Algorithm For Numerical Solution Research Articles

A simple algorithm for numerical solution of nonlinear parabolic partial differential equations

In this paper, numerical solution of nonlinear two-dimensional parabolic partial differential equations with initial and Dirichlet boundary conditions is considered. The time derivative is approximated using finite difference scheme whereas space derivatives are approximated using Haar wavelet collocation method. The proposed method is developed for semilinear and quasilinear cases, however, it can easily be extended to other types of nonlinearities as well. The proposed method is also illustrated for nonlinear heat equation and Burgers’ equation. The proposed method is implemented upon five test problems and the numerical results are shown using tables and figures. The numerical results validate the accuracy and efficiency of the proposed method.

An efficient hyperbolic relaxation system for dispersive non-hydrostatic water waves and its solution with high order discontinuous Galerkin schemes

In this paper we propose a novel set of first-order hyperbolic equations that can model dispersive non-hydrostatic free surface flows. The governing PDE system is obtained via a hyperbolic approximation of the family of non-hydrostatic free-surface flow models recently derived by Sainte-Marie et al. in [1]. Our new hyperbolic reformulation is based on an augmented system in which the divergence constraint of the velocity is coupled with the other conservation laws via an evolution equation for the depth-averaged non-hydrostatic pressure, similar to the hyperbolic divergence cleaning applied in generalized Lagrangian multiplier methods (GLM) for magnetohydrodynamics (MHD). We suggest a formulation in which the divergence errors of the velocity field are transported with a large but finite wave speed that is directly related to the maximal eigenvalue of the governing PDE.We then use arbitrary high order accurate (ADER) discontinuous Galerkin (DG) finite element schemes with an a posteriori subcell finite volume limiter in order to solve the proposed PDE system numerically. The final scheme is highly accurate in smooth regions of the flow and very robust and positive preserving for emerging topographies and wet-dry fronts. It is well-balanced making use of a path-conservative formulation of HLL-type Riemann solvers based on the straight line segment path. Furthermore, the proposed ADER-DG scheme with a posteriori subcell finite volume limiter adapts very well to modern GPU architectures, resulting in a very accurate, robust and computationally efficient computational method for non-hydrostatic free surface flows. The new model proposed in this paper has been applied to idealized academic benchmarks such as the propagation of solitary waves, as well as to more challenging physical situations that involve wave runup on a shore including wave breaking in both one and two space dimensions. In all cases the achieved agreement with analytical solutions or experimental data is very good, thus showing the validity of both, the proposed mathematical model and the numerical solution algorithm.

Robust computational framework for wire-mesh sensor potential field calculations

With the development of the next generation of nuclear reactor safety system codes fast underway, increased importance has been placed on enhancing physical closure correlations and amassing representative benchmark-quality experimental data for validation purposes. Wire-mesh sensors, a reputable experimental measurement technique with sufficient spatial and temporal resolution to serve such goals, and related data reconstruction algorithms have been the subject of renewed interest as researchers attempt to characterize their measurement uncertainty. To assist in such investigations, the present work establishes a comprehensive numerical framework with which to quantify the electric potential field around wire-mesh sensors. Using the finite-volume foundations of OpenFOAM, a numerical solution algorithm is developed to predict the transmitted electric current between transmitter and receiver electrodes for both homogeneous and heterogeneous electrical conductivity fields. A detailed verification against seminal numerical calculations and robust validation procedure is included to ensure the accuracy of the proposed methodology. Parametric studies of spherical bubble diameter, lateral crossing position, and spheroidal shape influence are conducted to provide preliminary insights into wire-mesh sensor operation and the suitability of various calibration approaches. Observed trends in the transmitted currents reveal overshoots relative to calibration conditions, which are fundamentally linked to the maldistributed electric potential field in heterogeneous bubbly flows. The present investigation offers a vital first step towards a comprehensive multi-physics model of multiphase flow around a wire-mesh sensor.

