Numerical Solution Techniques Research Articles

Large deformation analysis of 2D hyperelastic bodies based on the compressible nonlinear elasticity: A numerical variational method

A numerical solution technique named as variational differential quadrature (VDQ) is adopted herein for the compressible nonlinear elasticity problems. The governing equations are obtained based on the virtual work principle by considering displacement as the unknown field. The neo-Hookean model is also considered for the hyperelastic behavior of material. In the solution method, an efficient vector–matrix formulation is developed from which the discretized governing equations are achieved from the weak form of equations in a direct approach. Simplicity in implementation and accuracy are among the features of the proposed approach. Moreover, it does not suffer from the locking problem and unphysical instabilities. Fast convergence rate and computational efficiency are other advantages of this method. A number of numerical examples are given to reveal the good performance of VDQ in the large deformation analysis of compressible and nearly-incompressible bodies.

A spectral perspective about the accuracy of numerical predictions of flow instabilities

The paper discusses the usefulness of information provided by the eigenvalues, eigenfunctions and eigenvectors of flow stability problems, aiming to propose in a spectral perspective an evaluation of the accuracy of the related numerical solution schemes. The presented material further develops the results of past applications of numerical techniques in the analysis of the stability of heated systems containing single-phase, two-phase and supercritical pressure fluids. Since in all these applications both nonlinear and linear stability analyses were performed adopting very similar techniques, information on the different modes of oscillation or excursion exhibited by these systems during unstable behaviour was also made available. These spectral data, whose usefulness was up to now not completely exploited, are now considered in a new perspective, by addressing the simplest of the previously addressed problems both by analytical and numerical means. Information on the level of accuracy obtained by numerical solution techniques in capturing the intrinsic dynamics of the addressed flow systems is presented.

Reduced Basis Method Applied to Eigenvalue Problems from Convection

The reduced basis method is a suitable technique for finding numerical solutions to partial differential equations that must be obtained for many values of parameters. This method is suitable when researching bifurcations and instabilities of stationary solutions for partial differential equations. It is necessary to solve numerically the partial differential equations along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions for a large number of bifurcation parameter values. In this paper, the reduced basis method has been used to solve eigenvalue problems derived from the linear stability analysis of stationary solutions in a two-dimensional Rayleigh–Bénard convection problem. The bifurcation parameter is the Rayleigh number, which measures buoyancy. The reduced basis considered belongs to the eigenfunction spaces derived from the eigenvalue problems for different types of solutions in the bifurcation diagram depending on the Rayleigh number. The eigenvalue with the largest real part and its corresponding eigenfunction are easily calculated and the bifurcation points are correctly captured. The resulting matrices are small, which enables a drastic reduction in the computational cost of solving the eigenvalue problems.

An efficient iterative method for solving Bratu-type equations

In this paper, an efficient analytical iterative method is introduced for solving the initial value and boundary value Bratu problems. A quasilinearization method with an optimal parameter is derived and utilized to reduce the highly nonlinear Bratu equations to a sequence of linear problems. An optimal Picard iteration method is used to obtain approximate analytical solution of linearized versions of Bratu’s equations. The numerical results presented the effectiveness and high performance of the proposed method and comparisons are made with many other existing numerical solution techniques. The comparisons reveal the efficiency and applicability of the present work.

Trigonometric and cylindrical polynomials and their applications in electromagnetics

ABSTRACT A development of the theory of trigonometric polynomials (TPs) is considered that involves generalization of the notion of TP and extension of the methods and results to cylindrical polynomials (CPs). On the basis of the proposed augmentation of TPs and CPs, a general approach is presented to the analysis of guided waves and resonances in electromagnetics and beyond. The method employs the known explicit forms of dispersion equations (DEs) describing eigenoscillations and normal waves in layered structures and is based on the development of the theory of generalized TPs and CPs performed in the study. The approach enables one to complete rigorous proofs of existence and determine domains of localization of the TP and CP zeros and the DE roots and validate iterative numerical solution techniques.

