Set Of Linear Algebraic Equations Research Articles

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.

Electromagnetic Wave Scattering by a Multiple Core Model of Composite Cylindrical Wires at Oblique Incidence

A complex cylindrical structure consisting of a group of parallel stratified circular lossy dielectric cylinders, embedded in a dielectric circular cylindrical region and surrounded by unbounded dielectric space, is considered in this paper. The scattering of electromagnetic (EM) plane waves by the aforementioned configuration was studied; the EM waves impinged obliquely upon the structure and were arbitrarily polarized. The formulation used was based on the boundary-value approach coupled with the generalized separation of variables method. The EM field in each region of space was expanded in cylindrical wave-functions. Furthermore, the translational addition theorem of these functions was applied in order to match the EM field components on any cylindrical interface and enforce the boundary conditions. The end result of the analysis is an infinite set of linear algebraic equations with the wave amplitudes as unknowns. The system is solved by the truncation of series and unknowns and then matrix inversion; thus, we provide a semi-analytical solution for the scattered far-field and, as a consequence, for the scattering cross section of the complex cylindrical structure. The numerical results focus on calculations of the electric- and magnetic-field intensity of the far-field as well as of the total scattering cross section of several geometric configurations that fall within the aforementioned general structure. The effect of the geometrical and electrical characteristics of the structure on the scattered field was investigated. Specifically, the cylinders’ size and spacing, their conductivity and permittivity as well as the incidence direction were modified in order to probe how these variations are imprinted on scattering. Moreover, comparisons with previously published results, as well as convergence tests, were performed; all tests and comparisons proved to be successful.

Spark range data reduction using a Gauss collocation method

A new method is proposed for reducing spark range data using a Gauss pseudospectral collocation. Existing methods use numerical integration to propagate the differential equation model and its sensitivities forward in time. Here, a collocation method is used to transcribe the differential equations into algebraic ones that are easily solved. Realizing that the auxiliary equations used to find model sensitivities are linear, a second transcription renders a set of linear algebraic equations that solve exactly for the needed sensitivities without iteration. Thus, measurement sensitivities with respect to initial conditions and aerodynamic parameters are easily found through solution of a set of linear algebraic equations. The method is demonstrated on a set of actual data from the M898 155-mm projectile. For models using fixed-plane coordinates, and only the 10 most influential aerodynamic coefficients, the new method produces results that rival established commercial codes in accuracy.

Elastic wave scattering by flat-bottomed indentations on a plate

• Method for multiple scattering by scatterers with unmatched midplanes is proposed. • The method can analyze coupled flexural and in-plane extensional motions. • Mode conversions between different elastic waves are observed. • Stop bands of shear and flexural waves are captured simultaneously. • Broadband scattering polarization of flexural waves is achieved by multiple scattering. Out-of-plane flexural and in-plane extensional motions are independently analyzed in available studies of multiple scattering due to the assumption of the consistent mid-plane between the scatterer and host medium. When mid-planes of the scatterer and the host medium are inconsistent, such as flat-bottomed indentations, the coupled out-of-plane flexural and in-plane extensional motions should be considered. This paper aims to investigate a general numerical procedure to solve the three-dimensional multiple scattering problem of elastic waves on a plate with cylindrically flat-bottomed indentations. Wave expansion functions derived from the Poisson and Mindlin plate theories are used to express three-dimensional displacements of the host plate and flat-bottomed indentations. A linear matrix representing analytical relationships between wave expansion coefficients of exciting and scattered fields is deduced by the continuity conditions of flat-bottomed indentations. A set of linear algebraic equations is then developed by the fact that the exciting fields around a scatterer are the sum of scattered fields coming from all other scatterers. The numerical collocation method is used to solve the preceding algebraic equations. Simulation results verify mode conversions between compressional, shear, and flexural waves. Details of stop-band formations of shear and flexural waves are captured simultaneously. Furthermore, far-field scattering polarizations of shear and flexural waves are found in the case of multiple scattering.

When multilayer links exchange their roles in synchronization.

Real world networks contain multiple layers of links whose interactions can lead to extraordinary collective dynamics, including synchronization. The fundamental problem of assessing how network topology controls synchronization in multilayer networks remains open due to serious limitations of the existing stability methods. Towards removing this obstacle, we propose an approximation method which significantly enhances the predictive power of the master stability function for stable synchronization in multilayer networks. For a class of saddle-focus oscillators, including Rössler and piecewise linear systems, our method reduces the complex stability analysis to simply solving a set of linear algebraic equations. Using the method, we analytically predict surprising effects due to multilayer coupling. In particular, we prove that two coupling layers-one of which would alone hamper synchronization and the other would foster it-reverse their roles when used in a multilayer network. We also analytically demonstrate that increasing the size of a globally coupled layer, that in isolation would induce stable synchronization, makes the multilayer network unsynchronizable.

