A solution method for static and dynamic analysis of three-dimensional contact problems with friction

A solution method is presented for the analysis of contact between two (or more) three-dimensional bodies. The surfaces of the contacting bodies are discretized using quadrilateral surface segments. A Lagrange multiplier technique is employed to impose that, in the contact area, the surface displacements of the contacting bodies are compatible with each other. Distributed contact tractions over the surface segments are calculated from the externally applied forces, inertia forces and internal element stresses. Using the segment tractions, Coulomb's law of friction is enforced in a global sense over each surface segment. The time integration of dynamic response is performed using the Newmark method with parameters and . Using these parameters the energy and momentum balance criteria for the contacting bodies are satisfied accurately when a reasonably small time step is used.

The applicability of the algorithm is illustrated by selected sample numerical solutions to static and dynamic contact problems.     

Consistent treatment of boundaries with mortar contact formulations using dual Lagrange multipliers

Computational modelling of contact problems raises two basic questions: Which method should be used to enforce the contact conditions and how should this method be discretised? The most popular enforcement methods are the Lagrange multiplier method, the penalty method and combinations of these two. A frequently used discretisation method is the so called node-to-segment approach. However, this approach might lead to problems like jumps in contact forces, loss of convergence or failure to pass the patch test. Thus in the last few years, several segment-to-segment contact algorithms based on the mortar method were proposed.Combination of a mortar discretisation with a penalty based enforcement of the contact conditions leads to unphysical penetrations. On the other hand, a Lagrange multiplier mortar method requires additional unknowns. Hence, condensation of the Lagrange multipliers is desirable to preserve the initial size of the system of equations. This can be achieved by interpolating the Lagrange multipliers with so-called dual shape functions.Discretising two contacting bodies leads to opposed contact surface representations of finite element edges, called slave and master elements, respectively. In current versions of dual Lagrange multiplier mortar formulations an inconsistency at the boundary appears when only a part of a slave element (instead of the entire element) belongs to the contact area. We present a modified definition of the dual shape functions in such slave elements. The basic idea is to construct dual shape functions that fulfill the so-called biorthogonality condition within the contact area. This leads to consistent mortar matrices also in the boundary region. To avoid ill-conditioning of the stiffness matrix, the modified mortar matrices are weighted with appropriate weighting factors. In doing so, the corresponding modified Lagrange multiplier nodal values are of the same order as the unmodified ones. Various examples demonstrate the performance of the modified mortar contact algorithm.     

A Dirichlet–Neumann type algorithm for contact problems with friction

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. We introduce a new algorithm for the numerical solution of a nonlinear contact problem with Coulomb friction between linear elastic bodies. The discretization of the nonlinear problem is based on mortar techniques. We use a dual basis Lagrange multiplier space for the coupling of the different bodies. The boundary data transfer at the contact zone is essential for the algorithm. It is realized by a scaled mass matrix which results from the mortar discretization on non-matching triangulations. We apply a nonlinear block Gauss–Seidel method as iterative solver which can be interpreted as a Dirichlet–Neumann algorithm for the nonlinear problem. In each iteration step, we have to solve a linear Neumann problem and a nonlinear Signorini problem. The solution of the Signorini problem is realized in terms of monotone multigrid methods. Numerical results illustrate the performance of our approach in 2D and 3D. Received: 20 March 2001 / Accepted: 1 February 2002 Communicated by P. Deuflhard     

A two-level iteration method for solution of contact problems

The merits and limitations of some existing procedures for the solution of contact problems, modeled by the finite element method, are examined. Based on the Lagrangian multiplier method, a partitioning scheme can be used to obtain a small system of equation for the Lagrange multipliers which is then solved by the conjugate gradient method. A two-level contact algorithm is employed which first linearizes the nonlinear contact problem to obtain a linear contact problem that is in turn solved by the Newton method. The performance of the algorithm compared to some existing procedures is demonstrated on some test problems.     