Solving optimal control problems of Fredholm constraint optimality via the reproducing kernel Hilbert space method with error estimates and convergence analysis

Modeling of dynamic systems of optimal control problems (OCPs) is very important issue in applied sciences and engineering. In this analysis, by developed the reproducing kernel Hilbert space (RKHS) method within the calculus of variations, the OCP is solved with respect to initial conditions and Fredholm operator optimality. The solution methodology involves the use of two generalized Hilbert spaces (HSs) for both range and domain spaces. Numerical algorithm and procedure of solution are assembled compatibility with the optimal formulation of the problem. The convergence analysis and error rating of the utilized method are considered under some presumptions, which provide the theoretical structure behind the technique. The optimal profiles show the performance of the numerical solutions and the effect of the Fredholm operator in the optimal results. In this approach, computational simulations are introduced to delineate the suitability, straightforwardness, and relevance of the calculations created.

Numerical modeling of adiabatic heat generation during rock fracture under dynamic loading

SummaryThis paper considers adiabatic heat generation during rock fracture under dynamic loading. To this end, a thermo‐viscoplastic constitutive model for rock based on Mohr‐Coulomb and Rankine criteria augmented with the adiabatic heat equation is developed. Young's modulus and the compressive and tensile strengths are taken as the temperature‐dependent parameters. A numerical solution algorithm based on the Newton‐Raphson iteration and the cutting plane algorithm at the material point level is presented. Furthermore, there is a description of an explicit staggered scheme for solving the heat conduction at the specimen level in rock material. Then, the model is applied in 2D and 3D numerical simulations of compression and tension tests on a rock‐like material with compressive and tensile strengths of 120 and 10 MPa, respectively. The simulations predict that the temperature rise because of adiabatic heat generation during rock fracture is in the order of a few degrees centigrade, even at a strain rate of 100 s−1. In conclusion, the effect of including heat conduction is shown to be insignificant because of the short duration of dynamic loading processes.

Numerical method for 3D two-component isothermal compressible flows with application to digital rock physics

Abstract We propose a numerical algorithm for simulations of two-component viscous compressible isothermal flows with surface effects in 3D domains of complex shape with voxel representation of geometry. The basic mathematical model is the regularized system of Navier–Stokes–Cahn–Hilliard equations. Simulations of droplet spreading over a flat base and displacement of one liquid by another one in a pore space of real rock sample are carried out. The simulation results demonstrate the applicability and good efficiency of the used system of equations, the corresponding difference scheme, and its implementation algorithms for numerical solution of the considered class of problems.

On the distribution of local current density along a PEM fuel cell cathode channel

We report analysis of a recent model for local current density distribution along the air channel in a PEM fuel cell. Analytical solution for the case of small cell ohmic resistivity is obtained. An algorithm for numerical solution of the model equations is developed for practically important case when the cell polarization curve and ohmic resistivity are known. Good agreement between the calculated and experimental shapes of the local current density is demonstrated. Cell ohmic resisitivty greatly homogenizes the distribution of local current. A Python code for local current calculation is available for download.

Development and Application of Cast Steel Numerical Simulation System for Heat Treatment

A numerical simulation system for heat treatment of cast steel was developed using C++ environment, which has been integrated into the InteCAST platform as a new feature module. The system mainly includes heat treatment process inquiry module, initialization and property parameter setting module, automatic import and display module of time-temperature-transformation (TTT) and continuous cooling transformation (CCT) curves, heat treatment process design module, organization, performance and stress–strain prediction module. This paper introduces the mathematical model and numerical solution algorithm in the process of numerical simulation of the heat treatment organization, performance and stress–strain as well as the interface operation process and application examples of the system.

Approximate solution of time-fractional fuzzy partial differential equations

In this study, we develop perturbation–iteration algorithm (PIA) for numerical solutions of some types of fuzzy fractional partial differential equations (FFPDEs) with generalized Hukuhara derivative. We also present the convergence analysis of the method. The proposed approach reveals fast convergence rate and accuracy of the present method when compared with exact solutions of crisp problems. The main efficiency of this method is that while scaling support zone of uncertainty for the fractional partial differential equations, it eliminates over calculation and produces highly approximate and accurate results. Error analysis of the PIA for the FFPDEs is also illustrated within examples.

Quartic trigonometric B-spline algorithm for numerical solution of the regularized long wave equation

In this paper, an application of the quartic trigonometric B-spline finite element method is used to solve the regularized long wave equation numerically. This approach involves a Galerkin method based on the quartic trigonometric B-spline function in space discretization together with second and fourth order schemes in time discretization. The accuracy of the proposed methods are demonstrated by test problems and numerical results are compared with the exact solution and some previous results.

