Analytical Solutions of Shock Fitting Equations for the Multishock Compaction of Die-Contained Powder Media

1992 ◽  
Vol 114 (1) ◽  
pp. 63-70
Author(s):  
Yukio Sano ◽  
Koji Tokushima ◽  
Yuji Inoue ◽  
Yoshihito Tomita
Keyword(s):  
Closed Form ◽  
Analytical Solutions ◽  
Specific Volume ◽  
Linear Equations ◽  
Closed Form Solution ◽  
Form Solution ◽  
Material Constants ◽  
The Difference ◽  
Volume Relation ◽  
Compaction Processes

In an earlier paper [4], two sets of equations which governed the processes of propagation of shock waves reflected from the punch and plug surfaces in a die-contained copper powder medium were presented. The pressure-specific volume relation included in the sets of equations was composed of three partial relations having different material constants. In the present paper the sets of equations are simplified by assuming that the pressure and specific volume at the front and back sides of the shock front are always related by the same material constants, and linear equations are obtained by introducing a further minor assumption into the simplified nonlinear equations included in the sets of equations. Two sorts of analytical solutions of the linear equations are obtained. One is a general-form solution, while the other is a closed-form solution. The general-form solution calculated is compared satisfactorily with the difference solution computed in the previous study, confirming that the assumption introduced into the simplified equations is minor. Furthermore, calculated characteristics of the general-form solution are revealed by the consideration of the simplified equations and the linear equations, giving greater insight into the compaction processes. The closed-form solution, which is obtained only for the propagation of the shock wave starting from the punch surface and returning from the plug surface, agrees well with the general-form solution.

An Analytical Study of Skull Deformation During Head Impacts

2013 ◽  
Author(s):  
Shahab Mansoor-Baghaei ◽  
Ali M. Sadegh
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Realistic Model ◽  
Form Solution ◽  
Ellipsoidal Shell ◽  
Complex Differential Equation ◽  
Head Impacts ◽  
The Difference ◽  
The Impact ◽  
Skull Deformation

Spherical shells have been employed to model impacts to human heads; however, an ellipsoidal shell is that is more realistic model of the head has not fully investigated. In this paper, impact of an elastic ellipsoidal shell with an elastic flat half space is analytically analyzed and a closed-form solution is derived which led to a complex differential equation. Due to the complexity of the impact equation it could not be solved by standard solutions. Therefore, the Newtonian method and a linearization scheme are employed to simplify this equation in order to obtain the response of the impact problem and the closed-form solution. The analytical solutions are validated by finite element method. Good agreement between the closed form solution and the FE results is observed. To show the difference, the ellipsoidal solutions are also compared to the spherical solutions. To the best of our knowledge, this method and its closed-form solution have not been addressed in the literature. It is concluded that the closed-form solution is trustworthy and can be used to investigate the impact of the skull (as an elastic ellipsoidal shell) with a rigid or elastic plate, including the skull deformation and parametric studies. This solution could be expanded to include the brain materials inside the ellipsoidal shell.

scholarly journals Closed-Form Solution of a Rational Difference Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Tarek F. Ibrahim ◽  
Abdul Qadeer Khan ◽  
Burak Oğul ◽  
Dağistan Şimşek
Keyword(s):  
Difference Equation ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Real Numbers ◽  
Positive Real ◽  
Rational Difference Equation ◽  
The Difference

In this paper, we study the solution of the difference equation Ω m + 1 = Ω m − 7 q + 6 / 1 + ∏ t = 0 5 Ω m − q + 1 t − q , where the initials are positive real numbers.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

2021 ◽  
Vol 10 (7) ◽  
pp. 435
Author(s):  
Yongbo Wang ◽  
Nanshan Zheng ◽  
Zhengfu Bian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Simulated Data ◽  
Real Data ◽  
Point Clouds ◽  
Form Solution ◽  
Spatial Transformation ◽  
Dual Quaternions ◽  
Feature Based ◽  
Planar Feature

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

2021 ◽  
pp. 1-10
Author(s):  
Puneet Pasricha ◽  
Anubha Goel
Keyword(s):  
Closed Form ◽  
Stochastic Volatility ◽  
Closed Form Solution ◽  
Form Solution ◽  
Stochastic Volatility Model ◽  
Strike Price ◽  
Volatility Model ◽  
Exchange Options ◽  
Pricing Formula ◽  
Exchange Option

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

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