Active Set Strategy Research Articles

An Improved Normal Compliance method for dynamic hyperelastic problems with energy conservation property

The purpose of this work is to present an improved energy conservation method for hyperelastodynamic contact problems based on specific normal compliance conditions. In order to determine this Improved Normal Compliance (INC) law, we use a Moreau–Yosida α-regularization to approximate the unilateral contact law. Then, based on the work of Hauret and Le Tallec (2006), we propose in the discrete framework a specific approach allowing to respect the energy conservation of the system in adequacy with the continuous case. This strategy (INC) is characterized by a conserving behavior for frictionless impacts and admissible dissipation for friction phenomena while limiting penetration. Then, we detail the numerical treatment within the framework of the semi-smooth Newton method and primal–dual active set strategy for the normal compliance conditions with friction. We finally provide some numerical experiments to bring into light the energy conservation and the efficiency of the INC method by comparing with different classical methods from the literature through representative contact problems.

A trust-region based an active-set interior-point algorithm for fuzzy continuous Static Games

<abstract><p>In this paper, a novel treatment for fuzzy continuous static games (FCSGs) is introduced. This treatment is based on the fact that, as well as having a fuzzy number, the fuzziness is applied to the control vectors to deal with high vagueness and imprecision in a continuous static game. The concept of the $ \alpha $-level set used for converting the FCSGs to a deterministic problem $ \alpha $-FCSGs. An active-set strategy is used with Newton's interior point method and a trust-region strategy to insure global convergence for deterministic $ \alpha $-FCSGs problems from any starting point. A reduced Hessian technique is used to overcome the difficulty of having an infeasible trust-region subproblem. The active-set interior-point trust-region algorithm has new features; it is easy to implement and has rapid convergence. Preliminary numerical results are reported.</p></abstract>

Improving the evaluation efficiency of failure probability for large and complex systems using an innovative active set strategy

Reliability-based design optimization (RBDO) is a well-known design procedure for solving uncertainties and seeking better design, but evaluation of its failure probabilistic constraints involves an unbearable computational burden. In this study, the innovative active set strategy (ASS) for the reliability index approach is proposed to improve the efficiency and robustness of evaluating probabilistic constraints for RBDO by establishing an ASS and defining an update factor. The innovative ASS is defined using inequality to effectively perform reliability analysis and obtain the feasible most probable failure point in the inner loop. Meanwhile, a defined update factor is applied to eliminate the unnecessary probability constraints and renew the active constraints in the outer loop. The inner loop and outer loop are synchronized to enhance computational efficiency during the whole optimization process. Five commonly used examples are presented, demonstrating that the ASS is a superior efficient and robust method for a large-scale system.

A novel reliability index approach and applied it to cushioning packaging design

This paper is aimed at proposing a new approach to Reliability-based Design Optimization (RBDO) and applying it to cushioning packaging design with a highly nonlinear system. The problem is formulated as an RBDO problem, which is included a minimizing cost function and probabilistic constraints. Here, the thickness of the cushion material is dealt with uncertainty and uncontrolled parameters. The traditional reliability index approach (RIA) has evolved as a powerful tool to solve the RBDO problem; however, due to its convergence problem, the modified reliability index approach (MRIA) is proposed. Although the MRIA method solves the problems of the traditional RIA, it inherits the low efficiency of searching for the most probability point (MPP). Thus, we developed a novel RIA based on MRIA to improve the efficiency and robustness during the RBDO process. The innovation active set strategy is developed in reliability assessment, which is a strict inequality to determine whether the current constraint is active or inactive. An application example is presented, and the results are compared with MRIA to assess cost-effectiveness and efficiency. Results indicate the proposed method is feasible to solve the uncertainty problem of packaging materials in the processing process and is also an efficient RBDO method.

