Journal articles: 'Approximate boundary conditions'

1

Karlsson, Anders. "Approximate Boundary Conditions for Thin Structures." IEEE Transactions on Antennas and Propagation 57, no. 1 (January 2009): 144–48. http://dx.doi.org/10.1109/tap.2008.2009720.

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2

Roberts, A. J. "Boundary conditions for approximate differential equations." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 34, no. 1 (July 1992): 54–80. http://dx.doi.org/10.1017/s0334270000007384.

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AbstractA large number of mathematical models are expressed as differential equations. Such models are often derived through a slowly-varying approximation under the assumption that the domain of interest is arbitrarily large; however, typical solutions and the physical problem of interest possess finite domains. The issue is: what are the correct boundary conditions to be used at the edge of the domain for such model equations? Centre manifold theory [24] and its generalisations may be used to derive these sorts of approximations, and higher-order refinements, in an appealing and systematic fashion. Furthermore, the centre manifold approach permits the derivation of appropriate initial conditions and forcing for the models [25, 7]. Here I show how to derive asymptotically-correct boundary conditions for models which are based on the slowly-varying approximation. The dominant terms in the boundary conditions typically agree with those obtained through physical arguments. However, refined models of higher order require subtle corrections to the previously-deduced boundary conditions, and also require the provision of additional boundary conditions to form a complete model.

3

Wang, Lian Wen. "Approximate Controllability of Boundary Control Systems with Nonlinear Boundary Conditions." Applied Mechanics and Materials 538 (April 2014): 408–12. http://dx.doi.org/10.4028/www.scientific.net/amm.538.408.

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The paper deals with the approximate controllability of a class of nonlinear delayed control systems in which the control is acted through the boundary of the region and the boundary conditions are nonlinear. The approximate controllability result is established provided the approximate controllability of the corresponding linear systems.

4

Codina, Ramon, and Joan Baiges. "Approximate imposition of boundary conditions in immersed boundary methods." International Journal for Numerical Methods in Engineering 80, no. 11 (June 19, 2009): 1379–405. http://dx.doi.org/10.1002/nme.2662.

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5

Senior, T. B. A. "Approximate boundary conditions for homogeneous dielectric bodies." Journal of Electromagnetic Waves and Applications 9, no. 10 (January 1, 1995): 1227–39. http://dx.doi.org/10.1163/156939395x00019.

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6

Berdnyk, Serhii, Andrey Gomozov, Dmitriy Gretskih, Viktor Kartich, and Mikhail Nesterenko. "Approximate boundary conditions for electromagnetic fields in electrodmagnetics." RADIOELECTRONIC AND COMPUTER SYSTEMS, no. 3 (October 4, 2022): 141–60. http://dx.doi.org/10.32620/reks.2022.3.11.

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The results of an analytical review of literature sources on the use of approximate boundary conditions for electromagnetic fields of impedance type in solving boundary value problems of electromagnetism for more than 80 recent years are presented. During this period, the impedance approach was generalized to various electrodynamic problems, in which its use made it possible to significantly expand the limits of mathematical modeling, which adequately considers the physical properties of real boundary surfaces. More than eighty years have passed since the publication of approximate boundary conditions for electromagnetic fields. The meaning and value of these conditions lies in the fact that they allow solving diffraction problems about fields outside well-conducting bodies without considering the fields inside them, which greatly simplifies the solution. Since then, numerous publications have been devoted to the application of impedance boundary conditions, the main of which (according to the authors) are presented in this paper. Particular attention is paid to the characteristics of electrically thin impedance vibrators and film-type surface structures as a personal contribution of the authors to the theory of impedance boundary conditions in electromagnetism. The subject of research in this article is the analysis of the limits and conditions for the correct application of impedance boundary conditions. The goal is to systematize the results of using the concept of approximate impedance boundary conditions for electromagnetic fields in problems of electrodynamics based on an analytical review of literature sources. The following results were obtained. The types of metal-dielectric structures are presented, for which methods of theoretical determination of the values of surface impedances for film-type structures are currently known, which are the most promising for creating technological control elements on their basis in centimeter and millimeter wavelength devices. Conclusions. The materials of this paper do not pretend to be a complete reference book covering all the results and aspects of the development of the concept of approximate impedance type boundary conditions in problems of electromagnetism over the past decades. Simultaneously, the authors hope that the information presented in this paper will be useful to specialists in the field of theoretical and applied electrodynamics, as well as graduate students, young scientists and students who are just mastering radiophysics and radio engineering specialties.

