Relevant bibliographies by topics / Dirac delta function

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Dirac delta function'

Author: Grafiati
Published: 4 June 2021
Last updated: 7 February 2022

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Journal articles on the topic "Dirac delta function"

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Dhanuk, B. B., K. Pudasainee, H. P. Lamichhane, and R. P. Adhikari. "Dirac Delta Function from Closure Relation of Orthonormal Basis and its Use in Expanding Analytic Functions." Journal of Nepal Physical Society 6, no. 2 (2020): 158–63. http://dx.doi.org/10.3126/jnphyssoc.v6i2.34872.

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One of revealing and widely used concepts in Physics and mathematics is the Dirac delta function. The Dirac delta function is a distribution on real lines which is zero everywhere except at a single point, where it is infinite. Dirac delta function has vital role in solving inhomogeneous differential equations. In addition, the Dirac delta functions can be used to explore harmonic information’s imbedded in the physical signals, various forms of Dirac delta function and can be constructed from the closure relation of orthonormal basis functions of functional space. Among many special functions, we have chosen the set of eigen functions of the Hamiltonian operator of harmonic oscillator and angular momentum operators for orthonormal basis. The closure relation of orthonormal functions used to construct the generator of Dirac delta function which is used to expand analytic functions log(x + 2),exp(-x2) and x within the valid region of arguments.
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Katz, Mikhail G., and David Tall. "A Cauchy-Dirac Delta Function." Foundations of Science 18, no. 1 (2012): 107–23. http://dx.doi.org/10.1007/s10699-012-9289-4.

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Feng, Zaiyong, Linghua Ye, and Yi Zhang. "On the Fractional Derivative of Dirac Delta Function and Its Application." Advances in Mathematical Physics 2020 (October 14, 2020): 1–7. http://dx.doi.org/10.1155/2020/1842945.

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The Dirac delta function and its integer-order derivative are widely used to solve integer-order differential/integral equation and integer-order system in related fields. On the other hand, the fractional-order system gets more and more attention. This paper investigates the fractional derivative of the Dirac delta function and its Laplace transform to explore the solution for fractional-order system. The paper presents the Riemann-Liouville and the Caputo fractional derivative of the Dirac delta function, and their analytic expression. The Laplace transform of the fractional derivative of the Dirac delta function is given later. The proposed fractional derivative of the Dirac delta function and its Laplace transform are effectively used to solve fractional-order integral equation and fractional-order system, the correctness of each solution is also verified.
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Meghwal, Ram Swaroop. "Fundamental Study of Dirac Delta Function." International Journal for Research in Applied Science and Engineering Technology 8, no. 12 (2020): 918–24. http://dx.doi.org/10.22214/ijraset.2020.32665.

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Simons, Stuart. "90.41 Introducing the Dirac delta function." Mathematical Gazette 90, no. 518 (2006): 292–93. http://dx.doi.org/10.1017/s0025557200179756.

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Chu, J. L., and Sanboh Lee. "Dirac delta function — an alternative expression." International Journal of Mathematical Education in Science and Technology 27, no. 5 (1996): 753–83. http://dx.doi.org/10.1080/0020739960270515.

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Balakrishnan, V. "All about the dirac delta function(?)." Resonance 8, no. 8 (2003): 48–58. http://dx.doi.org/10.1007/bf02866759.

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Khairi, Fathul, and Malahayati. "Penerapan Fungsi Green dari Persamaan Poisson pada Elektrostatika." Quadratic: Journal of Innovation and Technology in Mathematics and Mathematics Education 1, no. 1 (2021): 56–79. http://dx.doi.org/10.14421/quadratic.2021.011-08.

