Direct Solution Method Research Articles

A Spectral Method for Two-Dimensional Ocean Acoustic Propagation

The accurate calculation of the sound field is one of the most concerning issues in hydroacoustics. The one-dimensional spectral method has been used to correctly solve simplified underwater acoustic propagation models, but it is difficult to solve actual ocean acoustic fields using this model due to its application conditions and approximation error. Therefore, it is necessary to develop a direct solution method for the two-dimensional Helmholtz equation of ocean acoustic propagation without using simplified models. Here, two commonly used spectral methods, Chebyshev–Galerkin and Chebyshev–collocation, are used to correctly solve the two-dimensional Helmholtz model equation. Since Chebyshev–collocation does not require harsh boundary conditions for the equation, it is then used to solve ocean acoustic propagation. The numerical calculation results are compared with analytical solutions to verify the correctness of the method. Compared with the mature Kraken program, the Chebyshev–collocation method exhibits higher numerical calculation accuracy. Therefore, the Chebyshev–collocation method can be used to directly solve the representative two-dimensional ocean acoustic propagation equation. Because there are no model constraints, the Chebyshev–collocation method has a wide range of applications and provides results with high accuracy, which is of great significance in the calculation of realistic ocean sound fields.

Dynamic Performance of Vehicle–Guideway Bridge Systems for Low–Medium-Speed Maglev Trains Under Earthquakes

This paper developed a numerical model for predicting the seismic responses of vehicle–guideway bridge systems for low–medium-speed (LMS) maglev trains. Each vehicle was characterized as a multi-rigid-body with 50 degree of freedoms (DOFs), and the guideway bridge was modeled by the finite element method. The actively controlled electromagnetic forces were considered in simulating the vehicle–guideway interaction relationship. Subsequently, the equations of motion for the vehicle–guideway coupled system under earthquake were, respectively, established in relative and absolute coordinate systems to quantify the effect of structural pseudo-static components, so that the seismic effect can be taken into account. Case study was then conducted to thoroughly discuss the seismic responses of the vehicle–guideway coupled system in both time and frequency domains. Furthermore, parametric study was carried out to determine the effect of key parameters (i.e. vehicle speed, stiffness of guideway) on the system’s responses. The results show that the conventional seismic analysis method relative motion method (RMM) (ignoring the structural pseudo-static component) will considerably underestimate the seismic responses of the coupled system, especially of the vehicle. It is suggested that the formulation be established in the absolute coordinate system (i.e. using direct solution method, DSM) for more actual prediction. The frequency responses indicate that the vibrations of vehicle–guideway coupled system under earthquake relate significantly to the natural frequencies of vehicle and bridge, while the same is not true for the vehicle-induced excitation.

Interactions of Soliton in Weakly Nonlocal Nonlinear Media

Solitary waves or solitons is a nonlinear phenomenon which has been studied intensively due to its application in solid-state matter such as Bose-Einstein condensates state, plasma physics, optical fibers and nematic liquid crystal. In particular, the study of nonlinear phenomena occurs in the structure of waves gained interest of scholars since their discovery by John Russell in 1844. The Nonlinear Schrödinger Equation (NLSE) is the theoretical framework for the investigation of nonlinear pulse propagation in optical fibers. Nonlocality can be found in an underlying transport mechanisms or long-range forces like electrostatic interactions in liquid crystals and many-body interactions with matter waves in Bose-Einstein condensate or plasma waves. The length of optical beam width and length of response function are used to classify nonlocality in optical materials. The nonlocality can be categorized as weak nonlocal if the width of the optical beam broader than the length of response function and if the width of the optical beam is narrower than the length of response function, it is considered as highly nonlocal. This work investigates the interactions of solitons in a weakly nonlocal Cubic NLSE with Gaussian external potential. The variational approximation (VA) method was employed to solve non integrable NLSE to ordinary differential equation (ODE). The soliton parameters and the computational program are used to simulate the propagation of the soliton width and its center-of-mass position. In the presence of Gaussian external potential, the soliton may be transmitted, reflected or trapped based on the critical velocity and potential strength. Direct numerical simulation of Cubic NLSE is programmed to verify the results of approximation method. Good agreement is achieved between the direct numerical solution and VA method results.

