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 unified approach to the mathematical analysis of generalized RKPM,gradient RKPM,and GMLS

It is well-known that the conventional reproducing kernel particle method (RKPM) is unfavorable when dealing with the derivative type essential boundary conditions [1], [2], [3]. To remedy this issue a group of meshless methods in which the derivatives of a function can be incorporated in the formulation of the corresponding interpolation operator will be discussed. Formulation of generalized moving least squares (GMLS) on a domain and GMLS on a finite set of points will be presented. The generalized RKPM will be introduced as the discretized form of GMLS on a domain. Another method that helps to deal with derivative type essential boundary conditions is the gradient RKPM which incorporates the first gradients of the function in the reproducing equation. In present work the formulation of gradient RKPM will be derived in a more general framework. Some important properties of the shape functions for the group of methods under consideration are discussed. Moreover error estimates for the corresponding interpolants are derived. By generalizing the concept of corrected collocation method, it will be seen that in the case of employing each of the proposed methods to a BVP, not only the essential boundary conditions involving the function, but also the essential boundary conditions which involve the derivatives could be satisfied exactly at particles which are located on the boundary.     

A steepest descents method for reentry optimization

A steepest descents optimization program is applied to the problem of a lifting vehicle entering the earth's atmosphere. The program employs penalty functions representing terminal conditions and inflight inequality constraints. During each iteration, it reduces a single performance measure which is the sum of the performance index and the penalty functions. Therefore, only one set of adjoint equations must be integrated per iteration.Values of weight factors, multiplying the penalty functions, are automatically adjusted before each iteration in order that the penalty functions will approach acceptable values. This method is shown to be a form of the classical Lagrange multiplier methods.     

Topology optimization of fluid channels with flow rate equality constraints

This note presents topology optimization of fluid channels with flow rate equality constraints. The equality constraints on the specified boundaries are implemented using the lumped Lagrange multiplier method. The quadratic penalty term and cut-off sensitivity are used to maintain the stability of optimization.     

Using augmented Lagrangian particle swarm optimization for constrained problems in engineering">Using augmented Lagrangian particle swarm optimization for constrained problems in engineering

The comparatively new stochastic method of particle swarm optimization (PSO) has been applied to engineering problems especially of nonlinear, non-differentiable, or non-convex type. Its robustness and its simple applicability without the need for cumbersome derivative calculations make PSO an attractive optimization method. However, engineering optimization tasks often consist of problem immanent equality and inequality constraints which are usually included by inadequate penalty functions when using stochastic algorithms. The simple structure of basic particle swarm optimization characterized by only a few lines of computer code allows an efficient implementation of a more sophisticated treatment of such constraints. In this paper, we present an approach which utilizes the simple structure of the basic PSO technique and combines it with an extended non-stationary penalty function approach, called augmented Lagrange multiplier method, for constraint handling where ill conditioning is a far less harmful problem and the correct solution can be obtained even for finite penalty factors. We describe the basic PSO algorithm and the resulting method for constrained problems as well as the results from benchmark tests. An example of a stiffness optimization of an industrial hexapod robot with parallel kinematics concludes this paper and shows the applicability of the proposed augmented Lagrange particle swarm optimization to engineering problems.     

A solution method for dynamic contact problems

An efficient method is presented for analyzing the transient dynamic contact problems of elastic bodies in this paper. This approach exploits the Lagrange multiplier concept and a special time integration algorithm. Due to the introduced high-frequency dissipation in this time integration algorithm, this method can lead to the effective analysis of real response of elastic bodies with dynamic surface contact constraints. The results of numerical examples show that this method can avoid the weakness of the classical Lagrange multiplier method in dealing with dynamic contact problems with relatively high inertial forces. Stable results can be provided when the time integration step size is small. The properties of this method have also been discussed in this paper.     