Exact Solution to Axisymmetric Interface Crack Problem in Magneto-Piezo-Electric Transversely Isotropic Materials

Summary This seems to be the first exact closed-form solution to the problem of a penny-shaped interface crack, subjected to an axisymmetric normal and tangential loading. The crack is located at the boundary between two bonded magneto-piezo-electric transversely isotropic half-spaces, made of different materials. We use the combination of Green’s functions for two different half-spaces and Fourier transform. We derive first the governing equations, which are valid for a crack of arbitrary shape. The usual approach leads to five hypersingular integral equations, we arrive at four integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications or not noticed at all. In the particular case of axial symmetry, the problem is reduced to just one singular equation, for which an exact closed-form solution is known. We are not aware of any other publication with which our results can be compared.

Improvement to the fictitious pile method for pile-soil interaction problems

Dynamic pile-soil interaction analysis is often a key task in many civil engineering problems and several methods have been developed and applied. To many researchers, the fictitious pile method is a good candidate because of its high computational efficiency compared with the three-dimensional finite element method (FEM) and boundary element method (BEM). However, due to the assumptions made in the method, application of the method to dynamic pile-soil interaction at sufficiently high frequency may be problematic. A study on this issue is performed in this paper by comparing with FEM. It is found that the assumption that the vertical load (termed the patch load) applied by the fictitious pile to the ground is uniformly distributed on the cross-section is an important factor invalidating the method at high frequencies. Upon this founding, the original fictitious pile method is then improved by replacing the uniformly distributed patch load with a series of ring loads which make the vertical displacements at the intermediate radii of the ring loads identical. A set of linear algebraic equations are established to determine the ring loads. Results from both the improved and original fictitious pile methods are compared with those from FEM as benchmark. The comparisons show that the improved fictitious pile method has a much better accuracy (or for a given accuracy, works to higher frequencies) than the original one but the calculation efficiency remains similar. Although for the cases studied in this paper the improved fictitious pile method is much more accurate than the original one, the effectiveness of former may be still unsatisfactory for other cases. Therefore, the effect of parameters, including soil Young's modulus, pile radius and pile length, on the accuracy of the fictitious pile method is also investigated. It turns out that the pile radius and length have adverse effects on the accuracy of the fictitious pile method while stiffer soil makes the method work better. Based on the parameters of the pile and soil, a critical frequency is derived below which the fictitious pile method can be well applied.

Calculation of Critical Load of Axially Functionally Graded and Variable Cross-Section Timoshenko Beams by Using Interpolating Matrix Method

In this paper, the interpolation matrix method (IMM) is proposed to solve the buckling critical load of axially functionally graded (FG) Timoshenko beams. Based on Timoshenko beam theory, a set of governing equations coupled by the deflection function and rotation function of the beam are obtained. Then, the deflection function and rotation function are decoupled and transformed into an eigenvalue problem of a variable coefficient fourth-order ordinary differential equation with unknown deflection function. According to the theory of interpolation matrix method, the eigenvalue problem of the variable coefficient fourth-order ordinary differential equation is transformed into an eigenvalue problem of a set of linear algebraic equations, and the critical buckling load and the corresponding deflection function of the axially functionally graded Timoshenko beam can be calculated by the orthogonal triangular (QR) decomposition method, which is the most effective and widely used method for finding all eigenvalues of a matrix. The numerical results are in good agreement with the existing results, which shows the effectiveness and accuracy of the method.

Algorithmic Advancements and a Comparative Investigation of Left and Right Looking Sparse LU Factorization on GPU Platform for Circuit Simulation

Sparse LU factorization is a key tool in the solution of large linear set of algebraic equations encompassing a wide range of computing applications. Recent advances in this field exploit the massively parallel architecture of the GPUs via left-looking algorithm (LLA) and right-looking algorithm (RLA). In this paper, <i>adaptive cluster</i> mode is proposed to improve the state-of-the-art in LLA for GPU platforms. The proposed method takes into consideration of varying sparsity at different levels during cluster mode execution, to adaptively configure the GPU block size and the number of parallel columns. The new refinements for LLA are also integrated with the dynamic parallelism that is available in modern GPU architectures. The paper also provides a comprehensive performance comparison of the LLA and hybrid RLA along with state-of-the-art advances on the same GPU platform. The results indicate that, when implemented with similar refinements and on a same platform, LLA provides better performance compared to the hybrid-RLA. The results would be useful to the scientific community while making decision on adopting LLA or RLA algorithms for sparse LU factorization.