GLS methods for Stokes equations under boundary condition of friction type: formulation-analysis-numerical schemes and simulations

In this paper, we present in two and three dimensional space Galerkin least squares (GLS) methods allowing the use of equal order approximation for both the velocity and pressure modeling the Stokes equations under Tresca’s boundary condition. We propose and analyse two finite element formulations in bounded domains. Firstly, we construct the unique weak solution for each problem by using the method of regularization combined with the monotone operators theory and compactness properties. Secondly, we study the convergence of the finite element approximation by deriving a priori error estimate. Thirdly, we formulate three numerical algorithms namely; projection-like algorithm couple with Uzawa’s iteration, the alternative direction method of multiplier and an active set strategy. Finally some numerical experiments are performed to confirm the theoretical findings and the efficiency of the schemes formulated.

An isogeometric finite element formulation for frictionless contact of Cosserat rods with unconstrained directors

This paper presents an isogeometric finite element formulation for nonlinear beams with impenetrability constraints, based on the kinematics of Cosserat rods with unconstrained directors. The beam cross-sectional deformation is represented by director vectors of an arbitrary order. For the frictionless lateral beam-to-beam contact, a surface-to-surface contact algorithm combined with an active set strategy and a penalty method is employed. The lateral boundary surface of the beam is parameterized by its axis and cross-sectional boundary curves with NURBS basis functions having at least C^2-continuity, which yields a continuous surface metric and curvature for the closest point projection. Three-dimensional constitutive laws of hyperelastic materials are considered. Several numerical examples verify the accuracy and efficiency of the proposed beam contact formulation in comparison to brick element solutions. The lateral contact pressure distribution of the beam formulation is in excellent agreement with the contact pressure of the brick element formulation while requiring much less degrees-of-freedom.

Unified primal-dual active set method for dynamic frictional contact problems

In this paper, we propose a semi-smooth Newton method and a primal-dual active set strategy to solve dynamical contact problems with friction. The conditions of contact with Coulomb’s friction can be formulated in the form of a fixed point problem related to a quasi-optimization one thanks to the semi-smooth Newton method. This method is based on the use of the primal-dual active set (PDAS) strategy. The main idea here is to find the correct subset mathcal{A} of nodes that are in contact (active) opposed to those which are not in contact (inactive). For each case, the nonlinear boundary condition is replaced by a suitable linear one. Numerical experiments on both hyper-elastic problems and rigid granular materials are presented to show the efficiency of the proposed method.

Minimization over the ell _1-ball using an active-set non-monotone projected gradient

The ell _1-ball is a nicely structured feasible set that is widely used in many fields (e.g., machine learning, statistics and signal analysis) to enforce some sparsity in the model solutions. In this paper, we devise an active-set strategy for efficiently dealing with minimization problems over the ell _1-ball and embed it into a tailored algorithmic scheme that makes use of a non-monotone first-order approach to explore the given subspace at each iteration. We prove global convergence to stationary points. Finally, we report numerical experiments, on two different classes of instances, showing the effectiveness of the algorithm.

An active set strategy to address the ill-conditioning of smoothing methods for solving finite linear minimax problems

An active set strategy to address the ill-conditioning of smoothing methods for solving finite linear minimax problems

Active set strategy-based sequential approximate programming for reliability-based design optimization

To improve the evaluation efficiency of failure probability in RBDO models with uncertainty, many RIA-based, PMA-based methods have evolved as a powerful procedure, including the modified reliability index approach (MRIA), PMA two-level, PMA with sequential approximate programming (SAP). However, MRIA may encounter inefficiency and instability when applied to complex concave performance functions, and so does PMA two-level, not for PMA with SAP. The active set strategy-based SAP (ASS-based SAP) for PMA is proposed to accelerate computational efficiency through establishing an active set strategy and a deciding factor. The active set strategy defined by using inequality is to identify the feasible most probable target point (MPTP) in the inner loop. The decision factor integrates the reliability index and the active set strategy to quickly renew the active constraints in the outer loop. The reliability assessment and outer optimization are driven simultaneously, thereby the computational efficiency is strengthened. Numerical examples are compared with other reliability methods to demonstrate the excellent performance of the proposed method in efficiency and robustness. Results also show that the proposed method has the ability to solve complex RBDO problems.

