A Duhamel approach for forced vibrations of rotating axisymmetric bodies

In this work a novel approach for the determination of the dynamic behaviour of a rotating device subject to forces fixed with respect to a stationary reference system is developed. The procedure can be applied to any axisymmetric rotating structure and it is particularly convenient for components of complex geometry for which an accurate dynamic characterization is fundamental. It is based on the application of Duhamel's integral. The time evolution of the system is obtained with the determination of the impulse response of the body using one single finite element transient analysis. By combining the results of the transient analysis with the time history of the forces applied to the rotating body, it is possible to determine the ground‐based vibration response of the component. Copyright © 2012 John Wiley & Sons, Ltd.     

A NURBS‐based interface‐enriched generalized finite element method for problems with complex discontinuous gradient fields

A non‐uniform rational B‐splines (NURBS)‐based interface‐enriched generalized finite element method is introduced to solve problems with complex discontinuous gradient fields observed in the structural and thermal analysis of the heterogeneous materials. The presented method utilizes generalized degrees of freedom and enrichment functions based on NURBS to capture the solution with non‐conforming meshes. A consistent method for the generation and application of the NURBS‐based enrichment functions is introduced. These enrichment functions offer various advantages including simplicity of the integration, possibility of different modes of local solution refinement, and ease of implementation. In addition, we show that these functions well capture weak discontinuities associated with highly curved material interfaces. The convergence, accuracy, and stability of the method in the solution of two‐dimensional elasto‐static problems are compared with the standard finite element scheme, showing improved accuracy. Finally, the performance of the method for solving problems with complex internal geometry is highlighted through a numerical example. Copyright © 2015 John Wiley & Sons, Ltd.     

A method for modeling the transition of weak discontinuities to strong discontinuities: from interfaces to cracks

Cohesive zone models are widely used to model interface debonding problems; however, these models engender some significant drawbacks, including the need for a conforming mesh to delimit the interfaces between different materials or components and that penalty or other constraint methods necessary to enforce initially perfect adhesion at interfaces degrade the critical time step for stability in explicit time integration. This article proposes a new technique based on the extended finite element method that alleviates these shortcomings by representing the transition from perfect interfacial adhesion to debonding by switching the enriched approximation basis functions from weakly discontinuous to strongly discontinuous. At this transition, the newly activated degrees of freedom are initialized to satisfy a point‐wise consistency condition at the interface for both displacement and velocity. Analysis of the stable time step for one‐dimensional elements with mass lumping is presented, which shows the increase of the stable time step compared with a cohesive zone model. Both one‐dimensional and two‐dimensional verification examples are presented, illustrating the potential of this new approach. Copyright © 2015 John Wiley & Sons, Ltd.     

An extended equivalent domain integral method for mixed mode fracture problems by the p-version of FEM

A new path-independent contour integral formula is presented to estimate the crack-tip integral parameter, J-value, for two-dimensional cracked elastic bodies which may quantify the severity of the crack-tip stress fields. The conventional J-integral method based on a line integral has been converted to an equivalent area or domain integral (EDI) by the divergence theorem. It is noted that the EDI method is very attractive because all the quantities necessary for computation of domain integrals are readily available in a finite element analysis. The details and its implementation are extended to the p-version FE model with hierarchic elements using integrals of Legendre polynomials. By decomposing the displacement field obtained from the p-version finite element analysis into symmetric and antisymmetric displacement fields with respect to the crack line, the Mode-I and Mode-II non-dimensional stress intensity factors can be determined by using the decomposition method. The example problems for validating the proposed techniques are centrally oblique cracked plates under tensile loading. The numerical results associated with the variation of oblique angles show very good agreement with the existing solutions. Also, the selective distribution of polynomial orders and the corner elements for automatic mesh generation are applied to improve the numerical solution in this paper. © 1998 John Wiley & Sons, Ltd.     

