Dynamic analysis of second kind soft-matter quasicrystals of point group 14mm

2019 ◽  
Vol 33 (13) ◽  
pp. 1950154
Author(s):  
Fang Wang ◽  
Tian You Fan ◽  
Hui Cheng ◽  
Zhu Feng Sun
Keyword(s):  
Time Domain ◽  
Soft Matter ◽  
Equations Of Motion ◽  
Point Group ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Group 14 ◽  
Elementary Excitations ◽  
The Finite Difference Method ◽  
Initial Boundary Value

The hydrodynamic model for soft-matter quasicrystals with 14-fold symmetry is investigated. The 14-fold symmetry soft-matter quasicrystals belong to the second kind of soft-matter quasicrystals. The most distinction between the first and the second kinds of the soft-matter quasicrystals lies in the four elementary excitations including two phason elementary excitations exist for the latter. The equations of motion for point group 14 mm soft-matter qusicrystals are given, and an initial-boundary value problem of the equations is solved by applying the finite difference method. The effects of the phonons, first phasons, second phasons, fluid phonon and their interactions in space-time domain are explored.

TWO DISCRETIZATION SCHEMES FOR A TIME-DOMAIN DISSIPATIVE ACOUSTICS PROBLEM

2006 ◽  
Vol 16 (10) ◽  
pp. 1559-1598 ◽  
Author(s):  
ALFREDO BERMÚDEZ ◽  
RODOLFO RODRÍGUEZ ◽  
DUARTE SANTAMARINA
Keyword(s):  
Time Domain ◽  
Existence And Uniqueness ◽  
Energy Equation ◽  
Time Discretization ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Temperature Fields ◽  
Displacement Fields ◽  
Discretization Schemes ◽  
Initial Boundary Value

This paper deals with a time-domain mathematical model for dissipative acoustics and is organized as follows. First, the equations of this model are written in terms of displacement and temperature fields and an energy equation is obtained. The resulting initial-boundary value problem is written in a functional framework allowing us to prove the existence and uniqueness of solution. Next, two different time-discretization schemes are proposed, and stability and error estimates are proved for both. Finally, numerical results are reported which were obtained by combining these time-schemes with Lagrangian and Raviart–Thomas finite elements for temperature and displacement fields, respectively.

On the Vibrations of an Axially Moving String With a Time-Dependent Velocity

2015 ◽  
Author(s):  
Rajab A. Malookani ◽  
Wim T. van Horssen
Keyword(s):  
Perturbation Method ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Approximate Solutions ◽  
Axially Moving String ◽  
Fourier Sine Series ◽  
Axially Moving ◽  
Initial Boundary Value

The transverse vibrations of an axially moving string with a time-varying speed is studied in this paper. The governing equations of motion describing an axially moving string is analyzed using two different techniques. At first, the initial-boundary value problem is discretized using the Fourier sine series, and then the two timescales perturbation method is employed in search of infinite mode approximate solutions. Secondly, a new approach based on the two timescales perturbation method and the method of characteristics is used. It is found that there are infinitely many values of the velocity fluctuation frequency yielding infinitely many resonance conditions in the system. The response of the system with harmonically varying velocity function is computed for particular harmonic initial conditions.

scholarly journals INVESTIGATION OF HYPERELASTIC SOFT SHELLS NONSTATIONARY DYNAMICS PROBLEMS SOLUTION FEATURES

2021 ◽  
Vol 83 (2) ◽  
pp. 151-159
Author(s):  
E.A. Korovaytseva
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Ordinary Differential Equations ◽  
Boundary Value ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Difference Approximation ◽  
Analytical Studies ◽  
Initial Boundary Value

Results of hyperelastic soft shells nonlinear axisymmetric dynamic deforming problems solution algorithm testing are represented in the work. Equations of motion are given in vector-matrix form. For the nonlinear initial-boundary value problem solution an algorithm which lies in reduction of the system of partial differential equations of motion to the system of ordinary differential equations with the help of lines method is developed. At this finite-difference approximation of partial time derivatives is used. The system of ordinary differential equations obtained as a result of this approximation is solved using parameter differentiation method at each time step. The algorithm testing results are represented for the case of pressure uniformly distributed along the meridian of the shell and linearly increasing in time. Three types of elastic potential characterizing shell material are considered: Neo-hookean, Mooney – Rivlin and Yeoh. Features of numerical realization of the algorithm used are pointed out. These features are connected both with the properties of soft shells deforming equations system and with the features of the algorithm itself. The results are compared with analytical solution of the problem considered. Solution behavior at critical pressure value is investigated. Formulations and conclusions given in analytical studies of the problem are clarified. Applicability of the used algorithm to solution of the problems of soft shells dynamic deforming in the range of displacements several times greater than initial dimensions of the shell and deformations much greater than unity is shown. The numerical solution of the initial boundary value problem of nonstationary dynamic deformation of the soft shell is obtained without assumptions about the limitation of displacements and deformations. The results of the calculations are in good agreement with the results of analytical studies of the test problem.