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The Dirac delta function is a function that mathematically does not meet the criteria as a function, this is because the function has an infinite value at a point. However, in physics the Dirac Delta function is an important construction, one of which is in constructing the Green function. This research constructs the Green function by utilizing the Dirac Delta function and Green identity. Furthermore, the construction is directed at the Green function of the Poisson's equation which is equipped with the Dirichlet boundary condition. After the form of the Green function solution from the Poisson's equation is obtained, the Green function is determined by means of the expansion of the eigen functions in the Poisson's equation. These results are used to analyze the application of the Poisson equation in electrostatic.
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Crosbie, A. L., and G. W. Davidson. "Dirac-delta function approximations to the scattering phase function." Journal of Quantitative Spectroscopy and Radiative Transfer 33, no. 4 (1985): 391–409. http://dx.doi.org/10.1016/0022-4073(85)90200-6.

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Nedeljkov, Marko, and Michael Oberguggenberger. "Ordinary differential equations with delta function terms." Publications de l'Institut Math?matique (Belgrade) 91, no. 105 (2012): 125–35. http://dx.doi.org/10.2298/pim1205125n.

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This article is devoted to nonlinear ordinary differential equations with additive or multiplicative terms consisting of Dirac delta functions or derivatives thereof. Regularizing the delta function terms produces a family of smooth solutions. Conditions on the nonlinear terms, relating to the order of the derivatives of the delta function part, are established so that the regularized solutions converge to a limiting distribution.
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Dissertations / Theses on the topic "Dirac delta function"

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Öhrn, Håkan, and Adam Lindell. "Discretization of the Dirac delta function for application in option pricing." Thesis, Uppsala universitet, Avdelningen för beräkningsvetenskap, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-297648.

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This paper compares two different approximations of the Dirac deltafunction used in a Fokker-Planck equation. Both methods deal with the singularity problem in the initial condition. The Dirac delta approximation, constructed in MATLAB with a method derived by Tornberg and Engquist, was compared to an already given method Aït-Sahalia. The methods were implemented as the initial condition in the Fokker-Planck equation, e.g approximating a probability density function. In most cases Aït-Sahalia and Tornberg-Engquist were interchangeable. During specific circumstances one method was significantly more accurate than the other. Increasing the amount of time/spatial steps enhanced the differences in error while having less time/spatial steps made the difference in error converge. The Aït-Sahalia method produces slightly more accurate results in more cases.
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Amato, Marco. "The Galerkin method for vibrational problem on stepped structures." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2020.

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Molti fenomeni fisici studiati in ingegneria sono modellati attraverso equazioni differenziali. In generale il problema ottenuto difficilmente trova soluzione analitica in forma chiusa. Ecco perché si è iniziato a cercare approcci numerici approssimati per risolvere le equazioni differenziali. Gli approcci più diffusi sono: il metodo degli elementi finiti, il metodo delle differenze finite e il metodo dei residui pesati. Quest'ultimo, in particolare, è una famiglia di metodi. Un membro di tale gruppo è l'argomento principale di questa tesi, ovvero il metodo Galerkin. Il metodo di Galerkin è stato proposto nel 1915. Molti articoli sono dedicati alla sua applicazione su strutture elastiche e in letteratura è anche possibile trovare diversi studi che riportano l'applicazione del metodo di Galerkin su strutture a gradini. In questi lavori la procedura di Galerkin viene applicata nella forma naïve. In questa tesi ci occupiamo del problema vibrazionale di strutture a gradini fornendo due versioni del metodo Galerkin. La prima, naïve, consiste nell'integrazione lungo ogni gradino, dove la rigidezza e la massa rimangono costanti. La seconda versione, denominata rigorous, consiste nel rappresentare la rigidezza e la massa come funzioni generalizzate. Questa implementazione utilizza la funzione di Heaviside, nonché il delta di Dirac e la sua derivata. Per verificare la solidità del metodo studiamo diversi elementi strutturali: bielle, travi e piastre in diverse condizioni di vincolo. Emerge che l’implementazione rigorous porta a termini aggiuntivi che non compaiono nell’implementazione naïve. Entrambe le versioni del metodo di Galerkin sono confrontate con la soluzione esatta. Il risultato ottenuto è che, contrariamente alla versione naïve, l'implementazione rigorous tende alla soluzione analitica. Questo lavoro dimostra che dobbiamo prestare attenzione quando implementiamo il metodo Galerkin per le strutture a gradini perché solo l’implementazione rigorous converge.
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Zsitva, Norbert. "Aproximace LTI SISO systémů s dopravním zpožděním pomocí zobecněných Laguerrových funkcí." Master's thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2018. http://www.nusl.cz/ntk/nusl-376971.

