Journal articles: 'Elastic boundary condition'

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Randall, C. J. "Absorbing boundary condition for the elastic wave equation." GEOPHYSICS 53, no. 5 (May 1988): 611–24. http://dx.doi.org/10.1190/1.1442496.

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Extant absorbing boundary conditions for the elastic wave equation are generally effective only for waves nearly normally incident upon the boundary. High reflectivity is exhibited for waves traveling obliquely to the boundary. In this paper, a new and efficient absorbing boundary condition for two‐dimensional and three‐dimensional finite‐difference calculations of elastic wave propagation is presented. Compressional and shear components of the incident vector displacement fields are separated by calculating intermediary scalar potentials, allowing the use of Lindman’s boundary condition for scalar fields, which is highly absorbing for waves incident at any angle. The elastic medium is assumed to be homogeneous in the region immediately adjacent to the boundary. The reflectivity matrix of the resulting absorbing boundary for elastic waves is calculated, including the effects of finite‐difference truncation error. For effectively all angles of incidence, reflectivities are much smaller than those of the commonly employed paraxial absorbing boundaries, and the boundary condition is stable for any physical Poisson’s ratio. The nearly complete absorption predicted by the reflectivity matrix calculations, even at near grazing incidence, is demonstrated in a finite‐difference application.

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Zhao, Zhencong, Jingyi Chen, Xiaobo Liu, and Baorui Chen. "Frequency-domain elastic wavefield simulation with hybrid absorbing boundary conditions." Journal of Geophysics and Engineering 16, no. 4 (August 1, 2019): 690–706. http://dx.doi.org/10.1093/jge/gxz038.

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Abstract The frequency-domain seismic modeling has advantages over the time-domain modeling, including the efficient implementation of multiple sources and straightforward extension for adding attenuation factors. One of the most persistent challenges in the frequency domain as well as in the time domain is how to effectively suppress the unwanted seismic reflections from the truncated boundaries of the model. Here, we propose a 2D frequency-domain finite-difference wavefield simulation in elastic media with hybrid absorbing boundary conditions, which combine the perfectly matched layer (PML) boundary condition with the Clayton absorbing boundary conditions (first and second orders). The PML boundary condition is implemented in the damping zones of the model, while the Clayton absorbing boundary conditions are applied to the outer boundaries of the damping zones. To improve the absorbing performance of the hybrid absorbing boundary conditions in the frequency domain, we apply the complex coordinate stretching method to the spatial partial derivatives in the Clayton absorbing boundary conditions. To testify the validity of our proposed algorithm, we compare the calculated seismograms with an analytical solution. Numerical tests show the hybrid absorbing boundary condition (PML plus the stretched second-order Clayton absorbing condition) has the best absorbing performance over the other absorbing boundary conditions. In the model tests, we also successfully apply the complex coordinate stretching method to the free surface boundary condition when simulating seismic wave propagation in elastic media with a free surface.

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Keller, Joseph B., and Marcus J. Grote. "Exact Nonreflecting Boundary Condition For Elastic Waves." SIAM Journal on Applied Mathematics 60, no. 3 (January 2000): 803–19. http://dx.doi.org/10.1137/s0036139998344222.

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Taylor, Adam G., and Jae H. Chung. "Analysis of tangential contact boundary value problems using potential functions." Royal Society Open Science 6, no. 3 (March 2019): 182106. http://dx.doi.org/10.1098/rsos.182106.

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This paper presents an analysis technique of high-order contact potential problems and its application to an elastic settlement analysis of a shallow foundation system subjected to a combined traction boundary condition. Closed-form solutions of potential functions are derived for an elastic half-space subjected to bilinear tangential traction boundary conditions over rectangular surface regions. Using the principle of superposition, the present solutions provide a means to form an approximate and continuous solution of elastic contact problems with higher-order tangential boundary conditions. As an application example, an elastic settlement analysis of a rigid footing founded on a dense granular soil is performed under a tangential traction boundary condition prescribed in an analogy with the stress equilibrium states of static sandpiles. A generalized solution approach to combined normal and tangential traction boundary value problems is discussed in the context of foundation engineering.

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LANGE, A., J. ZHOU, and N. SAFFARI. "OPTIMISED ABSORBING BOUNDARY CONDITIONS FOR ELASTIC-WAVE PROPAGATION." Journal of Computational Acoustics 09, no. 03 (September 2001): 1005–14. http://dx.doi.org/10.1142/s0218396x01001091.

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Second-order absorbing boundary conditions for numerical modeling of elastic-wave propagation are studied. The corresponding reflection coefficients are derived, from which a necessary and sufficient condition for complete absorption at normal incidence is deduced. We define a family of absorbing boundary conditions from symmetrically specified zero reflection incidences. Conditions to avoid singular reflection coefficients are given for this case, these ensure that the solutions of the elastic wave equation also satisfy the boundary conditions. These are then optimised over a wide range of materials, and absorbing boundary conditions that give an efficient absorption for the whole range are obtained. We also compare the results with absorbing boundary conditions developed from the least-squares solution of the system requiring complete absorption at all incidences. The best set of conditions are presented and compared with Clayton and Engquist6 (A2) condition.

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Peng, Chengbin, and M. Nafi Toksöz. "An optimal absorbing boundary condition for elastic wave modeling." GEOPHYSICS 60, no. 1 (January 1995): 296–301. http://dx.doi.org/10.1190/1.1443758.

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Absorbing boundary conditions are widely used in numerical modeling of wave propagation in unbounded media to reduce reflections from artificial boundaries (Lindman, 1975; Clayton and Engquist, 1977; Reynolds, 1978; Liao et al., 1984; Cerjan et al., 1985; Randall, 1988; Higdon, 1991). We are interested in a particular absorbing boundary condition that has maximum absorbing ability with a minimum amount of computation and storage. This is practical for 3-D simulation of elastic wave propagation by a finite‐difference method. Peng and Toksöz (1994) developed a method to design a class of optimal absorbing boundary conditions for a given operator length. In this short note, we give a brief introduction to this technique, and we compare the optimal absorbing boundary conditions against those by Reynolds (1978) and Higdon (1991) using examples of 3-D elastic finite‐difference modeling on an nCUBE-2 parallel computer. In the Appendix, we also give explicit formulas for computing coefficients of the optimal absorbing boundary conditions.