Modeling Current Distribution Within Conductors and Between Parallel Conductors in High-Frequency Magnetics

In both planar and wire-wound transformers, large copper cross-sections and parallel windings are often used to increase conduction area and decrease copper loss. However, at high frequency, current is not guaranteed to spread out maximally over the cross-section of a single conductor or to split evenly between parallel conductors. Finite element analysis (FEA) and SPICE-based systems have been used to analyze current distribution within magnetic components, but these methods are computationally intensive. In this paper, we show that Maxwell's equations, in the high frequency limit, yield a set of linear algebraic equations that are rapidly solvable to yield both the current and magnetic field distribution and hence can be used to predict loss and leakage inductance. Due to its simplicity, this method is easily applied to cases with a one-dimensional or two-dimensional distribution of current. We show that predicted results match both FEA simulations and experimental measurements very accurately for a variety of cases. This paper is accompanied by several software implementations of the method. This method can be used to rapidly analyze high frequency current distribution in transformers and can easily be integrated into numerical optimization algorithms.

Φ-Haar Wavelet Operational Matrix Method for Fractional Relaxation-Oscillation Equations ContainingΦ-Caputo Fractional Derivative

This paper proposes a numerical method for solving fractional relaxation-oscillation equations. A relaxation oscillator is a type of oscillator that is based on how a physical system returns to equilibrium after being disrupted. The primary equation of relaxation and oscillation processes is the relaxation-oscillation equation. The fractional derivatives in the relaxation-oscillation equations under consideration are defined in theΦ-Caputo sense. The numerical method relies on a novel type of operational matrix method, namely, theΦ-Haar wavelet operational matrix method. The operational matrix approach has a lower computational complexity. The proposed scheme simplifies the main problem to a set of linear algebraic equations. Numerical examples demonstrate the validity and applicability of the proposed technique.

Statistical learning for fluid flows: Sparse Fourier divergence-free approximations

We reconstruct the velocity field of incompressible flows given a finite set of measurements. For the spatial approximation, we introduce the Sparse Fourier divergence-free approximation based on a discrete L2 projection. Within this physics-informed type of statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. We regularize our minimization problems with the seminorm of the fractional Sobolev space in a Tikhonov fashion. In the Fourier setting, the incompressibility (divergence-free) constraint becomes a finite set of linear algebraic equations. We couple our spatial approximation with the truncated singular-value decomposition of the flow measurements for temporal compression. Our computational framework thus combines supervised and unsupervised learning techniques. We assess the capabilities of our method in various numerical examples arising in fluid mechanics.

An efficient approach for solving nonlinear multidimensional Schrödinger equations

An efficient numerical method is proposed for the solution of the nonlinear cubic Schrödinger equation. The proposed method is based on the Fréchet derivative and the meshless method with radial basis functions. An important characteristic of the method is that it can be extended from one-dimensional problems to multi-dimensional ones easily. By using the Fréchet derivative and Newton–Raphson technique, the nonlinear equation is converted into a set of linear algebraic equations which are solved iteratively. Numerical examples reveal that the proposed method is efficient and reliable with respect to the accuracy and stability.

Theoretical modeling of heat transfer in a multilayer rectangular body with spatially-varying convective heat transfer boundary condition

Theoretical analysis of heat transfer in a multilayer body is relevant for several engineering applications where heat transfer occurs through a stack of thermally dis-similar materials. In most of such literature, a constant convective heat transfer coefficient is assumed as a boundary condition at one end of the body. In contrast, there is a lack of work to model a problem with spatial variation in the convective heat transfer coefficient. This paper presents an analytical solution for thermal conduction in a multilayer rectangular body with spatially varying convective heat transfer coefficient at one end. In addition, this solution also accounts for internal heat generation and inter-layer thermal contact resistance and spatially varying heat flux on the other end of the body. The solution is derived in a series form, and it is shown that the coefficients of the series can be determined by solving a well-defined set of linear algebraic equations. The analytical solution is verified by comparison with numerical simulations, as well as with standard solution for a simplified special case. The impact of spatially varying cooling provided by jet impingement is analyzed using the analytical solution. A resource optimization problem pertaining to the optimal use of jet cooling is solved. Results presented here may benefit thermal management analysis of a broad variety of engineering applications involving heat transfer in multi-layer bodies.