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This final thesis deals with the approximation of time delay in time invariant systems. First, the generalized Laguerre functions and their characteristics are presented. After this, the approximation of the Dirac delta function with the help of these functions is shown. Also, the choice of the free parameters is discussed and the results are evaluated with the help of energy. In the final part of the thesis, the approximations of systems with generalized and simple Laguerre functions are compared.
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Ghaderi, Hazhar. "Triangle Loop in Scalar Decay and Cutting Rules." Thesis, Uppsala universitet, Institutionen för fysik och astronomi, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-212511.

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In this report we will calculate the amplitude for a scalar-to-scalars (φ3 <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Crightarrow" /> φ2φ2) decay which involves a triangle loop. We compute the real and imaginary part of the amplitude separately and will argue that this is much more straightforward and practical in this case rather than having to deal with or worry about branch cuts of logarithms. We will derive simple cutting rules closely related to the imaginary part of the amplitude. In doing this, we derive a formula that deals with expressions of the form δ[f(x,y)]δ[g(x,y)], containing two Dirac delta functions.
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Books on the topic "Dirac delta function"

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S, Lokanathan, ed. Quantum mechanics: Theory and applications. Kluwer Academic Pulbishers, 2004.

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Ghatak, A. K. Quantum mechanics: Theory and applications. Kluwer Academic Publishers, 2004.

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Horing, Norman J. Morgenstern. Graphene. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0012.

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Chapter 12 introduces Graphene, which is a two-dimensional “Dirac-like” material in the sense that its energy spectrum resembles that of a relativistic electron/positron (hole) described by the Dirac equation (having zero mass in this case). Its device-friendly properties of high electron mobility and excellent sensitivity as a sensor have attracted a huge world-wide research effort since its discovery about ten years ago. Here, the associated retarded Graphene Green’s function is treated and the dynamic, non-local dielectric function is discussed in the degenerate limit. The effects of a quantizing magnetic field on the Green’s function of a Graphene sheet and on its energy spectrum are derived in detail: Also the magnetic-field Green’s function and energy spectrum of a Graphene sheet with a quantum dot (modelled by a 2D Dirac delta-function potential) are thoroughly examined. Furthermore, Chapter 12 similarly addresses the problem of a Graphene anti-dot lattice in a magnetic field, discussing the Green’s function for propagation along the lattice axis, with a formulation of the associated eigen-energy dispersion relation. Finally, magnetic Landau quantization effects on the statistical thermodynamics of Graphene, including its Free Energy and magnetic moment, are also treated in Chapter 12 and are seen to exhibit magnetic oscillatory features.
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Mann, Peter. Noether’s Theorem for Fields. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0028.

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This is a unique chapter that discusses classical path integrals in both configuration space and phase space. It examines both Lagrangian and Hamiltonian formulations before qualitatively discussing some interesting features of gauge fixing. This formulation is then linked to superspace and Grassmann variables for a fermionic field theory. The chapter then shows that the corresponding operatorial formulation is none other than the Koopman–von Neumann theory. In parallel to quantum theory, the classical propagator or the transition amplitude between two classical states is given exactly by the phase space partition function. The functional Dirac delta is discussed, and the chapter closes by briefly mentioning Faddeev–Popov ghosts, which were introduced earlier in the chapter.
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Bueno, Otávio, and Steven French. Applying Problematic Mathematics, Interpreting Successful Structures. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198815044.003.0007.

