Triangular Cellular Automata for Computing Two-Dimensional Elastodynamic Response on Arbitrary Domains

2010 ◽  
Vol 78 (2) ◽  
Author(s):  
Ryan K. Hopman ◽  
Michael J. Leamy
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Cellular Automata ◽  
Differential Equations ◽  
Numerical Experiments ◽  
Two Dimensional ◽  
Wave Fronts ◽  
Rule Set ◽  
Spurious Oscillations ◽  
Elastodynamic Response

This study extends a recently developed cellular automata (CA) modeling approach (Leamy, 2008, “Application of Cellular Automata Modeling to Seismic Elastodynamics,” Int. J. Solids Struct., 45(17), pp. 4835–4849) to arbitrary two-dimensional geometries via the development of a rule set governing triangular automata (cells). As in the previous rectangular CA method, each cell represents a state machine, which updates in a stepped manner using a local “bottom-up” rule set and state input from neighboring cells. Notably, the approach avoids the need to develop and solve partial differential equations and the complexity therein. The elastodynamic responses of several general geometries and loading cases (interior, Neumann, and Dirichlet) are computed with the method and then compared with results generated using the earlier rectangular CA and finite element approaches. Favorable results are reported in all cases with numerical experiments indicating that the extended CA method avoids, importantly, spurious oscillations at the front of sharp wave fronts.

A Difference Method for Plane Problems in Magnetoelastodynamics

1972 ◽  
Vol 39 (3) ◽  
pp. 689-695 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Difference Method ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Symmetric Hyperbolic System ◽  
Von Neumann ◽  
First Order ◽  
Independent Variables

The two-dimensional equations of magnetoelastodynamics are considered as a symmetric hyperbolic system of linear first-order partial-differential equations in three independent variables. The characteristic properties of the system are determined and a numerical method for obtaining the solution to mixed initial and boundary-value problems in plane magnetoelastodynamics is presented. Results on the von Neumann necessary condition are presented. Application of the method to a problem which has a known solution provides further numerical evidence of the convergence and stability of the method.

FINITE ELEMENT METHODS FOR NONLINEAR ACOUSTICS IN FLUIDS

2007 ◽  
Vol 15 (03) ◽  
pp. 353-375 ◽  
Author(s):  
TIMOTHY WALSH ◽  
MONICA TORRES
Keyword(s):  
Finite Element ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Acoustics ◽  
Time Step ◽  
Partial Differential ◽  
Time Step Size ◽  
Finite Element Discretizations ◽  
Acoustic Fluid ◽  
Element Discretizations

In this paper, weak formulations and finite element discretizations of the governing partial differential equations of three-dimensional nonlinear acoustics in absorbing fluids are presented. The fluid equations are considered in an Eulerian framework, rather than a displacement framework, since in the latter case the corresponding finite element formulations suffer from spurious modes and numerical instabilities. When taken with the governing partial differential equations of a solid body and the continuity conditions, a coupled formulation is derived. The change in solid/fluid interface conditions when going from a linear acoustic fluid to a nonlinear acoustic fluid is demonstrated. Finite element discretizations of the coupled problem are then derived, and verification examples are presented that demonstrate the correctness of the implementations. We demonstrate that the time step size necessary to resolve the wave decreases as steepening occurs. Finally, simulation results are presented on a resonating acoustic cavity, and a coupled elastic/acoustic system consisting of a fluid-filled spherical tank.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

A Simple Three-States Cellular Automaton for Modelling Excitable Media

1998 ◽  
Vol 12 (05) ◽  
pp. 601-607 ◽  
Author(s):  
M. Andrecut
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Cellular Automata ◽  
Differential Equations ◽  
Cellular Automaton ◽  
Cellular Automaton Model ◽  
Excitable Media ◽  
Self Organization ◽  
Partial Differential ◽  
Bz Reaction

Wave propagation in excitable media provides an important example of spatiotemporal self-organization. The Belousov–Zhabotinsky (BZ) reaction and the impulse propagation along nerve axons are two well-known examples of this phenomenon. Excitable media have been modelled by continuous partial differential equations and by discrete cellular automata. Here we describe a simple three-states cellular automaton model based on the properties of excitation and recovery that are essential to excitable media. Our model is able to reproduce the dynamics of patterns observed in excitable media.

scholarly journals Design of a Hybrid Programmable 2-D Cellular Automata Based Pseudo Random Number Generator

2020 ◽  
Vol 8 (6) ◽  
pp. 5741-5748
Keyword(s):  
Cellular Automata ◽  
Random Number ◽  
Hamming Distance ◽  
Random Number Generator ◽  
Number Generator ◽  
Two Dimensional ◽  
Pseudo Random Number ◽  
Rule Set ◽  
Initial Seed ◽  
Pseudo Random Number Generator

This paper proposes a hybrid programmable two-dimensional Cellular Automata (CA) based pseudo-random number generator which includes a newly designed rule set. The properties and evolution of one and two dimensional CA are revisited. The various metrics for evaluating CA as a Pseudo-Random Number Generator (PRNG) are discussed. It is proved that the randomness is high irrespective of the initial seed by applying this newly designed rule set. The PRNG is tested against a popular statistical test called Diehard test suite and the results show that the PRNG is highly random. The chaotic measures like entropy, hamming distance and cycle length have been measured

Sign in / Sign up

Export Citation Format

Share Document