Augmented Lagrangian Formulation Research Articles

A Direct Approach Toward Global Minimization for Multiphase Labeling and Segmentation Problems

This paper intends to extend the minimization algorithm developed by Bae, Yuan and Tai [IJCV, 2011] in several directions. First, we propose a new primal-dual approach for global minimization of the continuous Potts model with applications to the piecewise constant Mumford-Shah model for multiphase image segmentation. Different from the existing methods, we work directly with the binary setting without using convex relaxation, which is thereby termed as a direct approach. Second, we provide the sufficient and necessary conditions to guarantee a global optimum. Moreover, we provide efficient algorithms based on a reduction in the intermediate unknowns from the augmented Lagrangian formulation. As a result, the underlying algorithms involve significantly fewer parameters and unknowns than the naive use of augmented Lagrangian-based methods; hence, they are fast and easy to implement. Furthermore, they can produce global optimums under mild conditions.

Comparison of Alternative Strategies for Multilevel Optimization of Hierarchical Systems

The augmented Lagrangian penalty formulation and four different coordination strategies are used to examine the nu- merical behavior of Analytical Target Cascading (ATC) for multilevel optimization of hierarchical systems. The coordination strategies considered include augmented Lagrangian using the method of multipliers and alternating direction method of multipliers, diagonal quadratic approximation, and truncated diagonal quadratic approximation. Properties examined include computational cost and solution accuracy based on the selected values for the different parameters that appear in each formulation. The different strategies are implemented using two- and three-level decomposed example problems. While the results show the interaction between the selected ATC formulation and the values of associated parameters, they clearly highlight the impact they could have on both the solution accuracy and computational cost.

An upper bound-based solution for the shape factors of bearing capacity of footings under drained conditions using a parallelized mixed f.e. formulation with quadratic velocity fields

A strict upper-bound limit analysis finite element formulation is used to estimate shape factors sγ and sq for determining the bearing capacity of shallow foundations using the classic bearing capacity formula. The finite element formulation uses a quadratic approximation for the velocity field, an extension of a previously published Augmented Lagrangian formulation with a linear velocity field, and was implemented for a parallel processing environment. Results from determining the limit loads under three-dimensional conditions are presented and compared with previously published data. The results obtained allow a strict upper-bound determination of the shape factors. Furthermore, a practical proposal for these factors is made and compared with other proposals made by other authors.

Impact-induced vibration in complex flexible multi-body systems: numerical and experimental investigation of press machines

This article is concerned with the numerical and experimental investigations of impact-induced vibrations in highly non-linear complex multibody system applications. In order to accurately model the impact response of such complex systems, a computational procedure based on the integration of finite element and multibody system algorithms is used. This procedure, which employs the finite element floating frame of reference formulation, allows for defining arbitrary joint constraint and forcing functions, for accurately capturing the coupling between the rigid body motion and the elastic deformation, and for modeling the impact forces between rigid and flexible bodies in the multibody system model. The system equations of motion are developed using the augmented Lagrangian formulation that allows for the use of sparse matrix techniques. The algorithm used in this study ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The impact-induced vibration of a highly non-linear press machine is numerically and experimentally examined in this study. In order to accurately model the dynamic behavior of such a press machine, it is necessary to model the transfer feeder elastic deformations resulting from the impulsive impact forces. The deformations of the transfer press system as well as those of its frame are modeled in this investigation using the floating frame of reference formulation that allows for effectively reducing the number of degrees of freedom by filtering out very high-frequency modes of vibration using component mode synthesis techniques. The formulations of several relative angle and angular velocity kinematic constraints that are required to model this multibody system are developed in order to model transfer feeder complex rotational components. The numerical results obtained using the three-dimensional multibody system transfer press model developed in this study are validated using experimental results.