An Active-Set Fischer–Burmeister Trust-Region Algorithm to Solve a Nonlinear Bilevel Optimization Problem

In this paper, the Fischer–Burmeister active-set trust-region (FBACTR) algorithm is introduced to solve the nonlinear bilevel programming problems. In FBACTR algorithm, a Karush–Kuhn–Tucker (KKT) condition is used with the Fischer–Burmeister function to transform a nonlinear bilevel programming (NBLP) problem into an equivalent smooth single objective nonlinear programming problem. To ensure global convergence for the FBACTR algorithm, an active-set strategy is used with a trust-region globalization strategy. The theory of global convergence for the FBACTR algorithm is presented. To clarify the effectiveness of the proposed FBACTR algorithm, applications of mathematical programs with equilibrium constraints are tested.

A survey of numerical methods for hemivariational inequalities with applications to contact mechanics

In this paper we present an abstract nonsmooth optimization problem for which we recall existence and uniqueness results. We show a numerical scheme to approximate its solution. The theory is later applied to a sample static contact problem that describes an elastic body in frictional contact with a foundation. This problem leads to a hemivariational inequality which we solve numerically. Finally, we compare three computational methods of solving contact mechanical problems: the direct optimization method, the augmented Lagrangian method and the primal–dual active set strategy.

A truncated Newton algorithm for nonconvex sparse recovery

Some second-order sufficient conditions are established for problems with ℓ1, SCAD and MCP penalties respectively and then an algorithm with active set strategy is proposed for solving ℓ1, SCAD and MCP penalties based on an approximation of the second order sufficient conditions. The active sets are estimated by an identification technique to identify accurately zero components in a neighborhood of a critical point of ℓ1, SCAD and MCP penalties. In the algorithm, a truncated Newton direction is used to update the free variables, while dk=−xk is used to update the active variables. To accelerate the convergence, a nonmonotone line search strategy is used to guarantee global convergence. Numerical results illustrate that the proposed algorithm is competitive with several well-known methods.

An Augmented Lagrangian Method Exploiting an Active-Set Strategy and Second-Order Information

In this paper, we consider nonlinear optimization problems with nonlinear equality constraints and bound constraints on the variables. For the solution of such problems, many augmented Lagrangian methods have been defined in the literature. Here, we propose to modify one of these algorithms, namely ALGENCAN by Andreani et al., in such a way to incorporate second-order information into the augmented Lagrangian framework, using an active-set strategy. We show that the overall algorithm has the same convergence properties as ALGENCAN and an asymptotic quadratic convergence rate under suitable assumptions. The numerical results confirm that the proposed algorithm is a viable alternative to ALGENCAN with greater robustness.

Fast and reliable transient simulation and continuous optimization of large-scale gas networks

We are concerned with the simulation and optimization of large-scale gas pipeline systems in an error-controlled environment. The gas flow dynamics is locally approximated by sufficiently accurate physical models taken from a hierarchy of decreasing complexity and varying over time. Feasible work regions of compressor stations consisting of several turbo compressors are included by semiconvex approximations of aggregated characteristic fields. A discrete adjoint approach within a first-discretize-then-optimize strategy is proposed and a sequential quadratic programming with an active set strategy is applied to solve the nonlinear constrained optimization problems resulting from a validation of nominations. The method proposed here accelerates the computation of near-term forecasts of sudden changes in the gas management and allows for an economic control of intra-day gas flow schedules in large networks. Case studies for real gas pipeline systems show the remarkable performance of the new method.

LMBOPT: a limited memory method for bound-constrained optimization

Recently, Neumaier and Azmi gave a comprehensive convergence theory for a generic algorithm for bound constrained optimization problems with a continuously differentiable objective function. The algorithm combines an active set strategy with a gradient-free line search CLS along a piecewise linear search path defined by directions chosen to reduce zigzagging. This paper describes LMBOPT, an efficient implementation of this scheme. It employs new limited memory techniques for computing the search directions, improves CLS by adding various safeguards relevant when finite precision arithmetic is used, and adds many practical enhancements in other details. The paper compares LMBOPT and several other solvers on the unconstrained and bound constrained problems from the CUTEst collection and makes recommendations on which solver to use and when. Depending on the problem class, the problem dimension, and the precise goal, the best solvers are LMBOPT, ASACG, and LMBFG-EIG-MS.

