System Of Linear Algebraic Equations Research Articles

Reduction in the Computational Complexity of Calculating Losses on Eddy Currents in a Hydrogen Fuel Cell Using the Finite Element Analysis

This paper considers the issues of the reduction in the computational complexity of the numerical problem and the requirements of the computer memory size when calculating the power of the eddy currents in a hydrogen fuel cell based on a proton exchange membrane. The study was performed on a model problem based on a geometric pattern of a hydrogen fuel cell of typical dimensions and characteristics operating at a nominal load in steady-state conditions. The power of the eddy currents was calculated using a numerical model based on a tetrahedral finite element mesh and a mathematical model of a quasi-stationary electromagnetic field in the classical formulation through the vector magnetic potential. The requirements for the minimum degree of the geometric model approximation by a finite element mesh to achieve mesh stability of the computational problem results were defined. The paper considers issues of finite element order selection, the method of ordering, and the solution of the systems of linear algebraic equations (SLAEs). It shows the techniques for determining the accuracy of the calculated SLAE solution obtained by the direct method, as well as the effectiveness of the low-rank approximation of the SLAE to reduce the computational complexity of the computational problem solution and reduce the requirements for the amount of computer memory needed, considering the reduced accuracy of the SLAE solutions.

ОПРЕДЕЛЕНИЕ ЧАСТОТ АНТИСИММЕТРИЧНЫХ ФОРМ СВОБОДНЫХ КОЛЕБАНИЙ ПРЯМОУГОЛЬНОЙ CFCF-ПЛАСТИНЫ

To find the natural frequencies of antisymmetric oscillation patterns of a rectangular plate, two opposite sides of which are pinched and the other two are free (CFCF plate), a method of sorting frequency values («shooting method») in combination with the method of successive approximations is proposed. The deflection function is represented by two hyperbola-trigonometric series along two coordinates. When all the conditions of the boundary value problem are met, a homogeneous infinite system of linear algebraic equations is obtained with respect to the indeterminate coefficients of these series, which contains the oscillation frequency as a parameter. To find nontrivial solutions of the corresponding reduced system, an iterative process of parameter iteration is organized. Numerical results are obtained for the first two antisymmetric eigenforms. The influence of the number of series terms and the number of iterations on the accuracy of the results is investigated. A comparison with the results of other authors and experimental data is given.

Wire-Grid and Sparse MoM Antennas: Past Evolution, Present Implementation, and Future Possibilities

Since the end of the 19th century, radioelectronic devices (REDs) have actively penetrated into all modern community spheres. Achievements in the fields of radio engineering and electronics, as well as computing, information, telecommunications, and other technologies, have greatly contributed to this. The main elements of REDs are antennas and microwave devices. For example, linear (wire) antennas are the basis of long-distance communication agency networks of various law enforcement agencies and departments. The manufacturing of REDs requires the regular and rapid appearance of more and more advanced types with minimal costs. At the same time, the design complexity of REDs and the tightening of EMC requirements caused by the growth of upper frequencies of useful and interfering signals, the mounting density, as well as the capabilities of generators of intentional electromagnetic impacts, together with the need to take into account inter-element, inter-unit, and inter-system interference, require more and more accurate designs of REDs. However, this becomes impossible without computer modeling, which saves the time and financial resources required for their development, as well as to evaluate the correctness of the proposed technical solutions. During the design process, as a rule, a multivariate analysis or optimization of the product is performed. In this case, methods of computational electrodynamics (one of which is the method of moments) are used. They are based on the replacement of continuous functions with their discrete analogues (construction of a grid), which reduces the problem to the solution of a system of linear algebraic equations (SLAE). The problem’s complexity depends on the complexity of the SLAE solution, which is determined by its order (which in turn is determined by the complexity of the simulated object and its surrounding area) and by the number of the required SLAE solutions for each problem (determined by the upper frequency of the signal, the number, and range of the optimized parameters). This dramatically increases the computational cost, which becomes the main constraint for the optimal design. Therefore, reducing the computational cost for the analysis and optimization of RED elements (in particular, linear antennas) is an important scientific problem. Meanwhile, finding new antenna structures that meet all the desired features (low price, required characteristics, manufacturable design with small dimensions and windage, etc.) is no less important today. One of the promise solutions for these problems is using a wire grid and sparse antennas for modeling and constructing antennas. Since the last century, a lot of research has been performed on them. The aim of this paper is to review their history and the main related aspects such as computational, acceleration, and optimization used methods, the fields of their application, and their evolution to this moment. In addition, this paper provides a possible future implementation of wire-grid and sparse antennas from the authors’ point of view by presenting a new method that is under research to obtain effective wire sparse antennas.