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Yin, Yuhan, and Juan Liu. "Hybrid Method for Inverse Elastic Obstacle Scattering Problems." Mathematics 11, no. 8 (April 20, 2023): 1939. http://dx.doi.org/10.3390/math11081939.

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The problem of determining the shape of an object from knowledge of the far-field of a single incident wave in two-dimensional elasticity was considered. We applied an iterative hybrid method to tackle this problem. An advantage of this method is that it does not need a forward solver, and therefore, the exact boundary condition is not essential. By deriving the Fréchet derivatives of two boundary operators, we established reconstruction algorithms for objects with Dirichlet, Neumann, and Robin boundary conditions; by introducing a general boundary condition, we also established the reconstruction algorithm for objects with unknown physical properties. Numerical experiments showed the effectiveness of the proposed method.

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Ben-Haim, Yakov, and H. G. Natke. "Sequential Adaptation in Estimating Elastic Boundary-Condition Influence Matrices." Journal of Dynamic Systems, Measurement, and Control 115, no. 3 (September 1, 1993): 370–78. http://dx.doi.org/10.1115/1.2899112.

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A method is discussed for adaptive estimation of the boundary-condition influence matrix of a linear elastic system, based on modal measurements (eigenvalues and eigenvectors) and on knowledge of the influence matrix of the system with respect to reference boundary conditions. The estimation is terminated adaptively by an algorithm motivated by the concept of sequential analysis. The boundary-condition influence matrix is re-estimated with measurement of each additional mode, until a termination criterion indicates that adequate accuracy has been attained. The advantage of adaptive termination of the estimation is the enhanced computational (and possibly instrumental) efficiency of estimating with minimal modal data. An analytical technique for comparing the adaptive termination with a reasonable non-adaptive method has been developed and demonstrated by application to a uniform beam. When uncertainty in the boundary conditions is represented by convex models, it is shown that the adaptive estimation can terminate much earlier than the non-adaptive procedure.

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Mittet, Rune. "Implementation of the Kirchhoff integral for elastic waves in staggered‐grid modeling schemes." GEOPHYSICS 59, no. 12 (December 1994): 1894–901. http://dx.doi.org/10.1190/1.1443576.

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Implementation of boundary conditions in finite‐difference schemes is not straightforward for the elastic wave equation if a staggered grid formulation is used. Reverse time migration of VSP data requires a proper description of the recording surface so as not to excite false P‐ and S‐waves. Such contributions may cause artifacts in the imaging procedure. The boundary conditions for the elastic stress tensor can be implemented numerically in a staggered coarse grid modeling scheme by using band‐limited spatial delta‐functions and band‐limited first‐order derivatives of these spatial delta‐functions. A representation theorem for elastic waves is derived to test the implementation of the spatial part of the boundary condition. The implementation is tested in a 2-D numerical experiment for a closed, but curved, boundary S enclosing a volume V. The test condition is that within the volume V, the difference between the forward modeled field and the retropropagated field should be equal to zero. Both P‐ and S‐waves are properly recovered in a 2-D reverse time modeling example. The numerical artifacts related to the proposed spatial approximation of the boundary condition are found to be negligible.

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Yurchuk, V. M., and S. V. Sinchilo. "Torsional elastic waves. Some aspects of nonlinear analysis." Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics, no. 2 (2023): 172–75. http://dx.doi.org/10.17721/1812-5409.2023/2.31.

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The features of the use of boundary conditions in the nonlinear problem of torsional wave propagation for an elastically deformable medium with an external boundary are analyzed. The formulation and wave analysis in the linear (classical) approach are briefly described, since the linear solution is used in the work as a first approximation in the nonlinear approach. The first feature for a torsional wave is a significant complication in the nonlinear approach of the boundary conditions due to the difference between the shape of the boundary before and after the deformation (in the linear approach, the shape of the boundary does not change). The second feature is the significant complication of the mathematical representation of the boundary conditions due to the appearance of additional nonlinear terms. For a torsional wave, it was found that the use of the condition of absence of stresses on the boundary surface (assumption of a free boundary) may not be completely correct.

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Stoyan, Volodymyr, Dmytro Cherniy, Sergii Voloshchuk, and Arsen Stoian. "ON DEGENARATE CASE OF THREE-DIMENTIONAL PROBLEM OF THICK ELASTIC PLATESʼ DYNAMICS." Journal of Automation and Information sciences 1 (January 1, 2022): 11–24. http://dx.doi.org/10.34229/1028-0979-2022-1-2.

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The three-dimensional problem of elastic membrane dynamics of arbitrary boundary outline with a given superficially distributed external force effect on it, is solved. The three-dimensional field of elastic dynamic displacements of membrane points is described by a differential model, which is a degenerate case of the previously constructed equations of the spatial theory of elasticity concerning thick elastic plates. It is assumed that the latter are supplemented with the information on the initial-boundary condition of the membrane, which in the general case is not of a force nature, and its amount is not sufficient for a mathematically correct formulation of the initial-boundary value problem. The solution of these incompletely defined initial-boundary problems of membrane dynamics became possible after root-mean square mathematical modeling of existing spatial-dynamic observations for the initial condition and current dynamics of its outline by a system of discretely and continuously defined modeling functions. The field of elastic-dynamic displacements of membraneʼ points constructed in the work, being an accurate solution of its differential mathematical model, with the available information on the initial-boundary condition of the latter, is consistent with the root-mean square criterion. Estimations of the accuracy of such modeling are made and conditions for its uniqueness are formulated in the paper. Cases of purely transverse and purely planar oscillations of the membrane are considered. The numerical realization of the obtained considered problemsʼ solution is conducted even with a minimum number of observations for the membraneʼs condition confirms the simplicity and correctness of the approach to these mathematically complex problemsʼ research of elastic-dynamics of thick elastic plates.

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12

Zhang, Wx. "Analysis on Boundary Conditions for Elastic Structures." E3S Web of Conferences 236 (2021): 05038. http://dx.doi.org/10.1051/e3sconf/202123605038.

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With the rapid development of modern industry, elastic materials and structures have been widely used in all walks of life, such as construction, machinery and so on. The traditional elastic mechanics method generally adopts the semi inverse method. However, this method ignores the local deformation caused by boundary constraints and can only find part of the solution of the problem. In this paper, the basic equations of symplectic system are established, and the solutions of various boundary condition problems are given.