Numerical Simulation of the Sedimentation of Rigid Bodies in an Incompressible Viscous Fluid by Lagrange Multiplier/Fictitious Domain Methods Combined with the Taylor–Hood Finite Element Approximation

In this work we discuss an application of a distributed Lagrange multiplier based fictitious domain method, to the numerical simulation of the motion of rigid bodies settling in an incompressible viscous fluid. The solution method combines a third order finite element approximation, and time integration by operator splitting. Convergence results are shown for a simple Stokes flow with a circular rigid body that rotates with constant angular velocity. Results of numerical experiments for two sedimenting cylinders in a two-dimensional channel are presented. We present also results for the sedimentation of 100 and 504 cylinders.     

A mortar-based frictional contact formulation for large deformations using Lagrange multipliers

In this work a Lagrange multiplier method is proposed to solve 2D Coulomb frictional contact problems in the context of large deformations. As the proposed formulation is based on the mortar method, the constraints are imposed in a weak integral sense along the contact surface. In order to compute the contact integrals, we use a numerical integration based on the definition of the kinematical variables (gap, slip and their variations) at the quadrature points. The linearization of non-linear equations (virtual work and contact constraints) is developed in order to apply a Newton’s method. The examples show that the numerical integration still preserves the optimal rate of convergence of the finite element solution.     

A primal–dual active set strategy for non-linear multibody contact problems

Non-conforming domain decomposition methods provide a powerful tool for the numerical approximation of partial differential equations. For the discretization of a non-linear multibody contact problem, we use the mortar approach with a dual Lagrange multiplier space. To handle the non-linearity of the contact conditions, we apply a primal–dual active set strategy to find the actual contact zone. The algorithm can be easily generalized to multibody contact problems. A suitable basis transformation guarantees the same algebraic structure in the multibody situation as in the one body case. Using an inexact primal–dual active set strategy in combination with a multigrid method yields an efficient iterative solver. Different numerical examples for one and multibody contact problems illustrate the performance of the method.     

Tube Methods for BV Regularization

In this paper tube methods for reconstructing discontinuous data from noisy and blurred observation data are considered. It is shown that discrete bounded variation (BV)-regularization (commonly used in inverse problems and image processing) and the taut-string algorithm (commonly used in statistics) select reconstructions in a tube. A version of the taut-string algorithm applicable for higher dimensional data is proposed. This formulation results in a bilateral contact problem which can be solved very efficiently using an active set strategy. As a by-product it is shown that the Lagrange multiplier of the active set strategy is an efficient parameter for edge detection.     

Solution to boundary shape optimization problem of linear elastic continua with prescribed natural vibration mode shapes

This paper presents a practical method of numerical analysis for boundary shape optimization problems of linear elastic continua in which natural vibration modes approach prescribed modes on specified sub-boundaries. The shape gradient for the boundary shape optimization problem is evaluated with optimality conditions obtained by the adjoint variable method, the Lagrange multiplier method, and the formula for the material derivative. Reshaping is accomplished by the traction method, which has been proposed as a solution to boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The validity of the presented method is confirmed by numerical results of three-dimensional beam-like and plate-like continua.     

FaSa: A fast and stable quadratic placement algorithm

Placement is a critical step in VLSI design because it dominates overall speed and quality of design flow.In this paper,a new fast and stable placement algorithm called FaSa is proposed.It uses quadratic programming model and Lagrange multiplier method to solve placement problems.And an incremental LU factorization method is used to solve equations for speeding up.The experimental results show that FaSa is very stable,much faster than previous algorithms and its total wire length is comparable with other algorithms.     

A Fictitious Domain Method with Distributed Lagrange Multiplier for Parabolic Problems With Moving Interfaces

In this paper, we study the fictitious domain method with distributed Lagrange multiplier for the jump-coefficient parabolic problems with moving interfaces. The equivalence between the fictitious domain weak form and the standard weak form of a parabolic interface problem is proved, and the uniform well-posedness of the full discretization of fictitious domain finite element method with distributed Lagrange multiplier is demonstrated. We further analyze the convergence properties for the fully discrete finite element approximation in the norms of \(L^2\), \(H^1\) and a new energy norm. On the other hand, we introduce a subgrid integration technique in order to allow the fictitious domain finite element method to be performed on the triangular meshes without doing any interpolation between the authentic domain and the fictitious domain. Numerical experiments confirm the theoretical results, and show the good performances of the proposed schemes.     

