Sliding contact conditions using the master–slave approach with application on geometrically non-linear beams

Frictionless sliding conditions between two bodies are usually defined using either the method of Lagrangian multipliers or by prescribing an artificial (penalty) stiffness which resists the penetration at the contact point. Both of these methods impose the condition that the contact force should be normal to the contact surface, with the Lagrangian multiplier or the penalty parameter serving as a measure of this force. In this work, an alternative approach is undertaken: the frictionless sliding condition is defined through a relationship between nodal parameters of the virtual displacements of a discretised principle of virtual work. This method, which does not involve additional force parameters or degrees of freedom, is known as the master–slave or the minimum-set method and is particularly convenient for displacement-based finite-element implementation. The method is analysed in detail in context of bilateral sliding constraints characteristic of prismatic and cylindrical joints in flexible beam assemblies undergoing large overall motion. Two numerical examples are presented and assessed against the results in the literature.     

CRACK FACE CONTACT PROBLEM ANALYSIS USING THE SCALED BOUNDARY FINITE ELEMENT METHOD

比例边界有限元侧面上有任意荷载时,将侧面载荷分解成关于径向方向局部坐标的多项式函数的和,推导给出了考虑侧面载荷存在的新型形函数,并基于该形函数推导了刚度矩阵和等效节点载荷列阵.首次对比例边界有限元法求解裂纹面接触问题进行了研究,运用Lagrange乘子引入接触界面约束条件,推导给出了比例边界有限元求解裂纹面接触问题的控制方程.将裂纹面单元分为非裂尖单元和含有侧面的裂尖单元.在非裂尖单元中的裂纹面,裂纹面作为多边形单元的边界,边界上的接触力可等效到节点上,通过在节点上构造Lagrange乘子,采用点对点接触约束进行处理.对于含有侧面的裂尖单元,在整个侧面上构造Lagrange乘子的插值场,采用边对边接触约束进行处理.对三个不同的接触约束状态下的算例进行了数值计算,通过与解析解及有限元软件ABAQUS计算结果的对比,验证了本文提出的比例边界有限元点对点和边对边接触求解裂纹面接触问题的精确性与有效性.     

An optimal method for seismic drift design of concrete buildings using gradient and Hessian matrix calculations

This paper describes a novel seismic optimal design method for the reinforced concrete frame. First, an optimal mathematical model with time-dependent constraints, i.e., inter-story drift constraints, is established for achieving minimum weight design. Second, the inequality constraint problem with time-dependent constraints is converted into a sequence of appropriately formed unconstrained problems using the integral interior point penalty function method. Third, an efficient algorithm of the first and second derivatives of the inter-story drift with respect to design variables is formulated based on Newmark-β method. Gradient and Hessian matrix of the integral interior penalty function are also computed. Fourth, Marquardt’s method is employed to solve a sequence of unconstrained problems. Finally, the minimum weight design of a three-story, two-bay planar frame is demonstrated using the new optimization method and the augmented Lagrange multiplier method. The comparative results show the seismic optimal design method presented in this paper is more efficient than the augmented Lagrange multiplier method in terms of computational time. The proposed new method is an effective and efficient approach for minimum weight design of the reinforced concrete frames subjected to earthquake excitation.     

A frictional mortar contact approach for the analysis of large inelastic deformation problems

A multibody frictional mortar contact formulation (Gitterle et al., 2010) is extended for the simulation of solids undergoing finite strains with inelastic material behavior. The framework includes the modeling of finite strain inelastic deformation, the numerical treatment of frictional contact conditions and specific finite element technology. Several well-established and recent models are employed for each of these building blocks to capture the distinct physical aspects of the deformation behavior. The approach is based on a mortar formulation and the enforcement of contact constraints is realized with dual Lagrange multipliers. The introduction of nonlinear complementarity functions into the frictional contact conditions combined with the global equilibrium leads to a system of nonlinear equations, which is solved in terms of the semi-smooth Newton method. The resulting method can be interpreted as a primal–dual active set strategy (PDASS) which deals with contact nonlinearities, material and geometrical nonlinearities in one iterative scheme. The consistent linearization of all building blocks of the framework yields a robust and highly efficient approach for the analysis of metal forming problems. The effect of finite inelastic strains on the solution behavior of the PDASS method is examined in detail based on the complementarity parameters. A comprehensive set of numerical examples is presented to demonstrate the accuracy and efficiency of the approach against the traditional node-to-segment penalty contact formulation.     