A mortar formulation for 3D large deformation contact using NURBS-based isogeometric analysis and the augmented Lagrangian method

NURBS-based isogeometric analysis is applied to 3D frictionless large deformation contact problems. The contact constraints are treated with a mortar-based approach combined with a simplified integration method avoiding segmentation of the contact surfaces, and the discretization of the continuum is performed with arbitrary order NURBS and Lagrange polynomial elements. The contact constraints are satisfied exactly with the augmented Lagrangian formulation proposed by Alart and Curnier, whereby a Newton-like solution scheme is applied to solve the saddle point problem simultaneously for displacements and Lagrange multipliers. The numerical examples show that the proposed contact formulation in conjunction with the NURBS discretization delivers accurate and robust predictions. In both small and large deformation cases, the quality of the contact pressures is shown to improve significantly over that achieved with Lagrange discretizations. In large deformation and large sliding examples, the NURBS discretization provides an improved smoothness of the traction history curves. In both cases, increasing the order of the discretization is either beneficial or not influential when using isogeometric analysis, whereas it affects results negatively for Lagrange discretizations.

On an augmented Lagrangian-based preconditioning of Oseen type problems

The paper deals with a general framework for constructing preconditioners for saddle point matrices, in particular as arising in the discrete linearized Navier-Stokes equations (Oseen’s problem). We utilize the so-called augmented Lagrangian framework, where the original linear system of equations is first transformed to an equivalent one, which latter is then solved by a preconditioned iterative solution method. The matrices in the linear systems, arising after the discretization of Oseen’s problem, are of two-by-two block form as are the best known preconditioners for these. In the augmented Lagrangian formulation, a scalar regularization parameter is involved, which strongly influences the quality of the block-preconditioners for the system matrix (referred to as outer), as well as the conditioning and the solution of systems with the resulting pivot block (referred to as inner) which, in the case of large scale numerical simulations has also to be solved using an iterative method. We analyse the impact of the value of the regularization parameter on the convergence of both outer and inner solution methods. The particular preconditioner used in this work exploits the inverse of the pressure mass matrix. We study the effect of various approximations of that inverse on the performance of the preconditioners, in particular that of a sparse approximate inverse, computed in an element-by-element fashion. We analyse and compare the spectra of the preconditioned matrices for the different approximations and show that the resulting preconditioner is independent of problem, discretization and method parameters, namely, viscosity, mesh size, mesh anisotropy. We also discuss possible approaches to solve the modified pivot matrix block.

A boundary elements formulation for 3D fretting-wear problems

Computational wear modeling is an extremely time-consuming problem, especially the 3D cases. In this work, a 3D boundary element method (BEM) formulation for wear modeling is proposed and applied to simulate 3D fretting-wear problems under gross sliding and partial slip conditions. The present formulation applies the BEM to approximate the elastic response of solids, and an augmented Lagrangian formulation to solve the contact problem. Contact restrictions fulfilment is established by a set of projection functions, and wear on contact surfaces is computed using the Archard wear law. The BEM proves to be a very suitable numerical method for this kind of mechanical interaction problems, considering only the boundary degrees of freedom involved in the problem and obtaining a very good approximation of contact tractions with a low number of elements. This is very interesting in terms of computational cost reductions of wear modeling, specially in 3D problems. In that regard, an acceleration strategy is applied to the proposed algorithm. It allows to obtain very important reductions on wear simulation times. The proposed methodology is therefore an efficient numerical tool for 3D fretting-wear problems modeling.

Field-of-Values Convergence Analysis of Augmented Lagrangian Preconditioners for the Linearized Navier–Stokes Problem

We study a block triangular preconditioner for finite element approximations of the linearized Navier-Stokes equations. The preconditioner is based on the augmented Lagrangian formulation of the problem and was introduced by the authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095-2113]. In this paper we prove field-of-values type estimates for the preconditioned system which lead to optimal convergence bounds for the GMRES algorithm applied to solve the system. Two variants of the preconditioner are considered: an ideal one based on exact solves for the velocity submatrix, and a more practical variant based on block triangular approximations of the velocity submatrix.