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In this chapter we consider whether, in applying mathematics in the ways described in previous chapters, there are grounds for taking the relevant mathematical entities to be indispensable. Our example here is Dirac’s introduction of the delta function into quantum mechanics and we point out, firstly, that Dirac himself was very clear about the function’s dispensability and, secondly, even if certain mathematical theories were indispensable, this wouldn’t justify a commitment to the existence of the associated entities. To illustrate this second point we use a further example, that of Dirac’s prediction of the existence of antimatter via the exploitation of surplus mathematical structure. We maintain here that commitment to the existence of the objects under consideration requires the satisfaction of certain criteria of existence and it is unclear whether mathematical entities meet these criteria.
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Horing, Norman J. Morgenstern. Retarded Green’s Functions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0005.

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Chapter 5 introduces single-particle retarded Green’s functions, which provide the probability amplitude that a particle created at (x, t) is later annihilated at (x′,t′). Partial Green’s functions, which represent the time development of one (or a few) state(s) that may be understood as localized but are in interaction with a continuum of states, are discussed and applied to chemisorption. Introductions are also made to the Dyson integral equation, T-matrix and the Dirac delta-function potential, with the latter applied to random impurity scattering. The retarded Green’s function in the presence of random impurity scattering is exhibited in the Born and self-consistent Born approximations, with application to Ando’s semi-elliptic density of states for the 2D Landau-quantized electron-impurity system. Important retarded Green’s functions and their methods of derivation are discussed. These include Green’s functions for electrons in magnetic fields in both three dimensions and two dimensions, also a Hamilton equation-of-motion method for the determination of Green’s functions with application to a 2D saddle potential in a time-dependent electric field. Moreover, separable Hamiltonians and their product Green’s functions are discussed with application to a one-dimensional superlattice in axial electric and magnetic fields. Green’s function matching/joining techniques are introduced and applied to spatially varying mass (heterostructures) and non-local electrostatics (surface plasmons).
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Ghatak, Ajoy, and S. Lokanathan. Quantum Mechanics: Theory and Applications (Fundamental Theories of Physics, 137). Springer, 2004.

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Lokanathan, S., and Ajoy Kumar Ghatak. Quantum Mechanics: Theory and Applications. Kluwer Academic Publishers, 2004.

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Ghatak, Ajoy, and S. Lokanathan. Quantum Mechanics: Theory and Applications (Fundamental Theories of Physics). Springer, 2004.

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Book chapters on the topic "Dirac delta function"

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Weik, Martin H. "Dirac-delta function." In Computer Science and Communications Dictionary. Springer US, 2000. http://dx.doi.org/10.1007/1-4020-0613-6_5112.

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Hassani, Sadri. "Dirac Delta Function." In Mathematical Methods. Springer New York, 2000. http://dx.doi.org/10.1007/978-0-387-21562-4_7.

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Ghatak, Ajoy, and S. Lokanathan. "The Dirac Delta Function." In Quantum Mechanics: Theory and Applications. Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-2130-5_1.

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Kanwal, Ram P. "The Dirac Delta Function and Delta Sequences." In Generalized Functions. Birkhäuser Boston, 2004. http://dx.doi.org/10.1007/978-0-8176-8174-6_1.

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Kanwal, Ram P. "The Dirac Delta Function and Delta Sequences." In Generalized Functions Theory and Technique. Birkhäuser Boston, 1998. http://dx.doi.org/10.1007/978-1-4684-0035-9_1.

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Buttkus, Burkhard. "The Dirac Delta Function and its Fourier Transform." In Spectral Analysis and Filter Theory in Applied Geophysics. Springer Berlin Heidelberg, 2000. http://dx.doi.org/10.1007/978-3-642-57016-2_4.

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Jovanović, B. S., J. D. Kandilarov, and L. G. Vulkov. "Construction and Convergence of Difference Schemes for a Modell Elliptic Equation with Dirac-delta Function Coefficient." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2001. http://dx.doi.org/10.1007/3-540-45262-1_50.