Fully nonlinear modeling of radiated waves generated by floating flared structures

The nonlinear radiated waves generated by a structure in forced motion, are simulated numerically based on the potential theory. A fully nonlinear numerical model is developed by using a higher-order boundary element method （HOBEM）. In this model, the instantaneous body position and the transient free surface are updated at each time step. A Lagrangian technique is employed as the time marching scheme on the free surface. The mesh regridding and interpolation methods are adopted to deal with the possible numerical instability. Several auxiliary functions are proposed to calculate the wave loads indirectly, instead of directly predicting the temporal derivative of the velocity potential. Numerical experiments are carried out to simulate the heave motions of a submerged sphere in infinite water depth, the heave and pitch motions of a truncated flared cylinder in finite depth. The results are verified against the published numerical results to ensure the effectiveness of the proposed model. Moreover, a series of higher harmonic waves and force components are obtained by the Fourier transformation to investigate the nonlinear effect of oscillation frequency. The difference among fully nonlinear, body-nonlinear and linear results is analyzed. It is found that the nonlinearity due to free surface and body surface has significant influences on the numerical results of the radiated waves and forces.     

Finite element analysis of the steady Navier-Stokes equations by a multiplier method

In this paper a fully explicit finite element method (FEFEM) is presented for solving steady incompressible viscous flow problems. This full explicitness is achieved by combining the multiplier (or augmented Lagrangian) method with a pseudo-time-iteration method. FEFEM needs no global matrix at all and is of great advantage to large-scale problems because they can be solved within the limit of core memory. The optimum choice of a time increment and a penalty parameter is discussed and the driven cavity flow at a Reynolds number of 1000 is computed with a refined mesh (60 × 60 elements).     

岩质圆形隧洞围岩应力场弹塑性新解

针对动态接触问题的有限元并行计算，提出了一种新的接触算法. 新算法引入局部拉氏 乘子技术来计算接触力. 由于同时考虑了无穿透的接触约束条件和相邻接触对的相互影响， 较之广泛使用的罚参数法，新算法使接触约束条件和系统平衡方程得到更充分的满足. 虽然 为提高接触计算精度而在局部采用了迭代技术，但算法仍然具有较高的效率，且与显式时间 积分方案完全相容. 此外，通过构造专门的区域分解方案，实现了将现有为串行程序开发的 搜索算法平滑移植到并行环境的目标. 数值算例表明，所提出的接触算法具有很好的并行性， 在保证了接触问题并行计算精度的同时，取得了满意的并行效率.     

Contact stresses: a short survey of models and methods of computations

Contact stresses are identified as normal and tangential forces between contacting solids. The normal stresses are modeled using unilateral and complementary conditions, elastic response and normal compliance. Friction laws describe the tangential traction. Friction of materials depends on pressure, sliding velocity, surface temperature, time of contact, surface roughness and presence of wear debris. Phenomenological, micro-mechanical and atomic-scale models as well as non-classical models of anisotropic and heterogeneous friction are important steps in the development of friction modeling. Sophisticated friction models are desirable in vibrating systems, materials processing, rolling contacts, rubber and polymers, geomechanics, bioengineering and living systems. Main numerical methods in contact mechanics are: finite element method, boundary element method and discrete element method. To include specific contact constraints, the following computing techniques are applied: Lagrange multipliers, penalty function, perturbated and augmented Lagrangian methods, mathematical programming methods. The advances of adhesion and impact modeling are outlined in this paper.     

Springback analysis for sheet forming processes by explicit finite element method in conjunction with the orthogonal regression analysis

A new method to evaluate the amount of springback in sheet forming processes based on the explicit finite element method and the orthogonal regression analysis is presented in this paper. To calculate springback accurately, a simple but effective contact searching algorithm is described and Lagrangian multiplier method was used to evaluate the contact force. The loading and unloading process could be simulated within one code. The numerical results by the present method were compared with the results by the commercial dynamic explicit code LS-DYNA3D, also with the experimental results and very good agreement was drawn. In order to obtain the springback conveniently for the purpose of practical use, the orthogonal regression analysis was implemented to establish the explicit relationship between the springback and some design parameters. The present method has been applied to the analysis of some actual sheet forming processes and very good agreement between the numerical results and the experimental results in the final geometry was obtained.     