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Wang, Wei Dong, Xiang Yu Niu, Kang Qi Fan, and Qing Yi Wang. "Stokes' Second Problem with Velocity Slip Boundary Condition." Key Engineering Materials 483 (June 2011): 287–92. http://dx.doi.org/10.4028/www.scientific.net/kem.483.287.

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The damping effect of microfluidics has great effect on the dynamic characteristics of MEMS devices. Based on the separation of variables and the integral transform methods, the Stokes' second problem is solved at the microscale velocity slip boundary condition and the analytical solution for velocity distribution is obtained. Furthermore, the expression of the penetration depth is gotten for Stokes model in this article. Through analysis, it’s found that due to the effect of the microscale velocity slip boundary condition, the velocity oscillation amplitude and the penetration depth have been reduced. Then the shear stress, the damping force and elastic force on the plate have been investigated. It’s shown that both the elastic coefficient and the damping coefficient increases as the oscillation frequency rises.

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14

Randall, C. J. "Absorbing boundary condition for the elastic wave equation: Velocity‐stress formulation." GEOPHYSICS 54, no. 9 (September 1989): 1141–52. http://dx.doi.org/10.1190/1.1442749.

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This paper describes an absorbing boundary condition for finite‐difference modeling of elastic wave propagation in two and three dimensions. The boundary condition is particularly effective for obliquely incident waves, typically quite troublesome for absorbing boundaries. Analytical predictions of the boundary reflection coefficients of a few percent or less for angles of incidence up to 89° are verified in example finite‐difference applications. The algorithm is appropriate for use in a velocity‐stress finite‐difference (vs‐fd) formulation. It is computationally simpler than a similar absorbing boundary given previously for the standard displacement formulation. A second algorithm is presented which may be advantageous when the boundary of interest is exposed to strong evanescent waves. Both algorithms require that the adjacent elastic medium be locally homogeneous.

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D’Ovidio, Mirko. "Fractional boundary value problems." Fractional Calculus and Applied Analysis 25, no. 1 (February 2022): 29–59. http://dx.doi.org/10.1007/s13540-021-00004-0.

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AbstractWe study some functionals associated with a process driven by a fractional boundary value problem (FBVP for short). By FBVP we mean a Cauchy problem with boundary condition written in terms of a fractional equation, that is an equation involving time-fractional derivative in the sense of Caputo. We focus on lifetimes and additive functionals characterizing the boundary conditions. We show that the corresponding additive functionals are related to the fractional telegraph equations. Moreover, the fractional order of the derivative gives a unified condition including the elastic and the sticky cases among the others.

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Wu, Yi. "An Equivalent Far-Field Dynamic-Artificial-Boundary Condition and its Accuracy Studies." Advanced Materials Research 243-249 (May 2011): 3795–803. http://dx.doi.org/10.4028/www.scientific.net/amr.243-249.3795.

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In order to eliminate some problems of ordinary artificial boundary conditions in soil-structure dynamic interaction analysis at present, such as plenty amount of work required in establishing viscous boundary and viscous-elastic boundary and the expensive computing hours cost in traditional far-field artificial boundary condition, an equivalent far-field artificial boundary condition for dynamic analysis was proposed based on the wave propagating theory and the reflection and transmission principle of wave on the interface of medium by setting a fully-transmitting layer and lower the speed of wave considerably propagating within it. Then numerical simulations and studies of factors affecting accuracy for equivalent far-field artificial boundary condition were conducted with FEM. Researches show that equivalent far-field artificial boundary is as nearly accurate as viscous boundary; it can overcome the problems of viscous boundary, viscous-elastic boundary and traditional far-field boundary mentioned above; it is very convenient to operate and realize; it is very efficient and low-cost in workforce. The equivalent far-field boundary proposed can be applied to engineering dynamic field such as anti-seismic of structures, vibration and impact etc.

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Diaz, Julien, and Victor Péron. "Equivalent Robin boundary conditions for acoustic and elastic media." Mathematical Models and Methods in Applied Sciences 26, no. 08 (June 7, 2016): 1531–66. http://dx.doi.org/10.1142/s0218202516500378.

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We present equivalent conditions and asymptotic models for a diffraction problem of acoustic and elastic waves. The mathematical problem is set with a Robin boundary condition. Elastic and acoustic waves propagate in a solid medium surrounded by a thin layer of fluid medium. Due to the thinness of the layer with respect to the wavelength, this problem is well suited for the notion of equivalent conditions and the effect of the fluid medium on the solid is as a first approximation local. This approach leads to solve only elastic equations. We derive and validate equivalent conditions up to the third order for the elastic displacement. The construction of equivalent conditions is based on a multiscale expansion in power series of the thickness of the layer for the solution of the transmission problem.

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Duan, Jin Xi, and Z. Shen. "Elastic Seismic Response of Steel-Concrete Composite Frames with Partial Interaction." Applied Mechanics and Materials 268-270 (December 2012): 729–32. http://dx.doi.org/10.4028/www.scientific.net/amm.268-270.729.

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The finite element formulations of steel-concrete composite (SCC) beams considering interlayer slip with end shear restraint is established. Elastic seismic response of SCC frame structures under different shear connection stiffness and slip boundary conditions are examined. The influences of the shear connection stiffness and the slip boundary condition on elastic seismic response are analyzed. With the shear connection stiffness increasing, the free vibration frequencies increase and the seismic responses decrease. The natural vibration properties of SCC frame structures and seismic responses are also significantly affected by the slip boundary condition, and it should be properly imposed on all composite beams in seismic response analysis.

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Mahbobian, Kamyar, and Y. Tadi Beni. "Investigation of Rotation Elastic Boundary Condition on the Pull-In Instability of Electrostatic Torsional Actuators." Advanced Materials Research 445 (January 2012): 1070–75. http://dx.doi.org/10.4028/www.scientific.net/amr.445.1070.

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In this paper, the static pull-in instability of torsional nanoelectromechanical systems (NEMS) with coupling effect between torsion and bending is investigated considering the effect of translation elastic boundary condition. A set of normalized equations governing the static actuation properties of the torsional actuator is derived to demonstrate the relationships between the parameters of static characteristics, such as torsion angle, vertical displacement, and applied voltage. The results from this model demonstrated that the real model of boundary condition (elastic boundary condition) is very important issue which must be considered in manufacturing process.

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Lv, Xiao Tang. "Surface Displacement of a Semi-Cylindrical Hill above a Subsurface Elastic Cylindrical Inclusion under SH-Wave." Advanced Materials Research 243-249 (May 2011): 4037–40. http://dx.doi.org/10.4028/www.scientific.net/amr.243-249.4037.