Base position optimization for mobile painting robot manipulators with multiple constraints

This paper proposes an efficient base position (BP) optimization method for mobile painting robot manipulators (MPRMs). An approximate decoupled model is first established to overcome the coupling problem of painting robots. And the manipulating characteristics are summarized as three constraints: positioning, orientation and singularity avoidance constraints. Then, joint-level performance criteria of one manipulating point and a painting path, which reflect the manipulability and dexterity, were constructed successively. Considering multiple constraints, the BP optimization problem is translated into a standard inequality constrained optimization problem of the path criterion. Two algorithms are designed to solve this problem: one is based on the internal penalty function method used to obtain an initial BP; the other is based on the generalized Lagrange multiplier method used to get the near-optimal BP. This method was applied to a real MPRM system painting three typical surfaces: flat, cylindrical and truncated conical surfaces. Application results demonstrate the effectiveness as well as the availability of the approximate decoupled model. Simultaneously, compared with previous methods, the efficiency is improved by hundreds of times.     

Comparison of Extended Jacobian and Lagrange Multiplier Based Methods for Resolving Kinematic Redundancy

Several methods have been proposed in the past for resolving the control of kinematically redundant manipulators by optimizing a secondary criterion. The extended Jacobian method constrains the gradient of this criterion to be in the null space of the Jacobian matrix, while the Lagrange multiplier method represents the gradient as being in the row space. In this paper, a numerically efficient form of the Lagrange multiplier method is presented and is compared analytically, computationally, and operationally to the extended Jacobian method. This paper also presents an improved method for tracking algorithmic singularities over previous work.     

一种新的非线性规划神经网络模型

提出一种新型的求解非线性规划问题的神经网络模型.该模型由变量神经元、Lagrange 乘子神经元和Kuhn-Tucker乘子神经元相互连接构成.通过将Kuhn-Tucker乘子神经元限 制在单边饱和工作方式,使得在处理非线性规划问题中不等式约束时不需要引入松弛变量,避 免了由于引入松弛变量而造成神经元数目的增加,有利于神经网络的硬件实现和提高神经网 络的收敛速度.可以证明,在适当的条件下,文中提出的神经网络模型的状态轨迹收敛到与非 线性规划问题的最优解相对应的平衡点.     

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.     

Numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques

We consider incompressible flow problems with defective boundary conditions prescribing only the net flux on some inflow and outflow sections of the boundary. As a paradigm for such problems, we simply refer to Stokes flow. After a brief review of the problem and of its well posedness, we discretize the corresponding variational formulation by means of finite elements and looking at the boundary conditions as constraints, we exploit a penalty method to account for them. We perform the analysis of the method in terms of consistency, boundedness and stability of the discrete bilinear form and we show that the application of the penalty method does not affect the optimal convergence properties of the finite element discretization. Since the additional terms introduced to account for the defective boundary conditions are non-local, we also analyze the spectral properties of the equivalent algebraic formulation and we exploit the analysis to set up an efficient solution strategy. In contrast to alternative discretization methods based on Lagrange multipliers accounting for the constraints on the boundary, the present scheme is particularly effective because it only mildly affects the structure and the computational cost of the numerical approximation. Indeed, it does not require neither multipliers nor sub-iterations or additional adjoint problems with respect to the reference problem at hand.     

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.     

On penalty methods for interelement constraints

The problem of enforcing constraints across interelement boundaries can be treated directly in the construction of the basis or by using multiplier or penalty methods. We demonstrate that the multiplier approach can induce linear dependence in the constraints and lead to a singular system of equations if appropriate precautions are not exercised. The penalty method, on the other hand, is shown to be directly applicable to the overconstrained¹ problem. There are, however, some subtle points regarding loss of accuracy in the penalty approach. We present some new results and corroborating numerical experiments to support the arguments. In particular, we show that there is progressive erosion of accuracy in the approximate solution using finite precision arithmetic as the penalty factor is increased beyond an optimal value. This implies that a control must be applied on the size of the penalty factor. These ideas are quite broadly pertinent to computation of solutions to other constrained optimization problems in which multiplier and penalty methods are used as well as the specific class of problems we consider here.     