scholarly journals The Laguerre spectral method as applied to numerical solution of a two−dimensional linear dynamic seismic problem for porous media

2016 ◽  
Vol 6 (1) ◽  
pp. 208-212
Author(s):  
Abdumauvlen Berdyshev ◽  
Kholmatzhon Imomnazarov ◽  
Jian-Gang Tang ◽  
Aleksander Mikhailov
Keyword(s):  
Porous Media ◽  
Numerical Solution ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Two Dimensional ◽  
Multiprocessor Computer ◽  
Hydrodynamic Approximation ◽  
Laguerre Transform ◽  
Initial Boundary Value

AbstractThe initial boundary value problem of the dynamics of fluid saturated porous media, described by three elastic parameters in the reversible hydrodynamic approximation, is numerically solved. A linear two-dimensional problem as dynamic equations of porous media for components of velocities, stresses and pore pressure is considered. The equations of motion are based on conservation laws and are consistent with thermodynamic conditions. In this case, a medium is considered to be ideally isotropic (in the absence of energy dissipation) and twodimensional heterogeneous with respect to space. For a numerical solution of the dynamic problem of poroelasticity we use the Laguerre transform with respect to time and the finite difference technique with respect to spatial coordinates on the staggered grids with fourth order of accuracy. The description of numerical implementation of the algorithm offered is presented, and its characteristics are analyzed. Numerical results of the simulation of seismic wave fields for the test layered models have been obtained on the multiprocessor computer.

Blow-up of error estimates in time-fractional initial-boundary value problems

2020 ◽  
Author(s):  
Hu Chen ◽  
Martin Stynes
Keyword(s):  
Numerical Methods ◽  
Boundary Value Problems ◽  
Error Bounds ◽  
Time Domain ◽  
Blow Up ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value Problems ◽  
Initial Boundary Value

Abstract Time-fractional initial-boundary value problems of the form $D_t^\alpha u-p \varDelta u +cu=f$ are considered, where $D_t^\alpha u$ is a Caputo fractional derivative of order $\alpha \in (0,1)$ and the spatial domain lies in $\mathbb{R}^d$ for some $d\in \{1,2,3\}$. As $\alpha \to 1^-$ we prove that the solution $u$ converges, uniformly on the space-time domain, to the solution of the classical parabolic initial-boundary value problem where $D_t^\alpha u$ is replaced by $\partial u/\partial t$. Nevertheless, most of the rigorous analyses of numerical methods for this time-fractional problem have error bounds that blow up as $\alpha \to 1^-$, as we demonstrate. We show that in some cases these analyses can be modified to obtain robust error bounds that do not blow up as $\alpha \to 1^-$.

Trajectory Planning of Tracked Vehicles

1995 ◽  
Vol 117 (4) ◽  
pp. 619-624 ◽  
Author(s):  
Zvi Shiller ◽  
William Serate
Keyword(s):  
Trajectory Planning ◽  
Nonholonomic Constraint ◽  
Equations Of Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Slip Angle ◽  
Planning Problem ◽  
Tracked Vehicles ◽  
Initial Boundary Value ◽  
Inclined Planes

This paper presents a method for computing the track forces and track speeds of planar tracked vehicles, required to follow a given path at specified speeds on horizontal and inclined planes. It is shown that the motions of a planar tracked vehicle are constrained by a velocity dependent nonholonomic constraint, derived from the force equation perpendicular to the tracks. This reduces the trajectory planning problem to determining the slip angle between the vehicle and the path tangent that satisfies the nonholonomic constraint along the entire path. Once the slip angle has been determined, the track forces are computed from the remaining equations of motion. Computing the slip angle is shown to be an initial boundary-value problem, formulated as a parameter optimization. This computational scheme is demonstrated numerically for a planar vehicle moving along circular paths on horizontal and inclined planes.

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