Complementarity Conditions Research Articles

Assembly simulation of Skin Model Shapes: a comparison of two methods taking into account external forces

Assembly simulation of parts with form defects using Skin Model Shapes (SMS) is much more difficult than the usual assembly of ideal models in a CAD system. Some research studies try to tackle this issue, but very few authors focus on form defect, multibody system and external forces and torques, all at once. Yet external forces are preponderant when considering the relative positioning of parts with form defects.The aim of the paper is to compare two different assembly simulation methods based on: (1) Linear Complementarity Condition (LCC), (2) elimination of contact configurations which are not in mechanical equilibrium. First, the two methods are introduced, and a theoretical comparison shows their differences. Next, several assembly cases are studied. Simulation results are compared to evaluate the proposed methods. The influences of form defect and assembly loads are emphasized through these cases.

Efficiently solving linear bilevel programming problems using off-the-shelf optimization software

Many optimization models in engineering are formulated as bilevel problems. Bilevel optimization problems are mathematical programs where a subset of variables is constrained to be an optimal solution of another mathematical program. Due to the lack of optimization software that can directly handle and solve bilevel problems, most existing solution methods reformulate the bilevel problem as a mathematical program with complementarity conditions (MPCC) by replacing the lower-level problem with its necessary and sufficient optimality conditions. MPCCs are single-level non-convex optimization problems that do not satisfy the standard constraint qualifications and therefore, nonlinear solvers may fail to provide even local optimal solutions. In this paper we propose a method that first solves iteratively a set of regularized MPCCs using an off-the-shelf nonlinear solver to find a local optimal solution. Local optimal information is then used to reduce the computational burden of solving the Fortuny-Amat reformulation of the MPCC to global optimality using off-the-shelf mixed-integer solvers. This method is tested using a wide range of randomly generated examples. The results show that our method outperforms existing general-purpose methods in terms of computational burden and global optimality.

Posing Multibody Dynamics With Friction and Contact as a Differential Complementarity Problem

This technical brief revisits the method outlined in Tasora and Anitescu 2011 [“A Matrix-Free Cone Complementarity Approach for Solving Large-Scale, Nonsmooth, Rigid Body Dynamics,” Comput. Methods Appl. Mech. Eng., 200(5–8), pp. 439–453], which was introduced to solve the rigid multibody dynamics problem in the presence of friction and contact. The discretized equations of motion obtained here are identical to the ones in Tasora and Anitescu 2011 [“A Matrix-Free Cone Complementarity Approach for Solving Large-Scale, Nonsmooth, Rigid Body Dynamics,” Comput. Methods Appl. Mech. Eng., 200(5–8), pp. 439–453]; what is different is the process of obtaining these equations. Instead of using maximum dissipation conditions as the basis for the Coulomb friction model, the approach detailed uses complementarity conditions that combine with contact unilateral constraints to augment the classical index-3 differential algebraic equations of multibody dynamics. The resulting set of differential, algebraic, and complementarity equations is relaxed after time discretization to a cone complementarity problem (CCP) whose solution represents the first-order optimality condition of a quadratic program with conic constraints. The method discussed herein has proven reliable in handling large frictional contact problems. Recently, it has been used with promising results in fluid–solid interaction applications. Alas, this solution is not perfect, and it is hoped that the detailed account provided herein will serve as a starting point for future improvements.

Compliant contact versus rigid contact: A comparison in the context of granular dynamics.

