thomson scattering, however, the effect of ion-component separation is significant, which means that ion-component separation should be included in the study of laser plasma interaction.

Froude Number Effects on Two-Dimensional Hydrofoils

The performance of a two-dimensional hydrofoil of arbitrary camber, moving at arbitrary Froude number at a constant depth below a free surface, is considered. The treatment is based upon the use of singularity distributions and thin foil theory. By assuming an appropriate series form for the vortex distribution representing the hydrofoil, it is shown that the problem can be reduced to the solution of a set of linear algebraic equations. These are solved by a collocation procedure. Numerical results for the performance characteristics are then given for several hydrofoil configurations, submergence depths, and Froude numbers. These indicate that operation at Froude numbers greater than about ten is practically equivalent to operation at infinite Froude number. However, at lower values of the Froude number and for all the configurations considered, Froude number effects are important, even at submergence depths of several chord lengths.

ANALYTICAL SOLUTION OF MODIFIED FORCED VAN DER POL VIBRATION EQUATION USING MODIFIED HARMONIC BALANCE METHOD

For obtaining analytical solutions to the modified forced Van der Pol equation, the modified harmonic balance method has been developed. In classical harmonic balance method, the numerical procedure is used for solving a set of nonlinear algebraic equations. But it requires laborious computational attempt and accurate primary guesses values which makes it very hard-working to calculate. According to our method, a set of nonlinear algebraic equations has been converted to a set of linear algebraic equations by using a nonlinear one instead of a set. As a result, it reduces the massive computational work. This method provides not only better results than the existing harmonic balance method but also provides very close solutions to the corresponding numerical results. It is noticeable that there is substantial similarity between the approximate and the numerical results attained by the fourth order Runge-Kutta method. Moreover, the method is facile and straightforward. This technique may play great role to handle strongly nonlinear damped systems with external forces.