A general solution procedure for the scaled boundary finite element method via shooting technique

The scaled boundary finite element method (SBFEM) is known for its inherent ability to simulate unbounded domains and singular fields, and its flexibility in the meshing procedure. Keeping the analytical form of the field variables along one coordinate intact, it transforms the governing partial differential equations of the problem into a system of one-dimensional (initial–)boundary value problems. However, closed-form solution of the said system is not available for most cases (e.g. transient heat transfer, acoustics, ultrasonics, etc.) since the system cannot be diagonalized in general. This paper aims to establish a numerical tool within the context of the shooting technique to evaluate the coefficient matrices of the subdomains without a priori knowledge of the analytical solution of the semi-discretized system. With proper choice of boundary conditions, the technique uses the strong form of the scaled boundary finite element equations to pass the required information and with the desired accuracy from one boundary to another. Due to generality of the technique, its procedure can be adjusted for any field equations. Since this technique is presented here for the first time, linear elastostatics, for which the closed-form solution is well-established, is formulated to provide valid comparisons. In addition, any direct solution method can be used for integrating the scaled boundary equations. Thus, without loss of generality, a Nyström extension of the classical fourth-order Runge–Kutta method is employed. A quantitative sensitivity analysis is also conducted, and efficiency of the classical and proposed solution techniques is compared in terms of computational time. Finally, some numerical examples, including bounded and unbounded domains, as well as singular stress fields are simulated based on the classical and proposed solution techniques.

A practical method for recovering Sturm–Liouville problems from the Weyl function

In the paper we propose a direct method for recovering the Sturm–Liouville potential from the Weyl–Titchmarsh m-function given on a countable set of points. We show that using the Fourier–Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from Kravchenko and Torba (2021 Inverse Problems 37 015015) is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.

A multi-GPU method for ADI-based fractional-step integration of incompressible Navier-Stokes equations

A computational method for GPU-accelerated fractional-step integration of incompressible Navier-Stokes equations based on the Alternating Direction Implicit (ADI) method is presented. Non-iterative, direct solution methods used in the semi-implicit fractional-step method take advantage of tridiagonal systems and Fourier transform whose solution can be computed using fast algorithms on a single GPU. However, when data is distributed to multiple GPUs, all-to-all matrix transposition is required, which increases computational cost significantly. In this work, a new strategy that does not require all-to-all transposition is proposed. The computational domain is divided in the wall-normal direction, and decoupled tridiagonal systems are obtained using Parallel Diagonal Dominant (PDD) and Parallel Partition (PPT) methods. An optimal batch size is determined to maximize the performance of PDD and PPT methods within a given amount of GPU memory. Strengths and weaknesses of this type of domain decomposition are investigated in comparison to conventional ways of dividing the domain along streamwise or spanwise directions. Using 8 NVIDIA Tesla P100 GPUs, the utility of the present method is demonstrated in a direct numerical simulation (DNS) of a canonical zero-pressure-gradient turbulent boundary layer and a DNS of a K-type boundary-layer transition on 1.4 billion grid cells.

Direct approach to optimize PID controller parameters of hydropower plants

As a core component of hydroturbine regulating systems (HTRS), a PID controller is essential to ensure system stability and regulate quality, although its parameter selection remains challenging. This study presents a state space model of an HTRS with a surge tank, as well as an explicit formula derived for the dimensionless turbine rotating speed, using system decoupling with the state matrix in the Jordan canonical form. The regulating quality parameters (RQPs) in the entire stable domain can be readily obtained using the direct solving method (DSM) without time-domain simulation, and the optimized PID controller parameters can be selected based on the RQP. A comparative analysis of the DSM and a traditional time-domain simulation was conducted based on a practical hydropower plant. Both methods yielded the same results; the DSM was considerably more efficient in providing the full RQP domain, where the effect of the surge tank subsystem was clearly demonstrated. In the proposed DSM, only state parameters were required for selecting the PID controller parameters. As hydropower plants operate under varying conditions, the DSM was effective for tuning purposes.