A stochastic analysis method of transient responses using harmonic wavelets, part 2: Time-dependent vehicle-bridge systems

Conventional stochastic response analysis methods of vehicle-bridge (VB) systems involve a brute-force calculation based on a large number of simulations, which imposes huge computational burdens because the dynamic analysis of a time-dependent VB system requires repeated assembles of system matrices. This paper proposes a novel stochastic analysis method of transient responses for time-dependent VB systems based on periodic generalized harmonic wavelets (GHW). We establish the equations of motion of the VB system and reveal the orthogonal characteristic relationships between the time-dependent system matrices and the wavelet functions. Then, two sets of linear algebraic equations of wavelet coefficients are established for describing the time-dependent VB system, and the time-varying power spectral density functions of system responses are obtained in a semi-analytical form. The stochastic response analysis of the time-dependent VB system is therefore converted into solving a set of linear algebraic equations, which significantly simplifies the stochastic response analysis. The accuracy and efficiency of the proposed method are validated via comparison against the Monte Carlo simulations of nonstationary stochastic analyses of a highway bridge and a railway bridge, respectively. The proposed method provides a high-efficient way to estimate the time-varying power spectral density functions of stochastic responses of time-dependent VB systems.

New approach to interface crack problems in transversely isotropic materials

The novelty of the approach contains several aspects: the method of derivation of the governing equations, the form and number of equations, the optimal choice of elastic constants. While majority of published articles derives the governing equations from the Green’s function for the compound space and the use of the reciprocal theorem, we use the combination of Green’s functions for 2 different half-spaces and Fourier transform. The usual approach leads to 3 hypersingular integral equations, we arrive at 2 integro-differential equations (one of them being complex). While the coefficients of the governing equations in existing publications are presented in terms of the results of the solution of a set of linear algebraic equations and are too cumbersome to be written explicitly in terms of the basic elastic constants, our choice of the basic constants leads to quite elegant explicit expressions for these coefficients and reveals certain symmetry, which was noticed only numerically in previous publications. The new form of the governing equations allows us to obtain an exact closed form solution for an axisymmetric interface crack problem by elementary means. The same method can be applied to obtain the exact solution for a non-axisymmetric problem. A comparison is made with the existing exact solutions and some are shown to be incorrect.

Multistage Linear Gauss Pseudospectral Method for Piecewise Continuous Nonlinear Optimal Control Problems

This article aims at proposing a multistage linear Gauss pseudospectral method (MS-LGPM) for solving the piecewise continuous nonlinear optimal control problem (OCP) with interior-point constraints and terminal constraints. First, the first-order necessary conditions for the multistage nonlinear OCP are derived, and a typical multipoint boundary value problem is obtained. Second, the original problem can be solved iteratively by integral prediction and quasi-linearization. Then Lagrange interpolation polynomials are used to approximate the state, control, and costate variables, to transfer those differential equations into a set of linear algebraic equations. Therefore, the control update coming closer to the optimal solution can be derived in an analytical manner. Additionally, those linear algebraic equations are formulated in regular banded matrix forms to make the code as concise as possible. Finally, the proposed method is applied to the midcourse guidance design for a dual-pulse missile to evaluate its performance. Simulation results show that the proposed method performs well in providing the optimal control with high accuracy and computational efficiency. Furthermore, a comparison with other typical guidance method and Monte Carlo simulations are also provided. Results demonstrate, even with some random uncertainties, the proposed method still has strong robustness and superior performance.

Conjugate Heat Transfer Analysis Using the Discrete Green's Function

Abstract Conjugate heat transfer problems generally require a coupled solution of the temperature fields in the fluid and solid domains. Implementing the boundary condition at the surface of the solid using a discrete Green's function (DGF) decouples the solutions. A DGF is determined first considering only the fluid domain with prescribed thermal boundary conditions, then the temperature distribution in the solid is calculated using standard numerical methods. The only compatibility requirement is that the DGF must be specified with the same discretization as the surface of the solid. The method is demonstrated for both steady-state and transient heating of a thin plate with laminar boundary layers flowing over both sides. The resulting set of linear algebraic equations for the steady-state problem or linear ordinary differential equations for the transient problem are easily solved using conventional scientific programming packages. The method converges with nearly second-order accuracy as the discretization resolution is increased.