Diffraction of TE polarized electromagnetic waves by a nonlinear layer: saturated and Kerr nonlinearities

The diffraction of a TE wave by a dielectric layer filled with a nonlinear medium is considered. The layer is located between two half-spaces with constant permittivities. Two types of nonlinearities (nonlinearity with saturation and Kerr nonlinearity) are considered and compared. In this paper, we show the difference between solvability results of the problem in these cases. Namely the diffraction problem with saturated nonlinearity has a unique solution under some additional conditions. At the same time, the diffraction problem in the case of Kerr nonlinearity has an infinite number of solutions. The analytical and numerical solution techniques are developed. Numerical results are presented.

Diffraction of a TE‐Polarized Wave by a Nonlinear Goubau Line

AbstractThe diffraction of a cylindrical wave by a nonlinear metal‐dielectric waveguide filled with nonlinear medium is considered. Two widely used types of nonlinearities (Kerr nonlinearity and nonlinearity with saturation) are considered. The problem is to find amplitudes of the reflected and the transmitted fields when the amplitude of the incident field is known. The analytical and numerical solution techniques are developed. Numerical results are presented.

Low Velocity Normal Impact Performance of Functionally Graded Conical Shell with Simple Power Law

The current study deals with the problem of functionally graded untwisted and pretwisted conical shells under to low-velocity normal impact by a spherical mass striker. A numerical solution technique is developed to solve and predict the impact performance in terms of contact force, impactor displacement, impactor velocity and target displacement of the FG conical shells with different simple power law exponent. A modified Hertzian contact theory considering permanent depression is used to calculate the contact force along with other impact performance parameters. Using the Newmark’s time integration scheme the dynamic equations are solved. Isoparametric shell element with eight nodes is considered for the current finite element model. Parametric observations are performed to see the effects of prime parameters like impactor initial velocity, twist angle and different power law exponent for Ti–6Al–4V-ZrO2 functionally graded conical shell under low velocity normal impact condition. The effects of impactor initial velocity, twist angle of conical shell and the constituent volume fraction of FGM configuration through power law index on the transient impact performance of the conical shell are examined and analyzed.

Dynamic Modeling of Slider-Crank Mechanism for Selecting Input Parameters for Desired Piston Speeds: Lumped Mass Approach

Here the dynamic behavior of slider-crank mechanism with a driving force applied at crank-pin center, has been modeled to formulate the piston speed.  In view of the abundance of the parameters involved, a lumped parameter approach has been preferred to obtain a compact equation in the form of a second order nonlinear differential equation. The complexity of the resulting equation has mandated implementing a numerical solution technique by which the effects of the selected parameters on the piston speed have been investigated. Under similar conditions an experimental model has been prepared and the tests have been carried out to compare the results. Low error levels achieved in two results have demonstrated the validity of the developed model.  

A MULTI-MODEL APPROACH TO SIMULATE THE STORM SURGE AT PUERTO RICO DUE TO HURRICANES IRMA AND MARIA

In this paper we present the results of a multimodel approach to simulating the recent storm surges due to hurricanes Irma and Maria. The study focuses on Puerto Rico which, as a consequence of hurricane Maria, experienced storm surge around the entire perimeter of the island. In this study the storm tide is modeled using a variety of state-of-the-art 2DH numerical models. All models are based on the long wave assumption and employ the Non-Linear Shallow Water (NLSW) equations. The models vary according to the form of the governing NLSW equations that they employ. Differences include linearization and primitive variable or conserved variable (divergence) form. The numerical solution techniques used to solve the governing equations, as well as the options available for the wind, pressure, tidal forcing terms and wetting/drying techniques also vary between (and within) the models.

Reduced spline method based on a proper orthogonal decomposition technique for fractional sub-diffusion equations

In this paper, a reduced spline (RS) method based on a proper orthogonal decomposition (POD) technique for numerical solution of time fractional sub-diffusion equations is investigated. Combining POD with polynomial and non-polynomial spline methods yield a new model with lower dimensions and sufficiently high accuracy, so that the amount of computations and the calculation time decrease in comparison with usual spline methods. Although, the classical L1 scheme on uniform meshes is one of the most successful methods for approximation of the Caputo fractional derivative in time, due to the fact that the solution of time fractional sub-diffusion equations typically has a singularity at the origin, a weak convergence will conclude for such scheme. To overcome this difficulty, we use the L1 scheme on graded meshes in time, such that considered time steps are very small near the origin which compensate the singularity of the solution. Also, the error estimates between the POD approximate solution of the RS scheme and the exact solution of fractional sub-diffusion equation are established for two types of uniform and graded meshes in time. Numerical examples are given to illustrate the feasibility and efficiency of the proposed method.