Optimization of one-step hybrid method for direct solution of fifth order ordinary differential equations of initial value problems

This paper focuses on the derivation, analysis and implementation of a hybrid method by optimizing the order of the method by introduction of six-hybrid points for direct solution of fifth order ordinary differential equations of initial value problems (IVPs). Power series was used as the basis function for the solution of the IVP. The basis function was interpolated at some selected hybrid points whereas the fifth derivative of the approximate solution was collocated at all the interval of integration of the method to generate a system of linear equations for the determination of the unknown parameters. The derived method was tested for consistency, zero stability, convergence and absolute stability. The method was tested with two linear test problems to confirm its accuracy and usability. The comparison of the results with some existing methods shows the superiority of the accuracy of the method.

Optimization of one-step hybrid method for direct solution of fifth order ordinary differential equations of initial value problems

Optimization of one-step hybrid method for direct solution of fifth order ordinary differential equations of initial value problems

Power flow analytical solutions and multi-dimensional voltage stability boundary based on the multivariate quotient-difference method

This paper proposes a novel Multivariate Quotient-Difference (MQD) method to obtain the approximate analytical solution for AC power flow equations. Hence, in the online environment, the power flow solutions covering different operating conditions can be directly obtained by plugging in values into multiple symbolic variables, such that the power injections and consumptions of selected buses or areas can be independently adjusted. This method first derives a power flow solution by Multivariate Power Series (MPS). Next, the MQD method is applied to transform the obtained MPS to Multivariate Padé Approximants (MPA) to expand the Radius of Convergence (ROC), so that the accuracy of the derived analytical solution can be significantly increased. Besides, the hypersurface of voltage stability boundary can be identified by an analytical formula obtained from the coefficients of MPA. This direct method for power flow solution and voltage stability boundary is fast for many online applications, since such analytical solutions can be derived offline and evaluated online by only plugging values into the symbolic variables according to the actual operating conditions. The proposed method is validated in detail on the New England 39-bus and IEEE 118-bus systems with independent load variations in multi-regions.

New Hybrid Method for Direct Numerical Solution of Nonlinear Second, Third and Fourth Orders Ordinary Differential Equations

New Hybrid Method for Direct Numerical Solution of Nonlinear Second, Third and Fourth Orders Ordinary Differential Equations

A hybrid two-step method for direct solutions of general second order initial value problem

The study is concerned with the development of a two-step hybrid linear multi-step method for direct solution of general second order initial value problem. Power series is adopted as the basis function, and the differential system arising from it are collocated at all grid and off-grid points, while the approximate solution is interpolated at the selected points. The scheme developed is tested for consistency and zero stability, and it was found to be convergent.The scheme is expanded term by term by Taylor series approach, also the efficiency of the methods is tested on some test problems, and the accuracy is compared with some existing result in literature.

Numerical Method for Direct Solution to Form-Finding Problem in Convex Gridshell

AbstractElastic gridshell is a class of net-like structure formed by an ensemble of elastically deforming rods coupled through joints, such that the structure can cover large areas with low self-weight and allow for a variety of aesthetic configurations. Gridshells, also known as X-shells or Cosserat Nets, are a planar grid of elastic rods in its undeformed configuration. The end points of the rods are constrained and positioned on a closed curve—the final boundary—to actuate the structure into a 3D shape. Here, we report a discrete differential geometry-based numerical framework to study the geometrically nonlinear deformation of gridshell structures, accounting for non-trivial bending-twisting coupling at the joints. The form-finding problem of obtaining the undeformed planar configuration given the target convex 3D topology is then investigated. For the forward (2D to 3D) physically based simulation, we decompose the gridshell structure into multiple one-dimensional elastic rods and simulate their deformation by the well-established discrete elastic rods (DER) algorithm. A simple penalty energy between rods and linkages is used to simulate the coupling between two rods at the joints. For the inverse problem associated with form-finding (3D to 2D), we introduce a contact-based algorithm between the elastic gridshell and a rigid 3D surface, where the rigid surface describes the target shape of the gridshell upon actuation. This technique removes the need of several forward simulations associated with conventional optimization algorithms and provides a direct solution to the inverse problem. Several examples—hemispherical cap, paraboloid, and hemi-ellipsoid—are used to show the effectiveness of the inverse design process.