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Estrada, Ricardo, and Ram P. Kanwal. "Series of Dirac Delta Functions." In Asymptotic Analysis. Birkhäuser Boston, 1994. http://dx.doi.org/10.1007/978-1-4684-0029-8_6.

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Estrada, Ricardo, and Ram P. Kanwal. "Series of Dirac Delta Functions." In A Distributional Approach to Asymptotics. Birkhäuser Boston, 2002. http://dx.doi.org/10.1007/978-0-8176-8130-2_7.

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Burkhardt, Hans, Marco Reisert, and Hongdong Li. "Invariants for Discrete Structures – An Extension of Haar Integrals over Transformation Groups to Dirac Delta Functions." In Lecture Notes in Computer Science. Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-540-28649-3_17.

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Conference papers on the topic "Dirac delta function"

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Wilcox, Bethany R., and Steven J. Pollock. "Student Difficulties with the Dirac Delta Function." In 2014 Physics Education Research Conference. American Association of Physics Teachers, 2015. http://dx.doi.org/10.1119/perc.2014.pr.064.

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Özc̣ağ, E. "Results on compositions involving Dirac-delta function." In APPLICATION OF MATHEMATICS IN TECHNICAL AND NATURAL SCIENCES: 9th International Conference for Promoting the Application of Mathematics in Technical and Natural Sciences - AMiTaNS’17. Author(s), 2017. http://dx.doi.org/10.1063/1.5007379.

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ROSAS-ORTIZ, O. "ON THE DIRAC-INFELD-PLEBAŃSKI DELTA FUNCTION." In Proceedings of 2002 International Conference. WORLD SCIENTIFIC, 2006. http://dx.doi.org/10.1142/9789812772732_0031.

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Lee, Sang-Chul, Young-Hun Lim, and Hyo-Sung Ahn. "Iterative learning control with dirac-delta function under state alignment condition." In 2013 13th International Conference on Control, Automaton and Systems (ICCAS). IEEE, 2013. http://dx.doi.org/10.1109/iccas.2013.6703934.

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Adebola, E., O. Olabiyi, and A. Annamalai. "Some remarks on the Dirac delta function approximation for ASER analysis of digital modulations over fading channels." In MILCOM 2012 - 2012 IEEE Military Communications Conference. IEEE, 2012. http://dx.doi.org/10.1109/milcom.2012.6415874.

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Marghitu, Dan B., and Yildirim Hurmuzlu. "Longitudinal Impact of a Rectilinear Elastic Link in Kinematics Chains." In ASME 1993 Design Technical Conferences. American Society of Mechanical Engineers, 1993. http://dx.doi.org/10.1115/detc1993-0276.

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Abstract In this article we consider the longitudinal impact of a rectilinear elastic link against a solid using Hertz contact theory. The main objective is to develop an analytical model that incorporates the effect of the general motion on the vibration of elastic elements in kinematic mechanisms. Equations for the translational and rotational motions of the link are developed by applying Hamilton’s principle. Kinetic energy that is required for the application of this principle has been derived by utilizing a generalized velocity field theory for elastic solids. This approach provides means to include the inertia terms directly in the equations of motion. Effects such as centrifugal stiffening and vibrations induced by Coriolis forces are accommodated automatically, rather than with the aid of ad hoc provisions. With the Dirac delta function defined as the limit of a sequence of functions we solve the discontinuities due to the impact.
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Yoo, David, and Ikjin Lee. "Sampling-Based Approach for Design Optimization in the Presence of Interval Variables." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-13151.