An active-set with barrier method and trust-region mechanism to solve a nonlinear Bilevel programming problem

<abstract><p>Nonlinear Bilevel programming (NBLP) problem is a hard problem and very difficult to be resolved by using the classical method. In this paper, Karush-Kuhn-Tucker (KKT) condition is used with Fischer-Burmeister function to convert NBLP problem to an equivalent smooth single objective nonlinear programming (SONP) problem. An active-set strategy is used with Barrier method and trust-region technique to solve the smooth SONP problem effectively and guarantee a convergence to optimal solution from any starting point. A global convergence theory for the active-set barrier trust-region (ACBTR) algorithm is studied under five standard assumptions. An applications to mathematical programs are introduced to clarify the effectiveness of ACBTR algorithm. The results show that ACBTR algorithm is stable and capable of generating approximal optimal solution to the NBLP problem.</p></abstract>

An innovation active set strategy reliability optimization method for cushioning design based on dynamic stress–strain curve

The uncertainty of cushioning volume in the process of manufacturing is the main cause of products damaged during transport. Many methods have been presented in protecting the products from damage. However, the uncertainty factors are still difficult to overcome in practical engineering. In order to handle those issues, an innovation active set strategy (IASS) reliability method with probabilistic constraints is proposed. This proposed method is tested through a packaging uncertainty problem, and the results obtained are compared with the modified reliability index approach (MRIA) by evaluating the probability constraints to demonstrate the cost‐effectiveness and efficiency. The results obtained have been indicated that the IASS not only solves the uncertainty problems in packaging engineering but also an efficient and faster reliability analysis method in the reliability‐based design optimization (RBDO) process without sacrificing numerical accuracy and stability. Furthermore, the proposed IASS provides a new design method for cushioning packaging design based on the dynamic stress–strain curve of cushioning material with minimum materials cost. In addition, the application of the RBDO methods to handle the uncertainty problems in packaging engineering has not been studied so far.

A quasi-monolithic phase-field description for orthotropic anisotropic fracture with adaptive mesh refinement and primal–dual active set method

In this work, thermodynamically consistent phase-field fracture frameworks for transversely isotropic and orthotropic settings are proposed. We formulate an anisotropic crack phase-field via a penalization approach for each family of fibers. The resulting model is augmented with thermodynamical arguments and then carefully analyzed from a mechanical perspective. The fracture dissipation inequality to prevent crack healing is imposed via a primal–dual active set strategy. Predictor–corrector mesh adaptivity allows to work with small length-scale parameters at a reasonable computational cost. Due to the importance of laminated structures for industrial applications, fracture responses for transversely isotropic and orthotropic materials are performed. Therein, several studies are conducted that include comparisons of anisotropic formulations with Griffith’s critical elastic energy release rate and with specific critical fracture energy formulations, as well as non-split and split approaches.

A novel framework of surrogate-based feasibility analysis for establishing design space of twin-column continuous chromatography

Multi-column periodic counter-current chromatography (PCC) has attracted wide attention for the primary capture for the purpose of achieving continuous biomanufacturing. Consequently, determining the design space of the continuous capture process is very important to facilitate process understanding and improving product quality. In this work, we proposed a novel approach to identify the design space of continuous chromatography to balance the computational complexity and model predictions. Specifically, surrogate-based feasibility analysis with adaptive sampling is applied to establish the design space of twin-column CaptureSMB process. The surrogate model is constructed based on the developed mechanistic model for the identification of the design space. The effects of process variables (including interconnected loading time, interconnected flowrate, and batch flowrate) on the design space are comprehensively examined based on an active set strategy. Besides, essential factors like recovery-regeneration time and constraints of column performance parameters (yield, productivity, and capacity utilization) are thoroughly investigated. The impact of design variables such as column length is also studied.