Numerical analysis of fluid squeezed between two parallel plates by meshless method

This paper presents the steady-state and transient analysis of the fluid squeezed between two long parallel plates. The governing coupled partial differential equations have been discretized by element free Galerkin method and implemented using variational approach. Penalty and Lagrange multiplier techniques have been utilized to enforce the essential boundary conditions. Four point Gauss quadrature has been used to evaluate the viscous terms in the coefficient matrix whereas reduced integration scheme (i.e. one point Gauss quadrature) has been used to evaluate the penalty terms over two-dimensional domain (Ω). Cubicspline, exponential and rational weight functions have been used in the present work. The results obtained by EFG method are compared with those obtained by finite element and analytical methods. The effect of scaling and penalty parameters on EFG results has been discussed in detail.     

Nonconvex plus quadratic penalized low-rank and sparse decomposition for noisy image alignment针对含噪图像对准问题的非凸加二次罚项约束的低秩和稀疏分解方法

This paper proposes a general method for dealing with the problem of recovering the low-rank structure, in which the data can be deformed by some unknown transformations and corrupted by sparse or nonsparse noises. Nonconvex penalization method is used to remedy the drawbacks of existing convex penalization method and a quadratic penalty is further used to better tackle the nonsparse noises in the data. We exploits the local linear approximation (LLA) method for turning the resulting nonconvex penalization problem into a series of weighted convex penalization problems and these subproblems are efficiently solved via the augmented Lagrange multiplier (ALM). Besides comparing with the method of robust alignment by sparse and low-rank decomposition for linearly correlated images (RASL), we also propose a nonconvex penalized lowrank and sparse decomposition (NLSD) model as comparison. Numerical experiments are conducted on both controlled and uncontrolled data to demonstrate the outperformance of the proposed method over RASL and NLSD.     

Lagrange神经网络的稳定性分析

若重新定义与不等式约束相关的乘子为正定函数,则在构造Lagrange神经网络时,可直接使用处理等式约束的方法处理不等式约束,不需再用松驰变量将不等式约束转换为等式约束,减小了网络实现的复杂程度.利用Liapunov一阶近似原理,严格分析了这类Lagrange神经网络的局部稳定性;并采用LaSalle不变集原理,讨论其大范围稳定性.     

改进Lagrange乘子法及收敛性分析

将与不等式约束相关的乘子重新定义为原乘子的正定函数,则Karush-Kuhn-Tucker必要条件中关于不等式约束乘子的非负约束可以去掉,并能构造出直接处理不等式约束的Lagrange乘子法.分析了算法的收敛性,利用LaSalle不变集原理揭示其稳定机制,并讨论如何减弱收敛条件和扩大收敛域.     

Binary level set methods for topology and shape optimization of a two-density inhomogeneous drum

We optimize eigenvalues in optimal shape design using binary level set methods. The interfaces of subregions are represented implicitly by the discontinuities of binary level set functions taking two values 1 or ?1 at convergence. A binary constraint is added to the original model problems. We propose two variational algorithms to solve the constrained optimization problems. One is a hybrid type by coupling the Lagrange multiplier approach for the geometry constraint with the augmented Lagrangian method for the binary constraint. The other is devised using the Lagrange multiplier method for both constraints. The two iterative algorithms are both largely independent of the initial guess and can satisfy the geometry constraint very accurately during the iterations. Intensive numerical results are presented and compared with those obtained by level set methods, which demonstrate the effectiveness and robustness of our algorithms.     

An augmented Lagrangian method for a class of LMI-constrained problems in robust control theory

We present a new approach to a class of non-convex LMI-constrained problems in robust control theory. The problems we consider may be recast as the minimization of a linear objective subject to linear matrix inequality (LMI) constraints in tandem with non-convex constraints related to rank deficiency conditions. We solve these problems using an extension of the augmented Lagrangian technique. The Lagrangian function combines a multiplier term and a penalty term governing the non-convex constraints. The LMI constraints, due to their special structure, are retained explicitly and not included in the Lagrangian. Global and fast local convergence of our approach is then obtained either by an LMI-constrained Newton type method including line search or by a trust-region strategy. The method is conveniently implemented with available semi-definite programming (SDP) interior-point solvers. We compare its performance to the wellknown D - K iteration scheme in robust control. Two test problems are investigated and demonstrate the power and efficiency of our approach.