Parallel model order reduction methods based on structured matrix analysis for discrete-time systems with parametric uncertainty

This paper explores two novel parallel parametric model order reduction methods for discrete-time systems with parametric uncertainty. The proposed methods allow that the parametric dependence is non-affine. With the shift-transformation matrix of the discrete orthogonal polynomials, including Charlier polynomials and Krawtchouk polynomials, the expansion coefficients of state variable in discrete orthogonal polynomials space can be obtained by solving the system of linear algebraic equations (SLAE). By applying the structured matrix analysis of the SLAE, two parallel strategies based on the equivalent transformation of block bi-diagonal matrices and the block discrete Fourier transform of block ϵ-circulant matrices are proposed to solve the SLAE. Then, the projection matrix is constructed to reduce discrete-time parametric systems. Moreover, we analyze the properties of the proposed parallel methods, including the invariable coefficients, the invertibility, and the error analysis. Finally, the effectiveness of the proposed methods is illustrated by the numerical experiments.

A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution

In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions.

Realization of the inverse scattering transform method for the Korteweg–de Vries equation

A method for practical realization of the inverse scattering transform method for the Korteweg–de Vries equation is proposed. It is based on analytical representations for Jost solutions and for integral kernels of transformation operators obtained recently. The representations have the form of functional series in which the first coefficient plays a crucial role both in solving the direct scattering and the inverse scattering problems. The direct scattering problem reduces to computation of a number of the coefficients following a simple recurrent integration procedure with a posterior calculation of scattering data by well known formulas. The inverse scattering problem reduces to a system of linear algebraic equations from which the first component of the solution vector leads to the recovery of the potential. We prove the applicability of the finite section method to the system of linear algebraic equations and discuss numerical aspects of the proposed method. Numerical examples are given, which reveal the accuracy and speed of the method.

Soliton solutions of the (2+1)-dimensional Kadomtsev–Petviashvili equation via two different integration schemes

In this research paper, we take into account the ([Formula: see text])-dimensional Kadomtsev–Petviashvili equation which is important in the soliton theory of nonlinear physics. To get the desired soliton solutions, the modified F-expansion method using the Riccati equation which has many solution functions, as well as the modified generalized Kudryashov’s method, had been effectively implemented. One of the reasons for the preference of the methods is that the proposed methods have been widely used before and they have not been applied to this problem. First, the wave transform is applied to the considered nonlinear partial differential equation (NLPDE), the nonlinear ordinary differential equation (NODE) form and the balancing constant are determined. The next step is to use the auxiliary equation depending on the proposed method to find the solution of the NODE form and to obtain the linear algebraic equation system. The solution of this system gives different solution sets for unknown parameter values. Then, soliton solution functions are constructed by using the suitable solution sets. After testing and confirming that the obtained solution functions satisfy the main equation, the three- and two-dimensional illustrations are depicted.

The exact Riemann Solver to the Shallow Water equations for natural channels with bottom steps