7

Puska, P. P., S. A. Tretyakov, and A. H. Sihvola. "Approximate impedance boundary conditions for isotropic multilayered media." IEE Proceedings - Microwaves, Antennas and Propagation 146, no. 2 (1999): 163. http://dx.doi.org/10.1049/ip-map:19990561.

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8

Borggaard, J., and T. Iliescu. "Approximate deconvolution boundary conditions for large eddy simulation." Applied Mathematics Letters 19, no. 8 (August 2006): 735–40. http://dx.doi.org/10.1016/j.aml.2005.08.022.

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9

Lill, Georg. "Exact and approximate boundary conditions at artificial boundaries." Mathematical Methods in the Applied Sciences 16, no. 10 (October 1993): 691–705. http://dx.doi.org/10.1002/mma.1670161003.

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10

Huddleston, P. L. "Scattering by finite, open cylinders using approximate boundary conditions." IEEE Transactions on Antennas and Propagation 37, no. 2 (1989): 253–57. http://dx.doi.org/10.1109/8.18715.

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11

Johansson, M., P. D. Folkow, A. M. Hägglund, and P. Olsson. "Approximate boundary conditions for a fluid-loaded elastic plate." Journal of the Acoustical Society of America 118, no. 6 (December 2005): 3436–46. http://dx.doi.org/10.1121/1.2126927.

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12

Hagstrom, Thomas, and John Goodrich. "Experiments with approximate radiation boundary conditions for computational aeroacoustics." Applied Numerical Mathematics 27, no. 4 (August 1998): 385–402. http://dx.doi.org/10.1016/s0168-9274(98)00021-x.

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13

Johansson, Gunnar, and A. Jonas Niklasson. "Approximate dynamic boundary conditions for a thin piezoelectric layer." International Journal of Solids and Structures 40, no. 13-14 (June 2003): 3477–92. http://dx.doi.org/10.1016/s0020-7683(03)00151-3.

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14

Garazo, A. N., V. A. Kuz, and M. A. Vila. "Exact and approximate boundary conditions of a fluid interface." Journal of Colloid and Interface Science 119, no. 1 (September 1987): 49–54. http://dx.doi.org/10.1016/0021-9797(87)90243-8.

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15

Balaras, Elias, Carlo Benocci, and Ugo Piomelli. "Two-layer approximate boundary conditions for large-eddy simulations." AIAA Journal 34, no. 6 (June 1996): 1111–19. http://dx.doi.org/10.2514/3.13200.

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16

Rawlins, A. D. "Approximate boundary conditions for diffraction by thin transmissive media." ZAMM 87, no. 10 (October 29, 2007): 711–14. http://dx.doi.org/10.1002/zamm.200510346.

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17

Tretyakov, Sergei A. "Thin pseudochiral layers: Approximate boundary conditions and potential applications." Microwave and Optical Technology Letters 6, no. 2 (February 1993): 112–15. http://dx.doi.org/10.1002/mop.4650060209.

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18

Maksimov, V. P., and A. L. Chadov. "The constructive investigation of boundary-value problems with approximate satisfaction of boundary conditions." Russian Mathematics 54, no. 10 (August 26, 2010): 71–74. http://dx.doi.org/10.3103/s1066369x10100105.

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19

Lemke, Mathias, and Julius Reiss. "Approximate acoustic boundary conditions in the time-domain using volume penalization." Journal of the Acoustical Society of America 153, no. 2 (February 2023): 1219–28. http://dx.doi.org/10.1121/10.0017347.

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This paper presents an immersed boundary method for modeling complex impedance boundary conditions in wave-based finite-difference time-domain simulations. The fully parallelizable and physically motivated Brinkman method allows for the representation of complex geometries on simple Cartesian grids as porous material by introducing a friction term and an effective volume. The parameters are specified using blending functions, enabling impedance boundary conditions without the need for grid fitting or special boundary treatment. Representative acoustic configurations are analyzed to assess the method. In detail, acoustic materials on and in front of a rigid wall, a reacting surface as well as fully reflecting walls are examined. Comparison with analytical solutions shows satisfactory agreement of the resulting impedances in the range from 20 Hz up to 4 kHz. The method is derived for the (non-)linear Euler equations and the acoustic wave equation. An extensive stability analysis is carried out.