Application of a modified Lindstedt–Poincaré method in coupled TDOF systems with quadratic nonlinearity and a constant external excitation

An analytical approach is developed for the nonlinear oscillation of a conservative, two-degree-of-freedom (TDOF) mass-spring system with serial combined linear–nonlinear stiffness excited by a constant external force. The main idea of the proposed approach lies in two categories, the first one is the transformation of two nonlinear differential equations of a two-mass system using suitable intermediate variables into a single nonlinear differential equation. Another is the treatment a quadratic nonlinear oscillator (QNO) by the modified Lindstedt–Poincaré (L-P) method presented recently by the authors. The first-order and second-order analytical approximations for the modified L-P method are established for the QNOs with satisfactory results. After solving the nonlinear differential equation, the displacements of two-mass system can be obtained directly from the governing linear second-order differential equation. Unlike the common perturbation method, the modified L-P method is valid for weak as well as strong nonlinear oscillation systems. On the other hand, the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation. In short, this new approach yields extended scope of applicability, simplicity, flexibility in application, and avoidance of complicated numerical integration as compared to the previous approaches such as the perturbation and classical harmonic balance methods. Two examples of nonlinear TDOF mass-spring systems excited by a constant external force are selected and the approximate solutions are verified with the exact solutions derived from the Jacobi elliptic function and also the numerical fourth-order Runge–Kutta solutions.     

The Lagrangian approach of advective term treatment and its application to the solution of Navier—Stokes equations

A one-dimensional transport test applied to some conventional advective Eulerian schemes shows that linear stability analyses do not guarantee the actual performances of these schemes. When adopting the Lagrangian approach, the main problem raised in the numerical treatment of advective terms is a problem of interpolation or restitution of the transported function shape from discrete data. Several interpolation methods are tested. Some of them give excellent results and these methods are then extended to multi-dimensional cases. The Lagrangian formulation of the advection term permits an easy solution to the Navier-Stokes equations in primitive variables V, p, by a finite difference scheme, explicit in advection and implicit in diffusion. As an illustration steady state laminar flow behind a sudden enlargement is analysed using an upwind differencing scheme and a Lagrangian scheme. The importance of the choice of the advective scheme in computer programs for industrial application is clearly apparent in this example.     

Transient dynamic response analysis of 3-D patterned tire rolling over cleat

Transient dynamic response of a rolling tire impacting with a small cleat is analyzed by an explicit finite element method. A 3-D tire model considering detailed tread blocks is used to accurately simulate the local tire-cleat impact process. The frictional dynamic contact problem is formulated by making use of total Lagrangian scheme and the penalty method. By imposing mass-proportional damping to only the tire parts showing the significant lateral deformation, the dynamic viscosity effect is artificially reflected. Time-history and frequency responses of the dynamic forces exerted on the tire axis are numerically predicted and assessed through the comparison with experimental results. As well as, the effects of the tire rolling speed and the inflation pressure on the transient dynamic response are parametrically investigated.     

Simulating surface tension with smoothed particle hydrodynamics

A method for simulating two‐phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high‐order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid‐based volume of fluid method for two‐dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two‐phase system and readily extend to three‐dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free‐surface flows are not addressed. Copyright © 2000 John Wiley & Sons, Ltd.     

A method for the analysis of dynamic response of structure containing non-smooth contactable interfaces

A novel single-step method is proposed for the analysis of dynamic response of visco-elastic structures containing non-smooth contactable interfaces. In the method, a two-level algorithm is employed for dealing with a nonlinear boundary condition caused by the dynamic contact of interfaces. At the first level, and explicit method is adopted to calculate nodal displacements of global viscoelastic system without considering the effect of dynamic contact of interfaces and at the second level, by introducing contact conditions of interfaces, a group of equations of lower order is derived to calculate dynamic contact normal and shear forces on the interfaces. The method is convenient and efficient for the analysis of problems of dynamic contact. The accuracy of the method is of the second order and the numerical stability condition is wider than that of other explicit methods. The project supported by the National Natural Science Foundation of China (59578032) and the Key Project of the Ninth Five-Year Plan (96221030202)     

Using mesh-based methods to solve nonlinear problems of statics for thin shells

The paper outlines a numerical procedure for solving physically and geometrically nonlinear problems of statics for thin shells based on three mesh-based methods: finite-difference, variational difference, and finite-element methods. The methodological, algorithmic, and analytical aspects of implementing the Kirchhoff–Love hypotheses are analyzed. The algorithmic approach employs Lagrangian multipliers. The advantages and disadvantages of these methods are evaluated Translated from Prikladnaya Mekhanika, Vol. 45, No. 1, pp. 41–70, January 2009.     

A new computational algorithm for contact friction modeling of large plastic deformation in powder compaction processes

In this paper, the large deformation frictional contact of powder forming process is modeled based on a new computational algorithm by imposing the contact constraints and modifying the contact properties of frictional slip. A simple and efficient numerical algorithm is presented for imposing the contact constraints and frictional contact properties based on the node-to-surface contact technique to simulate the large deformation contact problem in the compaction process of powder. The Coulomb friction law is used to simulate the friction between the rigid punch and the workpiece by the use of penalty approach. A double-surface cap plasticity model is employed together with the nonlinear contact friction algorithm within the framework of large FE deformation in order to predict the non-uniform relative density distribution during large deformation of powder die-pressing. Finally, the numerical schemes are examined for accuracy and efficiency in modeling of a set of powder components.     