Response of multiferroic composites inferred from a fast-Fourier-transform-based numericalscheme

The effective response and the local fields within periodic magneto-electric multiferroiccomposites are investigated by means of a numerical scheme based on fast Fouriertransforms. This computational framework relies on the iterative resolution of coupledseries expansions for the magnetic, electric and strain fields. By using an augmentedLagrangian formulation, a simple and robust procedure which makes use of the uncoupledGreen operators for the elastic, electrostatics and magnetostatics problems is proposed. Itsaccuracy is assessed in the cases of laminated and fibrous two-phase composites for whichanalytical solutions exist.

A FEM–BEM fast methodology for 3D frictional contact problems

This paper presents a new and efficient methodology for 3D frictional contact problems based on the Generalized Newton Method with line search (GNMls). The influence coefficients modeling the behavior of solids are approached by means of the Finite Element Method (FEM) and/or the Boundary Element Method (BEM). The contact problem is based on an Augmented Lagrangian formulation and the use of projection functions to establish the contact restrictions. A new and more simplified projection operator is presented, which makes it possible to obtain a quasi-complementarity of the contact variables. As a consequence, the number of contact problem unknowns is reduced. This, together with the use of iterative resolution techniques for computing the Newton line search, results in a fast and reliable algorithm, as shown in the examples solved in this paper.

Sensitivity analysis for frictional contact problems in the augmented Lagrangian formulation

Direct differentiation method of sensitivity analysis is developed for frictional contact problems. As a result of the augmented Lagrangian treatment of contact constraints, the direct problem is solved simultaneously for the displacements and Lagrange multipliers using the Newton method. The main purpose of the paper is to show that this formulation of the augmented Lagrangian method is particularly suitable for sensitivity analysis because the direct differentiation method leads to a non-iterative exact sensitivity problem to be solved at each time increment. The approach is applied to a general class of three-dimensional frictional contact problems, and numerical examples are provided involving large deformations, multibody contact interactions, and contact smoothing techniques.

A three-field augmented Lagrangian formulation of unilateral contact problems with cohesive forces

We investigate unilateral contact problems with cohesive forces, leading to the constrained minimization of a possibly nonconvex functional. We analyze the mathematical structure of the minimization problem. The problem is reformulated in terms of a three-field augmented Lagrangian, and sufficient conditions for the existence of a local saddle-point are derived. Then, we derive and analyze mixed finite element approximations to the stationarity conditions of the three-field augmented Lagrangian. The finite element spaces for the bulk displacement and the Lagrange multiplier must satisfy a discrete inf-sup condition, while discontinuous finite element spaces spanned by nodal basis functions are considered for the unilateral contact variable so as to use collocation methods. Two iterative algorithms are presented and analyzed, namely an Uzawa-type method within a decomposition-coordination approach and a nonsmooth Newton's method. Finally, numerical results illustrating the theoretical analysis are presented.

A Duality-Based Splitting Method for $\ell^1$-$TV$ Image Restoration with Automatic Regularization Parameter Choice

A novel splitting method is presented for the $\ell^1$-$TV$ restoration of degraded images subject to impulsive noise. The functional is split into an $\ell^2$-$TV$ denoising part and an $\ell^1$-$\ell^2$ deblurring part. The dual problem of the relaxed functional is smooth with convex constraints and can be solved efficiently by applying an Arrow–Hurwicz-type algorithm to the augmented Lagrangian formulation. The regularization parameter is chosen automatically based on a balancing principle. The accuracy, the fast convergence, and the robustness of the algorithm and the use of the parameter choice rule are illustrated on some benchmark images and compared with an existing method.

Computational approach for composite materials with coupled constitutive laws

A numerical scheme based on fast Fourier transform is presented to compute the effective response and the local fields within a heterogeneous material which exhibits a coupled constitutive law. It consists in the iterative resolution of periodic coupled Lippmann–Schwinger equations. This approach is illustrated in the case of electroelastic composite materials. By using an augmented Lagrangian formulation, a simple iterative scheme relying on the uncoupled Green operators for the elastic and electrostatics problems is proposed. This computational framework, which allows to consider composite materials with an infinite contrast on the local properties, is assessed in the case of porous and fiber-reinforced piezoelectric materials.

