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Showing papers on "Boundary value problem published in 2012"


Book
17 Jan 2012
TL;DR: In this paper, the authors present an extension of the Dirichlet problem for the case of Perforated Domains with a Non-Periodic Structure, where the boundary value problem is solved with Neumann conditions on the outer part of the boundary and on the surface of the Cavities.
Abstract: Some Mathematical Problems of the Theory of Elasticity. Some Functional Spaces and Their Properties. Auxiliary Propositions. Korn's Inequalities. Boundary Value Problems of Linear Elasticity. Perforated Domains with a Periodic Structure. Extension Theorems. Estimates for Solutions of Boundary Value Problems of Elasticity in Perforated Domains. Periodic Solutions of Boundary Value Problems for the System of Elasticity. Saint-Venant's Principle for Periodic Solutions of the Elasticity System. Estimates and Existence Theorems for Solutions of the Elasticity System in Unbounded Domains. Strong G -Convergence of Elasticity Operators. Homogenization of the System of Linear Elasticity. Composites and Perforated Materials. The Mixed Problem in a Perforated Domain with the Dirichlet Boundary Conditions on the Outer Part of the Boundary and the Neumann Conditions on the Surface of the Cavities. The Boundary Value Problem with Neumann Conditions in a Perforated Domain. Asymptotic Expansions for Solutions of Boundary Value Problems of Elasticity in a Perforated Layer. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Elasticity System in a Perforated Domain. Asymptotic Expansions for Solutions of the Dirichlet Problem for the Biharmonic Equation. Some Generalizations for the Case of Perforated Domains with a Non-Periodic Structure. Homogenization of the System of Elasticity with Almost-Periodic Coefficients. Homogenization of Stratified Structures. Estimates for the Rate of G -Convergence of Higher-Order Elliptic Operators. Spectral Problems . Some Theorems from Functional Analysis. Spectral Problems for Abstract Operators. Homogenization of Eigenvalues and Eigenfunctions of Boundary Value Problems for Strongly Non-Homogeneous Elastic Bodies. On the Behaviour of Eigenvalues and Eigenfunctions of the Dirichlet Problem for Second Order Elliptic Equations in Perforated Domains. Third Boundary Value Problem for Second Order Elliptic Equations in Domains with Rapidly Oscillating Boundary. Free Vibrations of Bodies with Concentrated Masses. On the Behaviour of Eigenvalues of the Dirchlet Problem in Domains with Cavities Whose Concentration is Small. Homogenization of Eigenvalues of Ordinary Differential Operators. Asymptotic Expansion of Eigenvalues and Eigenfunctions of the Sturm-Liouville Problem for Equations with Rapidly Oscillating Coefficients. On the Behaviour of the Eigenvalues and Eigenfunctions of a G -Convergent Sequence of Non-Self-Adjoint Operators. References.

838 citations


Journal ArticleDOI
TL;DR: It is shown that expansion in DMD modes is unique under certain conditions, and an “optimized” DMD is introduced that computes an arbitrary number of dynamical modes from a data set and is superior at calculating physically relevant frequencies, and is less numerically sensitive.
Abstract: Dynamic mode decomposition (DMD) is an Arnoldi-like method based on the Koopman operator. It analyzes empirical data, typically generated by nonlinear dynamics, and computes eigenvalues and eigenmodes of an approximate linear model. Without explicit knowledge of the dynamical operator, it extracts frequencies, growth rates, and spatial structures for each mode. We show that expansion in DMD modes is unique under certain conditions. When constructing mode-based reduced-order models of partial differential equations, subtracting a mean from the data set is typically necessary to satisfy boundary conditions. Subtracting the mean of the data exactly reduces DMD to the temporal discrete Fourier transform (DFT); this is restrictive and generally undesirable. On the other hand, subtracting an equilibrium point generally preserves the DMD spectrum and modes. Next, we introduce an “optimized” DMD that computes an arbitrary number of dynamical modes from a data set. Compared to DMD, optimized DMD is superior at calculating physically relevant frequencies, and is less numerically sensitive. We test these decomposition methods on data from a two-dimensional cylinder fluid flow at a Reynolds number of 60. Time-varying modes computed from the DMD variants yield low projection errors.