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Scattering of SH-wave by a semi-cylindrical hill above a subsurface elastic cylindrical inclusion in half-space is studied by complex variable function. Firstly, the whole solution domain is divided into two parts, and the solutions that satisfied the boundary conditions are constructed in two parts respectively. Then according to the “conjunction” condition of junction interface and the boundary condition around the subsurface elastic cylindrical inclusion, a set of infinite algebraic equations about the problem can be obtained. Finally the computational results of surface displacement were provided and discussed.

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Li, Ming, Hui Ming Zheng, Luo Xia, and Liu Yang. "Small Scale Effect on Boundary Conditions of Cantilever Single Carbon Nanotubes." Applied Mechanics and Materials 275-277 (January 2013): 33–37. http://dx.doi.org/10.4028/www.scientific.net/amm.275-277.33.

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In this paper, the boundary condition on free vibration of cantilever single-walled carbon nanotubes (SWCNTs) with Winkler elastic medium is investigated. The Euler beam theory with nonlocal elasticity is modeled as SWCNT. The analytical solution is derived and the numerical results show that the additional boundary conditions from small scale do not change natural frequencies. The reason is that the additional work made by the moment and shear force at the free end from small scale effect cancel each other, the boundary conditions due to local elasticity and nonlocal elasticity are also equivalent. Thus for simplicity, one can apply the local boundary condition to replace the small scale boundary condition.

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Modestov, Konstantin, Sergey Saiyan, Artyom Erokhin, and Oleg Brygar. "Derivation of the one-dimensional radiation condition in elasticity theory by introducing infinitesimal viscosity." E3S Web of Conferences 410 (2023): 03025. http://dx.doi.org/10.1051/e3sconf/202341003025.

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With the rapid development of engineering science and technology, tasks related to the calculation of wave processes in elastic medium are becoming more and more relevant. One of the important problems is the search for effective methods for calculating the propagation of harmonic oscillations in elastic mediums, in particular, in semi-infinite spaces. One of these methods is the use of asymptotic boundary conditions of the “radiation conditions” type, which allow us to describe the processes of wave propagation outside the computational domain. In this paper we consider the application of infinitesimal viscosity to derive an asymptotic boundary condition of the “radiation condition” type for the propagation of harmonic oscillations in a semi-infinite elastic space. The results of the study can be used in the design of various devices and structures subjected to the influence of wave processes in elastic mediums, in particular, in the field of soil mechanics.

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Wang, Yong, and Wenpei Wu. "Initial boundary value problems for the three-dimensional compressible elastic Navier-Stokes-Poisson equations." Advances in Nonlinear Analysis 10, no. 1 (January 1, 2021): 1356–83. http://dx.doi.org/10.1515/anona-2020-0184.

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Abstract We study the initial-boundary value problems of the three-dimensional compressible elastic Navier-Stokes-Poisson equations under the Dirichlet or Neumann boundary condition for the electrostatic potential. The unique global solution near a constant equilibrium state in H 2 space is obtained. Moreover, we prove that the solution decays to the equilibrium state at an exponential rate as time tends to infinity. This is the first result for the three-dimensional elastic Navier-Stokes-Poisson equations under various boundary conditions for the electrostatic potential.

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Cao, Yuan, Rui Zhong, Dong Shao, Qingshan Wang, and Dongtao Wu. "Free In-Plane Vibrations of Orthotropic Rectangular Plates by Using an Accurate Solution." Mathematical Problems in Engineering 2019 (November 11, 2019): 1–20. http://dx.doi.org/10.1155/2019/4687082.

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Many numerical methods have been developed for in-plane vibration of orthotropic rectangular plates with various boundary conditions; however, the exact results for such structures with elastic boundary conditions are very scarce. Therefore, the object of this paper is to present an accurate solution for free in-plane vibration of orthotropic rectangular plates with various boundary conditions by the method of reverberation ray matrix (MRRM) and improved golden section search (IGSS) algorithm. The boundary condition studied in this paper is defined as that a set of opposite edges is with one kind of simply supported boundary conditions, while the other set is with any kind of classical and general elastic boundary conditions or their combination. Its accuracy, reliability, and efficiency are verified by some numerical examples where the results are compared with other exact solutions in the published literature and the FEA results based on the ABAQUS software. Finally, some new accurate results for free in-plane vibration of orthotropic rectangular plates with elastic boundary conditions are examined and further can be treated as the reference data for other approximate methods or accurate solutions.

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Jin, Zhong Kun, and Tong Qing Wang. "Calculation on Acoustic Scattering of Viscoelastic Layer Coupling with Elastic Shell." Applied Mechanics and Materials 248 (December 2012): 107–13. http://dx.doi.org/10.4028/www.scientific.net/amm.248.107.

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This paper is devoted to numerical research on coupling between elastic spherical shell and the coated viscoelastic layer as well as the scattering of incident plane wave by the double-layer spherical shell. The scattering sound field is solved based on impedance boundary condition by boundary element method (BEM). Dynamic finite element method (FEM) is used to numerically simulate the acoustic impedance boundary condition which involved in the coupled spherical shell. Impedance distribution for elastic spherical shell and elastic spherical shell coated viscoelastic layer is calculated and its effect on the target strength (TS) is discussed finally.

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Belan, S., A. Chernykh, and V. Lebedev. "Boundary layer of elastic turbulence." Journal of Fluid Mechanics 855 (September 21, 2018): 910–21. http://dx.doi.org/10.1017/jfm.2018.662.

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We investigate theoretically the near-wall region in elastic turbulence of a dilute polymer solution in the limit of large Weissenberg number. As has been established experimentally, elastic turbulence possesses a boundary layer where the fluid velocity field can be approximated by a steady shear flow with relatively small fluctuations on the top of it. Assuming that at the bottom of the boundary layer the dissolved polymers can be considered as passive objects, we examine analytically and numerically the statistics of the polymer conformation, which is highly non-uniform in the wall-normal direction. Next, imposing the condition that the passive regime terminates at the border of the boundary layer, we obtain an estimate for the ratio of the mean flow to the magnitude of the flow fluctuations. This ratio is determined by the polymer concentration, the radius of gyration of polymers and their length in the fully extended state. The results of our asymptotic analysis reproduce the qualitative features of elastic turbulence at finite Weissenberg numbers.

