Dynamic crack analysis in 2D elastic solids with the singular edge-based smoothed finite element method

The singular edge-based smoothed finite element method (sES-FEM) is developed for stationary dynamic crack analysis in two-dimensional (2D) elastic solids. The paper aims at providing a better understanding of the dynamic fracture behaviors in linear elastic solids by means of the strain smoothing technique. The strains are smoothed and the system stiffness matrix is performed using the strain smoothing over the smoothing domains associated with the element edges. A two-layer singular five-node crack-tip element is employed while the standard implicit time integration scheme is used for solving the discrete sES-FEM equation system. Dynamic stress intensity factors (DSIFs) are extracted using the domain-form of interaction integrals in terms of the smoothing technique. The normalized DSIFs are compared with reference solutions showing a high accuracy of the sES-FEM. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A micromechanics model for FRC composites

A multiscale model for FRC composite structures taking into consideration the complex interactions at the scales of the fiber and microcracks is proposed. At the scale of the single fiber, a semi-analytical model characterizes the microslip behavior at the interface between the matrix and the fiber in terms of the overall composite stresses. The influence of fiber bundles on microcrack bridging and arrest is taken into account within the framework of linear elastic fracture mechanics. Upscaling to the macroscopic level using continuum micromechanics shows that the macroscopic deformation of the FRC composite is governed by a ’TERZAGHI’ like effective stress. For the finite element analyses of failure behavior at the scale of the composite structure, an ’interface solid element’ technique is used to consider localized cracking. Selected numerical and semi-analytical results together with experimental validations are provided. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

介质与结构三维动力相互作用分析的半解析边界元—半解析有限元结合法

本文将半解析边界元一半解析有限无结合法用于介质与结构的动力相互作用研究:用半解析边界元法分析具有复杂地表面的半无限介质,用半解析有限元法分析具有任意截面形状的柱体结构,利用介质与结构交界面上的位移相容条件和力平衡条件,将介质与结构联系起来。联立京解上述半解析边界元方程和半解析有限元方程,对应每一时间步进,可同时求出介质与结构交界面上的位移、速度、加速度和相互作用力以及地表面的运动情况.与目前广泛研究的边界元—有限元结合法相比,本方法在介质与结构二个个区域各降低了一维空间,因而离散单元数和计算工作量大幅度减少,人工输入数据非常简单.文中还考虑了地下结构的长跨比效应、厚度效应和介质效应.     

On the coupling of the homotopy perturbation method and Laplace transformation

In this paper, a Laplace homotopy perturbation method is employed for solving one-dimensional non-homogeneous partial differential equations with a variable coefficient. This method is a combination of the Laplace transform and the Homotopy Perturbation Method (LHPM). LHPM presents an accurate methodology to solve non-homogeneous partial differential equations with a variable coefficient. The aim of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in other semi-analytical methods such as HPM, VIM, and ADM. The approximate solutions obtained by means of LHPM in a wide range of the problem’s domain were compared with those results obtained from the actual solutions, the Homotopy Perturbation Method (HPM) and the finite element method. The comparison shows a precise agreement between the results, and introduces this new method as an applicable one which it needs fewer computations and is much easier and more convenient than others, so it can be widely used in engineering too.     

Coupled FEM-BEM for elastoplastic contact problems using Lagrange multipliers

The coupling of the elastoplastic finite element and elastic boundary element methods for two-dimensional frictionless contact stress analysis is presented. Interface traction matching (boundary element approach), which involves the force terms in the finite element analysis being transformed to tractions, is chosen for the coupling method. The analysis at the contact region is performed by the finite element method, and the Lagrange multiplier approach is used to apply the contact constraints. Since the analyses of elastoplastic problems are non-linear and involve iterative solution, the reduced size of the final system of equations introduced by combining the two methods is very advantageous, especially for contact problems where the nature of the problem also involves an iterative scheme.     