672 citations


Book
22 Dec 2012
TL;DR: In this paper, the Explicit Method and the Explicit Methods of the Implicit Methods are discussed, as well as several other methods, such as adorption and convection of two-dimensional systems.
Abstract: Introduction. Basic Equations. Approximations to Derivatives. Ordinary Differential Equations. The Explicit Method. Boundary Conditions. Unequal Intervals. The Commonly Used Implicit Methods. Other Methods. Adsorption. Effects Due to Uncompensated Resistance and Capacitance. Two-Dimensional Systems. Convection. Performance. Programming. Simulation Packages. Some Mathematical Proofs. Useful Procedures. Example Programs.

548 citations


Journal ArticleDOI
TL;DR: A new formulation of the boundary condition at static and moving solid walls in SPH simulations based on a local force balance between wall and fluid particles and applies a pressure boundary condition on the solid particles to prevent wall penetration.

539 citations


BookDOI
01 Jan 2012
TL;DR: Sobolev spaces with weights and applications to the boundary value problems are discussed in this article, where the authors consider the case of irregular domains and elliptic problems with variable coefficients and show that these problems can be solved by Rellich inequalities.
Abstract: 1.Introduction to the problem.- 2.Sobolev spaces.- 3.Exitence, Uniqueness of basic problems.- 4.Regularity of solution.- 5.Applications of Rellich's inequalities and generalization to boundary value problems.- 6.Sobolev spaces with weights and applications to the boundary value problems.- 7.Regularity of solutions in case of irregular domains and elliptic problems with variable coefficients.

487 citations


Journal ArticleDOI
TL;DR: In this paper, the classical Nitsche type weak boundary conditions are extended to a fictitious domain setting and an additional penalty term, acting on the jumps of the gradients over element faces in the interface zone, is added to ensure that the conditioning of the matrix is independent of how the boundary cuts the mesh.

430 citations


Journal ArticleDOI
TL;DR: In this article, an original hyperbolic sine shear deformation theory for the bending and free vibration analysis of functionally graded plates is presented, which accounts for through-the-thickness deformations.

412 citations


Book
27 Sep 2012
TL;DR: In this article, the authors present a flow in a tube and demonstrate the properties of the flow in terms of velocity, acceleration, and balance of forces on a fluid element, as well as its properties.
Abstract: 1 Preliminary Concepts.- 1.1 Flow in a Tube.- 1.2 What Is a Fluid?.- 1.3 Microscopic and Macroscopic Scales.- 1.4 What Is Flow?.- 1.5 Eulerian and Lagrangian Velocities.- 1.6 Acceleration in a Flow Field.- 1.7 Is Blood a Newtonian Fluid?.- 1.8 No-Slip Boundary Condition.- 1.9 Laminar and Turbulent Flow.- 1.10 Problems.- 1.11 References and Further Reading.- 2 Equations of Fluid Flow.- 2.1 Introduction.- 2.2 Equations at a Point.- 2.3 Equations and Unknowns.- 2.4 Conservation of Mass: Equation of Continuity.- 2.5 Momentum Equations.- 2.6 Forces on a Fluid Element.- 2.7 Constitutive Equations.- 2.8 Navier-Stokes Equations.- 2.9 Problems.- 2.10 References and Further Reading.- 3 Steady Flow in Tubes.- 3.1 Introduction.- 3.2 Simplified Equations.- 3.3 Steady-State Solution: Poiseuille Flow.- 3.4 Properties of Poiseuille Flow.- 3.5 Balance of Energy Expenditure.- 3.6 Cube Law.- 3.7 Arterial Bifurcation.- 3.8 Arterial Tree.- 3.9 Entry Length.- 3.10 Noncircular Cross Section.- 3.11 Problems.- 3.12 References and Further Reading.- 4 Pulsatile Flow in a Rigid Tube.- 4.1 Introduction.- 4.2 Oscillatory Flow Equations.- 4.3 Fourier Analysis.- 4.4 Bessel Equation.- 4.5 Solution of Bessel Equation.- 4.6 Oscillatory Velocity Profiles.- 4.7 Oscillatory Flow Rate.- 4.8 Oscillatory Shear Stress.- 4.9 Pumping Power.- 4.10 Oscillatory Flow at Low Frequency.- 4.11 Oscillatory Flow at High Frequency.- 4.12 Tubes of Elliptic Cross Sections.- 4.13 Problems.- 4.14 References and Further Reading.- 5 Pulsatile Flow in an Elastic Tube.- 5.1 Introduction.- 5.2 Bessel Equations and Solutions.- 5.3 Balance of Forces.- 5.4 Equations of Wall Motion.- 5.5 Coupling with Fluid Motion.- 5.6 Matching at the Tube Wall.- 5.7 Wave Speed.- 5.8 Arbitrary Constants.- 5.9 Flow Properties.- 5.10 Problems.- 5.11 References and Further Reading.- 6 Wave Reflections.- 6.1 Introduction.- 6.2 One-Dimensional Wave Equations.- 6.3 Basic Solution of Wave Equation.- 6.4 Primary Wave Reflections in a Tube.- 6.5 Secondary Wave Reflections in a Tube.- 6.6 Pressure-Flow Relations.- 6.7 Effective Admittance.- 6.8 Vascular Tree Structure.- 6.9 Problems.- 6.10 References and Further Reading.- Appendix B. Solutions to Problems.