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Chen, Hui, Xin Yue Bi, Li Jia Zhang, and Yan Shen. "Temperature-Graded Ferroelectric Thin Films under Two Boundary Conditions." Advanced Materials Research 750-752 (August 2013): 1910–13. http://dx.doi.org/10.4028/www.scientific.net/amr.750-752.1910.

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In the framework of the mean field approximation, a transverse Ising model (TIM) was adopted to analyze the polarization properties of temperature-graded ferroelectric films under two boundary conditions, free boundary condition (FBC) and clamped boundary condition (CBC). Due to the ferroelectric distortion aroused by temperature gradient across the film, the elastic thermal stress increased. A distribution function was introduced to characterize the different boundary conditions. The results show that boundary conditions have very important influence on film properties. Polarizations under FBC are larger than that under CBC, polarization variations aroused by changed film thickness and temperature gradient under FBC are not as obvious as that under CBC, and films under different boundary conditions present obviously different physical behavior.

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Ting, T. C. T., Y. Hu, and H. O. K. Kirchner. "Anisotropic Elastic Materials With a Parabolic or Hyperbolic Boundary: A Classical Problem Revisited." Journal of Applied Mechanics 68, no. 4 (January 2, 2001): 537–42. http://dx.doi.org/10.1115/1.1381393.

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When an anisotropic elastic material is under a two-dimensional deformation that has a hole of given geometry Γ subjected to a prescribed boundary condition, the problem can be solved by mapping Γ to a circle of unit radius. It is important that (i) each point on Γ is mapped to the same point for the three Stroh eigenvalues p1,p2,p3 and (ii) the mapping is one-to-one for the region outside Γ. In an earlier paper it was shown that conditions (i) and (ii) are satisfied when Γ is an ellipse. The paper did not address to the case when Γ is an open boundary, such as a parabola or hyperbola that was studied by Lekhnitskii. We examine the mappings employed by Lekhnitskii for a parabola and hyperbola, and show that while the mapping for a parabola satisfies conditions (i) and (ii), the mapping for a hyperbola does not satisfy condition (i). Nevertheless, a valid solution can be obtained for the problem with a hyperbolic boundary, although the prescription of the boundary condition is restricted. We generalize Lekhnitskii’s solutions for general anisotropic elastic materials and for more general boundary conditions. Using known identities and new identities presented here, real form expressions are given for the displacement and hoop stress vector at the parabolic and hyperbolic boundary.

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Cerjan, Charles, Dan Kosloff, Ronnie Kosloff, and Moshe Reshef. "A nonreflecting boundary condition for discrete acoustic and elastic wave equations." GEOPHYSICS 50, no. 4 (April 1985): 705–8. http://dx.doi.org/10.1190/1.1441945.

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One of the nagging problems which arises in application of discrete solution methods for wave‐propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh. These undesired events eventually override the actual seismic signals which propagate in the modeled region. The solution to avoiding boundary effects used to be to enlarge the numerical mesh, thus delaying the side reflections and wraparound longer than the range of times involved in the modeling. Obviously this solution considerably increases the expense of computation. More recently, nonreflecting boundary conditions were introduced for the finite‐difference method (Clayton and Enquist, 1977; Reynolds, 1978). These boundary conditions are based on replacing the wave equation in the boundary region by one‐way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh. This approach has been relatively successful, except that its effectiveness degrades for events which impinge on the boundaries at shallow angles. It is also not clear how to apply this type of boundary condition to global discrete methods such as the Fourier method for which all grid points are coupled.

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Meguid, S. A., and X. Zhao. "The Interface Crack Problem of Bonded Piezoelectric and Elastic Half-Space Under Transient Electromechanical Loads." Journal of Applied Mechanics 69, no. 3 (May 1, 2002): 244–53. http://dx.doi.org/10.1115/1.1460910.

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The interface crack problem of bonded piezoelectric and elastic half-space under transient electromechanical loads is considered. Both the permeable and impermeable boundary conditions are examined and discussed. Based on the use of integral transform techniques, the problem is reduced either to a singular integral equation for the permeable boundary condition or to two coupled singular integral equations for the impermeable boundary condition, which can be solved using Chebyshev polynomial expansions. Numerical results are provided to show the effect of the applied electric fields, the electric boundary conditions along the crack faces and a free surface on the resulting dynamic stress intensity factor and electric displacement intensity factor.

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Renping, Shao, Purong Jia, and Xiankun Qi. "3-D elastic coupling vibration and acoustical radiation characteristics of cracked gear under elastic support condition." Journal of Vibration and Control 23, no. 9 (October 23, 2015): 1548–68. http://dx.doi.org/10.1177/1077546315596482.

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According to the actual working condition of the gear, the supporting gear shaft is treated as an elastic support. Its impact on the gear body vibration is considered and investigated and the dynamic response of elastic teeth and gear body is analyzed. On this basis, the gear body is considered as a three-dimensional elastic disc and the gear teeth are treated as an elastic cantilever beam. Under the conditions of the elastic boundary (support shaft), combining to the elastic disk and elastic teeth, the influence of three-dimensional elastic discs on the meshing tooth response under an elastic boundary condition is also included. A dynamic model of the gear support system and calculated model of the gear tooth response are then established. The inherent characteristics of the gear support system and dynamics response of the meshing tooth are presented and simulated. It was shown by the results that it is correct to use the elastic support condition to analyze the gear support system. Based on the above three-dimensional elastic dynamics analysis, this paper set up a dynamics coupling model of a cracked gear structure support system that considered the influence of a three-dimensional elastic disc on a cracked meshing tooth under elastic conditions. It discusses the dynamic characteristic of the cracked gear structure system and coupling dynamic response of the meshing tooth, offering a three-dimensional elastic body model of the tooth root crack and pitch circle crack with different sizes, conducting the three-dimensional elastic dynamic analysis to the faulty crack. ANSYS was employed to carry out dynamic responses, as well as to simulate the acoustic field radiation orientation of a three-dimensional elastic crack body at the tooth root crack and pitch circle with different sizes.

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32

Khludnev, Alexander. "On Equilibrium Problem for T-Shape Elastic Structure." Axioms 14, no. 1 (January 10, 2025): 49. https://doi.org/10.3390/axioms14010049.

