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In this paper we will present a review of recent advances in the application of the augmented Lagrange multiplier method as a general approach for generating multiplier-free stabilised methods. The augmented Lagrangian method consists of a standard Lagrange multiplier method augmented by a penalty term, penalising the constraint equations, and is well known as the basis for iterative algorithms for constrained optimisation problems. Its use as a stabilisation methods in computational mechanics has, however, only recently been appreciated. We first show how the method generates Galerkin/Least Squares type schemes for equality constraints and then how it can be extended to develop new stabilised methods for inequality constraints. Application to several different problems in computational mechanics is given.

Mortar finite element methods allow for a flexible and efficient coupling of arbitrary nonconforming interface meshes and are by now quite well established in nonlinear contact analysis. In this paper, a mortar method for three-dimensional (3D) finite deformation contact is presented. Our formulation is based on so-called dual Lagrange multipliers, which in contrast to the standard mortar approach generate coupling conditions that are much easier to realize, without impinging upon the optimality of the method. Special focus is set on second-order interpolation and on the construction of novel discrete dual Lagrange multiplier spaces for the resulting quadratic interface elements (8-node and 9-node quadrilaterals, 6-node triangles). Feasible dual shape functions are obtained by combining the classical biorthogonality condition with a simple basis transformation procedure. The finite element discretization is embedded into a primal-dual active set algorithm, which efficiently handles all types of nonlinearities in one single iteration scheme and can be interpreted as a semismooth Newton method. The validity of the proposed method and its efficiency for 3D contact analysis including Coulomb friction are demonstrated with several numerical examples.

In this paper, we suggest a new framework for analyzing primal subgradient methods for nonsmooth convex optimization problems. We show that the classical step-size rules, based on normalization of subgradient, or on knowledge of the optimal value of the objective function, need corrections when they are applied to optimization problems with constraints. Their proper modifications allow a significant acceleration of these schemes when the objective function has favorable properties (smoothness, strong convexity). We show how the new methods can be used for solving optimization problems with functional constraints with a possibility to approximate the optimal Lagrange multipliers. One of our primal-dual methods works also for unbounded feasible set.

The purpose of this study was to examine the impact of Gross Domestic Product (GDP) and Logistics Performance Index (LPI) on international trade of nations across each continent and worldwide. Secondary data collected on 142 countries—37 Asian, 41 European, 41 African, 3 Oceania, 14 Middle East, 11 North American and 9 South American–were analysed across the years 2007, 2010, 2012, 2014, 2016, and 2018. Panel regression technique was applied and the random effect (RE) model was chosen based on the results of the Hausman tests and Breusch–Pagan Lagrange Multiplier test. The findings revealed that the LPI has a positive relationship with net exports globally and specifically within the continents of Asia, Europe, and Oceania. Moreover, while the GDP appears to have a significant negative impact on net exports, specifically within Asia, in contrast, countries in Oceania and the Middle East present a positive relationship. Also on the African continent, GDP has a significant negative impact on the net exports. Findings provide a holistic picture of the impact of LPI & GDP on net exports, which will assist governments in the formulation and revision of its strategies and policies to expedite the growth of exports and in turn, the economy. This study was the first of its kind to explore the impact of GDP and LPI on international trade of nations across worldwide.

In this paper, we proposed an approach for computing the Hausdorff distance between convex semialgebraic sets. We exploit the KKT conditions to rewrite the Hausdorff distance as polynomial maximization problems under some assumptions, in which polynomial and rational expressions of Lagrange multipliers are used. Then, polynomial maximization problems are solved by Lasserre’s hierarchy of semi-definite relaxations. Finally, some numerical examples are reported.

We present a general framework for the analysis and modelling of frictional contact involving composite materials. The study has focused on composite materials formed by a matrix of rubber and synthetic or metallic fibres, which is the case of standard tires. We detail the numerical treatment of incompressibility at large deformations that rubber can experience, as well as the stiffening effect that properly oriented fibres will induce within the rubber. To solve the frictional contact between solids, a Dual Augmented Lagrangian Multiplier Method is used together with the Mortar method. This ensures a variationally consistent estimation of the contact forces. A modified Serial-Parallel Rule of Mixtures is employed to model the behaviour of composite materials. This is a simple and novel methodology that allows the blending of constitutive behaviours as diverse as rubber (very low stiffness and incompressible behaviour) and steel (high stiffness and compressible behaviour) taking into account the orientation of the fibres within the material. The locking due to the incompressibility constraint in the rubber material has been overcome by using Total Lagrangian mixed displacement-pressure elements. A collection of numerical examples is provided to show the accuracy and consistency of the methodology presented when solving frictional contact, incompressibility and composite materials under finite strains.