384 citations


Journal ArticleDOI
TL;DR: NHWAVE as mentioned in this paper is a shock-capturing non-hydrostatic model for simulating wave refraction, diffraction, shoaling, breaking and landslide-generated tsunami in finite water depth.

380 citations


Journal ArticleDOI
TL;DR: It is proved that the proposed adaptive scheme leads to a sequence of discrete solutions, for which the corresponding error estimators tend to zero, under a saturation assumption for the non-perturbed problem which is observed empirically.

369 citations


Journal ArticleDOI
TL;DR: In this paper, the nonlinear free vibration of microbeams made of functionally graded materials (FGMs) is investigated based on the modified couple stress theory and von Karman geometric nonlinearity.


Journal ArticleDOI
TL;DR: It is shown that the basic scheme is inconsistent when moving surfaces are allowed to approach closer than twice the step size, and a remedy is developed based on excluding from the force computation all surface markers whose stencil overlaps with the stencil of a marker located on the surface of a collision partner.

Journal ArticleDOI
TL;DR: In this article, various higher-order shear deformation beam theories for bending and free vibration of functionally graded beams are developed, which have strong similarities with Euler-Bernoulli beam theory in some aspects such as equations of motion, boundary conditions and stress resultant expressions.

Journal ArticleDOI
TL;DR: In this paper, a trigonometric shear deformation theory for isotropic and composite laminated and sandwich plates is developed, which accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required.

Journal ArticleDOI
TL;DR: In this paper, the shifted Jacobi operational matrix (JOM) of fractional derivatives was derived and applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs).

Journal ArticleDOI
TL;DR: In this article, a new shear deformation theory for sandwich and composite plates is developed, which accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required.
Abstract: A new shear deformation theory for sandwich and composite plates is developed. The proposed displacement field, which is “m” parameter dependent, is assessed by performing several computations of the plate governing equations. Therefore, the present theory, which gives accurate results, is relatively close to 3D elasticity bending solutions. The theory accounts for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surface, thus a shear correction factor is not required. Plate governing equations and boundary conditions are derived by employing the principle of virtual work. The Navier-type exact solutions for static bending analysis are presented for sinusoidally and uniformly distributed loads. The accuracy of the present theory is ascertained by comparing it with various available results in the literature.

Journal ArticleDOI
TL;DR: In this article, the existence of positive solutions for a class of nonlinear boundary value problems with integral boundary conditions was studied, based on the known Guo-Krasnoselskii fixed point theorem.

Journal ArticleDOI
TL;DR: In this article, a (1 + 1)-dimensional directed polymers with log-gamma distributed weights were studied and the expected upper bounds on the exponents of the free energy and the polymer path were derived.
Abstract: We study a (1 + 1)-dimensional directed polymer in a random environment on the integer lattice with log-gamma distributed weights. Among directed polymers, this model is special in the same way as the last-passage percolation model with exponential or geometric weights is special among growth models, namely, both permit explicit calculations. With appropriate boundary conditions, the polymer with log-gamma weights satisfies an analogue of Burke’s theorem for queues. Building on this, we prove the conjectured values for the fluctuation exponents of the free energy and the polymer path, in the case where the boundary conditions are present and both endpoints of the polymer path are fixed. For the polymer without boundary conditions and with either fixed or free endpoint, we get the expected upper bounds on the exponents.

