scholarly journals High Order Godunov Type Multimesh Method for 3d Impact Problems of Elastoplastic Media

2021 ◽  
Keyword(s):  
Three Dimensional ◽  
Second Order ◽  
The Body ◽  
Godunov Scheme ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Sharp Interface Method ◽  
Impact Problems ◽  
Elastoplastic Media ◽  
Compact Stencil

A numerical method for calculating the three-dimensional processes of impact interaction of elastoplastic bodies under large displacements and deformations based on the multi mesh sharp interface method and modified Godunov scheme is presented. To integrate the equations of dynamics of an elastoplastic medium, the principle of splitting in space and in physical processes is used. The solutions of the Riemann problem for first and second order accuracy for compact stencil for an elastic medium in the case of an arbitrary stress state are obtained and presented, which are used at the “predictor” step of the Godunov scheme. A modification of the scheme is described that allows one to obtain solutions in smoothness domains with a second order of accuracy on a compact stencil for moving Eulerian-Lagrangian grids. Modification is performed by converging the areas of influence of the differential and difference problems for the Riemann’s solver. The “corrector” step remains unchanged for both the first and second order accuracy schemes. Three types of difference grids are used. The first – a moving surface grid – consists of a continuous set of triangles that limit and accompany the movement of bodies; the size and number of triangles in the process of deformation and movement of the body can change. The second – a regular fixed Eulerian grid – is limited to a surface grid; separately built for each body; integration of equations takes place on this grid; the number of cells in this grid can change as the body moves. The third grid is a set of local Eulerian-Lagrangian grids attached to each moving triangle of the surface from the side of the bodies and allowing obtain the parameters on the boundary and contact surfaces. The values of the underdetermined parameters in cell’s centers near the contact boundaries on all types of grids are interpolated. Comparison of the obtained solutions with the known solutions by the Eulerian-Lagrangian and Lagrangian methods, as well as with experimental data, shows the efficiency and sufficient accuracy of the presented three-dimensional methodology.

scholarly journals THE METHOD OF DECOMPOSITION OF GAPES IN THE THREE-DIMENSIONAL DYNAMICS OF ELASTOPLASTIC MEDIA

2020 ◽  
Vol 82 (3) ◽  
pp. 377-389
Author(s):  
K.M. Abuzyarov
Keyword(s):  
Three Dimensional ◽  
Sufficient Accuracy ◽  
The Body ◽  
Godunov Scheme ◽  
Physical Processes ◽  
Order Of Accuracy ◽  
The Third ◽  
Impact Interaction ◽  
Elastoplastic Media ◽  
Eulerian Grid

A numerical method for calculating the three-dimensional processes of impact interaction of elastoplastic bodies with large displacements and deformations based on the method of disintegration of discontinuities according to the Godunov scheme is presented. To integrate the equations of dynamics of an elastoplastic medium, the principle of splitting in space and in physical processes is used. The Riemann's solver for an elastic medium in the case of an arbitrary stress state are obtained and presented. A modification of the scheme is described that allows one to obtain solutions in smoothness domains with a second order of accuracy on a compact template for moving Eulerian – Lagrangian grids. Three types of difference grids are used. The first – a moving surface grid – consists of a continuous set of triangles that limit and accompany the movement of bodies; the size and number of triangles in the process of deformation and movement of the body can vary. The second – a regular fixed Eulerian grid – is limited to a surface grid; separately built for each body; integration of equations takes place on this grid; the number of cells in this grid can change as the body moves. The third grid is a set of local Eulerian – Lagrangian grids attached to each moving triangle of the surface from the side of the bodies and allowing to determine the parameters on the boundary and contact surfaces. The values of the underdetermined parameters near the contact boundaries on all types of grids are interpolated. Comparison of the obtained solutions with the known solutions and with the experimental data, shows the efficiency and sufficient accuracy of the presented three-dimensional methodology.

A three-dimensional computation of the force and torque on an ellipsoid settling slowly through a viscoelastic fluid

1995 ◽  
Vol 283 ◽  
pp. 1-16 ◽  
Author(s):  
J. Feng Feng ◽  
D. D. Joseph ◽  
R. Glowinski ◽  
T. W. Pan
Keyword(s):  
Reynolds Number ◽  
Viscoelastic Fluid ◽  
Tilt Angle ◽  
Three Dimensional ◽  
Major Axis ◽  
Weissenberg Number ◽  
Second Order ◽  
The Body ◽  
Fall Velocity ◽  
Elasticity Number