Numerical conjugate problem solution algorithm for fluid-deformable body system

This paper describes the numerical conjugate problem solution algorithm for fluid-deformable body system. It is based on respective subtasks according to physical principals. To solve these subtasks, independent “black box” program modules are used. These modules are connected by conjugate conditions. Fluid phase domain boundary interface is discrete movable i.e. it is not movable on the integration time step, but its position is adjusted on the next time step.

Nonlinear parametric oscillations of a viscoelastic shallow shell of variable thickness

The problem of parametric oscillations of an isotropic viscoelastic shallow shell of variable thickness under periodic load is considered. It is believed that under the influence of specified load, the shallow shell allows displacements (in particular, deflections), commensurate with its thickness. In a geometrically nonlinear statement, taking into account the viscoelastic properties of material, a mathematical model of the problem has been developed using the classical Kirchhoff-Love hypothesis. Using the Bubnov-Galerkin method based on the polynomial approximation of the deflections, the problem is reduced to the study of the system of integro-differential equations, where time is the independent variable. The solution of the system of integrodifferential equations is determined by the proposed numerical method. Based on this method, a numerical solution algorithm is described. The Koltunov-Rzhanitsyn kernel with three different rheological parameters is chosen as a weakly singular kernel. At the same time, the effect of geometric nonlinearity, viscoelastic properties of material, as well as other physicomechanical and geometric parameters and factors (rheological parameters, thickness, initial shape imperfections, aspect ratios, boundary conditions, excitation coefficient) on the area of dynamic instability is taken into account. The results obtained in this study are in good agreement with the results and data obtained by other authors.

РОЗРАХУНКОВІ МОДЕЛІ ДЛЯ АНАЛІЗУ ЕФЕКТИВНИХ ХАРАКТЕРИСТИК ТРИВИМІРНИХ НАНОКОМПОЗИТІВ ІЗ СИСТЕМАМИ ВКЛЮЧЕНЬ

Для опису механічних властивостей тривимірних композитів та нанокомпозитів використовуються підходи, засновані на співвідношеннях механіки суцільного середовища. За припущеннями лінійної пружності розглянуті механічні властивості композиційних матеріалів та нанокомпозитів. Застосовуються умови ідеального контакту і некласичні умови Гуртіна-Мердока для опису нанорозмірного контакту між матрицею та включеннями на поверхні розділу, яка розглядається як еластична мембрана з власними пружними характеристиками та заданим поверхневим натягом. Розглянуті різні типи представницьких комірок. Для числової симуляції механічних властивостей композитних та нанокомпозитних матеріалів використовуються методи скінченних та граничних елементів. У тривимірному формулюванні здійснено порівняння результатів, отриманих при використанні методу скінченних елементів (гексагональний репрезентативний об’ємний елемент) і методу граничних елементів (циліндричний репрезентативний об’ємний елемент). Метод скінченних елементів застосовано для визначення напружено-деформованого стану різних репрезентативних об'ємних елементів тривимірних нанокомпозитів. Розглядаються матриці у вигляді куба та циліндра скінченних розмірів та неоднорідності у вигляді куль, сфер, циліндрів, волокон та трубок. Базовані на МСЕ розрахункові моделі поширено на опис ефективних пружних властивостей композитів з розподіленими нановключеннями з випадковою та впорядкованою орієнтаціями. Метод скінченних елементів застосовано для визначення НДС різних репрезентативних об'ємних елементів тривимірних нанокомпозитів. Досліджено вплив форми і відносних розмірів неоднорідностей та матриць репрезентативного об’єму на ефективні модулі пружності нанокомпозитів. Отримані моделі створюють інформативну базу для технологій синтезування нанокомпозитів з вдосконаленими деформаційними і міцнісними характеристиками.