A 3D FEM–BEM rolling contact formulation for unstructured meshes

This work presents a new formulation for solving 3D steady-state rolling contact problems. The convective terms for computing the tangential slip velocities involved in the rolling problem, are evaluated using a new approximation inspired in numerical fluid dynamics techniques for unstructured meshes. Moreover, the elastic influence coefficients of the surface points in contact are approached by means of the finite element method (FEM) and/or the boundary element method (BEM). The contact problem is based on an Augmented Lagrangian Formulation and the use of projection functions to establish the contact restrictions. Finally, the resulting nonlinear equations set is solved using the generalized Newton method with line search (GNMls), presenting some acceleration strategies as: a new and more simplified projection operator, which makes it possible to obtain a quasi-complementarity of the contact variables, reducing the number of contact problem unknowns, and using iterative solvers. The presented methodology is validated solving some rolling contact problems and analyzed for some unstructured mesh examples.     

A new simple method of implicit time integration for dynamic problems of engineering structures

This paper presents a new simple method of implicit time integration with two control parameters for solving initial-value problems of dynamics such that its accuracy is at least of order two along with the conditional and unconditional stability regions of the parameters. When the control parameters in the method are optimally taken in their regions, the accuracy may be improved to reach of order three. It is found that the new scheme can achieve lower numerical amplitude dissipation and period dispersion than some of the existing methods, e.g. the Newmark method and Zhai’s approach, when the same time step size is used. The region of time step dependent on the parameters in the new scheme is explicitly obtained. Finally, some examples of dynamic problems are given to show the accuracy and efficiency of the proposed scheme applied in dynamic systems. The project supported by the National Key Basic Research and Development Foundation of the Ministry of Science and Technology of China (G2000048702, 2003CB716707), the National Science Fund for Distinguished Young Scholars (10025208), the National Natural Science Foundation of China (Key Program) (10532040), the Research Fund for Oversea Chinese (10228028). The English text was polished by Keren Wang.     

SECTIONAL FINITE ELEMENT ANALYSIS OF COUPLED DEFORMATION BETWEEN ELASTOPLASTIC SHEET METAL AND VISCO-ELASTOPLASTIC BODY

The present paper is devoted to developing a new numerical simulation method for the analysis of viscous pressure forming (VPF), which is a sheet flexible-die forming (FDF) process. The pressure-carrying medium used in VPF is one kind of semisolid, flowable and viscous material and its deformation behavior can be described by the visco-elastoplastic constitutive model. A sectional finite element model for the coupled deformation analysis between the visco- elastoplastic pressure-carrying medium and the elastoplastic sheet metal is proposed. The resolution of the Updated Lagrangian (UL) formulation is based on a static explicit approach. The frictional contact between sheet metal and visco-elastoplastic pressure-carrying medium is treated by the penalty function method. Coupled deformation between sheet metal and visco-elastoplastic pressure-carrying medium with large slip is analyzed to validate the developed algorithm. Finally, the viscous pressure bulging (VPB) process of DC06 sheet metal is simulated. Good agreement between numerical simulation results and experimental measurements shows the validity of the developed algorithm.     

Response of frictional contact problems in thermo-rheologically complex structures

This paper presents a comprehensive computational model for predicting the nonlinear response of frictional viscoelastic contact systems under thermo-mechanical loading and experience geometrical nonlinearity. The nonlinear viscoelastic constitutive model is expressed by an integral form of a creep function, whose elastic and time-dependent properties change with stresses and temperatures. The thermo-viscoelastic behavior of the contacting bodies is assumed to follow a class of thermo-rheologically complex materials. An incremental-recursive formula for solving the nonlinear viscoelastic integral equation is derived. Such formula necessitates data storage only from the previous time step. The contact problem as a variational inequality constrained model is handled using the Lagrange multiplier method for exact satisfaction of the inequality contact constraints. A local nonlinear friction law is adopted to model friction at the contact interface. The material and geometrical nonlinearities are modeled in the framework of the total Lagrangian formulation. The developed model is verified using available benchmarks. The effectiveness and accuracy of the developed computational model is validated by solving two thermo-mechanical contact problems with different natures. Moreover, obtained results show that the mechanical properties and the class of thermo-rheological behavior of the contacting bodies as well as the coefficient of friction have significant effects on the contact response of nonlinear thermo-viscoelastic materials.