A numerical technique for solving fractional optimal control problems

This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs). The fractional derivative in these problems is in the Caputo sense. The method is based upon the Legendre orthonormal polynomial basis. The operational matrices of fractional Riemann-Liouville integration and multiplication, along with the Lagrange multiplier method for the constrained extremum are considered. By this method, the given optimization problem reduces to the problem of solving a system of algebraic equations. By solving this system, we achieve the solution of the FOCP. Illustrative examples are included to demonstrate the validity and applicability of the new technique.     

Automated visco-elastic analysis of large scale inertia-variant spatial vehicles

This study is concerned with the dynamic analysis of spatial vehicles consisting of interconnected rigid, elastic, and visco-elastic components. The configuration of each elastic or viscoelastic component is identified using two sets of generalized coordinates; reference and elastic coordinates that, respectively, describe the motion of the component reference and the deformation relative to this reference. Nonlinear constraints between adjacent vehicle components are introduced to the formulation using a Lagrange multiplier technique. The finite element method is employed to introduce the elastic coordinates and component modes are used to reduce the number of degrees of freedom of the vehicle. A linear visco-elastic model is employed in this analysis, wherein the stress is assumed to be proportional to the strain and its time derivative (Kelvin-Voigt model). The resulting mathematical model is highly nonlinear and a strong coupling between the reference motion and elastic deformation appears in the kinetic energy expression. No coupling, however, appears in the stiffness and damping matrices. Nonlinearities in the suspension and tire models are also considered, and the change in the suspension force due to a large change in geometry of vehicle components is accounted for. The vehicle mathematical model, written in terms of a mixed set of physical reference and modal elastic coordinates, is integrated numerically using an explicit-implicit direct numerical integration method and the physical state of the vehicle is recovered using the modal transformation.     

A multi-functional dynamic state estimator for error validation: measurement and parameter errors and sudden load changes

We propose a new and efficient algorithm to detect, identify, and correct measurement errors and branch parameter errors of power systems. A dynamic state estimation algorithm is used based on the Kalman filter theory. The proposed algorithm also successfully detects and identifies sudden load changes in power systems. The method uses three normalized vectors to process errors at each sampling time: normalized measurement residual, normalized Lagrange multiplier, and normalized innovation vector. An IEEE 14-bus test system was used to verify and demonstrate the effectiveness of the proposed method. Numerical results are presented and discussed to show the accuracy of the method.     

Assessment of temporal integration schemes for the sensitivity analysis of frictional contact/impact response of axisymmetric composite structures

An assessment is made of three temporal integration schemes for the sensitivity analysis of the frictional contact/impact response of axisymmetric composite structures. The structures considered consist of an arbitrary number of perfectly-bonded homogeneous anisotropic layers. The material of each layer is assumed to be hyperelastic, and the effect of geometric non-linearity is included. Sensitivity coefficients measure the sensitivity of the response to variations in the different material, lamination and geometric parameters of the structure. A displacement finite element model is used for the spatial discretization. Normal contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with fundamental unknowns consisting of both the nodal displacements and the Lagrange multipliers associated with the contact conditions. The Lagrange multipliers are allowed to be discontinuous at interelement boundaries. Tangential contact conditions are incorporated by using either a penalty method or a Lagrange multiplier technique, in conjunction with the classical Coulomb's friction model. The three temporal integration schemes considered are: the implicit Newmark and Houbolt schemes, and the explicit central difference method. In the case of the implicit methods, the Newton-Raphson iterative technique is used for the solution of the resulting non-linear algebraic equations, and for the determination of the contact region, contact conditions (sliding or sticking), and the contact pressures. Sensitivity coefficients are evaluated by using a direct differentiation approach in conjunction with the incremental equations. Numerical results are presented for the frictional contact of a composite spherical cap impacting a rigid plate, showing the effects of each of the following factors on the accuracy of the predicted response and sensitivity coefficients: (a) incorporating the normal contact conditions, (b) the magnitude of the penalty parameter in the normal direction (for the perturbed Lagrangian method), and (c) the time step size for the response and the sensitivity analyses.     