The Shallow Water equations model is commonly used to reproduce a wide variety of environmental flows. Its applications include river hydraulics and flood forecasting.Therefore, an algorithm to solve the Riemann problem for the Shallow Water equations can be a considerable asset for the development of numerical schemes devoted to the solution of such equations. Nevertheless, for real world applications, it becomes mandatory to consider the influence of the irregular shape of the cross section of natural channels in the solution.This work presents an energy based exact Riemann solver for the Shallow Water equations for channels with an arbitrarily shaped cross section with the presence of a non flat bottom and dry states.The procedure for solving the Riemann problem involves the solution of a system of non linear algebraic equations, for which a CPU time efficient solution strategy based on nested Newton-Raphson methods is presented.The proposed solver can be used to develop Finite Volume or Discontinuous Galerkin schemes, based on the Godunov method. For a non flat bottom, the Shallow Water equations are a non conservative system, once the bottom elevation is included in the state variables. Many available schemes take advantage of this fact to use approximate path-conservative Riemann solvers. In other cases, the term related to the non flat bottom is treated as a source term. The use of this exact solver avoids the necessity of considering the bottom topography as a source term and of reverting to approximate solvers. These strategies can in facts pose numerical problems in situations such as transonic rarefactions and flows approaching abrupt changes in bottom topography.The solver is tested by reproducing the solution of Riemann problems with a Godunov finite volume scheme. An analysis of the performance of the solver shows that it requires less computational time than path-conservative schemes and that it has better stability and convergence properties, especially in situations involving resonant waves, such as transonic flows across a bottom step.

Numerical Construction of the Transform of the Kernel of the Integral Representation of the Poincaré–Steklov Operator for an Elastic Strip

For an isotropic stratified elastic strip we consider the Poincaré–Steklov operator that maps normal stresses into normal displacements on part of the boundary. A new approach is proposed for constructing the transform of the kernel of the integral representation of this operator. A variational formulation of the boundary value problem for the transforms of displacements is obtained. A definition is given and the existence and uniqueness are proved for a generalized solution of the problem. An iteration method for solving variational equations is constructed, and conditions for its convergence are obtained based on the contraction mapping principle. The variational equations are approximated by the finite element method. As a result, at each step of the iteration method, it is required to solve two independent systems of linear algebraic equations, which are solved using the tridiagonal matrix algorithm. A heuristic algorithm is proposed for choosing the sequence of parameters of the iteration method that ensures its convergence. Verification of the developed computational algorithm is carried out, and recommendations on the use of adaptive finite element grids are given

Generalized Shifted Airfoil Polynomials of the Second Kind to Solve a Class of Singular Electrohydrodynamic Fluid Model of Fractional Order

In this manuscript, we find the numerical solutions of a class of fractional-order differential equations with singularity and strong nonlinearity pertaining to electrohydrodynamic flow in a circular cylindrical conduit. The nonlinearity of the underlying model is removed by the quasilinearization method (QLM) and we obtain a family of linearized equations. By making use of the generalized shifted airfoil polynomials of the second kind (SAPSK) together with some appropriate collocation points as the roots of SAPSK, we arrive at an algebraic system of linear equations to be solved in an iterative manner. The error analysis and convergence properties of the SAPSK are established in the L2 and L∞ norms. Through numerical simulations, it is shown that the proposed hybrid QLM-SAPSK approach is not only capable of tackling the inherit singularity at the origin, but also produces effective numerical solutions to the model problem with different nonlinearity parameters and two fractional order derivatives. The accuracy of the present technique is checked via the technique of residual error functions. The QLM-SAPSK technique is simple and efficient for solving the underlying electrohydrodynamic flow model. The computational outcomes are accurate in comparison with those of numerical values reported in the literature.

Method for solving inverse spectral problems on quantum star graphs

Abstract A new method for solving inverse spectral problems on quantum star graphs is proposed. The method is based on Neumann series of Bessel function representations for solutions of Sturm–Liouville equations. The representations admit estimates for the series remainders which are independent of the real part of the square root of the spectral parameter. This feature makes them especially useful for solving direct and inverse spectral problems requiring calculation of solutions on large intervals in the spectral parameter. Moreover, the first coefficient of the representation is sufficient for the recovery of the potential. The method for solving the inverse spectral problem on the graph consists in reducing the problem to a two-spectra inverse Sturm–Liouville problem on each edge. Then a system of linear algebraic equations is derived for computing the first coefficient of the series representation for the solution on each edge and hence for recovering the potential. The proposed method leads to an efficient numerical algorithm that is illustrated by a number of numerical tests.

