Linear Complementarity Formulation Research Articles

Numerical investigation of the seismic response of a polar crane based on linear complementarity formulation

This paper numerically analyzes the seismic response of a polar crane operating at a nuclear plant, with derailment considered. In this case, the wheels and rails can no longer be treated as being bonded, and thus traditional methods in structural dynamics (e.g. response spectrum method) are no longer suitable for transient contact analysis. On the other hand, contact-impact analysis of flexible multibody systems will often face great difficulties attributed to the complexity of modeling and low computational efficiency. To overcome these problems, the wheel-rail interactions are modeled as multiple frictional unilateral constraints, and the resulting contact-impact problem is formulated as a linear complementarity problem (LCP). Later, a novel smoothing method is applied to smooth the LCP formulation by replacing the normal gap with the time-averaged normal gap over a short time period. A model smoothing method is then combined with modal synthesis to derive the equations of motion without high frequency components. Finally, the validity and feasibility of the proposed method are demonstrated through several numerical examples. The results show the different contact states during the seismic process, including smooth contact, detachment and re-contact.

On the pricing of multi-asset options under jump-diffusion processes using meshfree moving least-squares approximation

The moving least-squares (MLS) approximation is a powerful numerical scheme widely used in the meshfree literature to construct local multivariate polynomial basis functions for expanding the solution of a given differential or integral equation. For partial integro-differential equations arising from the valuation of multi-asset options written on correlated Lévy-driven assets, we propose here an MLS-based collocation scheme in conjunction with implicit-explicit (IMEX) temporal discretization to numerically solve the problem. We apply the method to price both European and American options and compute the option hedge parameters. In the case of American options, we use an operator splitting approach to solve the linear complementarity formulation of the problem. Our numerical experiments show the efficiency of the proposed scheme in comparison with some competing approaches, specially finite difference methods.

A simple linear complementarity approach for sliding cable modeling considering friction

• A LCP-based approach is proposed for sliding cable modeling with friction. • This approach is mathematically elegant, simple, and easy to implement. • Uniqueness of sliding length under a given configuration has been proved. • Examples are presented to demonstrate the robustness and versatility. • Influence of friction on deployment of a clustered tensegrity is investigated. Sliding cable system with friction is encountered in many engineering applications. This system is typically characterized by existence of complex and diverse motion states of sliding nodes (pulleys), which leads to significant difficulties for analysis. This paper aims at developing a simple method for analyzing the global response of a sliding cables system. A linear complementarity approach is proposed for both static and dynamic analysis of sliding cable with friction. Within this approach, the challenging and significant issue of determination of motion states of sliding nodes can be handled in a standard linear complementarity formulation. The proposed approach is mathematically elegant, simple, and easy to implement. It allows us to capture the complex and diverse motion states of pulleys in a rather direct way. It also offers a new way to study theory problems such as prove or disprove the existence and uniqueness of the solution by using complementarity theories. Multiple examples are presented to demonstrate the robustness and versatility of the proposed approach, as well as to reveal novel phenomena involved in sliding cable systems considering sliding friction.

Piecewise affine parameterized value-function based bilevel non-cooperative games

Generalizing certain network interdiction games communicated to us by Andrew Liu and his collaborators, this paper studies a bilevel, non-cooperative game wherein the objective function of each player’s optimization problem contains a value function of a second-level linear program parameterized by the first-level variables in a non-convex manner. In the applied network interdiction games, this parameterization is through a piecewise linear function that upper bounds the second-level decision variable. In order to give a unified treatment to the overall two-level game where the second-level problems may be minimization or maximization, we formulate it as a one-level game of a particular kind. Namely, each player’s objective function is the sum of a first-level objective function ± a value function of a second-level maximization problem whose objective function involves a difference-of-convex (dc), specifically piecewise affine, parameterization by the first-level variables. This non-convex parameterization is a major difference from the family of games with min-max objectives discussed in Facchinei et al. (Comput Optim Appl 59(1):85–112, 2014) wherein the convexity of the overall games is preserved. In contrast, the piecewise affine (dc) parameterization of the second-level objective functions to be maximized renders the players’ combined first-level objective functions non-convex and non-differentiable. We investigate the existence of a first-order stationary solution of such a game, which we call a quasi-Nash equilibrium, and study the computation of such a solution in the linear-quadratic case by Lemke’s method using a linear complementarity formulation.