The orientation of an ellipsoid falling in a viscoelastic fluid is studied by methods of perturbation theory. For small fall velocity, the fluid's rheology is described by a second-order fluid model. The solution of the problem can be expressed by a dual expansion in two small parameters: the Reynolds number representing the inertial effect and the Weissenberg number representing the effect of the non-Newtonian stress. Then the original problem is split into three canonical problems: the zeroth-order Stokes problem for a translating ellipsoid and two first-order problems, one for inertia and one for second-order rheology. A Stokes operator is inverted in each of the three cases. The problems are solved numerically on a three-dimensional domain by a finite element method with fictitious domains, and the force and torque on the body are evaluated. The results show that the signs of the perturbation pressure and velocity around the particle for inertia are reversed by viscoelasticity. The torques are also of opposite sign: inertia turns the major axis of the ellipsoid perpendicular to the fall direction; normal stresses turn the major axis parallel to the fall. The competition of these two effects gives rise to an equilibrium tilt angle between 0° and 90° which the settling ellipsoid would eventually assume. The equilibrium tilt angle is a function of the elasticity number, which is the ratio of the Weissenberg number and the Reynolds number. Since this ratio is independent of the fall velocity, the perturbation results do not explain the sudden turning of a long body which occurs when a critical fall velocity is exceeded. This is not surprising because the theory is valid only for slow sedimentation. However, the results do seem to agree qualitatively with ‘shape tilting’ observed at low fall velocities.

scholarly journals Study on Accuracy of the High-Resolution Schemes

2014 ◽  
Vol 6 ◽  
pp. 905053
Author(s):  
Yawen Tang ◽  
Bo Yu ◽  
Jianyu Xie ◽  
Jingfa Li ◽  
Peng Wang
Keyword(s):  
High Resolution ◽  
Numerical Solution ◽  
Richardson Extrapolation ◽  
Second Order ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Numerical Examples ◽  
High Resolution Schemes ◽  
Artificial Diffusion ◽  
Convection Dominated

The high-resolution (HR) schemes have been widely used as they can achieve the numerical solution without oscillation and artificial diffusion, especially for convection-dominated problems. However, there still have arguments about the order of accuracy of HR schemes, especially about the extreme value of the solution. In this paper, it is proved that any HR scheme designed in the NVD diagram has second-order accuracy when its combined segments totally locate in the BAIR region. In other words, it has been verified in our study that the segments, which have low-order accuracy when independently employed, have at least second-order accuracy when locate in BAIR region by analysis of two implementation methods of HR scheme and also a number of numerical examples. Meanwhile Richardson extrapolation has been used to estimate the order of accuracy of HR schemes which achieve the same conclusion.

Schemes of the second order accuracy for computing the dynamics of disturbances in an elastic body

2017 ◽  
Vol 12 (1) ◽  
pp. 44-50 ◽  
Author(s):  
A.A. Aganin ◽  
N.A. Khismatullina
Keyword(s):  
Elastic Body ◽  
Analytic Solutions ◽  
Second Order ◽  
The Body ◽  
Godunov Method ◽  
Tvd Scheme ◽  
Order Accuracy ◽  
Linear Waves ◽  
Approximation Errors ◽  
The Impact

The efficiency of the second-order accurate UNO- and TVD-modifications of the Godunov method is compared using a number of problems on the propagation of linear waves in an elastic body, their interaction with each other and with the surface of the body. In particular, one-dimensional problems having analytic solutions and a two-dimensional problem of the dynamics of a body in the vicinity of the impact domain on its free surface are considered. It is shown that if in the problems there are well-marked extrema or short waves, then the UNO-scheme is more effective, since in such cases decrease in the accuracy of the TVD-scheme becomes apparent due to strictly satisfying the TVD condition. Because of approximately satisfying the TVD condition, the UNO-scheme can lead to the appearance of oscillations of numerical nature at the level of approximation errors. However, this does not reduce the efficiency of the UNO scheme since the amplitude of those oscillations decreases with refinement of the grid.

Construction of Space-Time High Resolution Second Order Accurate Three Dimensional MP Difference Schemes

2006 ◽  
Vol 306-308 ◽  
pp. 685-690
Author(s):  
Kai-Teng Wu ◽  
Jian Guo Ning
Keyword(s):  
Hyperbolic Conservation Laws ◽  
Natural Extension ◽  
Three Dimensional ◽  
Second Order ◽  
Difference Schemes ◽  
Vector System ◽  
Maximum Principles ◽  
General Context ◽  
Three Dimension ◽  
Order Accuracy

In this paper, a new Riemann-solver-free class of difference schemes are constructed to scalar nonlinear hyperbolic conservation laws in the three dimension (3D). We proved that these schemes had second order accuracy in space and time, and satisfied maximum principles (marked as MPs) under an appropriate CFL condition. This results in a second-order accuracy, MP schemes a natural extension of the one (two)-dimensional second-order. In addition, these schemes can still be extended to the vector system of conservation law. We yet prove that these schemes satisfied the scalar and vector maximum principle, and in the more general context of systems.