Pragmatic solvers for 3D Stokes and elasticity problems with heterogeneous coefficients: evaluating modern incomplete LDL<sup><i>T</i></sup> preconditioners

Abstract. The need to solve large saddle point systems within computational Earth sciences is ubiquitous. Physical processes giving rise to these systems include porous flow (the Darcy equations), poroelasticity, elastostatics, and highly viscous flows (the Stokes equations). The numerical solution of saddle point systems is non-trivial since the operators are indefinite. Primary tools to solve such systems are direct solution methods (exact triangular factorization) or approximate block factorization (ABF) preconditioners. While ABF solvers have emerged as the state-of-the-art scalable option, they are invasive solvers requiring splitting of pressure and velocity degrees of freedom, a multigrid hierarchy with tuned transfer operators and smoothers, machinery to construct complex Schur complement preconditioners, and the expertise to select appropriate parameters for a given coefficient regime – they are far from being “black box” solvers. Modern direct solvers, which robustly produce solutions to almost any system, do so at the cost of rapidly growing time and memory requirements for large problems, especially in 3D. Incomplete LDLT (ILDL) factorizations, with symmetric maximum weighted-matching preprocessing, used as preconditioners for Krylov (iterative) methods, have emerged as an efficient means to solve indefinite systems. These methods have been developed within the numerical linear algebra community but have yet to become widely used in applications, despite their practical potential; they can be used whenever a direct solver can, only requiring an assembled operator, yet can offer comparable or superior performance, with the added benefit of having a much lower memory footprint. In comparison to ABF solvers, they only require the specification of a drop tolerance and thus provide an easy-to-use addition to the solver toolkit for practitioners. Here, we present solver experiments employing incomplete LDLT factorization with symmetric maximum weighted-matching preprocessing to precondition operators and compare these to direct solvers and ABF-preconditioned iterative solves. To ensure the comparison study is meaningful for Earth scientists, we utilize matrices arising from two prototypical problems, namely Stokes flow and quasi-static (linear) elasticity, discretized using standard mixed finite-element spaces. Our test suite targets problems with large coefficient discontinuities across non-grid-aligned interfaces, which represent a common challenging-for-solvers scenario in Earth science applications. Our results show that (i) as the coefficient structure is made increasingly challenging, by introducing high contrast and complex topology with a multiple-inclusion benchmark, the ABF solver can break down, becoming less efficient than the ILDL solver before breaking down entirely; (ii) ILDL is robust, with a time to solution that is largely independent of the coefficient topology and mildly dependent on the coefficient contrast; (iii) the time to solution obtained using ILDL is typically faster than that obtained from a direct solve, beyond 105 unknowns; and (iv) ILDL always uses less memory than a direct solve.

Integrability via Functional Expansion for the KMN Model

This paper considers issues such as integrability and how to get specific classes of solutions for nonlinear differential equations. The nonlinear Kundu–Mukherjee–Naskar (KMN) equation is chosen as a model, and its traveling wave solutions are investigated by using a direct solving method. It is a quite recent proposed approach called the functional expansion and it is based on the use of auxiliary equations. The main objectives are to provide arguments that the functional expansion offers more general solutions, and to point out how these solutions depend on the choice of the auxiliary equation. To see that, two different equations are considered, one first order and one second order differential equations. A large variety of KMN solutions are generated, part of them listed for the first time. Comments and remarks on the dependence of these solutions on the solving method and on form of the auxiliary equation, are included.

AstroSeis: A 3D Boundary Element Modeling Code for Seismic Wavefields in Irregular Asteroids and Bodies

Abstract We developed a 3D elastic boundary element method computer code, called AstroSeis, to model seismic wavefields in a body with an arbitrary shape, such as an asteroid. Besides the AstroSeis can handle arbitrary surface topography, it can deal with a liquid core in an asteroid model. Both the solid and liquid domains are homogenous in our current code. For seismic sources, we can use single forces or moment tensors. The AstroSeis is implemented in the frequency domain, and the frequency-dependent Q can be readily incorporated. The code is in MATLAB (see Data and Resources), and it is straightforward to set up the model to run the code. The frequency-domain calculation is advantageous to study the long-term elastic response of a celestial body due to a cyclic force, such as the tidal force, with no numerical dispersion issue suffered by many other methods requiring volume meshing. Our AstroSeis has been benchmarked with other methods such as normal-mode summation and the direct solution method. This open-source AstroSeis will be a useful tool to study the interior and surface processes of asteroids.