We summarize and numerically compare two approaches for modeling and simulating the dynamics of dry granular matter. The first one, the discrete-element method via penalty (DEM-P), is commonly used in the soft matter physics and geomechanics communities; it can be traced back to the work of Cundall and Strack [P.Cundall, Proc. Symp. ISRM, Nancy, France 1, 129 (1971); P. Cundall and O. Strack, Geotechnique 29, 47 (1979)GTNQA80016-850510.1680/geot.1979.29.1.47]. The second approach, the discrete-element method via complementarity (DEM-C), considers the grains perfectly rigid and enforces nonpenetration via complementarity conditions; it is commonly used in robotics and computer graphics applications and had two strong promoters in Moreau and Jean [J. J. Moreau, in Nonsmooth Mechanics and Applications, edited by J. J. Moreau and P. D. Panagiotopoulos (Springer, Berlin, 1988), pp. 1-82; J. J. Moreau and M. Jean, Proceedings of the Third Biennial Joint Conference on Engineering Systems and Analysis, Montpellier, France, 1996, pp.201-208]. The DEM-P and DEM-C are manifestly unlike each other: They use different (i) approaches to model the frictional contact problem, (ii) sets of model parameters to capture the physics of interest, and (iii) classes of numerical methods to solve the differential equations that govern the dynamics of the granular material. Herein, we report numerical results for five experiments: shock wave propagation, cone penetration, direct shear, triaxial loading, and hopper flow, which we use to compare the DEM-P and DEM-C solutions. This exercise helps us reach two conclusions. First, both the DEM-P and DEM-C are predictive, i.e., they predict well the macroscale emergent behavior by capturing the dynamics at the microscale. Second, there are classes of problems for which one of the methods has an advantage. Unlike the DEM-P, the DEM-C cannot capture shock-wave propagation through granular media. However, the DEM-C is proficient at handling arbitrary grain geometries and solves, at large integration step sizes, smaller problems, i.e., containing thousands of elements, very effectively. The DEM-P vs DEM-C comparison is carried out using a public-domain, open-source software package; the models used are available online.

An optimization approach for deriving upper and lower bounds of transportation network vulnerability under simultaneous disruptions of multiple links

This paper aims to develop an optimization approach for deriving the upper and lower bounds of transportation network vulnerability under simultaneous disruptions of multiple links without the need to evaluate all possible combinations as in the enumerative approach. Mathematically, we formulate the upper and lower bounds of network vulnerability asa binary integer bi-level program (BLP). The upper-level subprogram maximizes or minimizes the remaining network throughput under a given number of disrupted links, which corresponds to the upper and lower vulnerability bounds. The lower-level subprogram checks the connectivity of each origin-destination (O-D) pair under a network disruption scenario without path enumeration. Two alternative modeling approaches are provided for the lower-level subprogram: the virtual link capacity-based maximum flow problem formulation and the virtual link cost-based shortest path problem formulation. Computationally, the BLP model can be equivalently reformulated asa single-level mixed integer linear program by making use of the optimality conditions of the lower-level subprograms and linearization techniques for the complementarity conditions and bilinear terms. Numerical examples are also provided to systematically demonstrate the validity, capability, and flexibility of the proposed optimization model. The vulnerability envelope constructed by the upper and lower bounds is able to effectively consider all possible combinations without the need to perform a full network scan, thus avoiding the combinatorial complexity of enumerating multi-disruption scenarios. Using the vulnerability envelope asa network performance assessment tool, planners and managers can more cost-effectively plan for system protection against disruptions, and prioritize system improvements to minimize disruption risks with limited resources.

Dynamic contact of a thermoelastic Mindlin–Timoshenko beam with a rigid obstacle

We concentrate on the dynamics of a thermoelastic Mindlin–Timoshenko beam striking a rigid obstacle. We state classical formulations involving complementarity conditions. Weak formulations are in the form of systems consisting of a hyperbolic variational inequality for a deflection, a hyperbolic and a parabolic equation for an angle of rotation and a thermal strain, respectively. The penalization method is applied to solve the unilateral problem. The time derivative of the function representing the deflection of the beam’s middle line is not continuous due to the hitting the obstacle. The acceleration term has the form of a vector measure.

Complementarity and stability conditions

We discuss the issue of complementarity between the confining phase and the Higgs phase for gauge theories in which there are no light particles below the scale of confinement or spontaneous symmetry breaking. We show with a number of examples that even though the low energy effective theories are the same (and trivial), discontinuous changes in the structure of heavy stable particles can signal a phase transition and thus we can sometimes argue that two phases which have different structures of heavy particles that cannot be continuously connected and thus the phases cannot be complementary. We discuss what this means and suggest that such “stability conditions” can be a useful physical check for complementarity.

