Vibrations of angle-ply laminated circular cylindrical shells subjected to different sets of edge boundary conditions

This paper studies the free vibrations of finite, closed, circular cylindrical shells, made of one or more monoclinic layers. The study is based on the Love-type version of a unified shear-deformable shell theory. This theory enables the trial and testing of different through-thickness transverse shear-strain distributions and, among them, strain distributions that do not involve the undesirable implications of the transverse-shear correction factors. For flexural vibrations, the analytical solution of the corresponding axisymmetric solution is obtained, as a particular case, when it is assumed that the free-vibration pattern is independent of the circumferential co-ordinate parameter. If the appropriate material simplifications are employed, the present analysis yields, as a further particular case, the corresponding free-vibration solution that has already been presented elsewhere for cross-ply laminated cylindrical shells.     

轴向运动黏弹性Timoshenko梁横向非线性强受迫振动

研究轴向运动黏弹性Timoshenko梁横向非线性强受迫振动的稳态响应。由广义Hamilton变分原理推导出轴向运动黏弹性Timoshenko梁横向振动的控制方程及相应的边界条件。模型中考虑剪切模量、转动惯量对梁的影响。黏弹性本构关系中运用Kelvin模型并引入物质时间导数。对控制方程施用直接多尺度法,建立强受迫共振的可解性条件,得到稳态响应振幅与激励频率关系曲线。应用Routh-Hurwitz判据判断稳态响应振幅的稳定性。利用数值结果给出不同参数下,如非线性系数、激励振幅与黏弹性阻尼等对稳态幅频响应及稳定性影响。     

Coupled three‐layered analysis of smart piezoelectric beams with different electric boundary conditions

This paper presents a theoretical and finite element (FE) formulation of a three‐layered smart beam with two piezoelectric layers acting as sensors or actuators. For the definition of the mechanical model a partial layerwise theory is considered for the approximation of the displacement field of the core and piezoelectric face layers. An electrical model for different electric boundary conditions (EBC), namely, electroded layers with either closed‐ or open‐circuit electrodes with electric potential prescribed or layers without electrodes, is considered. Using a variational formulation, the direct piezoelectric effect for the different EBC is physically incorporated into the mechanical model through appropriate approximations of the electric field in the axial and transverse directions. An FE model of a three‐layered smart beam with different EBC is proposed considering a fully coupled electro‐mechanical theory through the use of effective stiffness parameters and a modified static condensation. FE solutions of the quasi‐static electrical and mechanical actuations and natural frequencies are presented. Comparisons with numerical FE and analytical solutions available in the literature demonstrate the representativeness of the developed theory and the effectiveness of the proposed FE model for different EBC. Copyright © 2005 John Wiley & Sons, Ltd.     

Inverse problem of fracture mechanics for a circular disc under mixed boundary conditions

Solution method for the problem of prevention of early brittle fracture of a circular disc with mixed boundary conditions was proposed. Theoretical analysis on determination of normal displacement of points on the boundary of a circular disc weakened by arbitrarily allocated rectilinear cracks was carried out. A closed system of algebraic equations permitting to provide minimization of fracture parameters (stress intensity factors) subject to geometrical and mechanical characteristics of a disc was constructed. Minimization of the stress intensity factors in a circular disc was carried out. They found normal displacement of points on the boundary of the circular disc increases carrying capacity of the disc.     

Approximate imposition of boundary conditions in immersed boundary methods

We analyze several possibilities to prescribe boundary conditions in the context of immersed boundary methods. As basic approximation technique we consider the finite element method with a mesh that does not match the boundary of the computational domain, and therefore Dirichlet boundary conditions need to be prescribed in an approximate way. As starting variational approach we consider Nitsche's methods, and we then move to two options that yield non‐symmetric problems but that turned out to be robust and efficient. The essential idea is to use the degrees of freedom of certain nodes of the finite element mesh to minimize the difference between the exact and the approximated boundary condition. Copyright © 2009 John Wiley & Sons, Ltd.     