Extension of the scaled boundary finite element method to geometrical and physical nonlinearity

The scaled boundary finite element method (SBFEM) has been used in many fields of engineering to solve the governing equations in bounded and unbounded 2D as well as 3D domains. In solid mechanics, the semi-analytical solution strategy of the SBFE formulation (numerical in circumferential direction, analytical in radial direction) is based on the assumption of linear elastic material behavior and only small geometrical changes. However, a large group of materials (e.g. rubber) shows geometrical and physical nonlinearity at mechanical loading. In this contribution, the extension of the SBFEM to geometrical and physical nonlinearity is examined. A plane finite element is developed which uses the concept of shape functions constructed by the SBFEM in the framework of a nonlinear finite element analysis. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Hybrid numerical method with adaptive overlapping meshes for solving nonstationary problems in continuum mechanics

Techniques that improve the accuracy of numerical solutions and reduce their computational costs are discussed as applied to continuum mechanics problems with complex time-varying geometry. The approach combines shock-capturing computations with the following methods: (1) overlapping meshes for specifying complex geometry; (2) elastic arbitrarily moving adaptive meshes for minimizing the approximation errors near shock waves, boundary layers, contact discontinuities, and moving boundaries; (3) matrix-free implementation of efficient iterative and explicit–implicit finite element schemes; (4) balancing viscosity (version of the stabilized Petrov–Galerkin method); (5) exponential adjustment of physical viscosity coefficients; and (6) stepwise correction of solutions for providing their monotonicity and conservativeness.     

Numerical studies of the stability of steady-state solutions to a contact problem in coupled thermoelasticity

A finite element approximation is used to study the stability of steady-state solutions and the erratic behavior that is present in a problem of heat conduction through an elastic rod that may come into contact with a rigid wall. The quasi-static fully coupled theory of linear thermoelasticity is assumed and a heat exchange coefficient that depends on the pressure and the gap size is imposed across the region of contact.     

Semi-analytic modelling of transversely isotropic magneto-electro-elastic materials under frictional sliding contact

Functionally graded magneto-electro-elastic (FGMEE) materials has been increasingly used in engineering applications, particularly in smart material or intelligent structure systems. This paper proposes a semi-analytical approach for sliding frictional contact problem between a rigid insulating sphere and a transversely isotropic FGMEE film and half-space based on frequency response functions (FRFs). Multilayered approximation is used to model the functionally graded material (FGM), and the FRFs for each MEE layer are derived explicitly. The unknown coefficients in FRFs are formulated by two matrix equations, and their efficient solution process is proposed. Based on the obtained FRFs, a highly efficient semi-analytical model (SAM) is developed which is able to solve the three-dimensional frictional contact of FGMEE materials with arbitrary layer designs. The model is validated with finite element method and the literature. Furthermore, the pressure/stress distribution and electric/magnetic potential are studied in different FGM designs to investigate the influence of material layout.     

Three-dimensional simulation of flat rolling through a combined finite element–boundary element approach

This paper presents an innovative approach for analysing three-dimensional flat rolling. The proposed approach is based on a solution resulting from the combination of the finite element method with the boundary element method. The finite element method is used to perform the rigid–plastic numerical modelling of the workpiece allowing the estimation of the roll separating force, rolling torque and contact pressure along the surface of the rolls. The boundary element method is applied for computing the elastic deformation of the rolls. The combination of the two numerical methods is made using the finite element solution of the contact pressure along the surface of the rolls to define the boundary conditions to be applied on the elastic analysis of the rolls. The validity of the proposed approach is discussed by comparing the theoretical predictions with experimental data found in the literature.     

Numerical analysis of a contact problem including bone remodeling

In this work, a contact problem between an elastic body and a deformable obstacle is numerically studied. The bone remodeling of the material is also taken into account in the model and the contact is modeled using the normal compliance contact condition. The variational problem is written as a nonlinear variational equation for the displacement field, coupled with a first-order ordinary differential equation to describe the physiological process of bone remodeling. An existence and uniqueness result of weak solutions is stated. Then, fully discrete approximations are introduced based on the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are obtained, from which the linear convergence of the algorithm is derived under suitable regularity conditions. Finally, some 2D numerical results are presented to demonstrate the behavior of the solution.     

An mode-matching approach for ridge-guided wave investigation

An analytical method based on partial wave decomposition, mode matching and transverse resonant analysis is proposed to solve the pure feature-guided modes in a rectangular elastic ridge embedded in a flat plate. Compared with the semi-analytical finite element (SAFE) models and experiment results, the effectiveness of the method on dispersion curve and some mode shape calculation are proven. Without the drawbacks of the SAFE, which induced by insufficient discretization at high frequencies and large additional absorption zones at low frequencies, the method can render more accurate results wherein. As a result, in a symmetrically embedded ridge, four more non-leaky ridge-guided modes, meaningful for nondestructive evaluation are founded, with the dispersion curves and most of the mode shapes precisely predicted.     