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This paper is concerned with an equilibrium problem for an elastic structure consisting of a plate and an elastic beam connected to each other at a given point. We consider two cases: In the first one, the elastic beam is connected to a rigid part of the elastic plate; in the second case, contact occurs between two elastic bodies. The elastic plate may contain a thin rigid delaminated inclusion. Neumann-type boundary conditions are considered at the external boundary of the plate. The existence of a solution to the considered problems is proven. A sufficient and necessary condition imposed onto the external forces for the solvability of the problems is found. Passages to the limit with respect to the rigidity parameter of the elastic beam are justified. For all problems, we analyze variational statements as well as differential ones.

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Ravasi, Matteo, and Andrew Curtis. "Nonlinear scattering based imaging in elastic media: Theory, theorems, and imaging conditions." GEOPHYSICS 78, no. 3 (May 1, 2013): S137—S155. http://dx.doi.org/10.1190/geo2012-0286.1.

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With the more widespread introduction of multicomponent recording devices in land and marine ocean-bottom seismic acquisition, elastic imaging may become mainstream in coming years. We have derived new, nonlinear, elastic imaging conditions. A correlation-type representation theorem for perturbed elastic media, commonly used in seismic interferometry to explain how a scattered wave response between two receivers/sources may be predicted given a boundary of sources/receivers, can be considered as a starting point for the derivation. Here, we use this theorem to derive and interpret imaging conditions for elastic migration by wavefield extrapolation (e.g., elastic reverse-time migration). Some approximations lead to a known, heuristically derived imaging condition that crosscorrelates P- and S-wave potentials that are separated in the subsurface after full-wavefield extrapolation. This formal connection reveals that the nonapproximated correlation-type representation theorem can be interpreted as a nonlinear imaging condition, that accounts also for multiply scattered and multiply converted waves, properly focusing such energy at each image point. We present a synthetic data example using either an ideal (acquisition on a full, closed boundary) or a real (partial boundary) seismic exploration survey, and we demonstrate the importance of nonlinearities in pure- and converted-mode imaging. In PP imaging, they result in better illumination and artifact reduction, whereas in PS imaging they show how zero time-lag and zero space-lag crosscorrelation imaging conditions are not ideal for imaging of converted-mode waves because no conversion arises from zero-offset experiments.

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Min, D. J. "Free-Surface Boundary Condition in Finite-Difference Elastic Wave Modeling." Bulletin of the Seismological Society of America 94, no. 1 (February 1, 2004): 237–50. http://dx.doi.org/10.1785/0120020116.

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Khabirov, S. V., and S. S. Khabirov. "Self-similar Elastic Condition of Filtration Through the Moving Boundary." Lobachevskii Journal of Mathematics 40, no. 11 (November 2019): 1950–58. http://dx.doi.org/10.1134/s1995080219110167.

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Liu, Yang, and Mrinal K. Sen. "A hybrid absorbing boundary condition for elastic staggered-grid modelling." Geophysical Prospecting 60, no. 6 (February 14, 2012): 1114–32. http://dx.doi.org/10.1111/j.1365-2478.2011.01051.x.

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Liu, Xin, Yang Liu, Zhi-Ming Ren, Xiao-Hui Cai, Bei Li, Shi-Gang Xu, and Le-Kai Zhou. "Hybrid absorbing boundary condition for three-dimensional elastic wave modeling." Applied Geophysics 14, no. 2 (June 2017): 270–78. http://dx.doi.org/10.1007/s11770-017-0623-z.

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Zhang, Zhaoyuan, Haohan Lin, Wei Li, and Luyan Ju. "Linear Displacement Boundary Condition Replacing Periodic Boundary Condition to Easily and Efficiently Homogenize the Elastic Properties of Composites." Integrated Ferroelectrics 240, no. 3 (March 23, 2024): 573–83. http://dx.doi.org/10.1080/10584587.2024.2324684.

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RABINOVICH, DANIEL, DAN GIVOLI, THOMAS HAGSTROM, and JACOBO BIELAK. "STRESS–VELOCITY COMPLETE RADIATION BOUNDARY CONDITIONS." Journal of Computational Acoustics 21, no. 03 (July 3, 2013): 1350003. http://dx.doi.org/10.1142/s0218396x13500033.

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A new high-order local Absorbing Boundary Condition (ABC) has been recently proposed for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. It is based on the stress–velocity formulation of the elastodynamics problem, and on the general Complete Radiation Boundary Condition (CRBC) approach, originally devised by Hagstrom and Warburton in 2009. The work presented here is a sequel to previous work that concentrated on the stability of the scheme; this is the first known high-order ABC for elastodynamics which is long-time stable. Stability was established both theoretically and numerically. The present paper focuses on the accuracy of the scheme. In particular, two accuracy-related issues are investigated. First, the reflection coefficients associated with the new CRBC for different types of incident and reflected elastic waves are analyzed. Second, various choices of computational parameters for the CRBC, and their effect on the accuracy, are discussed. These choices include the optimal coefficients proposed by Hagstrom and Warburton for the acoustic case, and a simplified formula for these coefficients. A finite difference discretization is employed in space and time. Numerical examples are used to experiment with the scheme and demonstrate the above-mentioned accuracy issues.

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Zhang, Xuqing, Yu Zhang, and Yidu Yang. "Guaranteed Lower Bounds for the Elastic Eigenvalues by Using the Nonconforming Crouzeix–Raviart Finite Element." Mathematics 8, no. 8 (July 31, 2020): 1252. http://dx.doi.org/10.3390/math8081252.

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This paper uses a locking-free nonconforming Crouzeix–Raviart finite element to solve the planar linear elastic eigenvalue problem with homogeneous pure displacement boundary condition. Making full use of the Poincaré inequality, we obtain the guaranteed lower bounds for eigenvalues. Besides, we also use the nonconforming Crouzeix–Raviart finite element to the planar linear elastic eigenvalue problem with the pure traction boundary condition, and obtain the guaranteed lower eigenvalue bounds. Finally, numerical experiments with MATLAB program are reported.

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Xu, Yixian, Jianghai Xia, and Richard D. Miller. "Numerical investigation of implementation of air-earth boundary by acoustic-elastic boundary approach." GEOPHYSICS 72, no. 5 (September 2007): SM147—SM153. http://dx.doi.org/10.1190/1.2753831.