A comparison of wave‐based discontinuous Galerkin,ultra‐weak and least‐square methods for wave problems

Several numerical methods using non‐polynomial interpolation have been proposed for wave propagation problems at high frequencies. The common feature of these methods is that in each element, the solution is approximated by a set of local solutions. They can provide very accurate solutions with a much smaller number of degrees of freedom compared to polynomial interpolation. There are however significant differences in the way the matching conditions enforcing the continuity of the solution between elements can be formulated. The similarities and discrepancies between several non‐polynomial numerical methods are discussed in the context of the Helmholtz equation. The present comparison is concerned with the ultra‐weak variational formulation (UWVF), the least‐squares method (LSM) and the discontinuous Galerkin method with numerical flux (DGM). An analysis in terms of Trefftz methods provides an interesting insight into the properties of these methods. Second, the UWVF and the LSM are reformulated in a similar fashion to that of the DGM. This offers a unified framework to understand the properties of several non‐polynomial methods. Numerical results are also presented to put in perspective the relative accuracy of the methods. The numerical accuracies of the methods are compared with the interpolation errors of the wave bases. Copyright © 2010 John Wiley & Sons, Ltd.     

Stochastic response determination of nonlinear structural systems with singular diffusion matrices: A Wiener path integral variational formulation with constraints

The Wiener path integral (WPI) approximate semi-analytical technique for determining the joint response probability density function (PDF) of stochastically excited nonlinear oscillators is generalized herein to account for systems with singular diffusion matrices. Indicative examples include (but are not limited to) systems with only some of their degrees-of-freedom excited, hysteresis modeling via additional auxiliary state equations, and energy harvesters with coupled electro-mechanical equations. In general, the governing equations of motion of the above systems can be cast as a set of underdetermined stochastic differential equations coupled with a set of deterministic ordinary differential equations. The latter, which can be of arbitrary form, are construed herein as constraints on the motion of the system driven by the stochastic excitation. Next, employing a semi-classical approximation treatment for the WPI yields a deterministic constrained variational problem to be solved numerically for determining the most probable path; and thus, for evaluating the system joint response PDF in a computationally efficient manner. This is done in conjunction with a Rayleigh-Ritz approach coupled with appropriate optimization algorithms. Several numerical examples pertaining to both linear and nonlinear constraint equations are considered, including various multi-degree-of-freedom systems, a linear oscillator under earthquake excitation and a nonlinear oscillator exhibiting hysteresis following the Bouc–Wen formalism. Comparisons with relevant Monte Carlo simulation data demonstrate a relatively high degree of accuracy.     

A novel singular node‐based smoothed finite element method (NS‐FEM) for upper bound solutions of fracture problems

It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node‐based smoothed finite element method (NS‐FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS‐FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five‐node singular crack tip element is used within the framework of NS‐FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix‐mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS‐FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

An application of Betti's reciprocal theorem for the analysis of an inclusion problem

This paper presents the application of Betti's reciprocal theorem for evaluation of displacements of an embedded rigid disc inclusion in an exact closed form.     

A practical stress analysis for predicting fatigue limit of metal with axisymmetric complex surface

In order to predict the fatigue limit of a specimen with an axisymmetric complex surface, a practical method to estimate a stress concentration factor (SCF) of its surface was proposed. The roughness is coarse-grained by removing high frequency components and approximated with a parallel row of a local notch and innumerable average notches. Then, the notches are each approximated with the elliptical holes in the infinite plate, and the SCF is calculated approximately by superposing the elastic solutions of the holes. Moreover, FEM analyses were carried out on the various notch models which consist of the local notch and innumerable average notches to examine the application limit of the present method. Then, the validity of the application limit was examined by using the real roughness and the infinite parallel row of the various notches, and it was shown that the present method was available for the real roughness.     

A self-regularized approach for rank-deficient systems in the BEM of 2D Laplace problems

The Laplace problem subject to the Dirichlet or Neumann boundary condition in the direct and indirect boundary element methods (BEM) sometimes both may result in a singular or ill-conditioned system (some special situations) for the interior problem. In this paper, the direct and indirect BEMs are revisited to examine the uniqueness of the solution by introducing the Fichera’s idea and the self-regularized technique. In order to construct the complete range of the integral operator in the BEM lacking a constant term in the case of a degenerate scale, the Fichera’s method is provided by adding the constraint and a slack variable to circumvent the problem of degenerate scale. We also revisit the Fredholm alternative theorem by using the singular value decomposition (SVD) in the discrete system and explain why the direct BEM and the indirect BEM are not indeed equivalent in the solution space. According to the relation between the SVD structure and Fichera’s technique, a self-regularized method is proposed in the matrix level to deal with non-unique solutions of the Neumann and Dirichlet problems which contain rigid body mode and degenerate scale, respectively, at the same time. The singularity and proportional influence matrices of 3 by 3 are studied by using the property of the symmetric circulant matrix. Finally, several examples are demonstrated to illustrate the validity and the effectiveness of the self-regularized method.     