Оценки сходимости итерационного алгоритма численного решения двумерных нестационарных задач магнитной гидродинамики

Оценки сходимости итерационного алгоритма численного решения двумерных нестационарных задач магнитной гидродинамики

Economic explicit-implicit schemes for solving multidimensional diffusion-convection problems

This work concerns the development of an efficient parallel algorithm for numerical solution of nonstationary diffusion-convection problems by means of a multiprocessor computer system with distributed memory. Economically explicit-implicit difference schemes and the method of splitting into physical processes are used as a basis. The original problem is replaced by a sequence of one-dimensional and two-dimensional difference problems using complex schemes that approximate the original problem in the general sense. Explicit-implicit difference schemes involve explicit approximation in horizontal directions and implicit approximation with weights in a vertical direction and require less time for solving diffusion-convection problems compared to explicit schemes while maintaining acceptable accuracy of solutions. The algorithm is proposed to find the optimal weight value and it yields the lowest approximation error in the solution of the diffusion-convection problem in the vertical direction for given time grid steps. The three-dimensional model problem of transport of suspensions in the water environment is considered. The model takes into account the following processes: advective transport caused by the movement of the water medium, microturbulent diffusion and gravitational deposition of suspended particles, as well as changes in the geometry of the bottom caused by the deposition of suspended particles or the rise of sediment particles. Application of the parallel algorithm developed for numerical modeling of suspension transport makes it possible to significantly improve the real-time forecast accuracy and the validity of the accepted engineering decisions used in designing the objects of coastal infrastructure.

An Adaptive, Highly Accurate and Efficient, Parker-Sochacki Algorithm for Numerical Solutions to Initial Value Ordinary Differential Equation Systems

An Adaptive, Highly Accurate and Efficient, Parker-Sochacki Algorithm for Numerical Solutions to Initial Value Ordinary Differential Equation Systems

Оценки сходимости итерационных алгоритмов численного решения трехмерных нестационарных задач магнитной гидродинамики

Оценки сходимости итерационных алгоритмов численного решения трехмерных нестационарных задач магнитной гидродинамики

Discrete Element Method applied to the simulation of the stress state in granular materials

This work examines the Discrete Element Method (DEM), also known as the particle method, for its application in soil mechanics, specifically to calculate the tension acting on granular materials without cohesion. First, theoretical aspects of soil mechanics and the physical properties of some types of granular materials are presented, and the material properties are used afterwards in numerical examples. Following this, the DEM formulation is described, corresponding to the force and movement equations acting on each particle. For that, Newton’s second law, the force–displacement law, Hertz’s contact law, and some concepts from particle mechanics are defined. The integration over time process and the numerical solution algorithm presented by T.I. Zohdi are also described. Additionally, a new optimisation process for contact detection is described, one which significantly diminishes computational costs and therefore analysis time. Finally, some basic physics examples necessary for the formulation validation and application in soil mechanics are presented. The results obtained with the software developed in this research are then compared with soil mechanics results, which are simulated using the GeoStudio software. The GiD program was utilised for graphical presentation of the results.

Transient Radiator Room Heating—Mathematical Model and Solution Algorithm

A mathematical model and robust numerical solution algorithm for radiator heating of an arbitrary room is presented in this paper. Three separate and coupled transient thermal energy equations are solved. A modified transient heat conduction equation is used for solving the heat transfer at multi-layer outer walls and room assembly. Heat exchange between the inner walls and the observed room are represented with their own transport equation and the transient thermal energy equation is solved for radiators as well. Explicit coupling of equations and linearization of source terms result in a simple, accurate, and stabile solution algorithm. Verification of the developed methodology is demonstrated on three carefully selected test cases for which an analytical solution can be found. The obtained results show that even for the small temperature differences between inner walls and room air, the corresponding heat flux can be larger than the transmission heat flux through outer walls or windows. The benefits of the current approach are stressed, while the plans for the further development and application of the methodology are highlighted at the end.

Experimental investigation and numerical modeling of destruction dynamics of thin silicone oil layers under the local heating

The processes of heat and mass transfer in systems with liquid-gas interface are of interest to a wide range of problems. Destruction dynamics of thin horizontal layers of silicone oils were investigated using confocal sensor on a 3D-positioning system. The numerical solution of the problem was obtained in the lubrication approximation theory for two-dimensional axisymmetric thermocapillary flow. The model takes into account the surface tension, viscosity, gravity and heat transfer in the substrate. The numerical algorithm for the joint solution of the energy equation and the evolution equation for the liquid layer thickness has been developed. The establishment method was used to obtain the stationary solutions. Experimental measurements and numerical calculations were made for silicone oils of different viscosities, heating power and initial thickness. The significant effect of the surface tension coefficient and its temperature coefficient on thermocapillary deformation was detected. It was experimentally established that the deformation value depends on a heat flux value. A liquid bump is formed at the boundary of the heating region that is also observed in numerically calculated profiles.