Finite horizon model predictive control of electrowetting on dielectric with pinning

A time-discrete spatially-continuous electrowetting on dielectric (EWOD) model with contact line pinning is considered as the state system in an optimal control framework. The pinning model is based on a complementarity condition. In addition to the physical variables describing velocity, pressure, and voltage, the solid-liquid-air interface, i.e., the contact line, arises as a geometric variable that evolves in time. Due to the complementarity condition, the resulting optimal control of a free boundary problem is thus a mathematical program with equilibrium constraints (MPEC) in function space. In order to cope with the geometric variable, a finite horizon model predictive control approach is proposed. Dual stationarity conditions are derived by applying a regularization procedure, exploiting techniques from PDE-constrained optimization, and then passing to the limit in the regularization parameters. Moreover, a function-space-based numerical procedure is developed by following the theoretical limit argument used in the derivation of the dual stationarity conditions. The performance of the algorithm is demonstrated by several examples; including barycenter matching and trajectory tracking.

A regularized method for solving constrained pseudoinverse problems

For a constrained pseudoinverse problem whose operators satisfy the complementarity condition we propose a one-parameter continuous regularization method of the second order. This method is based on stabilization of solutions to Cauchy problems for a linear differential equation of the second order in a Hilbert space which is obtained from the heavy ball method. We establish requirements to the parametric regularization function and perturbation levels that ensure the stability of the method in the class of all possible bounded perturbations.

Local and global convergence of a general inertial proximal splitting scheme for minimizing composite functions

This paper is concerned with convex composite minimization problems in a Hilbert space. In these problems, the objective is the sum of two closed, proper, and convex functions where one is smooth and the other admits a computationally inexpensive proximal operator. We analyze a family of generalized inertial proximal splitting algorithms (GIPSA) for solving such problems. We establish weak convergence of the generated sequence when the minimum is attained. Our analysis unifies and extends several previous results. We then focus on $$\ell _1$$ -regularized optimization, which is the ubiquitous special case where the nonsmooth term is the $$\ell _1$$ -norm. For certain parameter choices, GIPSA is amenable to a local analysis for this problem. For these choices we show that GIPSA achieves finite “active manifold identification”, i.e. convergence in a finite number of iterations to the optimal support and sign, after which GIPSA reduces to minimizing a local smooth function. We prove local linear convergence under either restricted strong convexity or a strict complementarity condition. We determine the rate in terms of the inertia, stepsize, and local curvature. Our local analysis is applicable to certain recent variants of the Fast Iterative Shrinkage–Thresholding Algorithm (FISTA), for which we establish active manifold identification and local linear convergence. Based on our analysis we propose a momentum restart scheme in these FISTA variants to obtain the optimal local linear convergence rate while maintaining desirable global properties.

Solving nearly-separable quadratic optimization problems as nonsmooth equations

An algorithm for solving nearly-separable quadratic optimization problems (QPs) is presented. The approach is based on applying a semismooth Newton method to solve the implicit complementarity problem arising as the first-order stationarity conditions of such a QP. An important feature of the approach is that, as in dual decomposition methods, separability of the dual function of the QP can be exploited in the search direction computation. Global convergence of the method is promoted by enforcing decrease in component(s) of a Fischer---Burmeister formulation of the complementarity conditions, either via a merit function or through a filter mechanism. The results of numerical experiments when solving convex and nonconvex instances are provided to illustrate the efficacy of the method.

Integration decision-making in technology-sourcing cross-border M&As: a mathematical model

Post-merger integration plays a key role in the success of mergers and acquisitions, but how to choose the appropriate integration mode to achieve potential synergies in mergers and acquisitions (M&As) still lacks widely accepted theoretical support and practical experience. Therefore, this article establishes a mathematical model considering the similarity and complementarity of resources in an attempt to explore the optimal integration strategy (including the choices of integration degree and autonomy granted to the target) under different conditions of resources similarity and complementarity, thus building a comprehensive framework that considers pre- and post-merger factors together to jointly explain innovation realization in technology-sourcing cross-border M&As. The results show that the higher the resource similarity, the higher the degree of integration and the lower the degree of target autonomy. Conversely, the higher the resource complementarity, the lower the degree of integration and the higher the degree of target autonomy. Resource similarity and resource complementarity have negative interactions for integration and target autonomy. The higher the resource similarity, the faster the convergence of optimal degree of integration, while the higher the resource complementarity, the slower the convergence. The study results add to our knowledge of M&A integration management, and provide some implication for practitioners.