Journal ArticleDOI
TL;DR: Recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles is described.
Abstract: In this article we describe recent progress on the design, analysis and implementation of hybrid numerical-asymptotic boundary integral methods for boundary value problems for the Helmholtz equation that model time harmonic acoustic wave scattering in domains exterior to impenetrable obstacles These hybrid methods combine conventional piecewise polynomial approximations with high-frequency asymptotics to build basis functions suitable for representing the oscillatory solutions They have the potential to solve scattering problems accurately in a computation time that is (almost) independent of frequency and this has been realized for many model problems The design and analysis of this class of methods requires new results on the analysis and numerical analysis of highly oscillatory boundary integral operators and on the high-frequency asymptotics of scattering problems The implementation requires the development of appropriate quadrature rules for highly oscillatory integrals This article contains a historical account of the development of this currently very active field, a detailed account of recent progress and, in addition, a number of original research results on the design, analysis and implementation of these methods

Journal ArticleDOI
TL;DR: An improved third order shear deformation theory is employed to formulate a governing equation for predicting free vibration of layered functionally graded beams in this article, where the Ritz method is adopted to solve the governing equation.

Journal ArticleDOI
TL;DR: In this paper, a continuous graded carbon nanotube-reinforced (CGCNTR) cylindrical panels based on the Eshelby-Mori-Tanaka approach is considered.
Abstract: In this paper, natural frequencies characteristics of a continuously graded carbon nanotube-reinforced (CGCNTR) cylindrical panels based on the Eshelby–Mori–Tanaka approach is considered. The volume fractions of oriented, straight single-walled carbon nanotubes (SWCNTs) are assumed to be graded in the thickness direction. In this research work, an equivalent continuum model based on the Eshelby–Mori–Tanaka approach is employed to estimate the effective constitutive law of the elastic isotropic medium (matrix) with oriented, straight carbon nanotubes (CNTs). The CGCNTR shell is assumed to be simply supported at one pair of opposite edges and arbitrary boundary conditions at the other edges such that trigonometric functions expansion can be used to satisfy the boundary conditions precisely at simply supported edges. The 2-D generalized differential quadrature method (GDQM) as an efficient and accurate numerical tool is used to discretize the governing equations and to implement the boundary conditions. The novelty of the present work is to exploit Eshelby–Mori–Tanaka approach in order to reveal the impacts of the volume fractions of oriented CNTs, different CNTs distributions, various mid radius-to-thickness ratio, shell angle, length-to-mean radius ratio and different combinations of free, simply supported and clamped boundary conditions on the vibrational characteristics of CGCNTR cylindrical panels. The interesting and new results show that continuously graded oriented CNTs volume fractions can be utilized for the management of vibrational behavior of structures so that the frequency parameters of structures made of such material can be considerably improved than that of the nanocomposites reinforced with uniformly distributed CNTs.

Journal ArticleDOI
TL;DR: In this paper, a nonlinear Langevin equation involving two fractional orders α ∈ ( 0, 1 ] and β ∈( 1, 2 ] with three-point boundary conditions was studied and the contraction mapping principle was applied to prove the existence of solutions.
Abstract: This paper studies a nonlinear Langevin equation involving two fractional orders α ∈ ( 0 , 1 ] and β ∈ ( 1 , 2 ] with three-point boundary conditions. The contraction mapping principle and Krasnoselskii’s fixed point theorem are applied to prove the existence of solutions for the problem. The existence results for a three-point third-order nonlocal boundary value problem of nonlinear ordinary differential equations follow as a special case of our results. Some illustrative examples are also discussed.

Journal ArticleDOI
TL;DR: The concept of piecewise continuous solutions for impulsive Cauchy problems and impulsive boundary value problems respectively are introduced by using a new fixed point theorem and many new existence, uniqueness and data dependence results of solutions are obtained via some generalized singular Gronwall inequalities.
Abstract: In this paper, the first purpose is treating Cauchy problems and boundary value problems for nonlinear impulsive differential equations with Caputo fractional derivative. We introduce the concept of piecewise continuous solutions for impulsive Cauchy problems and impulsive boundary value problems respectively. By using a new fixed point theorem, we obtain many new existence, uniqueness and data dependence results of solutions via some generalized singular Gronwall inequalities. The second purpose is discussing Ulam stability for impulsive fractional differential equations. Some new concepts in stability of impulsive fractional differential equations are offered from different perspectives. Some applications of our results are also provided.

Journal ArticleDOI
TL;DR: This work develops a new method based on polynomial interpolation that avoids the need of matching mesh condition on opposite RVE boundaries.

