The Exact Solution of the Translational Acceleration of a Low Modulus Elastic Medium in Rigid Spherical Shells—Implications for Head Injury Models

K. B. Chandran; Y. King Liu; D. U. von Rosenberg

doi:10.1115/1.3423700

The Exact Solution of the Translational Acceleration of a Low Modulus Elastic Medium in Rigid Spherical Shells—Implications for Head Injury Models

Journal of Applied Mechanics ◽

10.1115/1.3423700 ◽

1975 ◽

Vol 42 (4) ◽

pp. 759-762 ◽

Cited By ~ 1

Author(s):

K. B. Chandran ◽

Y. King Liu ◽

D. U. von Rosenberg

Keyword(s):

Exact Solution ◽

Head Injury ◽

Elastic Medium ◽

Closed Head Injury ◽

Delta Function ◽

Step Function ◽

Spherical Shells ◽

Dirac Delta ◽

Low Modulus ◽

Translational Acceleration

The exact solution in the form of a finite series has been obtained for the problem of low modulus elastic medium contained in rigid spherical shells subjected to translational acceleration about its diametrical axis. Laplace transformation technique and the shifting theorem were used to obtain the Green’s functions for the potentials when the external acceleration is a Dirac delta function. The solutions are formally extended to external accelerations which are general functions of time by the convolution integral. The shear stress distribution for a unit step function acceleration is illustrated. The results obtained are used to judge the adequacy of this and other similar models for the study of closed head injury mechanism.

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A Universal Adjoint-Weighted Algorithm for Geometric Sensitivity Analysis of K-Eigenvalue Based on Continuous-Energy Monte Carlo Method

Volume 9: Student Paper Competition ◽

10.1115/icone26-82494 ◽

2018 ◽

Author(s):

Hao Li ◽

Ganglin Yu ◽

Shanfang Huang ◽

Kan Wang

Keyword(s):

Monte Carlo ◽

Cross Section ◽

Geometric Parameter ◽

Neutron Transport ◽

Delta Function ◽

Step Function ◽

Universal Method ◽

Analytic Geometry ◽

Heaviside Step Function ◽

Dirac Delta

There exists a typical problem in Monte Carlo neutron transport: the effective multiplication factor sensitivity to geometric parameter. In several methods attempting to solve it, Monte Carlo adjoint-weighted theory has been proven to be quite effective. The major obstacle of adjoint-weighted theory is calculating derivative of cross section with respect to geometric parameter. In order to fix this problem, Heaviside step function and Dirac delta function are introduced to describe cross section and its derivative. This technique is crucial, and it establishes the foundation of further research. Based on above work, adjoint-weighted method is developed to solve geometric sensitivity. However, this method is limited to surfaces which are uniformly expanded or contracted with respect to its origin, such as vertical movement of plane or expansion of sphere. Rotation and translation are not allowed, while these two transformation types are more common and more important in engineering projects. In this paper, a more universal method, Cell Constraint Condition Perturbation (CCCP) method, is developed and validated. Different from traditional method, CCCP method for the first time explicitly articulates that the perturbed quantity is the parameter of spatial analytic geometry equations that used to describe surface. Thus, the CCCP can treat arbitrary one-parameter geometric perturbation of arbitrary surface as long as this surface can be described by spatial analytic geometry equation. Furthermore, CCCP can treat the perturbation of the whole cell, such as translation, rotation, expansion and constriction. Several examples are calculated to confirm the validity of CCCP method.

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Unified approach to the impulse response and green function in the circuit and field theory part II : multi–dimensional case

Journal of Electrical Engineering ◽

10.2478/v10187-012-0051-5 ◽

2012 ◽

Vol 63 (6) ◽

pp. 341-348

Author(s):

L’ubomír Šumichrast

Keyword(s):

Field Theory ◽

Impulse Response ◽

Transmission Lines ◽

Delta Function ◽

Step Function ◽

Unified Approach ◽

Dirac Delta ◽

Time Space ◽

Previous Part ◽

Heaviside Unit Step Function

In the circuit theory the concept of the impulse response of a linear system due to its excitation by the Dirac delta function δ(t) together with the convolution principle is widely used and accepted. The rigorous theory of symbolic functions, sometimes called distributions, where also the delta function belongs, is rather abstract and requires subtle mathematical tools [1-4]. Nevertheless, the most people intuitively well understand the delta function as a derivative of the (Heaviside) unit step function 1(t) without too much mathematical rigor. In the previous part [5] the concept of the impulse response of linear systems was approached in a unified manner and generalized to the time-space phenomena in one dimension (transmission lines). Here the phenomena in more dimensions (static and dynamic electromagnetic fields) are treated. It is shown that many formulas in the field theory, which are often postulated in an inductive way as results of the experiments, and therefore appear as “deux ex machina” effects, can be mathematically deduced from a few starting equations.