A cell-based smoothed finite element method for multi-body contact analysis using linear complementarity formulation

In this paper, the cell-based smoothed finite element using quadrilateral elements (CS-FEM) is used for 2D contact problems which are converted into linear complementarity problems (LCPs), which can be solved efficiently using the Lemke method. The modified Coulomb friction contact model with tangential strength and normal adhesion is considered, which models sticking-slipping, contacting-departing, and bonding-debonding processes, in a unified formulation. Smoothed Galerkin weak-form with contact boundary is deduced, in which the stiffness is implemented using the CS-FEM with 1 smoothing domain (1SD), 2SD, 3SD, 4SD, 8SD, and 16SD for each element. Contact interface equations are discretized through contact point-pairs that are constructed using a master-slave surface algorithm. Intensive numerical examples are given to investigate the effects of contact parameters on contact behaviors and examine the effectiveness of the proposed approach. The numerical results of CS-FEM models are compared with that of FEM-Q4 model, which demonstrates that all CS-FEM models are softer than FEM-Q4 model. The strain energy solutions, obtained using several CS-FEM models, are monotonically decreasing with the number of the SDs for each element increasing. The upper bound solutions in strain energy can be obtained using a CS-FEM-1SD model in our examples, while the lower bound solutions are obtained using CS-FEM-16SD model or FEM-Q4, with FEM-Q4 solution being the lowest.

Partial slip problem in frictional contact of orthotropic elastic half-plane and rigid punch

Planar partial-slip contact problem of a rigid indenter with flat end and an orthotropic elastic material is analytically and numerically investigated. In the analytical way, the coupled singular equations of this problem are reduced to a Fredholm integral equation with a regular kernel. The analytical solutions are derived in the forms of the Goodman’s and Spence’s approximations. In the numerical way, a linear complementarity formulation is developed by reformulating the governing equations are as coupled Volterra integral equations. And, a Newton-based optimization algorithm based on smoothing approximation is used to solve the problem. The validness of both the Goodman and Spence approximate solutions is verified by comparing with the numerical results for different orthotropic elastic materials.

A linear complementarity method for the solution of vertical vehicle–track interaction

ABSTRACTA new method is proposed for the solution of the vertical vehicle–track interaction including a separation between wheel and rail. The vehicle is modelled as a multi-body system using rigid bodies, and the track is treated as a three-layer beam model in which the rail is considered as an Euler-Bernoulli beam and both the sleepers and the ballast are represented by lumped masses. A linear complementarity formulation is directly established using a combination of the wheel–rail normal contact condition and the generalised-α method. This linear complementarity problem is solved using the Lemke algorithm, and the wheel–rail contact force can be obtained. Then the dynamic responses of the vehicle and the track are solved without iteration based on the generalised-α method. The same equations of motion for the vehicle and track are adopted at the different wheel–rail contact situations. This method can remove some restrictions, that is, time-dependent mass, damping and stiffness matrices of the coupled system, multiple equations of motion for the different contact situations and the effect of the contact stiffness. Numerical results demonstrate that the proposed method is effective for simulating the vehicle–track interaction including a separation between wheel and rail.

A linear complementarity formulation of meshfree method for elastoplastic analysis of gradient-dependent plasticity

This work presents a linear complementarity formulation for elastoplastic analysis of the gradient-dependent plasticity including large deformation problems. The formulation is based on the meshfree smoothed radial point interpolation method, where the parametric variational principle (PVP) is used in the form of linear complementarity and the gradient-dependent plasticity is represented by the linearization of yield criterion. The yield stress is linearly evolved through equivalent plastic strain as well as its Laplacian (namely second gradient). The global discretized system equations are transformed into a standard linear complementarity problem (LCP), which can be solved readily using the Lemke method. The proposed approach is capable of simulating material hardening/softening and strain localization. An extensive numerical study is performed to validate the proposed method and to investigate the effects of various parameters. The numerical results demonstrate that the proposed approach is accurate and stable for the elastoplastic analysis of 2D solids with gradient-dependent plasticity on strain localization.

A linear complementarity formulation for contact problems with regularized friction

A mathematical formulation in terms of a linear complementarity problem with regularized friction is introduced for multibody contact problems. In this approach, contacts are characterized based on kinematic constraints while the friction forces are simultaneously regularized and incorporated into the formulation. The variables of the resulting linear complementarity problem are only the normal forces. The main advantage of this formulation is that the dimension of the resulting linear complementarity problem is significantly less than its counterpart formulations in the literature, and hence, faster simulations can be achieved. The proposed formulation is examined for a set of benchmark examples yielding promising results.