不同边界条件和应变振幅对各向异性层合阻尼结构内耗的影响

制备了由纤维增强树脂复合材料层与黏弹阻尼材料层交替层合的7层各向异性层合阻尼结构, 借助动态热机械分析仪(DMA Q800)首次考察了不同的结构应变振幅和不同的边界条件对该结构内耗温度频率特性的影响, 为新型减振降噪阻尼结构的理论分析与应用提供相应的实验研究依据。结果表明: 在不同的边界条件和常温(25℃)下, 各向异性层合阻尼结构的内耗都随着结构应变振幅的增加而减少, 且结构内耗峰所对应的温度随应变振幅的增加而向低温方向移动; 当结构应变振幅相同时, 单悬臂梁模式下的结构内耗最高, 双悬臂梁模式下次之, 三点弯曲模式下最低。      

Imposition of boundary conditions by modifying the weighting coefficient matrices in the differential quadrature method

One of the important issues in the implementation of the differential quadrature method is the imposition of the given boundary conditions. There may be multiple boundary conditions involving higher‐order derivatives at the boundary points. The boundary conditions can be imposed by modifying the weighting coefficient matrices directly. However, the existing method is not robust and is known to have many limitations. In this paper, a systematic procedure is proposed to construct the modified weighting coefficient matrices to overcome these limitations. The given boundary conditions are imposed exactly. Furthermore, it is found that the numerical results depend only on those sampling grid points where the differential quadrature analogous equations of the governing differential equations are established. The other sampling grid points with no associated boundary conditions are not essential. Copyright © 2002 John Wiley & Sons, Ltd.     

Inverse problems for estimating bottom boundary conditions of natural waters

Unknown boundary conditions for natural waters are estimated using an inverse problem methodology. In the formulation of the inverse problem, expressed as a non‐linear constrained optimization problem, its objective function is given by a square difference term, between experimental and computed data, added to a regularization operator. The computed data are obtained by solving the radiative transfer direct problem using the LTS_N method. A key aspect to get a good reconstruction is played by observed data quality. The reconstruction strategy is examined for in situ radiance and irradiance data for many arrangements of the experimental grid of the measurement devices, in order to plan good designs for experimental works. Copyright © 2002 John Wiley & Sons, Ltd.     

Imposing Dirichlet boundary conditions in hierarchical Cartesian meshes by means of stabilized Lagrange multipliers

The use of Cartesian meshes independent of the geometry has some advantages over the traditional meshes used in the finite element method. The main advantage is that their use together with an appropriate hierarchical data structure reduces the computational cost of the finite element analysis. This improvement is based on the substitution of the traditional mesh generation process by an optimized procedure for intersecting the Cartesian mesh with the boundary of the domain and the use efficient solvers based on the hierarchical data structure. One major difficulty associated to the use of Cartesian grids is the fact that the mesh nodes do not, in general, lie over the boundary of the domain, increasing the difficulty to impose Dirichlet boundary conditions. In this paper, Dirichlet boundary conditions are imposed by means of the Lagrange multipliers technique. A new functional has been added to the initial formulation of the problem that has the effect of stabilizing the problem. The technique here presented allows for a simple definition of the Lagrange multipliers field that even allow us to directly condense the degrees of freedom of the Lagrange multipliers at element level. Copyright © 2014 John Wiley & Sons, Ltd.     

Stress-controlled weakly periodic boundary conditions: Axial stress under varying orientations

The accuracy of multiscale modeling approaches for the analysis of heterogeneous materials hinges on the representativeness of the micromodel. One of the issues that affects this representativeness is the application of appropriate boundary conditions. Periodic boundary conditions are the most common choice. However, when localization takes place, periodic boundary conditions tend to overconstrain the microscopic problem. Weakly periodic boundary conditions have been proposed to overcome this effect. In this study, the effectiveness of weakly periodic boundary conditions in restoring transverse isotropy of representative volume elements (RVE) for a fiber-reinforced composite with elastoplastic matrix is investigated. The formulation of weakly periodic boundary conditions is extended to allow for force-controlled simulations where a uniaxial stress can be applied. A series of simulations is performed where the orientation of applied stress is gradually varied and the influence of this orientation on the averaged response is examined. An original method is presented to test the correlation between the ultimate principal stress and average localization angle of shear bands within an RVE. It is concluded that weakly periodic boundary conditions alleviate anisotropy in the RVE response but do not remove it.     

