scholarly journals AN EFFICIENT FINITE DIFFERENCE SCHEME FOR THE NUMERICAL SOLUTION OF TIMOSHENKO BEAM MODEL

2021 ◽  
Vol 16 (5) ◽  
Author(s):  
Kamran Malik
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Timoshenko Beam ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Variable Load ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Step Size

We propose and implement a finite difference scheme for the numerical solution of the Timoshenko beam model without locking phenomenon. The averaging concept is used in approximating the function, and thus developing the scheme for elements. Finally, the system is discretized into the algebraic system using the proposed scheme and the numerical solution is attained. The numerical solutions are attained for a constant load and a variable load comprising linear and exponential functions. The mathematical model of the Timoshenko beam (TB) problem in the form of a boundary-value problem has been solved successfully for the rotation and displacement parameters. The results agree with other schemes in the literature for various values of the parameter and step size.

scholarly journals ENERGY-CONSERVING AND RECIPROCAL SOLUTIONS FOR HIGHER-ORDER PARABOLIC EQUATIONS

2001 ◽  
Vol 09 (01) ◽  
pp. 183-203 ◽  
Author(s):  
DMITRY MIKHIN
Keyword(s):  
Finite Difference ◽  
Energy Conservation ◽  
Difference Scheme ◽  
Numerical Solution ◽  
Finite Difference Scheme ◽  
Flow Reversal ◽  
Difference Operators ◽  
Energy Conserving ◽  
The One ◽  
Moving Media

The energy conservation law and the flow reversal theorem are valid for underwater acoustic fields. In media at rest the theorem transforms into well-known reciprocity principle. The presented parabolic equation (PE) model strictly preserves these important physical properties in the numerical solution. The new PE is obtained from the one-way wave equation by Godin12 via Padé approximation of the square root operator and generalized to the case of moving media. The PE is range-dependent and explicitly includes range derivatives of the medium parameters. Implicit finite difference scheme solves the PE written in terms of energy flux. Such formalism inherently provides simple and exact energy-conserving boundary condition at vertical interfaces. The finite-difference operators, the discreet boundary conditions, and the self-starter are derived by discretization of the differential PE. Discreet energy conservation and flow reversal theorem are rigorously proved as mathematical properties of the finite-difference scheme and confirmed by numerical modeling. Numerical solution is shown to be reciprocal with accuracy of 10–12 decimal digits, which is the accuracy of round-off errors. Energy conservation and wide-angle capabilities of the model are illustrated by comparison with two-way normal mode solutions including the ASA benchmark wedge.

scholarly journals C-N Difference Schemes for Dissipative Symmetric Regularized Long Wave Equations with Damping Term

2011 ◽  
Vol 2011 ◽  
pp. 1-16 ◽  
Author(s):  
Jinsong Hu ◽  
Bing Hu ◽  
Youcai Xu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Wave Equations ◽  
Numerical Solutions ◽  
Initial Boundary ◽  
Damping Term ◽  
Discrete Energy ◽  
Initial Boundary Problem ◽  
Long Wave

We study the initial-boundary problem of dissipative symmetric regularized long wave equations with damping term. Crank-Nicolson nonlinear-implicit finite difference scheme is designed. Existence and uniqueness of numerical solutions are derived. It is proved that the finite difference scheme is of second-order convergence and unconditionally stable by the discrete energy method. Numerical simulations verify the theoretical analysis.

scholarly journals A Multilevel Finite Difference Scheme for One-Dimensional Burgers Equation Derived from the Lattice Boltzmann Method

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiaojie Li ◽  
Zhoushun Zheng ◽  
Shuang Wang ◽  
Jiankang Liu
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Finite Difference Scheme ◽  
Numerical Solutions ◽  
Burgers Equation ◽  
One Dimensional ◽  
Explicit Finite Difference ◽  
Boltzmann Method

An explicit finite difference scheme for one-dimensional Burgers equation is derived from the lattice Boltzmann method. The system of the lattice Boltzmann equations for the distribution of the fictitious particles is rewritten as a three-level finite difference equation. The scheme is monotonic and satisfies maximum value principle; therefore, the stability is proved. Numerical solutions have been compared with the exact solutions reported in previous studies. TheL2, L∞and Root-Mean-Square (RMS) errors in the solutions show that the scheme is accurate and effective.

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