Modified augmented Lagrangian coordination and alternating direction method of multipliers with parallelization in non-hierarchical analytical target cascading

Analytical Target Cascading (ATC) is a decomposition-based optimization methodology that partitions a system into subsystems and then coordinates targets and responses among subsystems. Augmented Lagrangian with Alternating Direction method of multipliers (AL-AD), one of efficient ATC coordination methods, has been widely used in both hierarchical and non-hierarchical ATC and theoretically guarantees convergence under the assumption that all subsystem problems are convex and continuous. One of the main advantages of distributed coordination which consists of several non-hierarchical subproblems is that it can solve subsystem problems in parallel and thus reduce computational time. Therefore, previous studies have proposed an augmented Lagrangian coordination strategy for parallelization by eliminating interactions among subproblems. The parallelization is achieved by introducing a master problem and support variables or by approximating a quadratic penalty function to make subproblems separable. However, conventional AL-AD does not guarantee convergence in the case of parallel solving. Our study shows that, in parallel solving using targets and responses of the current iteration, conventional AL-AD causes mismatch of information in updating the Lagrange multiplier. Therefore, the Lagrange multiplier may not reach the optimal point, and as a result, increasing penalty weight causes numerical difficulty in the augmented Lagrangian coordination approach. To solve this problem, we propose a modified AL-AD with parallelization in non-hierarchical ATC. The proposed algorithm uses the subgradient method with adaptive step size in updating the Lagrange multiplier and also maintains penalty weight at an appropriate level not to cause oscillation. Without approximation or introduction of an artificial master problem, the modified AL-AD with parallelization can achieve similar accuracy and convergence with much less computational cost compared with conventional AL-AD with sequential solving.     

周期时变信道特性的宽带电力线比特功率分配算法

从宽带电力线的线性周期时变(蕴P栽V)性出发,提出了一种适应于宽带电力线的比特与功率分配算法.算法分为两个步骤,第一步通过拉格朗日乘子法对每个子信道进行比特预分配,第二步通过二分查找法的思想进行快速迭代.仿真结果表明:与传统算法相比,该算法在保证传输速率的同时能明显降低算法运算量,能克服宽带电力信道的时变特性,提高系统性能,是一种可用于实际通信的算法.     

基于区间直觉模糊集的C均值聚类算法

针对区间直觉模糊集(IVIFS)的聚类问题,提出了基于IVIFS的C均值聚类算法.算法首先应用IVIFS的欧氏距离,构造了聚类的目标函数;然后根据拉格朗日乘数法推导出聚类的迭代公式,得到IVIFS聚类算法;此外,还提出一种IVIFS聚类的有效性函数,并将此函数和聚类结合,给出可以确定最佳聚类类别数的聚类流程;最后通过实...     

Numerical Integrations and Unit Resolution Multipliers for Domain Decomposition Methods with Nonmatching Grids

In this paper, we are concerned with the non-overlapping domain decomposition method (DDM) with nonmatching grids for three-dimensional problems. The weak continuity of the DDM solution on the interface is imposed by some Lagrange multiplier. We shall first analyze the influence of the numerical integrations over the interface on the (non-conforming) approximate solution. Then we will propose a simple approach to construct multiplier spaces, one of which can be simply spanned by some smooth basis functions with local compact supports, and thus makes the numerical integrations on the interface rather simple and inexpensive. Also it is shown this multiplier space can generate an optimal approximate solution. Numerical results are presented to compare the new method with the point to point method widely used in engineering.     

Constraint variational schemes for analysis of rectangular plates on an elastic half space

The flexural interaction of a rectangular thin elastic plate resting in smooth contact with an isotropic homogeneous elastic half space is analysed by using constraint variational schemes. The deflected shape of the plate is represented by a double power series of spatial variables with a set of generalized coordinates. The contact stresses are expressed in terms of the generalized coordinates by discretizing the contact area into several rectangular regions and solving an appropriate flexibility equation based on generalized Boussinesq's solution. Using the representations adopted for displacement and contact stresses, a constraint energy functional is constructed to determine the generalized coordinates. The constraint term in the variational functional corresponds to plate edge boundary conditions and formulations corresponding to both Lagrange multiplier and penalty types are presented. It is noted that for the present class of problems, penalty type formulations are numerically efficient. The convergence and numerical stability of the solution scheme is confirmed. Selected numerical results are presented to illustrate the dependence of flexural response of plate on the governing parameters of the plate-half space system.