Oblique frictional unilateral contacts perceived in curved bridges

The pounding at the deck and abutment interface of a curved bridge, which occurs during earthquakes after closing the gap, directly leads to the deck unseating, in-plane deck rotation and torsional damage to the columns of the bridge. The study in hand assumes the deck of the curved bridge as a completely rigid body to analyse the multi-body oblique frictional impact at the deck–abutment interaction. Extending other authors’ work, a linear complementarity formulation is implemented for computing the unilateral contact to identify the effects of different impact states, such as stick and forward or backward slip, depending on the initial conditions during single or double impact. Stick and slip conditions during single impact are evaluated for different pre-impact rotations and ratios of transverse to normal pre-impact relative velocities. Also, the inward and outward slips during double impact, which represent the possibility of the deck being out-off lane, and the possibility of post-impact deck rotation are estimated for several curved geometries. The results show the curved bridge is very susceptible for in-plane deck rotation and out-off lane misalignment during the deck–abutment pounding.

Mixed linear complementarity formulation of discontinuous deformation analysis

In the classical Discontinuous Deformation Analysis (DDA) the contact conditions are enforced by what is referred to as open–close iteration. The open–close iteration involves repeatedly fixing and removing the artificial springs between blocks in contact and thus falls into the category of trial-and-error solution procedures that cannot guarantee the solution is always convergent. Meanwhile, improper stiffness parameters of the artificial springs might cause numerical problems. In order to avoid the introduction of the artificial parameters and to avoid the open–close iteration, this study reformulates DDA as a mixed linear complementarity problem (MLCP). Then, the Fisher–Burmeister Line Search Algorithm (FBLSA) is modified to solve MLCP. Some typical examples including those originally designed by the DDA inventor are reanalyzed, proving the procedure is feasible.

A contact analysis approach based on linear complementarity formulation using smoothed finite element methods

Based on the subdomain parametric variational principle (SPVP), a contact analysis approach is formulated in the incremental form for 2D solid mechanics problems discretized using only triangular elements. The present approach is implemented for the newly developed node-based smoothed finite element method (NS-FEM), the edge-based smoothed finite element method (ES-FEM) as well as standard FEM models. In the approach, the contact interface equations are discretized by contact point-pairs using a modified Coulomb frictional contact model. For strictly imposing the contact constraints, the global discretized system equations are transformed into a standard linear complementarity problem (LCP), which can be readily solved using the Lemke method. This approach can simulate different contact behaviors including bonding/debonding, contacting/departing, and sticking/slipping. An intensive numerical study is conducted to investigate the effects of various parameters and validate the proposed method. The numerical results have demonstrated the validity and efficiency of the present contact analysis approach as well as the good performance of the ES-FEM method, which provides solutions of about 10 times better accuracy and higher convergence rate than the FEM and NS-FEM methods. The results also indicate that the NS-FEM provides upper-bound solutions in energy norm, relative to the fact that FEM provides lower-bound solutions.

Comparison and survey of finite difference methods for pricing American options under finite activity jump-diffusion models

Partial integro-differential formulations are often used for pricing American options under jump-diffusion models. A survey on such formulations and their numerical methods is presented. A detailed description of six efficient methods based on a linear complementarity formulation and finite difference discretizations is given. Numerical experiments compare the performance of these methods for pricing American put options under finite activity jump models.

A Comparison and Survey of Finite Difference Methods for Pricing American Options Under Finite Activity Jump-Diffusion Models

Partial-integro differential formulations are often used for pricing American options under jump-diffusion models. A survey on such formulations and numerical methods for them is presented. A detailed description of six efficient methods based on a linear complementarity formulation and finite difference discretizations is given. Numerical experiments compare the performance of these methods for pricing American put options under finite activity jump models.

Linear complementarity formulation for 3D frictional sliding problems

Frictional sliding on quasi-statically deforming faults and fractures can be modeled efficiently using a linear complementarity formulation. We review the formulation in two dimensions and expand the formulation to three-dimensional problems including problems of orthotropic friction. This formulation accurately reproduces analytical solutions to static Coulomb friction sliding problems. The formulation accounts for opening displacements that can occur near regions of non-planarity even under large confining pressures. Such problems are difficult to solve owing to the coupling of relative displacements and tractions; thus, many geomechanical problems tend to neglect these effects. Simple test cases highlight the importance of including friction and allowing for opening when solving quasi-static fault mechanics models. These results also underscore the importance of considering the effects of non-planarity in modeling processes associated with crustal faulting.