20

Wang, Lianwen. "Approximate Boundary Controllability for Semilinear Delay Differential Equations." Journal of Applied Mathematics 2011 (2011): 1–10. http://dx.doi.org/10.1155/2011/587890.

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This paper considers the approximate controllability for a class of semilinear delay control systems described by a semigroup formulation with boundary control. Sufficient conditions for approximate controllability are established provided the approximate controllability of corresponding linear systems.

21

BABUŠKA, IVO, VICTOR NISTOR, and NICOLAE TARFULEA. "APPROXIMATE AND LOW REGULARITY DIRICHLET BOUNDARY CONDITIONS IN THE GENERALIZED FINITE ELEMENT METHOD." Mathematical Models and Methods in Applied Sciences 17, no. 12 (December 2007): 2115–42. http://dx.doi.org/10.1142/s0218202507002571.

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We propose a method for treating Dirichlet boundary conditions for the Laplacian in the framework of the Generalized Finite Element Method (GFEM). A particular interest is taken in boundary data with low regularity (possibly a distribution). Our method is based on using approximate Dirichlet boundary conditions and polynomial approximations of the boundary. The sequence of GFEM-spaces consists of nonzero boundary value functions, and hence it does not conform to one of the basic Finite Element Method (FEM) conditions. We obtain quasi-optimal rates of convergence for the sequence of GFEM approximations of the exact solution. We also extend our results to the inhomogeneous Dirichlet boundary value problem, including the case when the boundary data has low regularity (i.e. is a distribution). Finally, we indicate an effective technique for constructing sequences of GFEM-spaces satisfying our assumptions by using polynomial approximations of the boundary.

22

Maikov, A. R. "Approximate open boundary conditions for a class of hyperbolic equations." Computational Mathematics and Mathematical Physics 46, no. 6 (June 2006): 1007–22. http://dx.doi.org/10.1134/s0965542506060091.

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23

Osipov, O. V., and T. A. Panferova. "Approximate boundary conditions for thin chiral layers with curvilinear surfaces." Journal of Communications Technology and Electronics 55, no. 5 (May 2010): 532–34. http://dx.doi.org/10.1134/s1064226910050086.

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24

Pestov, Leonid, and Dmytro Strelnikov. "Approximate controllability of the wave equation with mixed boundary conditions." Journal of Mathematical Sciences 239, no. 1 (April 8, 2019): 75–85. http://dx.doi.org/10.1007/s10958-019-04289-8.

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25

South, Jerry C., Mohamed M. Hafez, and David Gottlieb. "Stability analysis of intermediate boundary conditions in approximate factorization schemes." Applied Numerical Mathematics 2, no. 3-5 (October 1986): 181–92. http://dx.doi.org/10.1016/0168-9274(86)90027-9.

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26

Goolin, Alexei V., Nikolai I. Ionkin, and Valentina A. Morozova. "Difference Schemes with Nonlocal Boundary Conditions." Computational Methods in Applied Mathematics 1, no. 1 (2001): 62–71. http://dx.doi.org/10.2478/cmam-2001-0004.

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AbstractThe paper deals with the stability, with respect to initial data, of difference schemes that approximate the heat-conduction equation with constant coefficients and nonlocal boundary conditions. Some difference schemes are considered for the one-dimensional heat-conduction equation, the energy norm is constructed, and the necessary and sufficient stability conditions in this norm are established for explicit and weighted difference schemes.

27

Qu, Liang-Hui, Lin Xing, Zhi-Yun Yu, Feng Ling, and Jian-Guo Xu. "An approximate method for solving a melting problem with periodic boundary conditions." Thermal Science 18, no. 5 (2014): 1679–84. http://dx.doi.org/10.2298/tsci1405679q.

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An effective thermal diffusivity method is used to solve one-dimensional melting problem with periodic boundary conditions in a semi-infinite domain. An approximate analytic solution showing the functional relation between the location of the moving boundary and time is obtained by using Laplace transform. The evolution of the moving boundary and the temperature field in the phase change domain are simulated numerically, and the numerical results are compared with previous results in open literature.

28

Li, Tatsien, and Bopeng Rao. "Approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary conditions." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 10. http://dx.doi.org/10.1051/cocv/2021006.