A selective smoothed finite element method with visco-hyperelastic constitutive model for analysis of biomechanical responses of brain tissues

Brain tissues are known for exhibiting complex nonlinear and time-dependent properties, which require visco-hyperelastic constitutive models for proper simulation. In this paper, a Total Lagrangian Explicit Selective Smoothed Finite Element Method (Selective S-FEM) is formulated to analyze the dynamic behavior of incompressible brain tissues undergoing extremely large deformation. The proposed Selective S-FEM deals with three-dimensional problems using four-node tetrahedron elements that can be automatically generated for geometrically complex soft tissues. It consists of the three key ingredients. (i) A visco-hyperelastic constitutive model is developed within the framework of S-FEM in the first time, allowing adequate modeling of the dynamic brain tissue behavior. (ii) Selective S-FEM strategy is used for overcome the mesh distortion and the volumetric locking that often occurs in soft tissues. (iii) Total Lagrangian formulation is used in an explicit algorithm allowing rigorous simulation of extreme large deformation. (iv) A combined implementation of Selective S-FEM with the visco-hyperelastic constitutive model for dynamic simulations. The shear deformation is calculated by Face/Edge-based S-FEM, and the volume deformation is calculated by NS-FEM. Numerical experiments show that Selective S-FEM is a robust solver with good accuracy, and excellent ability to reduce element distortion effects in simulate time-dependence behavior of bio-tissues.     

A new class of cells with embedded discontinuity for fracture analysis by the boundary element method

Material failure analysis are addressed by the continuum strong discontinuity approach within the implicit formulation of the boundary element method in this work. An automatic cell generation algorithm is used to track cracks during the loading processes in the nonlinear analysis. As a major novelty, a new class of cells with embedded discontinuity is developed in which nonuniform displacement jump components are considered. It is shown that this new approach eliminates the stress locking effect verified in propagation analyses using cells with embedded uniform discontinuous displacement field, since the relative rotational motion between the two portions of a cell can now be properly captured.     

A new algorithm for numerical solution of 3D elastoplastic contact problems with orthotropic friction law

3D elastoplastic frictional contact problems with orthotropic friction law belong to the unspecified boundary problems with nonlinearities in both material and geometric forms. One of the difficulties in solving the problem lies in the determination of the tangential slip states at the contact points. A great amount of computational efforts is needed so as to obtain high accuracy numerical results. Based on a combination of the well known mathematical programming method and iterative method, a finite element model is put forward in this paper. The problems are finally reduced to linear complementarity problems. A specially designed smoothing algorithm based on NCP-function is then applied for solving the problems. Numerical results are given to demonstrate the validity of the model and the algorithm proposed.The project is jointly supported by the National Natural Science Foundation (10225212, 50178016, 10302007), the National Key Basic Research Special Foundation (G1999032805), the Special Funds for Major State Basic Research Projects and the Foundation for University Key Teacher by the Ministry of Education of China. The authors are also grateful to the referees for their careful reading and detailed remarks on an earlier version of the paper.     

A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators

This paper presents the development of a semi-analytical axisymmetric shell finite element model with piezoelectric layers using the 3D linear elasticity theory. The piezoelectric effect of the material could be used as sensors and/or actuators in way to control shell deformation. In the present 3D axisymmetric model, the equations of motion are expressed by expanding the displacement field using Fourier series in the circumferential direction. Thus, the 3D elasticity equations of motion are reduced to 2D equations involving circumferential harmonics. In the finite element formulation the dependent variables, electric potential and loading are expanded in truncated Fourier series. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.     

A suitable computational strategy for the parametric analysis of problems with multiple contact

The aim of the present work is to develop an application of the LArge Time INcrement (LATIN) approach for the parametric analysis of static problems with multiple contacts. The methodology adopted was originally introduced to solve viscoplastic and large‐transformation problems. Here, the applications concern elastic, quasi‐static structural assemblies with local non‐linearities such as unilateral contact with friction. Our approach is based on a decomposition of the assembly into substructures and interfaces. The interfaces play the vital role of enabling the local non‐linearities, such as contact and friction, to be modelled easily and accurately. The problem on each substructure is solved by the finite element method and an iterative scheme based on the LATIN method is used for the global resolution. More specifically, the objective is to calculate a large number of design configurations. Each design configuration corresponds to a set of values of all the variable parameters (friction coefficients, prestress) which are introduced into the mechanical analysis. A full computation is needed for each set of parameters. Here we propose, as an alternative to carrying out these full computations, to use the capability of the LATIN method to re‐use the solution to a given problem (for one set of parameters) in order to solve similar problems (for the other sets of parameters). Copyright © 2003 John Wiley & Sons, Ltd.     