Accurate radiation boundary conditions for the time‐dependent wave equation on unbounded domains

Asymptotic and exact local radiation boundary conditions (RBC) for the scalar time‐dependent wave equation, first derived by Hagstrom and Hariharan, are reformulated as an auxiliary Cauchy problem for each radial harmonic on a spherical boundary. The reformulation is based on the hierarchy of local boundary operators used by Bayliss and Turkel which satisfy truncations of an asymptotic expansion for each radial harmonic. The residuals of the local operators are determined from the solution of parallel systems of linear first‐order temporal equations. A decomposition into orthogonal transverse modes on the spherical boundary is used so that the residual functions may be computed efficiently and concurrently without altering the local character of the finite element equations. Since the auxiliary functions are based on residuals of an asymptotic expansion, the proposed method has the ability to vary separately the radial and transverse modal orders of the RBC. With the number of equations in the auxiliary Cauchy problem equal to the transverse mode number, this reformulation is exact. In this form, the equivalence with the closely related non‐reflecting boundary condition of Grote and Keller is shown. If fewer equations are used, then the boundary conditions form high‐order accurate asymptotic approximations to the exact condition, with corresponding reduction in work and memory. Numerical studies are performed to assess the accuracy and convergence properties of the exact and asymptotic versions of the RBC. The results demonstrate that the asymptotic formulation has dramatically improved accuracy for time domain simulations compared to standard boundary treatments and improved efficiency over the exact condition. Copyright © 2000 John Wiley & Sons, Ltd.     

Time‐dependent absorbing boundary conditions for elastic wave propagation

This paper develops a finite element scheme to generate the spatial‐ and time‐dependent absorbing boundary conditions for unbounded elastic‐wave problems. This scheme first calculates the spatial‐ and time‐dependent wave speed over the cosine of the direction angle using the Higdon's one‐way first‐order boundary operator, and then this operator is used again along the absorbing boundary in order to simulate the behaviour of unbounded problems. Different from other methods, the estimation of the wave speed and directions is not necessary in this method, since the wave speed over the cosine of the direction angle is calculated automatically. Two‐ and three‐dimensional numerical simulations indicate that the accuracy of this scheme is acceptable if the finite element analysis is appropriately arranged. Moreover, only the displacements along absorbing boundary nodes need to be set in this method, so the standard finite element method can still be used. Copyright © 2001 John Wiley & Sons, Ltd.     

双声源法调制回热器边界声场的理论分析

为了方便获得热声系统较佳的回热器边界条件,提出了双声源法调制回热器边界声场的方案.基于Rott方程,推导出回热器以及有阻尼谐振管中的声场表达式,建立了回热器边界声场调制的理论模型,得到了回热器边界声场和双声源的声压之间的关系.理论分析表明双声源法可以实现回热器边界声场的任意调节,并实际讨论了各种回热器声场边界条件的调制方法.     

An analytical method for temperature-dependent free vibration analysis of functionally graded beams with general boundary conditions

Exact solutions are presented to study the free vibration of a beam made of symmetric functionally graded materials. The formulation used is based on a unified higher order shear deformation theory. Material properties are taken to be temperature-dependent, and vary continuously through the thickness according to a power law distribution (P-FGM), or an exponential law distribution (E-FGM) or a sigmoid law distribution (S-FGM). The beam is assumed to be initially stressed by a temperature rise through the thickness. Temperature field is considered constant in xy plane of the beam. Hamilton’s principle is used to derive the governing equations of motion. Free vibration frequencies are obtained by solving analytically a system of ordinary differential equations, for different boundary conditions.     