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The need for incorporating the traction-free condition at the air-earth boundary for finite-difference modeling of seismic wave propagation has been discussed widely. A new implementation has been developed for simulating elastic wave propagation in which the free-surface condition is replaced by an explicit acoustic-elastic boundary. Detailed comparisons of seismograms with different implementations for the air-earth boundary were undertaken using the (2,2) (the finite-difference operators are second order in time and space) and the (2,6) (second order in time and sixth order in space) standard staggered-grid (SSG) schemes. Methods used in these comparisons to define the air-earth boundary included the stress image method (SIM), the heterogeneous approach, the scheme of modifying material properties based on transversely isotropic medium approach, the acoustic-elastic boundary approach, and an analytical approach. The method proposed achieves the same or higher accuracy of modeled body waves relative to the SIM. Rayleigh waves calculated using the explicit acoustic-elastic boundary approach differ slightly from those calculated using the SIM. Numerical results indicate that when using the (2,2) SSG scheme for SIM and our new method, a spatial step of 16 points per minimum wavelength is sufficient to achieve 90% accuracy; 32 points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. When using the (2,6) SSG scheme for the two methods, a spatial step of eight points per minimum wavelength achieves 95% accuracy in modeled Rayleigh waves. Our proposed method is physically reasonable and, based on dispersive analysis of simulated seismographs from a layered half-space model, is highly accurate. As a bonus, our proposed method is easy to program and slightly faster than the SIM.

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Kamarudin, K. A., and Al Emran Ismail. "Prediction of Elastic Properties for Unidirectional Carbon Composites: Periodic Boundary Condition Approach." Applied Mechanics and Materials 773-774 (July 2015): 262–66. http://dx.doi.org/10.4028/www.scientific.net/amm.773-774.262.

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A three-dimensional finite element model of unidirectional fibre reinforced composites has been investigated numerically using periodic boundary condition method. This method was used to predict the elastic mechanical behaviour of a unit cell of such composites. Periodic boundary condition was used due to its capability to represent a single unit cell similar to the neighbouring unit cells with continuous physical elements. It is assumed that the paired nodes displaced continuously without separating or interrupting other nodes during the deformation step. From the study, it was revealed that the elastic modulus agreed well with the experimental results, indicating that the present model could be used effectively.

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Gahima, Stephan, Pedro Díez, Marco Stefanati, José Félix Rodríguez Matas, and Alberto García-González. "An Unfitted Method with Elastic Bed Boundary Conditions for the Analysis of Heterogeneous Arterial Sections." Mathematics 11, no. 7 (April 6, 2023): 1748. http://dx.doi.org/10.3390/math11071748.

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This manuscript presents a novel formulation for a linear elastic model of a heterogeneous arterial section undergoing uniform pressure in a quasi-static regime. The novelties are twofold. First, an elastic bed support on the external boundary (elastic bed boundary condition) replaces the classical Dirichlet boundary condition (i.e., blocking displacements at arbitrarily selected nodes) for elastic solids to ensure a solvable problem. In addition, this modeling approach can be used to effectively account for the effect of the surrounding material on the vessel. Secondly, to study many geometrical configurations corresponding to different patients, we devise an unfitted strategy based on the Immersed Boundary (IB) framework. It allows using the same (background) mesh for all possible configurations both to describe the geometrical features of the cross-section (using level sets) and to compute the solution of the mechanical problem. Results on coronary arterial sections from realistic segmented images demonstrate that the proposed unfitted IB-based approach provides results equivalent to the standard finite elements (FE) for the same number of active degrees of freedom with an average difference in the displacement field of less than 0.5%. However, the proposed methodology does not require the use of a different mesh for every configuration. Thus, it is paving the way for dimensionality reduction.

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Kong, Xiangsen, Hao Li, Xubin Zhou, Xiujuan Xiang, and Xing Shen. "Flywheel Vibration Isolation of Satellite Structure by Applying Structural Plates with Elastic Boundary Instead of Restrained Boundary." Applied Sciences 13, no. 23 (November 28, 2023): 12756. http://dx.doi.org/10.3390/app132312756.

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Flywheels play a critical role as core components in satellite attitude control systems. However, their high-speed rotation inevitably generates vibrations that have a detrimental impact on the in-orbit imaging capabilities of high-precision remote sensing payloads. This study focuses on the passive vibration isolation design of satellite flywheels. The flywheel-mounted structural plate and flywheel vibration isolation platform are considered as a whole system (termed a plate-isolator system). In this system, the structural plate is treated as an elastomer. By simplifying the plate-isolator system as a 2-degree-of-freedom vibration system, it becomes evident that obtaining an ideal vibration isolation effect through the optimization of the flywheel vibration isolation platform (FVIP) alone is difficult. In order to enhance the passive vibration isolation effect for satellite flywheels, this study introduces the concept of an elastic boundary applied to the flywheel-mounted structural plate, thus treating the elastic boundary as a design factor. Consequently, the plate-isolator system can be simplified as a 3-stage vibration isolation system. The optimization of the elastic boundary condition of the structural plate is performed using the kinetic model of the simplified 3-stage system. The vibration isolation effect of the plate-isolator system with an elastic boundary is further confirmed through finite element simulation. The calculation results demonstrate that, after establishing a reasonable elastic boundary for the satellite structural plate, the overall vibration/force transmission rate of the plate-isolator system becomes similar to that of a single-degree-of-freedom dynamic system. Finally, the proposed concept is validated through kinetic response analysis of a cube satellite. The results reveal that the vibration amplitude of the satellite’s top and side structural plates can be effectively lowered if the elastic boundary condition is set for the flywheel-mounted bottom structural plate.

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Shahzamanian, M. M., and W. J. Basirun. "Modeling of Cementitious Representative Volume Element with Various Water–Cement Ratios." Journal of Multiscale Modelling 11, no. 01 (May 13, 2019): 1950002. http://dx.doi.org/10.1142/s1756973719500021.

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This study uses the finite element method (FEM) to measure the mechanical properties of microstructure-based cementitious representative volume elements (RVEs) with various water–cement ratios (W/Cs) generated by CEMHYD3D. The finite element boundary condition effects that significantly and computationally change the elastic properties are studied and discussed. Various boundary conditions in ABAQUS are applied and compared with the results obtained using the variational asymptotic method for unit cell homogenization (VAMUCH). This comparison is conducted using ANSYS. This study aims to analyze and determine the effect of different boundary conditions in detail on the prediction of the elastic properties of cementitious RVE with various W/Cs and identify the best approach in this regard. Results show that Young’s, shear, and bulk moduli decrease with the increase in W/C and the boundary conditions in ABAQUS influence the outcomes, particularly on bulk modulus and Poisson’s ratio.

