Analysis of Elastic-Plastic Impact Involving Severe Distortions

A Lagrangian analysis technique is presented for two-dimensional axi-symmetric impact problems involving elastic-plastic flow. This technique is based on a triangular finite-element formulation rather than the quadrilateral formulation generally used in comparable finite-difference methods. For impact problems involving severe distortions, the triangular element formulation is better suited to represent the severe distortions than is the traditional quadrilateral finite-difference method. Included are the formulation of the technique and illustrative examples.

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Non-standard finite-difference methods for vibro-impact problems

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2004.1425 ◽

2005 ◽

Vol 461 (2058) ◽

pp. 1927-1950 ◽

Cited By ~ 19

Author(s):

Yves Dumont ◽

Jean M.-S Lubuma

Keyword(s):

Finite Difference ◽

Duffing Oscillator ◽

Finite Difference Methods ◽

Finite Difference Schemes ◽

Restitution Coefficient ◽

Conservation Of Energy ◽

Numerical Examples ◽

Impact Oscillators ◽

Difference Methods ◽

Impact Problems

Impact oscillators are non-smooth systems with such complex behaviours that their numerical treatment by traditional methods is not always successful. We design non-standard finite-difference schemes in which the intrinsic qualitative parameters of the system—the restitution coefficient, the oscillation frequency and the structure of the nonlinear terms—are suitably incorporated. The schemes obtained are unconditionally stable and replicate a number of important physical properties of the involved oscillator system such as the conservation of energy between two consecutive impact times. Numerical examples, including the Duffing oscillator that develops a chaotic behaviour for some positions of the obstacle, are presented. It is observed that the cpu times of computation are of the same order for both the standard and the non-standard schemes.

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Error estimates of finite difference methods for the Dirac equation in the massless and nonrelativistic regime

Numerical Algorithms ◽

10.1007/s11075-021-01159-w ◽

2021 ◽

Author(s):

Ying Ma ◽

Jia Yin

Keyword(s):

Dirac Equation ◽

Finite Difference ◽

Error Estimates ◽

Finite Difference Methods ◽

Difference Methods

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Biharmonic navigation using radial basis functions

Robotica ◽

10.1017/s0263574721000709 ◽

2021 ◽

pp. 1-12

Author(s):

Xu-Qian Fan ◽

Wenyong Gong

Keyword(s):

Finite Difference ◽

Boundary Conditions ◽

Radial Basis Functions ◽

Real World ◽

Biharmonic Equation ◽

Finite Difference Methods ◽

Basis Functions ◽

Large Size ◽

Radial Basis ◽

Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

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Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

Mathematics ◽

10.3390/math9030206 ◽

2021 ◽

Vol 9 (3) ◽

pp. 206

Author(s):

María Consuelo Casabán ◽

Rafael Company ◽

Lucas Jódar

Keyword(s):

Stochastic Process ◽

Finite Difference ◽

Coming Out ◽

Finite Difference Methods ◽

Diffusion Reaction ◽

Finite Difference Schemes ◽

Finite Degree ◽

Mean Square Stability ◽

Reaction Models ◽

Difference Methods

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

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