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This paper proposes a methodology for sampling-based design optimization in the presence of interval variables. Assuming that an accurate surrogate model is available, the proposed method first searches the worst combination of interval variables for constraints when only interval variables are present or for probabilistic constraints when both interval and random variables are present. Due to the fact that the worst combination of interval variables for probability of failure does not always coincide with that for a performance function, the proposed method directly uses the probability of failure to obtain the worst combination of interval variables when both interval and random variables are present. To calculate sensitivities of constraints and probabilistic constraints with respect to interval variables by the sampling-based method, the behavior of interval variables at the worst case is defined by utilizing the Dirac delta function. Then, Monte Carlo simulation is applied to calculate constraints and probabilistic constraints with the worst combination of interval variables, and their sensitivities. The important merit of the proposed method is that it does not require gradients of performance functions and transformation from X-space to U-space for reliability analysis after the worst combination of interval variables is obtained, thus there is no approximation or restriction in calculating the sensitivities of constraints or probabilistic constraints. Numerical results indicate that the proposed method can search the worst case probability of failure with both efficiency and accuracy and that it can perform design optimization with mixture of random and interval variables by utilizing the worst case probability of failure search.
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Hu, S. D., H. Li, and H. S. Tzou. "Static Nano-Control of Cantilever Beams Using the Inverse Flexoelectric Effect." In ASME 2011 International Mechanical Engineering Congress and Exposition. ASMEDC, 2011. http://dx.doi.org/10.1115/imece2011-65123.

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Flexoelectricity, an electromechanical coupling effect, exhibits two opposite electromechanical properties. One is the direct flexoelectric effect that mechanical strain gradient induces an electric polarization (or electric field); the other is the inverse flexoelectric effect that polarization (or electric field) gradient induces internal stress (or strain). The later can serve as an actuation mechanism to control the static deformation of flexible structures. This study focuses on an application of the inverse flexoelectric effect to the static displacement control of a cantilever beam. The flexoelectric layer is covered with an electrode layer on the bottom surface and an AFM probe tip on the top surface in order to generate an inhomogeneous electric field when powered. The control force induced by the inverse flexoelectric effect is evaluated and its spatial distribution resembles a Dirac delta function. Therefore, a “buckling” characteristic happens at the contact point of the beam under the inverse flexoelectric control. The deflection results of the cantilever beam with respect to the AFM probe tip radius indicate that a smaller AFM probe tip achieves a more effective control effect. To evaluate the control effectiveness, the flexoelectric deflections are also compared with those resulting from the converse piezoelectric effect. It is evident that the inverse flexoelectric effect provides much better localized static deflection control of.flexible beams.
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Li, Hao, Ganglin Yu, Shanfang Huang, and Kan Wang. "A Universal Adjoint-Weighted Algorithm for Geometric Sensitivity Analysis of K-Eigenvalue Based on Continuous-Energy Monte Carlo Method." In 2018 26th International Conference on Nuclear Engineering. American Society of Mechanical Engineers, 2018. http://dx.doi.org/10.1115/icone26-82494.

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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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Deng, Bolei, Huiyu Li, and Hornsen Tzou. "Flexoelectric Actuation and Vibration Control of Ring Shells." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-47994.

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The converse flexoelectric effect that the gradient of polarization (or electric field) induces internal stress (or strain) can be utilized to control the vibration of flexible structures. This study focuses on the microscopic actuation behavior and effectiveness of a flexoelectric actuator patch on an elastic ring. An atomic force microscope (AFM) probe is placed on the upper surface of the patch to implement the inhomogeneous electric field inducing stresses inside the actuation patch. The flexoelectric membrane force and bending moment, in turn, actuate the ring vibration and its actuation effect is studied. Actuator’s influence in the transverse and circumferential directions is respectively evaluated. For the transverse direction, the gradient of the electric field decays quickly along the ring thickness, resulting in a nonuniform transverse distribution of the induced stress and such distribution is not influenced by the patch thickness. The flexoelectric induced circumferential membrane force and bending moment resembles the Dirac delta function at the AFM contact point. The influence of the actuator can be regarded as a drastic bending on the ring. To evaluate the actuation effect, dynamic response of controllable displacements of the elastic ring under flexoelectric actuation is analyzed by adjusting the geometric parameters, such as the thickness of flexoelectric patch, AFM probe radius, ring thickness and ring radius. This study represents a thorough understanding of the flexoelectric actuation behavior and serves as a foundation of the flexoelectricity based vibration control.
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