Determination of a time-dependent heat source from nonlocal boundary conditions

In this paper, we investigate the inverse heat source problem of finding the time-dependent source function together with the temperature. Three general nonlocal conditions are considered for the boundary and overdetermination conditions resulting in six different cases. The boundary element method combined with Tikhonov regularization is employed in order to obtain an accurate and stable numerical solution.     

Vibration analyses of general thin and moderately thick laminated composite curved beams with variable curvatures and general boundary conditions

Abstract

An exact semianalytical method for vibration analysis of general thin and moderately thick laminated composite curved beams with variable curvatures and general boundary conditions is presented. In the framework of the first-order shear deformation theory, the method combines the variational principle and multilevel partition technique. As one of the innovation points, the general boundary conditions are enforced by using the virtual boundary spring technology. Each of the fundamental beam unknowns is then invariantly expanded as Jacobi polynomials. The convergence study and numerical verifications of the laminated composite curved beams with various boundary conditions are carried out.     

Application of essential boundary conditions in mesh‐free methods: a corrected collocation method

Collocation methods for applying essential boundary conditions are defined as those methods in which conditions are enforced exactly at a discrete set of boundary nodes. In mesh‐free methods, this is usually accomplished by replacing rows of the matrix equations which result from discretization of the weak form with equations which ensure the enforcement of boundary conditions. In this paper, an inconsistency in this method is pointed out, and a correction is derived. Numerical test are done on one‐ and two‐dimensional equations; it is shown that convergence rates decrease with the use of the invalid traditional collocation and are restored with the corrected method. Copyright © 2000 John Wiley & Sons, Ltd.     

High‐order artificial boundary conditions for ideal axisymmetric irrotational flow around 3D obstacles

In this paper the numerical solution for ideal irrotational incompressible flow around axisymmetric 3D obstacles is discussed with the artificial boundary method. By introducing an artificial boundary, we divide the exterior unbounded domain into a bounded part and an unbounded part. After setting up a proper artificial boundary condition, we get an approximate reduced problem defined on the bounded part. Both non‐local and local artificial boundary conditions are designed. Numerical experiment is also presented, and its result demonstrates the effectiveness of these artificial boundary conditions. Copyright © 2002 John Wiley & Sons, Ltd.     

Embedded kinematic boundary conditions for thin plate bending by Nitsche's approach

A stabilized variational formulation, based on Nitsche's method for enforcing boundary constraints, leads to an efficient procedure for embedding kinematic boundary conditions in thin plate bending. The absence of kinematic admissibility constraints allows the use of non‐conforming meshes with non‐interpolatory approximations, thereby providing added flexibility in addressing the C¹‐continuity requirements typical of these problems. Work‐conjugate pairs weakly enforce kinematic boundary conditions. The pointwise enforcement of corner deflections is key to good performance in the presence of corners. Stabilization parameters are determined from local generalized eigenvalue problems, guaranteeing coercivity of the discrete bilinear form. The accuracy of the approach is verified by representative computations with bicubic C² B‐splines, exhibiting optimal rates of convergence and robust performance with respect to values of the stabilization parameters. Copyright © 2012 John Wiley & Sons, Ltd.     

Hierarchical enrichment for bridging scales and mesh‐free boundary conditions

The finite‐element method, when used with a basis made up of piecewise polynomials, often requires the generation of a very fine computational mesh in order to capture localized solution phenomena such as boundary layers or near‐singularities. Enrichment of the basis with additional functions, obtained through analytical or experimental means, can allow for a coarser mesh and more accurate solution. We introduce an enrichment scheme in which an interaction or ‘bridging’ scale term is used to separate the basis formed by the enrichment functions from the original set of basis functions, in effect making the enrichment hierarchical. This separation of scales allows the simple application of essential boundary conditions. It also allows a quantification of the effects of the enrichment, leading to strategies for error estimation and control of the stiffness matrix condition number. We also find that this formulation allows for the simple application of essential boundary conditions for mesh‐free shape functions, which are notoriously problematic. We find that for multiple dimensions, care must be taken to derive a weak form which is truly consistent with the strong form on the essential boundary. Copyright © 2001 John Wiley & Sons, Ltd.