Novel Linear Implicit Integration Method and its Application to Linear/Nonlinear Structural System

2014 ◽  
Vol 1065-1069 ◽  
pp. 1535-1539
Author(s):  
Chuan Guo Jia ◽  
Yan Xing Liu ◽  
Ying Min Li ◽  
Min Mao Liao
Keyword(s):  
Computational Efficiency ◽  
Unconditional Stability ◽  
Second Order ◽  
Order Accuracy ◽  
Dynamic Simulations ◽  
Order Of Accuracy ◽  
Engineering Research ◽  
Time Integrators ◽  
Shear Type ◽  
The Stability

Dynamic simulations of structures to determine their seismic performance is an essential part of civil engineering research. Time integrators of increasing sophistication has been elaborated over the last few decades to achieve higher order accuracy, unconditional stability, computational efficiency and high-frequency dissipation. This paper tries to extend 1-stage Rosenbrock-based integrator to an integrator of second-order accuracy without losing computational efficiency and unconditional stability. Initially, 1-stage Rosenbrock integrator is introduced and its order of accuracy is studied theoretically. According to accuracy analysis, a force term at the end of current step is considered, resulting a novel integrator of second-order accuracy. Moreover, the stability of the proposed method is studied by means of the energy method. To investigate its performance for nonlinear structures, numerical simulations are conducted on a shear-type structure including a pendulum.

scholarly journals On second-order accuracy schemes for modeling of plasma oscillations

2020 ◽  
pp. 115-128
Author(s):  
Е.В. Чижонков
Keyword(s):  
Numerical Experiments ◽  
Second Order ◽  
Power Laser ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Plasma Oscillations ◽  
First Order ◽  
High Power Laser Pulse ◽  
Quantitative Characteristics ◽  
Free Plasma

Для моделирования колебаний холодной плазмы как в нерелятивистском случае, так и с учетом релятивизма предложены модификации классических разностных схем второго порядка точности: метода МакКормака и двухэтапного метода Лакса-Вендроффа. Ранее для подобных расчетов в эйлеровых переменных была известна только схема первого порядка точности. Для задачи о свободных плазменных колебаниях, инициированных коротким мощным лазерным импульсом, с целью тестирования представленных схем проведены численные эксперименты по сохранению энергии и других величин. Сделан вывод о достоверности численного анализа колебаний как на основе схемы МакКормака, так и на основе схемы Лакса-Вендроффа, однако для расчетов долгоживущих процессов первая схема более предпочтительна. Теоретическое исследование аппроксимации и устойчивости вместе с экспериментальным наблюдением за количественными характеристиками погрешности для наиболее чувствительных величин существенно повышает достоверность вычислений. Ключевые слова: численное моделирование, плазменные колебания, эффект опрокидывания, схемы МакКормака и Лакса-Вендроффа, порядок точности разностной схемы, законы сохранения. For modeling cold plasma oscillations in the non-relativistic and relativistic cases, some modifications of classical difference schemes of the second order of accuracy are proposed: the McCormack method and the two-stage Lax-Wendroff method. Previously, only the first-order accuracy scheme was known for calculations in Euler variables. For the problem of free plasma oscillations initiated by a short high-power laser pulse, the results of numerical experiments on energy conservation and other quantities were performed in order to test the proposed schemes. It is concluded that the numerical analysis of oscillations is reliable both for the McCormack scheme and for the Lax-Wendroff scheme however, for the calculation of long-lived processes, the first scheme is more preferable. The theoretical analysis of approximation and stability together with experimental observations of quantitative characteristics of errors for the most sensitive quantities significantly increases the reliability of calculations.

AN ACOUSTIC FINITE-DIFFERENCE TIME-DOMAIN ALGORITHM WITH ISOTROPIC DISPERSION

2005 ◽  
Vol 13 (02) ◽  
pp. 365-384 ◽  
Author(s):  
CHRISTOPHER L. WAGNER ◽  
JOHN B. SCHNEIDER
Keyword(s):  
Finite Difference ◽  
Time Domain ◽  
Finite Difference Time Domain ◽  
Phase Error ◽  
Three Dimensional ◽  
Second Order ◽  
Step Size ◽  
Order Accuracy ◽  
Discretization Step ◽  
Difference Time

The classic Yee Finite-Difference Time-Domain (FDTD) algorithm employs central differences to achieve second-order accuracy, i.e., if the spatial and temporal step sizes are reduced by a factor of n, the phase error associated with propagation through the grid will be reduced by a factor of n2. The Yee algorithm is also second-order isotropic meaning the error as a function of the direction of propagation has a leading term which depends on the square of the discretization step sizes. An FDTD algorithm is presented here that has second-order accuracy but fourth-order isotropy. This algorithm permits a temporal step size 50% larger than that of the three-dimensional Yee algorithm. Pressure-release resonators are used to demonstrate the behavior of the algorithm and to compare it with the Yee algorithm. It is demonstrated how the increased isotropy enables post-processing of the simulation spectra to correct much of the dispersion error. The algorithm can also be optimized at a specified frequency, substantially reducing numerical errors at that design frequency. Also considered are simulations of scattering from penetrable spheres ensonified by a pulsed plane wave. Each simulation yields results at multiple frequencies which are compared to the exact solution. In general excellent agreement is obtained.

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