A Two-Dimensional Numerical Solution for Elastic Waves in Variously Configured Rods

1971 ◽  
Vol 38 (1) ◽  
pp. 62-70 ◽  
Author(s):  
J. L. Habberstad
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Difference Equations ◽  
Pressure Pulse ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
First Order ◽  
Finite Difference Equations ◽  
Viscosity Term

The exact equations of motion governing elastic, axisymmetric wave propagation in a cylindrical rod are approximated by a first-order finite-difference scheme. This difference scheme is based on a displacement rather than a velocity formulation, thereby making it unnecessary to explicitly introduce an artificial viscosity term into the finite-difference equations. The resulting difference equations are used in conjunction with the boundary and initial conditions 10 study: (a) a pressure pulse applied to the end of a semi-infinite bar, (b) a bar composed of two materials joined together at some point along its length, and (c) a bar containing a discontinuity in cross section. The numerical results so obtained are compared to available experimental data and other analytical-numerical solutions.

A STUDY OF THE SHORT WAVE COMPONENTS IN COMPUTATIONAL ACOUSTICS

1993 ◽  
Vol 01 (01) ◽  
pp. 1-30 ◽  
Author(s):  
CHRISTOPHER K. W. TAM ◽  
JAY C. WEBB ◽  
ZHONG DONG
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Difference Scheme ◽  
Exact Solutions ◽  
Euler Equations ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Long Waves ◽  
Short Waves

It is shown by using a Dispersion-Relation-Preserving [Formula: see text] finite difference scheme that it is feasible to perform direct numerical simulation of acoustic wave propagation problems. The finite difference equations of the [Formula: see text] scheme have essentially the same Fourier-Laplace transforms and hence dispersion relations as the original linearized Euler equations over a broad range of wavenumbers (here referred to as long waves). Thus it is guaranteed that the acoustic waves, the entropy and the vorticity waves computed by the [Formula: see text] scheme are good approximations of those of the exact solutions of Euler equations as long as the wavenumbers are in the long wave range. Computed waves with higher wavenumber, or the short waves, generally have totally different propagation characteristics. There are no counterparts of such waves in the exact solutions. The short waves of a computation scheme are, therefore, contaminants of the numerical solutions. The characteristics of these short waves are analyzed here by group velocity consideration and standard dispersive wave theory. Numerical results of direct simulations of these waves are reported. These waves can be generated by discontinuous initial conditions. To purge the short waves so as to improve the quality of the numerical solution, it is suggested that artificial selective damping terms be added to the finite difference scheme. It is shown how the coefficients of such damping terms may be chosen so that damping is confined primarily to the high wavenumber range. This is important for then only the short waves are damped leaving the long waves basically unaffected. The effectiveness of the artificial selective damping terms is demonstrated by direct numerical simulations involving acoustic wave pulses with discontinuous wave fronts.

scholarly journals NUMERICAL PREDICTION ON TYPHOON TIDE IN TOKYO BAY

1964 ◽  
Vol 1 (9) ◽  
pp. 43
Author(s):  
Takeshi Ito ◽  
Mikio Hino ◽  
Jiro Wantanbe ◽  
Kazuko Hino
Keyword(s):  
Numerical Calculation ◽  
Finite Difference ◽  
Difference Equations ◽  
Equations Of Motion ◽  
Storm Surges ◽  
Tokyo Bay ◽  
Astronomical Tide ◽  
Finite Difference Equations ◽  
Mathematical Problems ◽  
Stable Forms

The paper discusses firstly mathematical problems on the numerical calculation of storm surges. The partial differential equations of motion adopted here take into account the Coriolis force and the nonlinear terms such as the inertial terms and a quadratic form of bottom friction. As a result, special care must be taken in order to obtain stable forms of finite-difference equations. It is shown that inadequate forms accumulate errors to cause divergence of the step by step calculations. A set of stable forms of the finite-difference equations of motion and continuity has been derived. Sometimes, it is convenient to divide the numerical integration region into two or more sub-regions, the mesh-dimensions of which are not equal. A method is described to calculate both regions by one procedure. Japan coasts were frequently damaged by severe storm surges (Typhoon Tides). To protect the metropolitan area from storm surges, a proposal has been made to construct a dike across Tokyo Bay. A numerical calculation has been made by means of IBM 7090 to estimate for several opening width of the proposed dike its effects on the reduction of surges. Interactions between daily tides (astronomical tide) and surges are also discussed.