Stable difference scheme for a nonlocal boundary value heat conduction problem

Abstract In this paper, a new finite difference method to solve nonlocal boundary value problems for the heat equation is proposed. The most important feature of these problems is the non-self-adjointness. Because of the non-self-adjointness, major difficulties occur when applying analytical and numerical solution techniques. Moreover, problems with boundary conditions that do not possess strong regularity are less studied. The scope of the present paper is to justify possibility of building a stable difference scheme with weights for mentioned type of problems above.

Free Vibration Analysis Of Horizontal Spinning Beams Using The Differential Quadrature Method

The governing differential equations in free are built vibration by using Hamilton’s principle for a horizontal spinning beam. The differential quadrature (DQ) method is a numerical solution technique for initial and/or boundary problems. By using the DQ method, the dimensionless natural frequencies at different rotational speeds are obtained for the cantilever beam. To ensure the accuracy of the present results by the DQ method, comparisons are made with those available in open literature and very good agreements are achieved.

Approximate analytical solutions for a class of nonlinear stochastic differential equations

An approximate analytical solution is derived for a certain class of stochastic differential equations with constant diffusion, but nonlinear drift coefficients. Specifically, a closed form expression is derived for the response process transition probability density function (PDF) based on the concept of the Wiener path integral and on a Cauchy–Schwarz inequality treatment. This is done in conjunction with formulating and solving an error minimisation problem by relying on the associated Fokker–Planck equation operator. The developed technique, which requires minimal computational cost for the determination of the response process PDF, exhibits satisfactory accuracy and is capable of capturing the salient features of the PDF as demonstrated by comparisons with pertinent Monte Carlo simulation data. In addition to the mathematical merit of the approximate analytical solution, the derived PDF can be used also as a benchmark for assessing the accuracy of alternative, more computationally demanding, numerical solution techniques. Several examples are provided for assessing the reliability of the proposed approximation.

CFD pretending of vapor and liquid refrigerant mixing in variable speed scroll compressor

The mixing of two-phase refrigerant dual ports through a variable speed scroll compressor has been simulated numerically using the shear stress transport turbulence model of the Ansys-CFX code taking into consideration the variable refrigerant flow (VRF) rate, inlet ports ratio range (0.1–0.3) at corresponding R407C flow pressure ratio of suction to secondary flow (1.5–1.8) at two variable speeds of the compressor. The investigation of the increased pressurized flow due to the mixing of the two-phase mixture with the main vapor flow port was performed. The proof of the numerical solution technique showed a percentage error within the orbit of (6–8%) with the mathematical and experimental works. The pressure ratio resultant related to refrigerant mixing volume fraction after reaching the steady state condition has been calculated, the and total mass flow rate ratio influence on the volumetric efficiency is introduced in this report. The injection technique with the designated flow ratio range has achieved an increment in the scroll compressor volumetric efficiency by 6% and achieved a decline in the outlet flow discharge temperature by (13–17) %.

Application of kinetic flux vector splitting scheme for solving viscous quantum hydrodynamical model of semiconductor devices

Abstract In this article, one-dimensional viscous quantum hydrodynamical model of semiconductor devices is numerically investigated. The model treats the propagation of electrons in a semiconductor device as the flow of a charged compressible fluid. It plays an important role in predicting the behavior of electron flow in semiconductor devices. The nonlinear viscous quantum hydrodynamic models contain Euler-type equations for density and current, viscous and quantum correction terms, and a Poisson equation for electrostatic potential. Due to high nonlinearity of model equations, numerical solution techniques are applied to obtain their solutions. The proposed numerical scheme is a splitting scheme based on the kinetic flux-vector splitting (KFVS) method for the hyperbolic step, and a semi-implicit Runge-Kutta method for the relaxation step. The KFVS method is based on the direct splitting of macroscopic flux functions of the system on the cell interfaces. The second order accuracy of the scheme is achieved by using MUSCL-type initial reconstruction and Runge-Kutta time stepping method. Several case studies are considered. For validation, the results of current scheme are compared with those obtained from the splitting scheme based on the NT central scheme. The effects of various parameters such as device length, viscosities, different doping and voltage are analyzed. The accuracy, efficiency and simplicity of the proposed KFVS scheme validates its generic applicability to the given model equations.

