Dimensional Analysis in Mathematical Modeling Systems: A Simple Numerical Method

1991 ◽  
Author(s):  
Hemant K. Bhargava
Keyword(s):  
Mathematical Modeling ◽  
Numerical Method ◽  
Dimensional Analysis ◽  
Modeling Systems

Analysis of a Soft Catch for Conventional Warheads

2005 ◽  
Author(s):  
Matthew Pope ◽  
Bradley Martin ◽  
David Lambert ◽  
Stanley E. Jones ◽  
Jonathan Muse
Keyword(s):  
Mathematical Modeling ◽  
Dimensional Analysis ◽  
Design Process ◽  
Average Velocity ◽  
Current Paper ◽  
Experimental Measurements ◽  
One Dimensional ◽  
Zero Velocity ◽  
Metallurgical Analysis ◽  
Significant Damage

A “soft catch” is a device with which an explosively formed projectile can be decelerated to zero velocity without sustaining significant damage. The recovered projectile provides data, via metallurgical analysis, on the deformation conditions found within the explosively formed projectile. At Eglin AFB, FL, the soft catch consists of a sequence of sections (Figures 1–3), each roughly one meter long, filled with various soft media. Velocity screens are placed at the entrance and exit of each section. This enables investigators to experimentally determine the time at which the projectile passes each station in the catch. Based on these experimental measurements, average velocity estimates for each section of the soft catch can be made. The purpose of this paper is to support the soft catch design process with a one-dimensional analysis. The mathematical modeling is based on observations presented in studies by Allen, Mayfield, and Morrison [1,2]. Their work addresses the penetration of sand, but their modeling is appropriate for materials in the soft catch. The current paper describes application of their model to interpreting three soft catch experiments where Tantalum projectiles with initial velocities of approximately 1400 m/s were successfully recovered.

scholarly journals Orthonormal Bernstein polynomials for Volterra integral equations of the second kind

2020 ◽  
Vol 9 (1) ◽  
pp. 9
Author(s):  
Jafar Biazar ◽  
Hamed Ebrahimi
Keyword(s):  
Mathematical Modeling ◽  
Integral Equation ◽  
Numerical Method ◽  
Integral Equations ◽  
Method Of Moments ◽  
Volterra Integral Equation ◽  
Bernstein Polynomials ◽  
Volterra Integral Equations ◽  
Linear Volterra Integral Equations ◽  
Relative Errors

The purpose of this research is to provide an effective numerical method for solving linear Volterra integral equations of the second kind. The mathematical modeling of many phenomena in various branches of sciences lead into an integral equation. The proposed approach is based on the method of moments (Galerkin- Ritz) using orthonormal Bernstein polynomials. To solve a Volterra integral equation, the ap-proximation for a solution is considered as an expansion in terms of Bernstein orthonormal polynomials. Ultimately, the usefulness and extraordinary accuracy of the proposed approach will be verified by a few examples where the results are plotted in diagrams, Also the re-sults and relative errors are presented in some Tables.  

scholarly journals Hyperreal Delta Functions as a New General Tool for Modeling Systems with Infinitely High Densities

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 244
Author(s):  
Marcoen J. T. F. Cabbolet
Keyword(s):  
Mathematical Modeling ◽  
Density Distribution ◽  
Special Functions ◽  
Real Numbers ◽  
Physical Systems ◽  
Modeling Systems ◽  
Nonzero Real Number ◽  
Isolated Points ◽  
Delta Functions ◽  
General Tool

In general, the state of a system in which a physical quantity such as mass is distributed over space can be modeled by a function that represents the density distribution. The purpose of this paper is to introduce special functions that can be applied when in the system to be modeled, where the quantity is distributed over isolated points. For that matter, the expanded real numbers are introduced as an ordered subring of the hyperreal number field that does not contain any infinitesimals, and hyperreal delta functions are defined as special functions from the real numbers to the expanded real numbers satisfying the condition that (i) the support is a singleton, and (ii) the integral over the reals is a nonzero real number. These newly defined hyperreal delta functions, and tensor products thereof, then provide a general tool, applicable for the mathematical modeling of physical systems in which infinitely high densities occur.

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