Derivation of First-Order Difference Equations for Dynamical Systems by Direct Application of Hamilton’s Principle

1970 ◽  
Vol 37 (2) ◽  
pp. 276-278 ◽  
Author(s):  
J. M. Vance ◽  
A. Sitchin
Keyword(s):  
Difference Equations ◽  
Equations Of Motion ◽  
Lagrangian Multiplier ◽  
Hamilton’S Principle ◽  
Hamilton's Principle ◽  
Order Difference ◽  
First Order ◽  
Finite Difference Equations ◽  
Lagrangian Multiplier Method ◽  
Dynamics Problems

In dynamics problems where the equations of motion are eventually reduced to finite-difference equations for numerical integration on a digital computer, an auxiliary condition exists that permits the application of the Lagrangian multiplier method to Hamilton’s principle in order to obtain directly a set of first-order difference equations. These equations are equivalent to Hamilton’s canonical equations and are derived without the necessity to obtain the Hamiltonian or take time derivatives.

scholarly journals Block algorithms to solve Zheng/Chen/Zhang's finite-difference equations

2021 ◽  
Vol 45 (3) ◽  
pp. 461-468
Author(s):  
D.L. Golovashkin ◽  
N.D. Morunov ◽  
L.V. Yablokova
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Experimental Research ◽  
Difference Equations ◽  
Fdtd Method ◽  
Finite Difference Equations ◽  
Implicit Finite Difference Scheme ◽  
Speed Up ◽  
Implicit Finite Difference ◽  
Block Algorithms

This paper is devoted to the design of multiblock algorithms of the FDTD-method intended for computations based on a Zheng-Chen-Zhang implicit finite-difference scheme. Special emphasis is placed on experimental research of the designed algorithms and detecting specific features of the multiblock computing based on implicit finite-difference equations. The efficiency of the proposed approaches is proved by a six-fold speed-up of computations.

scholarly journals Finite-difference equations of quasistatic motion of the shallow concrete shells in nonlinear setting

2020 ◽  
Vol 7 (1) ◽  
pp. 48-55 ◽  
Author(s):  
Bolat Duissenbekov ◽  
Abduhalyk Tokmuratov ◽  
Nurlan Zhangabay ◽  
Zhenis Orazbayev ◽  
Baisbay Yerimbetov ◽  
...  
Keyword(s):  
Finite Difference ◽  
Differential Equations ◽  
Difference Equations ◽  
Constant Load ◽  
Time Interval ◽  
Test Case ◽  
Algebraic Equations ◽  
Nonlinear Properties ◽  
Finite Difference Equations ◽  
Concrete Shells

AbstractThe study solves a system of finite difference equations for flexible shallow concrete shells while taking into account the nonlinear deformations. All stiffness properties of the shell are taken as variables, i.e., stiffness surface and through-thickness stiffness. Differential equations under consideration were evaluated in the form of algebraic equations with the finite element method. For a reinforced shell, a system of 98 equations on a 8×8 grid was established, which was next solved with the approximation method from the nonlinear plasticity theory. A test case involved computing a 1×1 shallow shell taking into account the nonlinear properties of concrete. With nonlinear equations for the concrete creep taken as constitutive, equations for the quasi-static shell motion under constant load were derived. The resultant equations were written in a differential form and the problem of solving these differential equations was then reduced to the solving of the Cauchy problem. The numerical solution to this problem allows describing the stress-strain state of the shell at each point of the shell grid within a specified time interval.

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