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Taylor, Adam G., and Jae H. Chung. "Application of low-order potential solutions to higher-order vertical traction boundary problems in an elastic half-space." Royal Society Open Science 5, no. 5 (May 2018): 180203. http://dx.doi.org/10.1098/rsos.180203.

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New solutions of potential functions for the bilinear vertical traction boundary condition are derived and presented. The discretization and interpolation of higher-order tractions and the superposition of the bilinear solutions provide a method of forming approximate and continuous solutions for the equilibrium state of a homogeneous and isotropic elastic half-space subjected to arbitrary normal surface tractions. Past experimental measurements of contact pressure distributions in granular media are reviewed in conjunction with the application of the proposed solution method to analysis of elastic settlement in shallow foundations. A numerical example is presented for an empirical ‘saddle-shaped’ traction distribution at the contact interface between a rigid square footing and a supporting soil medium. Non-dimensional soil resistance is computed as the reciprocal of normalized surface displacements under this empirical traction boundary condition, and the resulting internal stresses are compared to classical solutions to uniform traction boundary conditions.

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Zhao, J. X., and H. Qi. "Scattering of Plane Sh-Wave From a Partially Debonded Shallow Cylindrical Elastic Inclusion." Journal of Mechanics 25, no. 4 (December 2009): 411–19. http://dx.doi.org/10.1017/s1727719100002896.

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ABSTRACTThe scattering of plane SH-wave from a partially debonded shallow cylindrical elastic inclusion in half space is investigated in this paper by complex function method and expansion method of wave function. The debonding regions are considered as multiple arc-shaped interface cracks with non-contacting faces. Firstly, in the inclusion district, the standing wave function in the elastic inclusion with unknown coefficients which satisfies the boundary condition is constructed and generated into the Fourier series; in the half space, the stress and displacement boundary condition around the elastic inclusion can be modeled as the same as the standing wave function in the elastic inclusion. Then, a set of infinite algebraic equations can be obtained around the same boundary and the solution of problem can be gained. In the end, numerical examples of the surface displacement are provided and discussed. It is found that the interface cracks can raise the surface displacement amplitudes to a certain degree.

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Dudko, O. V., A. A. Lapteva, and V. E. Ragozina. "Nonstationary 1D Dynamics Problems for Heteromodular Elasticity with Piecewise-Linear Approximation of Boundary Conditions." PNRPU Mechanics Bulletin, no. 4 (December 15, 2019): 37–47. http://dx.doi.org/10.15593/perm.mech/2019.4.04.

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The paper provides the investigation of a heteromodular elastic medium under dynamic loading. The heteromodularity (when the stress - strain relation depends on the deformation direction) is a distinctive feature of many natural and structural materials: rocks, porous and cohesive bulk media, fibrous and granular composites, some metal alloys, etc. The fact that the listed materials show the heteromodular property at the stage of elastic deformation should be especially taken into account when solving problems of their shock dynamics. To describe the heteromodular behavior of an elastic medium in terms of small strains we use the physically nonlinear model of V.P. Myasnikov. The accepted assumption about the one-dimensional straining reduces the nonlinear relationship of stresses and small strains to piecewise linear equations. In the case of dynamic shock deformation, the initial nonlinearity of the model is concentrated in the equations which define the velocity of the shock wave abruptly transforming the heteromodular medium from a stretched to a compressed state. In this paper we investigate the processes of generation, motion, and possible interactions of plane one-dimensional deformation waves (including shock ones) in a heteromodular elastic half-space. The points of the half-space boundary undergo one-dimensional motions according to a given non-linear law corresponding to the “stretching-compression” mode. We suggest replacing the nonstationary boundary condition of the problem by its piecewise linear approximation and constructing a connected sequence of analytical solutions with a linear boundary condition at each local time interval. The proposed approach is the basis of the numerical solving algorithm for a boundary value problem with a given nonlinear condition. It is shown that the general solution behind the shock wave consists of several local layers, which number is related to the quantity of nodes in the piecewise linear decomposition of the boundary condition. In these layers, the compression deformation is defined by the relevant part of the boundary condition and simultaneously “stores” information on the preliminary tension, which should be considered an important feature of the heteromodular medium dynamics.

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Shekhan, Husain N., Erkan A. Gurdal, Lalitha Ganapatibhotla, Janna K. Maranas, Ron Staut, and Kenji Uchino. "Thermal Conductivities of PZT Piezoelectric Ceramics under Different Electrical Boundary Conditions." Insight - Material Science 3, no. 1 (March 17, 2020): 10. http://dx.doi.org/10.18282/ims.v3i1.301.

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Physical properties of polycrystalline lead-zirconate-titanate (PZT) changes according to electrical boundary conditions and poling. This paper reports the thermal properties of poled and unpoled PZT's in the poling direction for open circuit and short circuit conditions. The authors found that the short-circuit condition exhibited the largest thermal conductivity than the open-circuit condition. In the relationship between these two thermal properties, the authors propose the "electrothermal" coupling factor kκ33, which is similar to the electromechanical coupling factor k33 relating the elastic compliances under short- and open-circuit conditions. On the other hand, the thermal conductivity of the unpoled specimen exhibits the lowest thermal conductivity, in comparison with the poled specimens, which suggests the importance of phonon mode scattering on the thermal conductivity with respect to elastic compliance.

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Xia, Qi, and Tielin Shi. "Multiphysics Topology Optimization of Thermal Actuators by Using the Level Set-Based Multiple-Type Boundary Method." International Journal of Computational Methods 17, no. 08 (June 20, 2019): 1950044. http://dx.doi.org/10.1142/s0219876219500440.

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Thermal actuators use thermal expansion and contraction of an elastic body to produce motion at its output port. In the present study, a thermal actuator comprises an elastic body and heating/cooling devices. Such devices have a thin-layer shape and are installed on the surface of the elastic body. The design optimization of thermal actuator is a multiphysics problem, including both heat conduction and elastic deformation. The design variables include multiple types of boundaries, i.e., the temperature boundary (high temperature and low temperature) and the free boundary. In order to solve such a multiphysics optimization problem involving multiple types of boundaries, the level set-based multiple-type boundary method is employed. In the analysis for the shape derivative of the temperature boundary, the constrained variational principle is employed to explicitly include the temperature boundary condition into the weak form of heat conduction equation. Numerical examples in two dimensions are investigated.

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Journal articles on the topic 'Elastic boundary condition'

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