Load share and finite element stress analysis for double circular-arc helical gears

The authors analyze the tooth surface contact and stresses for double circular-arc helical gear drives. The geometry of such gear drives has been represented by the authors in their previous paper [1]. The proposed approach is based on application of (i) computerized simulation of meshing and contact of loaded gear drives, and (ii) the finite element method. Load share between the neighboring pairs of teeth is based on the analysis of position errors caused by surface mismatch and elastic deformation of teeth. The authors have investigated the conditions of load share under a load and determined the real contact ratio for aligned and misaligned gear drives, respectively. Elastic deformation of teeth and the stress analysis of the double circular-arc helical gears are accomplished by using the finite element method. The finite element models for the pinion and gear are constructed, respectively. Contact pressure is spread over elliptical area. The stress analysis for aligned and misaligned gear drives, respectively, has been performed. The numerical results have been compared with those obtained by other approaches.     

A posteriori error analysis for the normal compliance problem

In this work, a contact problem between a linear elastic material and a deformable obstacle is numerically analyzed. The contact is modeled using the well-known normal compliance contact condition. The weak formulation leads to a nonlinear variational equation which is approximated by using the finite element method. A priori error estimates are recalled. Then, we define an a posteriori error estimator of residual type to evaluate the accuracy of the finite element approximation of the problem. Upper and lower bounds of the discretization error are proved for this estimator.     

任意厚度具有自由边叠层板的精确解析解

自由边问题一直是三维弹性力学中的难题,通常很难满足自由边上一个正应力和两个剪应力都等于0．基于三维弹性力学基本方程和状态空间方法,引入自由边界位移函数并考虑全部弹性常数,建立了正交异性具有自由边单层和叠层板的状态方程．对状态方程中的变量以级数形式展开,通过边界条件的满足精确求解任意厚度具有自由边叠层板的位移和应力,此解满足层间应力和位移的连续条件．算例计算表明,采用引入的位移函数形式,简化了计算过程并且采用较少的级数项可以获得收敛解．与有限元方法计算结果进行了对比,可以得到较高精度的数值结果．其解可以作为其它数值方法和半解析方法的参考解．     

A Novel Numerical Simulations for Fornberg-Whitham and Modified Fornberg-Whitham Equations with Nonhomogeneous Boundary Conditions

In this study, the numerical solutions of the Fornberg-Whitham (FW) equation modeling the qualitative behavior of wave refraction and the modified Fornberg-Whitham (mFW) equation describing the solitary wave and peakon waves with a discontinuous first derivative at the peak have been obtained. To obtain numerical results, the collocation finite element method has been combined with quintic B-spline bases. Although there are solutions to these equations by semi-analytical and analytical methods in the literature, there are very few studies using numerical methods. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. We have considered four test problems with nonhomogeneous boundary conditions that have analytical solutions to show the performance of the method. The numerical results of the two problems are compared with some studies in the literature. Additionally, peakon wave solutions and some new numerical results of the mFW equation, which are not available in the literature, are given in the last two problems. No comparison has been made since there are no numerical results in the literature for the last two problems. The error norms $L_{2}$ and $L_{\infty }$ are calculated to demonstrate the presented numerical scheme''s accuracy and efficiency. The advantage of the scheme is that it produces accurate and reliable solutions even for modest values of space and time step lengths, rather than small values that cause excessive data storage in the computation process. In general, large step lengths in the space and time directions result in smaller matrices. This means less storage on the computer and results in faster outcomes. In addition, the present method gives more accurate results than some methods given in the literature.     

Semi-analytical method for solving nonlinear heat diffusion problems in spherical medium

A semi-analytical methodology, based on the finite integral transform technique, is proposed to solve the heat diffusion problem in a spherical medium subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The method proceeds by treating the nonlinearity term in the boundary condition as a source in the differential equation and keeping other conditions unchanged. The results obtained from this semi-analytical solutions are compared with those obtained from a numerical solution developed using an explicit finite difference method, which showed very good agreement.     

Damageable contact between an elastic body and a rigid foundation

In this work, the contact problem between an elastic body and a rigid obstacle is studied, including the development of material damage which results from internal compression or tension. The variational problem is formulated as a first-kind variational inequality for the displacements coupled with a parabolic partial differential equation for the damage field. The existence of a unique local weak solution is stated. Then, a fully discrete scheme is introduced using the finite element method to approximate the spatial variable and an Euler scheme to discretize the time derivatives. Error estimates are derived on the approximate solutions, from which the linear convergence of the algorithm is deduced under suitable regularity conditions. Finally, three two-dimensional numerical simulations are performed to demonstrate the accuracy and the behaviour of the scheme.