A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problems

In part I of this paper, we have established the G space theory and fundamentals for W² formulation. Part II focuses on the applications of the G space theory to formulate W² models for solid mechanics problems. We first define a bilinear form, prove some of the important properties, and prove that the W² formulation will be spatially stable, and convergent to exact solutions. We then present examples of some of the possible W² models including the SFEM, NS‐FEM, ES‐FEM, NS‐PIM, ES‐PIM, and CS‐PIM. We show the major properties of these models: (1) they are variationally consistent in a conventional sense, if the solution is sought in a proper H space (compatible cases); (2) They pass the standard patch test when the solution is sought in a proper G space with discontinuous functions (incompatible cases); (3) the stiffness of the discretized model is reduced compared with the finite element method (FEM) model and possibly to the exact model, allowing us to obtain upper bound solutions with respect to both the FEM and the exact solutions and (4) the W² models are less sensitive to the quality of the mesh, and triangular meshes can be used without any accuracy problems. These properties and theories have been confirmed numerically via examples solved using a number of W² models including compatible and incompatible cases. We shall see that the G space theory and the W² forms can formulate a variety of stable and convergent numerical methods with the FEM as one special case. Copyright © 2009 John Wiley & Sons, Ltd.     

A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective.     

A regularization method for the approximate particular solution of nonhomogeneous Cauchy problems of elliptic partial differential equations with variable coefficients

Radial basis functions (RBFs) have proved to be very flexible in representing functions. Based on the idea of the analog equation method and radial basis functions, in this paper, ill-posed Cauchy problems of elliptic partial differential equations (PDEs) with variable coefficients are considered for the first time using the method of approximate particular solutions (MAPS). We show that, using the Tikhonov regularization, the MAPS results an effective and accurate numerical algorithm for elliptic PDEs and irregular solution domains. Comparing the proposed MAPS with Kansa's method, numerical results show that the proposed MAPS is effective, accurate and stable to solve the ill-posed Cauchy problems.     

主动约束层阻尼圆柱壳动力学分析的新传递矩阵法

基于作者最近导出的被动约束层阻尼(PCLD)圆柱壳的一阶整合矩阵微分方程,结合压电材料本构关系和比例微分负增益反馈控制策略(PD),建立了一种求解主动约束层阻尼(ACLD)圆柱壳动力学问题的新传递矩阵方法。提出的ACLD圆柱壳的一阶矩阵微分方程,采用了简化的机电耦合模型。通过对ACLD圆柱壳自由振动及其在地震激励作用下的动力学响应分析,表明ACLD圆柱壳的阻尼特性和减振效果相对于PCLD圆柱壳具有明显优势,并且发现采用周向分块敷设ACLD,且施加与结构变形中的占优模态相匹配的控制电压分布方式对地震激励的抑制效果更好。     

求解地面运动激励下的部分覆盖约束层阻尼贮液圆柱壳耦振的整合状态传递矩阵法

摘要：基于一般情况下的线弹性薄壳方程和势流理论,考虑被动约束层阻尼（PCLD）的剪切变形的能量耗散和液固耦合相互作用,文中首先导出了PCLD圆柱层合壳的整合一阶矩阵微分方程,该方程的状态向量的每个元素都有明确的物理意义,更方便用于层合壳体在各种边界支承条件的动力学问题的求解。然后通过将流体动压力写成含待求系数的解析形式,借助流固交接条件、新型齐次扩容精细积分法和叠加原理,建立了一种分析该类结构耦振问题的高效率、高精度的半解析半数值方法。通过与无水、两端简支的全覆盖PCLD圆柱壳在轴对称情况下自由振动的解析解结果比较,验证了本文方法的有效性。最后基于文中提出的方法,研究了部分覆盖PCLD贮液圆柱容器在地面运动激励下的动力响应,研究了PCLD厚度、长度、敷设位置以及粘弹芯的复剪切模量模型对减振效果的影响。