Differential Quadrature Analysis For Free Vibration Of Horizontal Spinning Beams With Various Boundary Conditions

The governing differential equations in free vibration are built by using Hamilton’s principle for a horizontal spinning beam. The differential quadrature (DQ) method is a numerical solution technique for initial and/or boundary problems. By using the DQ method, the dimensionless natural frequencies at different rotational speeds are obtained for the beams with different boundary conditions. To ensure the accuracy of the present results by the DQ method, comparisons are made with those available in open literature and very good agreements are achieved.

Research on coupled characteristics of heat transfer and flow in the oil static storage process under periodic boundary conditions

With the increasing demand for oil storage, storage tanks are developing towards large-scale tanks better adaptability to extreme working conditions. The variation law of the oil temperature field and the flow field inside the tank should be accurately understood to guarantee safe and economical operation of an oil depot. Aimed at understanding the periodic variation in the ambient environment of the tank’s location and the physical properties of the crude oil, the theoretical model of unsteady heat transfer flow in a large floating roof tank was established by using the relative heat transfer theory in this paper. The numerical solution technique for the crude oil temperature field and the flow field was studied. This paper thoroughly analyzed the effect of heat transfer and flow coupling characteristics on the tank static storage process under the influences of the atmospheric temperature, solar radiation, etc. The results show that according to the variation law of the temperature field, the static storage cooling process was divided into a partial rapid cooling stage, overall rapid cooling stage, condensate reservoir growth stage and overall low-speed cooling stage. The heating process was divided into a partial low-speed heating stage and partial rapid heating stage. The solar radiation has a great influence on the temperature fluctuation, and the atmospheric temperature has a great influence on the rate of temperature decrease. The results can provide theoretical support for optimizing the design of the storage process and managing the manufacturing of a large floating roof tank.

3D magnetotelluric modeling by using finite-difference method: Comparison study of different forward modeling approaches

We have investigated different approaches to solve magnetotelluric forward modeling problems. Besides classical direct electromagnetic (EM) formulation, ungauged and gauged (Lorenz, Coulomb, and axial) vector and scalar potential formulations are discretized using the finite-difference (FD) numerical solution technique. Linear matrix equations obtained from FD discretization for each EM field formulation are solved by using the conjugate-gradient iterative solution method. We compared FD solutions of each approach with respect to accuracy and speed. We found that all approaches generate accurate results. However, an ungauged approach gave solutions with a lower conjugate-gradient iteration and accordingly less computer time. All stiffness matrices arising from each formulation are examined in terms of their size and type. FD solution of the Coulomb-gauge and ungauged approaches generates larger stiffness matrices than the other approaches. Direct EM, ungauged, and axial-gauge approaches generate a symmetric stiffness matrix. We recommend the use of the ungauged approach in inversion algorithms, which gives faster forward solution, amongst other things.

Spectral parameter power series for arbitrary order linear differential equations

Let L be the n‐th order linear differential operator Ly=ϕ0y(n)+ϕ1y(n−1)+⋯+ϕny with variable coefficients. A representation is given for n linearly independent solutions of Ly=λry as power series in λ, generalizing the SPPS (spectral parameter power series) solution that has been previously developed for n=2. The coefficient functions in these series are obtained by recursively iterating a simple integration process, beginning with a solution system for λ=0. It is shown how to obtain such an initializing system working upwards from equations of lower order. The values of the successive derivatives of the power series solutions at the basepoint of integration are given, which provides a technique for numerical solution of n‐th order initial value problems and spectral problems.