A collocation method for time‐fractional diffusion equation on a metric star graph with η$$ \eta $$ edges

A wavelet collocation method based on Haar wavelet is proposed to investigate the numerical solution of time fractional diffusion equation (TFDE) on a metric star graph. We have utilized the Riemann–Liouville definition of fractional integral operator together with Haar wavelet to obtain the Riemann–Liouville fractional integral operator for Haar wavelet (RLFIO‐H). The application of RLFIO‐H and Haar wavelet to TFDE on metric star graph returns a system of linear algebraic equations. Solving this system yields wavelet coefficients and subsequently the solution. The convergence analysis of the proposed method is discussed to establish the theoretical authenticity of the method. The proposed method is tested on four benchmark problems to validate the theoretical findings. Moreover, the comparison of the developed method with the existing method shows the superiority and accuracy of the proposed method. To the best of the author's knowledge, the proposed work is one of the first collocation method in the literature that deals with the solution of TFDE on metric star graph using wavelet numerically.

On non-uniqueness of pressures in problems of fluid filtration in fractured-porous media

The paper proposes a new approach to the numerical solution of the singular Neumann problem for a system of elliptic equations, in which, for example, the problem of filtration of a two-phase incompressible fluid in a fractured-porous medium is reduced within the framework of the dual porosity model. The one-dimensionality of the kernel of the corresponding operator is established, after which an extended generalized mixed formulation is considered in which the singularity is eliminated by introducing some additional bilinear form. The extended formulation is approximated by the mixed finite element method, and the properties of the resulting system of linear algebraic equations with a saddle point matrix are established. Moreover, some numerical examples illustrate the theoretical results.

Efficient computation of adjoint sensitivities at steady-state in ODE models of biochemical reaction networks.

Dynamical models in the form of systems of ordinary differential equations have become a standard tool in systems biology. Many parameters of such models are usually unknown and have to be inferred from experimental data. Gradient-based optimization has proven to be effective for parameter estimation. However, computing gradients becomes increasingly costly for larger models, which are required for capturing the complex interactions of multiple biochemical pathways. Adjoint sensitivity analysis has been pivotal for working with such large models, but methods tailored for steady-state data are currently not available. We propose a new adjoint method for computing gradients, which is applicable if the experimental data include steady-state measurements. The method is based on a reformulation of the backward integration problem to a system of linear algebraic equations. The evaluation of the proposed method using real-world problems shows a speedup of total simulation time by a factor of up to 4.4. Our results demonstrate that the proposed approach can achieve a substantial improvement in computation time, in particular for large-scale models, where computational efficiency is critical.

Wave radiation by a horizontal circular cylinder in a three-layer fluid

To construct cylindrical structures like circular pipe bridges or tunnels submerged in the almost still density-stratified ocean or seawater, the study of waves radiated by the cylinder is essential. This research solves the wave radiation problem by calculating non-dimensionalized added mass and damping coefficients to the mass of the fluid displaced by the submerged horizontal cylinder in either layer of a three-layer fluid, which is still otherwise. Under the linear theory of water waves, we investigate the circular cylinder's hydrodynamic forces by its swaying and heaving motion. The time-harmonic wave propagates with three distinct wavenumbers for a given frequency. The method of multipoles has been employed due to its rapid converging solutions by increasing the truncation limit. Potential functions are expressed into systems of linear algebraic equations, which are solved numerically for two sets of unknowns in each case by truncation. Then, the added mass and damping coefficients are obtained from the non-dimensionalized hydrodynamic forces when the horizontal circular cylinder is submerged in the lower, middle and upper layers, respectively. The obtained results are depicted graphically against wavenumber in numerous figures and analysed.

COMPUTATIONAL FEATURES OF CALCULAT-ING EXTENDED CONDUCTOR ANTENNAS BASED ON THE INTEGRAL EQUATIONS METHOD

The numerical solution of integral equations or a system of integral equations (for a system of conductors) consists in their discretization and reduction to a system of linear algebraic equations for the desired function. For discretization, it is possible to use projection methods, for example, the Galerkin method or the collocation method. A system of linear algebraic equations for the class of problems under consideration is characterized by a complete (filled) complex matrix. For conductor antenna structures that are quite arbitrary in geometry and size, systems of linear algebraic equations turn out to be of a high order, and special computational algorithms are required to solve them. The purpose of the work is to study adequate mathematical models for wire antennas with a subsequent description of uniform numerical algorithms based on methods of sampling and approximation of antenna current. With this method, the error in solving integral equations is determined by the error in calculating the elements of the matrix of a system of linear algebraic equations for a given piecewise polynomial approximation of the solution at step h. If the quadrature formula for numerical integration is chosen, then there are two ways to reduce the solution error. An increase in the number of collocation points leads to a rapid increase in the volume of calculations and, consequently, to a rapid increase in the amount of occupied computer memory. Therefore, the question arises of choosing the most profitable method of piecewise polynomial interpolation and discretization step h from the point of view of using computer resources, ensuring an acceptable number of solutions.