Studies have shown that the surrogate subgradient method, to optimize non-smooth dual functions within the Lagrangian relaxation framework, can lead to significant computational improvements as compared to the subgradient method. The key idea is to obtain surrogate subgradient directions that form acute angles toward the optimal multipliers without fully minimizing the relaxed problem. The major difficulty of the method is its convergence, since the convergence proof and the practical implementation require the knowledge of the optimal dual value. Adaptive estimations of the optimal dual value may lead to divergence and the loss of the lower bound property for surrogate dual values. The main contribution of this paper is on the development of the surrogate Lagrangian relaxation method and its convergence proof to the optimal multipliers, without the knowledge of the optimal dual value and without fully optimizing the relaxed problem. Moreover, for practical implementations, a stepsizing formula that guarantees convergence without requiring the optimal dual value has been constructively developed. The key idea is to select stepsizes in a way that distances between Lagrange multipliers at consecutive iterations decrease, and as a result, Lagrange multipliers converge to a unique limit. At the same time, stepsizes are kept sufficiently large so that the algorithm does not terminate prematurely. At convergence, the lower-bound property of the surrogate dual is guaranteed. Testing results demonstrate that non-smooth dual functions can be efficiently optimized, and the new method leads to faster convergence as compared to other methods available for optimizing non-smooth dual functions, namely, the simple subgradient method, the subgradient-level method, and the incremental subgradient method.

In this paper, efficient algorithms for contact problems with Tresca and Coulomb friction in three dimensions are presented and analyzed. The numerical approximation is based on mortar methods for nonconforming meshes with dual Lagrange multipliers. Using a nonsmooth complementarity function for the three-dimensional friction conditions, a primal-dual active set algorithm is derived. The method determines active contact and friction nodes and, at the same time, resolves the additional nonlinearity originating from sliding nodes. No regularization and no penalization are applied, and superlinear convergence can be observed locally. In combination with a multigrid method, it defines a robust and fast strategy for contact problems with Tresca or Coulomb friction. The efficiency and flexibility of the method is illustrated by several numerical examples.

We introduce a Nitsche-based finite element discretization of the unilateral contact problem in linear elasticity. It features a weak treatment of the nonlinear contact conditions through a consistent penalty term. Without any additional assumption on the contact set, we can prove theoretically its fully optimal convergence rate in the H¹ (Ω)-norm for linear finite elements in two dimensions, which is $O({h^{\\frac{1}{2} + v}})$ when the solution lies in ${H^{\\frac{3}{2} + v}}(\\Omega )$ , 0 < v ≤ 1/2. An interest of the formulation is that, as opposed to Lagrange multiplier-based methods, no other unknown is introduced and no discrete inf-sup condition needs to be satisfied.

Due to manufacturing and assembly errors, there are clearances in the mechanism, which reduces motion precision and working performance of the mechanism. Revolute joint is one of the most widely used joints in the mechanical system. Most researches only consider the radial clearance of revolute joint, but in fact there are both radial and axial clearances; this makes it impossible to master influence of revolute clearance joint on dynamic characteristics of the mechanism. As a kind of spatial parallel pointing mechanism, 3-RRRRR mechanism needs high precision. In order to accurately predict the dynamic characteristics of the mechanism, a modeling method of revolute joint considering both radial and axial clearances is derived, and a modeling method of spatial parallel mechanism considering spatial revolute clearance joint is proposed. On this basis, the dynamic model of 3-RRRRR mechanism with clearance is developed. The Runge–Kutta method is used to analyze the dynamic response of the mechanism when the axial clearance is considered or not. The necessity of the existence of the axial clearance is verified. Then, the dynamic response of the mechanism with different clearance sizes and numbers is analyzed, and the results are compared with virtual prototype simulation to verify the correctness of the mathematical model. Through the analysis, this paper verifies that the axial clearance cannot be ignored and provides a theoretical basis for predicting the effect of spatial revolute clearance joint on the dynamic characteristics of the mechanism and lays a good foundation for the manufacture and application of the mechanism.