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Application of Majorization-Minimization Method to Chan --- Vese Algorithm in the Image Segmentation Problem

Herald of the Bauman Moscow State Technical University Series Instrument Engineering ◽

10.18698/0236-3933-2019-6-19-29 ◽

2019 ◽

pp. 19-29

Author(s):

I.S. Druzhitskiy ◽

D.E. Bekasov

Keyword(s):

Image Segmentation ◽

Main Idea ◽

Operation Time ◽

Delta Function ◽

Optimization Method ◽

Step Function ◽

Minimization Method ◽

Heaviside Step Function ◽

Dirac Delta ◽

Wide Range

The purpose of the study was to modify Chan --- Vese algorithm in order to overcome its shortcomings, such as high computational complexity and the use of approximations. In the considered modification, optimization is carried out by the majorization-minimization method, the main idea of which is to reduce the complexity of the problem using the majority function. Due to the proposed optimization method, it is possible to use the Heaviside step function and Dirac delta function. This enabled the same or better saturation levels when optimization is done by the graph cut method in a smaller number of iterations, which reduced the operation time. The proposed algorithm was tested on a Caltech101 dataset. The algorithm is general, does not depend on the subject area and does not require prior training. This allows it to be used as the basis for a wide range of image segmentation algorithms.

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Unified approach to the impulse response and green function in the circuit and field theory, part I: one–dimensional case

Journal of Electrical Engineering ◽

10.2478/v10187-012-0040-8 ◽

2012 ◽

Vol 63 (5) ◽

pp. 273-280 ◽

Cited By ~ 1

Author(s):

L’Ubomír Šumichrast

Keyword(s):

Impulse Response ◽

Transmission Lines ◽

Delta Function ◽

Step Function ◽

Subsequent Paper ◽

Unified Approach ◽

One Dimensional ◽

Dirac Delta ◽

Time Space ◽

Heaviside Unit Step Function

In the circuit theory the concept of the impulse response of a linear system due to its excitation by the Dirac delta function ƍ(t) together with the convolution principle is widely used and accepted. The rigorous theory of symbolic functions, sometimes called distributions, where also the delta function belongs, is rather abstract and requires subtle mathematical tools [1], [2], [3], [4]. Nevertheless, the most people intuitively well understand the delta function as a derivative of the (Heaviside) unit step function 1(t) without too much mathematical rigor. The concept of the impulse response of linear systems is here approached in a unified manner and generalized to the time-space phenomena in one dimension (transmission lines), as well as in a subsequent paper [5] to the phenomena in more dimensions (static and dynamic electromagnetic fields).

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Approximation Techniques for Stationary and Dynamic Problems

Introduction to Quantum Nanotechnology ◽

10.1093/oso/9780192895073.003.0011 ◽

2021 ◽

pp. 181-208

Author(s):

Duncan G. Steel

Keyword(s):

Magnetic Field ◽

Exact Solution ◽

Perturbation Theory ◽

Golden Rule ◽

Delta Function ◽

Time Dependent ◽

Dynamic Problems ◽

Approximation Techniques ◽

Dirac Delta ◽

Physical Features

For many aspects of device design, an exact solution to Schrödinger’s equation is not needed. However, it may simultaneously be required that all of the physical features are clearly understood. The most important technique for approaching these problems is perturbation theory, since it is difficult to develop physical intuition by just numerical means. For the case of solutions to the time independent Schrödinger equation, such as where an electric or magnetic field is applied, time independent perturbation theory is very useful, and is typically adequate for many problems. In some cases, problems may need an exact solution, but it may not be necessary to consider all the levels, leading to the approximation of using just a few levels. If the Hamiltonian is time dependent, we use time dependent perturbation theory which leads to Fermi’s golden rule. The result leads to a Dirac delta-function which can be eliminated by using the density of states.

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An Analytic Exact form of Heaviside Function

10.20944/preprints201907.0218.v1 ◽

2019 ◽

Author(s):

John Venetis

Keyword(s):

Special Functions ◽

Error Function ◽

Operational Calculus ◽

Delta Function ◽

Step Function ◽

Heaviside Function ◽

Exact Form ◽

Dirac Delta ◽

Computational Procedures ◽

Engineering Practices

In this paper, the author obtains an analytic exact form of Heaviside function, which is also known as Unit Step function and constitutes a fundamental concept of the Operational Calculus.In particulat, this function is explicitly expressed in a very simple manner by the aid of purely algebraic representations. The novelty of this work is that the proposed explicit formula is not performed in terms of non – elementary special functions, e.g. Dirac delta function or Error function and also is neither the limit of a function, nor the limit of a sequence of functions with point wise or uniform convergence. Hence, it may be much more appropriate and useful in the computational procedures which are inserted into Operational Calculus techniques and other engineering practices.

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