On the properties of some inversion methods of the Laplace transform

The problem of inversion of the integral Laplace transform, which belongs to the class of ill-posed problems, is considered. Integral equations are reduced to ill-conditioned systems of linear algebraic equations, in which the unknowns are either the coefficients of the series expansion in terms of special functions, or the approximate values of the desired original at a number of points. Described method of inversion using special quadrature formulas of the highest degree of accuracy and the characteristics of the accuracy and stability of this method are indicated. Quadrature inversion formulas are constructed, which are adapted for the inversion of long-term and slow processes of linear viscoelasticity. A method of deformation of the integration contour in the Riemann-Mellin inversion formula is proposed, which leads the problem to the calculation of certain integrals and allows obtaining error estimates. A method is described for determining the possible breakpoints of the original and calculating the magnitude of the jump at these points.

On Problems in OpenBLAS Library Usage in Productized Code on RISC-V

The boundary element method usage for the numerical simulation in continuum mechanics problems leads to the need to solve a system of linear algebraic equations with a dense matrix. The de facto standards for the interface of functions over dense matrices and vectors software implementations are BLAS/LAPACK. Among the optimized open-source BLAS/LAPACK implementations, only the OpenBLAS library includes optimizations for the widest range of hardware platforms. This library is optimized for Intel, AMD, ARM and RISC-V architectures. The open RISC-V architecture ecosystem is currently actively developing. European supercomputing centers have opened RISC-V competence centers as part of the government's EuroHPC grant support, since solutions based on the ARM architecture are not recognized as part of the European initiative to develop its own technological independence. Currently, companies included in the international RISC-V consortium are developing not only high-performance RISC-V processors, but also AI accelerators, as well as video cards based on RISC-V architecture. OpenBLAS is actively supported and optimized for emerging RISC-V hardware and extensions. However, libraries used in product code are traditionally subject to strict requirements for stability and reliability in order to minimize possible errors and failures in the product. As it turned out, from this point of view, OpenBLAS has a number of problems that we had to solve in order to productize this library. In this article the OpenBLAS test system is described, the problems of testing the LAPACK functionality of the library and ways to solve them are discussed. In addition, the test coverage of the BLAS functionality is analyzed and the results achieved in increasing it are presented. It is planned to contribute the described changes to the OpenBLAS project.

Blade-like field cathode with a dielectric coating mathematical modeling

In this paper the results of a two-dimensional diode emission system based on a blade-like field cathode in a polar coordinate system modeling is presented. The top of the emitter is a circle with a dielectric coating. The anode is a circle coaxial to the top of the emitter. The boundary condition of the first kind is set on the cathode, and the first and second kind on the anode. The problem of calculating the electrostatic potential distribution is reduced to solving a system of linear algebraic equations with constant coefficients. All the geometric dimensions of the system and the values of the potentials on the electrodes are the parameters of the problem.

Diffraction of electromagnetic waves on one-dimensional diffraction gratings formed by slots in an absolutely absorbing screen

Two-sided approximate boundary conditions are obtained for an absolutely absorbing ("black") layer lying on a multilayer dielectric. Paired summation equations (PSEs) are obtained for the tangent components of the electric and magnetic field strengths at the slots. These equations are solved by the Galerkin method with basis functions in the form of Chebyshev and Legendre polynomials. The resulting system of linear algebraic equations has fast internal convergence. To control the accuracy of the obtained solution, a dual problem is solved --- a lattice of "black stripes". In this case, the unknowns in the PSU are the current density on the strips. The properties of lattices are analyzed. Keywords: approximate boundary conditions, "black" screen, diffraction grating, pair summation equations, Galerkin's method.