Algebraic Method of Solution of Schrödinger’s Equation ofa Quantum Model

This work is aiming to show the advantage of using the Lie algebraic decomposition technique to solvefor Schrödinger’s wave equation for a quantum model, compared with the direct method of solution. The advantageis a two-fold: one is to derive general form of solution, and, two is relatively manageable to deal with the case oftime-dependent system Hamiltonian. Specifically, we consider the model of 2-level optical atom and solve for thecorresponding Schrödinger’s wave equation using the Lie algebraic decomposition technique. The obtained formof solution for the wave function is used to examine computationally the atomic localization in the coordinate space.For comparison, the direct method of solution of the wave function is analysed in order to show its complicationwhen dealing with time-dependent Hamiltonian.The possibility of using the Lie algebraic method for a qubit model(a driven quantum dot model) is briery discussed, if Schrödinger’s wave function is to be examined for the qubitlocalization.

A Direct Solving Approach for High-Order Power Amplifier Matching Network Design

In this article, a direct solving method for power amplifier (PA) matching network design is explored from a theoretical perspective. Commonly, the input impedance function of a multivariable matching network is a transcendental equation and it is unsolvable. In the proposed new method, this problem can be solved to some extent by transforming the transcendental equations into solvable linear equations, where the unknown coefficients can be calculated easily. A family of impedance functions is obtained if the order of matching network is higher than the number of functions, and the best one can be selected according to the practical demand in PA design. As a design example, a triband PA with arbitrary frequency points (0.9/2.3/3.5 GHz) is implemented to validate the proposed approach. The calculation results show that the proposed method can solve the matching problem of high-order circuits. And the simulation and experiment results agree with the calculation results very well, which verified the validity of the proposed method.

Research on Error Estimations of the Interpolating Boundary Element Free-Method for Two-Dimensional Potential Problems

The interpolating boundary element-free method (IBEFM) is a direct solution method of the meshless boundary integral equation method, which has high efficiency and accuracy. The IBEFM is developed based on the interpolating moving least-squares (IMLS) method and the boundary integral equation method. Since the shape function of the IMLS method satisfies the interpolation characteristics, the IBEFM can directly and accurately impose the essential boundary conditions, which overcomes the shortcomings of the original boundary element-free method in enforcing the essential boundary approximately. This paper will study the error estimations of the IBEFM for two-dimensional potential problems and the relationship between the errors and the influence radius and the condition number of the coefficient matrix. Two numerical examples are presented to verify the correctness of the theoretical results in this paper.

Thermal conductivity of amorphous and crystalline GeTe thin film at high temperature: Experimental and theoretical study

Thermal transport properties bear a pivotal role in influencing the performance of phase change memory (PCM) devices, in which the PCM operation involves fast and reversible phase change between amorphous and crystalline phases. In this paper, we present a systematic experimental and theoretical study on the thermal conductivity of GeTe at high temperatures involving fast change from amorphous to crystalline phase upon heating. Modulated photothermal radiometry (MPTR) is used to experimentally determine thermal conductivity of GeTe at high temperatures in both amorphous and crystalline phases. Thermal boundary resistances are accurately taken into account for experimental consideration. To develop a concrete understanding of the underlying physical mechanism, rigorous and in-depth theoretical exercises are carried out. For this, first-principles density functional methods and linearized Boltzmann transport equations (LBTE) are employed using both direct and relaxation time based approach (RTA) and compared with that of the phenomenological Slack model. The amorphous phase experimental data has been described using the minimal thermal conductivity model with sufficient precision. The theoretical estimation involving direct solution and RTA method are found to retrieve well the trend of the experimental thermal conductivity for crystalline GeTe at high temperatures despite being slightly overestimated and underestimated, respectively, compared to the experimental data. A rough estimate of vacancy contribution has been found to modify the direct solution in such a way that it agrees excellently with the experiment. Umklapp scattering has been determined as the significant phonon-phonon scattering process. Umklapp scattering parameter has been identified for GeTe for the whole temperature range which can uniquely determine and compare Umklapp scattering processes for different materials