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In this paper, we first give an algebraic characterization of uniqueness of continuation for a coupled system of wave equations with coupled Robin boundary conditions. Then, the approximate boundary controllability and the approximate boundary synchronization by groups for a coupled system of wave equations with coupled Robin boundary controls are developed around this fundamental characterization.

29

Hagstrom, Thomas, Eliane Bécache, Dan Givoli, and Kurt Stein. "Complete Radiation Boundary Conditions for Convective Waves." Communications in Computational Physics 11, no. 2 (February 2012): 610–28. http://dx.doi.org/10.4208/cicp.231209.060111s.

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AbstractLocal approximate radiation boundary conditions of optimal efficiency for the convective wave equation and the linearized Euler equations in waveguide geometry are formulated, analyzed, and tested. The results extend and improve for the convective case the general formulation of high-order local radiation boundary condition sequences for anisotropic scalar equations developed in [4].

30

CHOBAN, MITROFAN M., and COSTICĂ N. MOROȘANU. "Well-posedness of a nonlinear second-order anisotropic reaction-diffusion problem with nonlinear and inhomogeneous dynamic boundary conditions." Carpathian Journal of Mathematics 38, no. 1 (November 15, 2021): 95–116. http://dx.doi.org/10.37193/cjm.2022.01.08.

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The paper is concerned with a qualitative analysis for a nonlinear second-order boundary value problem, endowed with nonlinear and inhomogeneous dynamic boundary conditions, extending the types of bounday conditions already studied. Under certain assumptions on the input data: $f_{_1}(t,x)$, $w(t,x)$ and $u_0(x)$, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a classical solution in the Sobolev space $W^{1,2}_p(Q)$. This extends previous works concerned with nonlinear dynamic boundary conditions, allowing to the present mathematical model to better approximate the real physical phenomena (the anisotropy effects, phase change in $\Omega$ and at the boundary $\partial\Omega$, etc.).

31

Barsan, Victor, and Mihaela-Cristina Ciornei. "Semiconductor quantum wells with BenDaniel–Duke boundary conditions: approximate analytical results." European Journal of Physics 38, no. 1 (December 6, 2016): 015407. http://dx.doi.org/10.1088/0143-0807/38/1/015407.

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32

Agouzal, A., and N. Debit. "Equilibrium method to approximate elliptic problems with non-standard boundary conditions." Applied Mathematics and Computation 95, no. 2-3 (September 1998): 165–71. http://dx.doi.org/10.1016/s0096-3003(97)10078-9.

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33

Boziev, O. L. "AN APPROXIMATE SOLUTION OF LOADED HYPERBOLIC EQUATION WITH HOMOGENIOS BOUNDARY CONDITIONS." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 8, no. 2 (2016): 14–18. http://dx.doi.org/10.14529/mmph160202.

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34

Folkow, Peter D., and Martin Johansson. "Dynamic equations for fluid-loaded porous plates using approximate boundary conditions." Journal of the Acoustical Society of America 125, no. 5 (2009): 2954. http://dx.doi.org/10.1121/1.3086267.

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35

Bogar, Gary A., and Ronald M. Jeppson. "Restricted range approximate solutions of nonlinear differential systems with boundary conditions." Journal of Approximation Theory 47, no. 1 (May 1986): 26–41. http://dx.doi.org/10.1016/0021-9045(86)90044-4.

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36

Barucq, H., R. Djellouli, and A. Saint-Guirons. "Three-dimensional approximate local DtN boundary conditions for prolate spheroid boundaries." Journal of Computational and Applied Mathematics 234, no. 6 (July 2010): 1810–16. http://dx.doi.org/10.1016/j.cam.2009.08.032.

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37

Itoyama, Hiroshi, and Alexei Morozov. "Boundary Ring or a Way to Construct Approximate NG Solutions with Polygon Boundary Conditions. II." Progress of Theoretical Physics 120, no. 2 (August 2008): 231–87. http://dx.doi.org/10.1143/ptp.120.231.

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38

Denche, M., and K. Bessila. "Quasi-boundary value method for non-well posed problem for a parabolic equation with integral boundary condition." Mathematical Problems in Engineering 7, no. 2 (2001): 129–45. http://dx.doi.org/10.1155/s1024123x01001570.