A 3D FEM–BEM rolling contact formulation for unstructured meshes

This work presents a new formulation for solving 3D steady-state rolling contact problems. The convective terms for computing the tangential slip velocities involved in the rolling problem, are evaluated using a new approximation inspired in numerical fluid dynamics techniques for unstructured meshes. Moreover, the elastic influence coefficients of the surface points in contact are approached by means of the finite element method (FEM) and/or the boundary element method (BEM). The contact problem is based on an Augmented Lagrangian Formulation and the use of projection functions to establish the contact restrictions. Finally, the resulting nonlinear equations set is solved using the generalized Newton method with line search (GNMls), presenting some acceleration strategies as: a new and more simplified projection operator, which makes it possible to obtain a quasi-complementarity of the contact variables, reducing the number of contact problem unknowns, and using iterative solvers. The presented methodology is validated solving some rolling contact problems and analyzed for some unstructured mesh examples.

Modeling and numerical treatment of elastic rods with frictionless self-contact

In this paper we present a hyperelastic rod model that takes into account self-contact forces. The model is based on Cosserat rod theory that incorporates shear, elongation, flexure and twist deformations. The problem of avoiding self-penetration of parts of the rod is handled by the introduction of a contact distance function and the incorporation of associated contact forces. We present a penalty method for the treatment of the multi-valued and non-differentiable contact law. We also describe an augmented Lagrangian formulation of this problem. We then give the details of the finite-element discretization of the elastic rod self-contact problem as well as some numerical examples.

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

Performance of an Energy-Conserving Algorithm for Multi-Body Dynamics

This work presents the application to the dynamics of multibody systems of three methods based on the augmented Lagrangian techniques, compares them, and gives some criteria for their use in realistic problems. The methods are: an augmented Lagrangian formulation with trapezoidal rule plus projections of velocities and accelerations, an augmented Lagrangian energy-conserving formulation, and an augmented Lagrangian formulation with conserving integrator plus projections of velocities and accelerations. The simulation of two mechanical systems is carried out in this work: a spherical compound pendulum, which is an academic example that permits to see the properties of the formulations; andthe whole model of a car, which is a realistic and demanding example to test the efficiency and robustness of the formulations.

A Modular Modeling Approach to Simulate Interactively Multibody Systems With a Baumgarte/Uzawa Formulation

In this paper, a modular modeling approach of multibody systems adapted to interactive simulation is presented. This work is based on the study of the stability of two differential algebraic equation solvers. The first one is based on the acceleration-based augmented Lagrangian formulation and the second one on the Baumgarte formulation. We show that these two solvers give the same results and have to satisfy the same criteria to stabilize the algebraic constraint acceleration error. For a modular modeling approach, we propose to use the Baumgarte formulation and an iterative Uzawa algorithm to solve external constraint forces. This work is also the first step to validate the concept of two types of numerical components for object-oriented programming.

Review of Contemporary Approaches for Constraint Enforcement in Multibody Systems

A hallmark of multibody dynamics is that most formulations involve a number of constraints. Typically, when redundant generalized coordinates are used, equations of motion are simpler to derive but constraint equations are present. Approaches to dealing with high index differential algebraic equations, based on index reduction techniques, are reviewed and discussed. Constraint violation stabilization techniques that have been developed to control constraint drift are also reviewed. These techniques are used in conjunction with algorithms that do not exactly enforce the constraints. Control theory forms the basis for a number of these methods. Penalty based techniques have also been developed, but the augmented Lagrangian formulation presents a more solid theoretical foundation. In contrast to constraint violation stabilization techniques, constraint violation elimination techniques enforce exact satisfaction of the constraints, at least to machine accuracy. Finally, as the finite element method has gained popularity for the solution of multibody systems, new techniques for the enforcement of constraints have been developed in that framework. The goal of this paper is to review the features of these methods, assess their accuracy and efficiency, underline the relationship among the methods, and recommend approaches that seem to perform better than others.