A primal-dual interior/exterior point method, with combined directions and quadratic test in reactive optimal power flow problems

In this paper, we propose a primal-dual interior/ exterior point method that uses the modified logarithmic barrier function together with a quadratic test strategy and new search direction procedures for solving the reactive optimal power flow problem. The quadratic test is proposed as an alternative strategy for the Cholesky procedure for calculating the positivity of the Hessian matrix of the problem. The new search directions investigated in the paper are determined by combining the search directions calculated in the predictor and corrector steps, and also by using information associated with the complementarity conditions. The method is implemented in Matlab and applied to solving the reactive optimal power flow problem for 9 and 39 bus systems, as well as for the IEEE 14, 30, 57 and 118 test systems.The results reinforce the efficiency of the method proposed for solving the problem.

Inference on Estimators Defined by Mathematical Programming

We propose an inference procedure for estimators defined by mathematical programming problems, focusing on the important special cases of linear programming (LP) and quadratic programming (QP). In these settings, the coefficients in both the objective function and the constraints of the mathematical programming problem may be estimated from data and hence involve sampling error. Our inference approach exploits the characterization of the solutions to these programming problems by complementarity conditions; by doing so, we can transform the problem of doing inference on the solution of a constrained optimization problem (a non-standard inference problem) into one involving inference based on a set of inequalities with pre-estimated coefficients, which is much better understood. We evaluate the performance of our procedure in several Monte Carlo simulations and an empirical application to the classic portfolio selection problem in finance.

Inference of Estimators Defined by Mathematical Programming

We propose an inference procedure for estimators defined by mathematical programming problems, focusing on the important special cases of linear programming (LP) and quadratic programming (QP). In these settings, the coefficients in both the objective function and the constraints of the mathematical programming problem may be estimated from data and hence involve sampling error. Our inference approach exploits the characterization of the solutions to these programming problems by complementarity conditions; by doing so, we can transform the problem of doing inference on the solution of a constrained optimization problem (a non-standard inference problem) into one involving inference based on a set of inequalities with pre-estimated coefficients, which is much better understood. We evaluate the performance of our procedure in several Monte Carlo simulations and an empirical application to the classic portfolio selection problem in finance.

Generic Properties for Semialgebraic Programs

In this paper we study genericity for the class of semialgebraic optimization problems with equality and inequality constraints, in which every problem of the class is obtained by linear perturbations of the objective function, while the “core” objective function and the constraint functions are kept fixed. Assume that the linear independence constraint qualification is satisfied at every point in the constraint set. It is shown that almost all problems in the class are such that (i) the restriction of the objective function on the constraint set is coercive and regular at infinity; (ii) there is a unique optimal solution, lying on a unique active manifold, at which the strict complementarity and second order sufficiency conditions, the quadratic growth condition, and the Hölder type global error bound hold, and (iii) all minimizing sequences converge. Furthermore, the active manifold is constant, and the optimal solution and the optimal value function depend analytically under local perturbations of the objective function. These results are combined with a standard result about the existence of sums of squares certificates to prove that we can build a sequence of semidefinite programs whose solutions give rise to a sequence of points converging to the optimal solution of the original problem in finitely many steps. It is worth emphasizing that the results of this study hold globally and we do not require the problem to be convex or the constraint set to be compact.

Existence and Uniqueness of a Global Solution for Reactive Transport with Mineral Precipitation-Dissolution and Aquatic Reactions in Porous Media

We consider a macroscopic (averaged) model of transport and reaction in the porous subsurface. The model consists of PDEs for the concentrations of the mobile (dissolved) species and of ODEs for the immobile (mineral) species. For the reactions, we assume the kinetic mass action law. The constant activity of the mineral species leads to set-valued rate functions or complementarity conditions coupled to the PDEs and ODEs. In this paper we first prove the equivalence of several formulations in a weak sense. Then we prove the existence and the uniqueness of a global solution for a multispecies multireaction setting with the method of a priori estimates. In addition to mineral precipitation-dissolution reactions, the model also allows for aquatic reactions, i.e., reactions among the mobile species. In both the mineral precipitation-dissolution rates and the aquatic reaction rates we consider polynomial nonlinearities of arbitrarily high order.