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In this paper we study the problem of control by the initial conditions of the heat equation with an integral boundary condition. This problem is ill-posed. Perturbing the final condition we obtain an approximate nonlocal problem depending on a small parameter. We show that the approximate problems are well posed. We also obtain estimates of the solutions of the approximate problems and a convergence result of these solutions. Finally, we give explicit convergence rates.

39

Wang, Zhi Hui, Yan Mei Gai, and Cheng Gang Zhao. "Discussion on the Accuracy of Approximate Spring-Dashpot Artificial Boundary in Fluid-Saturated Porous Media." Applied Mechanics and Materials 166-169 (May 2012): 1997–2000. http://dx.doi.org/10.4028/www.scientific.net/amm.166-169.1997.

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The accuracy of artificial boundary conditions plays an important role in the dynamic finite element analysis of unbounded media. By loading the solid skeleton an uniformly distributed triangular pulse on the approximate stress boundary and the original stress boundary respectively, the accuracy of the approximate artificial boundary is analysed. The calculation difference of radial displacement decreases with the increase of distance from artifical boundary to the scattering center, i.e. only when the artificial boundary are far enough can the accuracy be satisfied.

40

Azizi, Ehsan, and Yildiray Cinar. "Approximate Analytical Solutions for CO2 Injectivity Into Saline Formations." SPE Reservoir Evaluation & Engineering 16, no. 02 (May 8, 2013): 123–33. http://dx.doi.org/10.2118/165575-pa.

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Summary This paper presents new analytical models to estimate the bottomhole pressure (BHP) of a vertical carbon dioxide (CO2) injection well in a radial, homogeneous, horizontal saline formation. The new models include the effects of multiphase flow, CO2 dissolution in formation brine, and near-well drying out on the BHP. CO2 is injected into the formation at a constant rate. The analytical solutions are presented for three types of formation outer boundary conditions: closed boundary, constant-pressure boundary, and infinite-acting formation. The sensitivity of BHP computations to gas relative permeability, retardation factors, and CO2 compressibility is examined. The predictive capability of the analytical models is tested by use of numerical reservoir simulations. The results show a good agreement between the analytical and numerical computations for all three boundary conditions. Variations in gas compressibility, retardation factors, and gas relative permeability in the drying-out zone are found to have moderate effects on BHP computations. It is demonstrated for several hypothetical but realistic cases that the new models can estimate CO2 injectivity reliably.

41

Hue, Trinh Thi Thanh, Phan Thi Thu Phuong, and Pham Hong Anh. "Rayleigh waves in compressible orthotropic half-space overlaid by a thin un-coaxial orthotropic layer." Journal of Science and Technology in Civil Engineering (STCE) - HUCE 15, no. 4 (October 31, 2021): 54–64. http://dx.doi.org/10.31814/stce.huce(nuce)2021-15(4)-05.

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The problem of Rayleigh waves in compressible orthotropic elastic half-space overlaid by a thin elastic layer of which principal material axes are coincident have been researched by many scientists. However, the problem with the conditions that the half-space and the layer have only one common principal material axis that perpendicular to the layer while the remains are not identical has not gotten enough attention. This paper presents a traditional approach to obtain an approximate secular equation by approximately replacing the thin layer by effective boundary conditions of third-order. The wave then is considered as a Rayleigh wave propagating in an orthotropic half-space, without coating, subjected to the effective boundary conditions. This explicit approximate secular equation is potentially useful in non-damage assessment studies.

42

Chen, Yajun, and Qikui Du. "Exact Artificial Boundary Conditions for Quasi-Linear Problems in Semi-Infinite Strips." Journal of Mathematics 2021 (September 21, 2021): 1–10. http://dx.doi.org/10.1155/2021/1660711.

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In this paper, the exact artificial boundary conditions for quasi-linear problems in semi-infinite strips are investigated. Based on the Kirchhoff transformation, the exact and approximate boundary conditions on a segment artificial boundary are derived. The error estimate for the finite element approximation with the artificial boundary condition is obtained. Some numerical examples show the efficiency of this method.

43

Ton, Lan Hoang That. "Approximate natural frequencies of AFGM beam." Technical Journal of Daukeyev University 2, no. 2 (October 13, 2022): 66–72. http://dx.doi.org/10.52542/tjdu.2.2.66-72.