Journal ArticleDOI
TL;DR: This work studies the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation, included in the energy equation, and variable wall temperature.
Abstract: In this work, we study the flow and heat transfer characteristics of a viscous nanofluid over a nonlinearly stretching sheet in the presence of thermal radiation, included in the energy equation, and variable wall temperature. A similarity transformation was used to transform the governing partial differential equations to a system of nonlinear ordinary differential equations. An efficient numerical shooting technique with a fourth-order Runge-Kutta scheme was used to obtain the solution of the boundary value problem. The variations of dimensionless surface temperature, as well as flow and heat-transfer characteristics with the governing dimensionless parameters of the problem, which include the nanoparticle volume fraction ϕ, the nonlinearly stretching sheet parameter n, the thermal radiation parameter NR, and the viscous dissipation parameter Ec, were graphed and tabulated. Excellent validation of the present numerical results has been achieved with the earlier nonlinearly stretching sheet problem of Cortell for local Nusselt number without taking the effect of nanoparticles.

Journal ArticleDOI
TL;DR: In this paper, the boundary condition at the surface of the particles at the level of the discrete momentum and thermal energy equations of the fluid is incorporated with a second-order method.

Journal ArticleDOI
TL;DR: In this paper, a refined beam formulation with displacement variables is proposed, in which Lagrange-type polynomials are used to interpolate the displacement field over the beam cross-section.
Abstract: This paper proposes a refined beam formulation with displacement variables only. Lagrange-type polynomials, in fact, are used to interpolate the displacement field over the beam cross-section. Three- (L3), four- (L4), and nine-point (L9) polynomials are considered which lead to linear, quasi-linear (bilinear), and quadratic displacement field approximations over the beam cross-section. Finite elements are obtained by employing the principle of virtual displacements in conjunction with the Unified Formulation (UF). With UF application the finite element matrices and vectors are expressed in terms of fundamental nuclei whose forms do not depend on the assumptions made (L3, L4, or L9). Additional refined beam models are implemented by introducing further discretizations over the beam cross-section in terms of the implemented L3, L4, and L9 elements. A number of numerical problems have been solved and compared with results given by classical beam theories (Euler-Bernoulli and Timoshenko), refined beam theories based on the use of Taylor-type expansions in the neighborhood of the beam axis, and solid element models from commercial codes. Poisson locking correction is analyzed. Applications to compact, thin-walled open/closed sections are discussed. The investigation conducted shows that: (1) the proposed formulation is very suitable to increase accuracy when localized effects have to be detected; (2) it leads to shell-like results in case of thin-walled closed cross-section analysis as well as in open cross-section analysis; (3) it allows us to modify the boundary conditions over the cross-section easily by introducing localized constraints; (4) it allows us to introduce geometrical boundary conditions along the beam axis which lead to plate/shell-like cases.

Journal ArticleDOI
TL;DR: A general bounce-back scheme is proposed to implement concentration or thermal boundary conditions of convection-diffusion equation with the lattice Boltzmann method and the proposed scheme has second-order accuracy.
Abstract: In this paper, a general bounce-back scheme is proposed to implement concentration or thermal boundary conditions of convection-diffusion equation with the lattice Boltzmann method (LBM). Using this scheme, the general concentration boundary conditions, i.e., b1(∂Cw/∂n) + b2Cw = b3, can be easily implemented at boundaries with complex geometry structure like that in porous media. The numerical results obtained using the present scheme are in excellent agreement with the analytical solutions of flows with both stationary and moving interfaces. Furthermore, to better understand the halfway bounce-back scheme, an analytical study of the concentration jump is presented. The studies of theoretical analysis and numerical experiments demonstrate that the proposed scheme has second-order accuracy.

Journal ArticleDOI
TL;DR: In this paper, a bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories.
Abstract: Bending analysis of micro-sized beams based on the Bernoulli-Euler beam theory is presented within the modified strain gradient elasticity and modified couple stress theories. The governing equations and the related boundary conditions are derived from the variational principles. These equations are solved analytically for deflection, bending, and rotation responses of micro-sized beams. Propped cantilever, both ends clamped, both ends simply supported, and cantilever cases are taken into consideration as boundary conditions. The influence of size effect and additional material parameters on the static response of micro-sized beams in bending is examined. The effect of Poisson’s ratio is also investigated in detail. It is concluded from the results that the bending values obtained by these higher-order elasticity theories have a significant difference with those calculated by the classical elasticity theory.