An Optimization Approach for Deriving Upper and Lower Bounds of Transportation Network Vulnerability under Simultaneous Disruptions of Multiple Links

Abstract: This paper aims to develop an optimization approach for deriving the upper and lower bounds of transportation network vulnerability under simultaneous disruptions of multiple links without the need to evaluate all possible combinations as in the enumerative approach. Mathematically, we formulate the upper and lower bounds of network vulnerability as a binary integer bi-level program (BLP). The upper-level subprogram maximizes or minimizes the remaining network throughput under a given number of disrupted links, which corresponds to the upper and lower vulnerability bounds. The lower-level subprogram checks the connectivity of each origin-destination (O-D) pair under a network disruption scenario without path enumeration. Two alternative modeling approaches are provided for the lower-level subprogram: the virtual link capacity-based maximum flow problem formulation and the virtual link cost-based shortest path problem formulation. Computationally, the BLP model can be equivalently reformulated as a single-level mixed integer linear program by making use of the optimality conditions of the lower-level subprograms and linearization techniques for the complementarity conditions and bilinear terms. Numerical examples are also provided to systematically demonstrate the validity, capability, and flexibility of the proposed optimization model. The vulnerability envelope constructed by the upper and lower bounds is able to effectively consider all possible combinations without the need to perform a full network scan, thus avoiding the combinatorial complexity of enumerating multi-disruption scenarios. Using the vulnerability envelope as a network performance assessment tool, planners and managers can more cost-effectively plan for system protection against disruptions, and prioritize system improvements to minimize disruption risks with limited resources.

Investigation on Economic and Reliable Operation of Meshed MTDC/AC Grid as Impacted by Offshore Wind Farms

The influence of complementarity balancing connected grid and control of offshore wind power generation on the economic and reliable operation of hybrid multiterminal HVDC (MTDC) and AC power system is investigated in the paper. The variances of distribution curve of both transmission loss of AC/DC system and voltage of critical nodes assessed by cumulant-based method are used to demonstrate how the different complementarity conditions, generation balancing strategies, MTDC grid topologies, and control schemes affect power system operation. In this paper, a study case of 16-generator 68-bus power system connected with three offshore wind farms is presented. The variance indices of transmission loss and voltage magnitude are calculated and then compared under different complementarity levels, active power balancing strategies, HVDC transmission systems, and controls, respectively. The analysis reveals that higher complementarity and multi-machine balancing strategy can effectively reduce system operational cost and enhance operational reliability. Compared to the point-to-point topology, the control pattern and parameter setting of multi-terminal topology are two key factors to determine its impact on system operation.

A multisurface constitutive model for highly cross-linked polymers with yield data obtained from molecular dynamics simulations

Constitutive properties for highly cross-linked glassy polymers are currently determined by molecular dynamics (MD) simulations. This avoids the need for ad-hoc experimentation. Constitutive data in functional form, such as yield surfaces, still require identification and correspondence to an existing function. In addition, loss of information occurs with fitting procedures. The present alternative consists in directly defining piecewise-linear yield functions from a set of points obtained by MD simulations. To prevent the algorithmic issues of multisurface plasticity, we propose an alternative to active-set strategies by simultaneously including all yield functions regardless of being active. We smooth the complementarity conditions using the Chen–Mangasarian function. In addition, extrapolation is proposed for slowly-evolving quantities such as the effective plastic strain while fully implicit integration is adopted for rapidly-evolving constitutive quantities. Since polymers exhibit finite-strain behavior, we propose a semi-implicit integration algorithm which allows a small number of steps to be used up to very large strains. Experimentally-observed effects herein considered are: thermal effects on strain (i.e. thermal expansion), Young’s modulus dependence on temperature and the effects of strain rate and temperature on the yield stress. A prototype model is first studied to assess the performance of the integration algorithm, followed by a experimental validation and a fully-featured, thermally-coupled 2D example.