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This article gives the approximate natural frequencies of axial functionally graded material (AFGM) beams. Based on Matlab software and the traditional finite element method, we can estimate the natural frequencies for this structure under four types of boundary conditions. Some examples are given which prove that this way is simple and applicable.

44

Jourdon, Anthony, and Dave A. May. "An efficient partial-differential-equation-based method to compute pressure boundary conditions in regional geodynamic models." Solid Earth 13, no. 6 (June 29, 2022): 1107–25. http://dx.doi.org/10.5194/se-13-1107-2022.

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Abstract. Modelling the pressure in the Earth's interior is a common problem in Earth sciences. In this study we propose a method based on the conservation of the momentum of a fluid by using a hydrostatic scenario or a uniformly moving fluid to approximate the pressure. This results in a partial differential equation (PDE) that can be solved using classical numerical methods. In hydrostatic cases, the computed pressure is the lithostatic pressure. In non-hydrostatic cases, we show that this PDE-based approach better approximates the total pressure than the classical 1D depth-integrated approach. To illustrate the performance of this PDE-based formulation we present several hydrostatic and non-hydrostatic 2D models in which we compute the lithostatic pressure or an approximation of the total pressure, respectively. Moreover, we also present a 3D rift model that uses that approximated pressure as a time-dependent boundary condition to simulate far-field normal stresses. This model shows a high degree of non-cylindrical deformation, resulting from the stress boundary condition, that is accommodated by strike-slip shear zones. We compare the result of this numerical model with a traditional rift model employing free-slip boundary conditions to demonstrate the first-order implications of considering “open” boundary conditions in 3D thermo-mechanical rift models.

45

Itoyama, H., A. Mironov, and A. Morozov. "Boundary ring: A way to construct approximate NG solutions with polygon boundary conditions I. -symmetric configurations." Nuclear Physics B 808, no. 3 (February 2009): 365–410. http://dx.doi.org/10.1016/j.nuclphysb.2008.08.025.

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46

Cao, Shunhua, and Stewart Greenhalgh. "Attenuating boundary conditions for numerical modeling of acoustic wave propagation." GEOPHYSICS 63, no. 1 (January 1998): 231–43. http://dx.doi.org/10.1190/1.1444317.

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Four types of boundary conditions: Dirichlet, Neumann, transmitting, and modified transmitting, are derived by combining the damped wave equation with corresponding boundary conditions. The Dirichlet attenuating boundary condition is the easiest to implement. For an appropriate choice of attenuation parameter, it can achieve a boundary reflection coefficient of a few percent in a one‐wavelength wide zone. The Neumann‐attenuating boundary condition has characteristics similar to the Dirichlet attenuating boundary condition, but it is numerically more difficult to implement. Both the transmitting boundary condition and the modified transmitting boundary condition need an absorbing boundary condition at the termination of the attenuating region. The modified transmitting boundary condition is the most effective in the suppression of boundary reflections. For multidimensional modeling, there is no perfect absorbing boundary condition, and an approximate absorbing boundary condition is used.

47

Li, Yijun, Guanggan Chen, and Ting Lei. "Approximate dynamics of stochastic partial differential equations under fast dynamical boundary conditions." Mathematical Methods in the Applied Sciences 44, no. 11 (March 14, 2021): 8986–98. http://dx.doi.org/10.1002/mma.7326.

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48

Gao, Chao, Shuchi Yang, Shijun Luo, Feng Liu, and David M. Schuster. "Calculation of Airfoil Flutter by an Euler Method with Approximate Boundary Conditions." AIAA Journal 43, no. 2 (February 2005): 295–305. http://dx.doi.org/10.2514/1.5752.

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49

Piomelli, Ugo, Joel Ferziger, Parviz Moin, and John Kim. "New approximate boundary conditions for large eddy simulations of wall‐bounded flows." Physics of Fluids A: Fluid Dynamics 1, no. 6 (June 1989): 1061–68. http://dx.doi.org/10.1063/1.857397.

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50

Lerer, A. M., V. V. Makhno, P. V. Makhno, and A. A. Yachmenov. "Calculation of periodic metal nanostructures via the method of approximate boundary conditions." Journal of Communications Technology and Electronics 52, no. 4 (April 2007): 399–405. http://dx.doi.org/10.1134/s1064226907040043.

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Journal articles on the topic 'Approximate boundary conditions'

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