A Linear Complementarity formulation of rate-independent finite-strain elastoplasticity. Part II: Calculation of bifurcation and limit points

A methodology for the numerical solution of discretized boundary value problems that involve rate-independent, elastic-plastic finite-strain models is developed. The formulation is given in terms of a structural Linear Complementarity Problem. A methodology for the determination of bifurcation and limit points along an equilibrium path is described. The proposed method is suited particularly for plasticity models that involve yield surfaces with singular points (corners, edges, apexes, etc.).

A Linear Complementarity formulation of rate-independent finite-strain elastoplasticity. Part I: Algorithm for numerical integration

Abstract A methodology for the numerical integration of rate-independent, elastic–plastic finite-strain models is developed. The methodology is based on the idea of local linearization of the yield surface that was proposed in Maier (1969) , adopted as the basis for an integration scheme in Hodge (1977) , and developed further in Franchi and Genna, 1984 , Franchi and Genna, 1987 , so far for small-strain problems only. The proposed algorithm is based on the solution of a local Linear Complementarity Problem and is suited particularly for plasticity models that involve yield surfaces with singular points (corners, edges, apexes, etc.).

Valuation of European continuous-installment options

This paper is concerned with the valuation of European continuous-installment options where the aim is to determine the initial premium given a constant installment payment plan. The distinctive feature of this pricing problem is the determination, along with the initial premium, of an optimal stopping boundary since the option holder has the right to stop making installment payments at any time before maturity. Given that the initial premium function of this option is governed by an inhomogeneous Black–Scholes partial differential equation, we can obtain two alternative characterizations of the European continuous-installment option pricing problem, for which no closed-form solution is available. First, we formulate the pricing problem as a free boundary problem and using the integral representation method, we derive integral expressions for both the initial premium and the optimal stopping boundary. Next, we use the linear complementarity formulation of the pricing problem for determining the initial premium and the early stopping curve implicitly with a finite difference scheme. Finally, the pricing problem is posed as an optimal stopping problem and then implemented by a Monte Carlo approach.

A nonsmooth Newton solver for capturing exact Coulomb friction in fiber assemblies

We focus on the challenging problem of simulating thin elastic rods in contact, in the presence of friction. Most previous approaches in computer graphics rely on a linear complementarity formulation for handling contact in a stable way, and approximate Coulombs's friction law for making the problem tractable. In contrast, following the seminal work by Alart and Curnier in contact mechanics, we simultaneously model contact and exact Coulomb friction as a zero finding problem of a nonsmooth function. A semi-implicit time-stepping scheme is then employed to discretize the dynamics of rods constrained by frictional contact: this leads to a set of linear equations subject to an equality constraint involving a nondifferentiable function. To solve this one-step problem we introduce a simple and practical nonsmooth Newton algorithm which proves to be reasonably efficient and robust for systems that are not overconstrained. We show that our method is able to finely capture the subtle effects that occur when thin elastic rods with various geometries enter into contact, such as stick-slip instabilities in free configurations, entangling curls, resting contacts in braid-like structures, or the formation of tight knots under large constraints. Our method can be viewed as a first step towards the accurate modeling of dynamic fibrous materials.

New second-order cone linear complementarity formulation and semi-smooth Newton algorithm for finite element analysis of 3D frictional contact problem

We propose a new second-order cone linear complementarity problem (SOCLCP) formulation for the numerical finite element analysis of three-dimensional (3D) frictional contact problems by the parametric variational principle. Specifically, we develop a regularization technique to resolve the multi-valued difficulty involved in the frictional contact law, and use a second-order cone complementarity condition to handle the regularized Coulomb friction law in contact analysis. The governing equations of the 3D frictional contact problem is represented by an SOCLCP via the parametric variational principle and the finite element method, which avoids the polyhedral approximation to the Coulomb friction cone so that the problem to be solved has much smaller size and the solution has better accuracy. In this paper, we reformulate the SOCLCP as a semi-smooth system of equations via a one-parametric class of second-order cone complementarity functions, and then apply the non-smooth Newton method for solving this system. Numerical results confirm the effectiveness and robustness of the SOCLCP approach developed.