Hybrid method for parameter optimization with equality constraints

This article presents a hybrid method developed for solving an equality-constrained optimization problem. This method combines the co-evolutionary augmented Lagrangian method with the hybrid evolution strategy technique to overcome two major disadvantages of the evolutionary algorithm, i.e. poor constraint handling and a low convergence rate. Parameter and multiplier groups are evolved simultaneously to solve a zero-sum game transformed from a parameter optimization problem with equality constraints by the augmented Lagrangian method. Gradient individuals of parameters and multipliers are propagated by Newton’s method, and they play an important role in accelerating the speed of convergence. Ten test problems are solved to indicate that the proposed hybrid method supplies more accurate solutions with faster convergence than the co-evolutionary augmented Lagrangian method.     

A globally convergent numerical algorithm for damaging elasto‐plasticity based on the Multiplier method

In the paper it is proposed a new method for the solution of equilibrium problems based on a F.E. displacement formulation, that, at least in principle, is globally convergent as opposed to the classical Lagrangian method that presents only local convergence. The method, which appears to be particularly useful for plasticity models characterized by yield surfaces with regions of sharp curvature or corner points, is based on the Multiplier method. The structure of the procedure is presented and the consequent constrained optimization scheme is implemented for the case of associated plasticity coupled with damage. The main aspect of originality of the proposal is that it is not applied to the ‘return algorithm’, but to entire equilibrium iteration. At first, the local convergence properties of the constitutive equations are examined at the Gauss point level. It is proved that, also for involved constitutive models (a generalized Ottosen yield surface including isotropic hardening and damage is used in the applications), the convergence of a classical Newton's scheme is always reached with few iterations, ensuring a quadratic rate of convergence in the solution, provided a conversion of the inequality plastic constraint into an equality one is introduced, using an augmented Lagrangian functional for exactly evaluating the slack constraints. However, it is observed that the converged stresses are often attracted far from the initial trial point, towards regions with sharper curvature, and the main reason for the lack of convergence of the procedure is found in a divergence of the solution of the non‐linear equilibrium equations. It is shown that the Multiplier method allows to enlarge the radius of convergence with respect to classical iterations based on the Lagrangian method. The price for the enlargement of the convergence radius is a higher number of iterations, since the Multiplier method presents only a linear rate of convergence. Indeed, the exact fulfilment of the compatibility and admissibility equations is not attained simultaneously, once an equilibrated solution is reached, but it is iterative. In the closure of the paper a convergence analysis of an elastic–plastic problem characterized by a yield criterion resulting from the convex hull of crises surfaces, and as a consequence, presenting regions of non‐differentiability, is presented. It is shown how the ability of the Multiplier method in finding the solution of the structural problem for large loading step with respect to the classical Lagrangian technique compensate its slower convergence rate. Copyright © 2005 John Wiley & Sons, Ltd.     

On the formulation of closest‐point projection algorithms in elastoplasticity—part II: Globally convergent schemes

This paper presents the formulation of numerical algorithms for the solution of the closest‐point projection equations that appear in typical implementations of return mapping algorithms in elastoplasticity. The main motivation behind this work is to avoid the poor global convergence properties of a straight application of a Newton scheme in the solution of these equations, the so‐called Newton‐CPPM. The mathematical structure behind the closest‐point projection equations identified in Part I of this work delineates clearly different strategies for the successful solution of these equations. In particular, primal and dual closest‐point projection algorithms are proposed, in non‐augmented and augmented Lagrangian versions for the imposition of the consistency condition. The primal algorithms involve a direct solution of the original closest‐point projection equations, whereas the dual schemes involve a two‐level structure by which the original system of equations is staggered, with the imposition of the consistency condition driving alone the iterative process. Newton schemes in combination with appropriate line search strategies are considered, resulting in the desired asymptotically quadratic local rate of convergence and the sought global convergence character of the iterative schemes. These properties, together with the computational performance of the different schemes, are evaluated through representative numerical examples involving different models of finite‐strain plasticity. In particular, the avoidance of the large regions of no convergence in the trial state observed in the standard Newton‐CPPM is clearly illustrated. Copyright © 2001 John Wiley & Sons, Ltd.     

A contact algorithm for frictional crack propagation with the extended finite element method

We present an incremental quasi‐static contact algorithm for path‐dependent frictional crack propagation in the framework of the extended finite element (FE) method. The discrete formulation allows for the modeling of frictional contact independent of the FE mesh. Standard Coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity. The contact constraint is borrowed from non‐linear contact mechanics and embedded within a localized element by penalty method. Newton–Raphson iteration with consistent linearization is used to advance the solution. We show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called ‘LATIN’ for frictional crack propagation. Numerical examples include simulation of crack initiation and propagation in 2D plane strain with and without bulk plasticity. In the presence of bulk plasticity, the problem is also solved using an augmented Lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution. Copyright © 2008 John Wiley & Sons, Ltd.     

Solving short-term electric hydrothermal scheduling problems by exponential multiplier methods†

The short-term electric hydrothermal scheduling (STEHS) problem consists in optimizing the production of hydro and thermal electric generation units over a short time period (up to one week long). The problem described in this work can be modelled as a nonlinear network flow problem with linear and nonlinear side constraints. The minimization of this kind of problem can be performed by exploiting the efficiency of network flow techniques. It lies in minimizing approximately a series of augmented Lagrangian functions including only the side constraints, subject to balance constraints in the nodes and capacity bounds. One of the drawbacks of the multiplier methods with quadratic penalty function is that the augmented Lagrangian is not twice differentiable when it is applied to problems with inequality constraints. This article overcomes this difficulty by using the exponential multiplier method. In order to improve the performance some parameters are tuned. The efficiency of this method over STEHS test problems is illustrated by comparing its CPU-times with those of the quadratic multiplier method and with those of the general purpose codes MINOS, SNOPT, and KNITRO. Numerical results are promising.     

An augmented Lagrangian method for discrete large-slip contact problems

An augmented Lagrangian formulation is proposed for large-slip frictionless contact problems between deformable discretized bodies in two dimensions. Starting from a finite element discretization of the two bodies, a node-on-facet element is defined. A non-linear gap vector and its first variation are derived in terms of the nodal displacements. The relevant action and reaction principle is stated. The gap distance is then related to the conjugate pressure by a (multivalued non-differentiable) unilateral contact law. The resulting inequality constrained minimization problem is transformed into an unconstrained saddle point problem using an augmented Lagrangian function. Large slip over several facets is possible and the effects of target convexity or concavity are investigated. A generalized Newton method is used to solve the resulting piecewise differentiable equations necessary for equilibrium and contact. The proper tangent (Jacobian) matrices are calculated. The primal (displacements) and dual (contact forces) unknowns are simultaneously updated at each iteration.     

Thermo‐elasto‐plastic finite element analysis of quasi‐state processes in Eulerian reference frames

An Eulerian finite element formulation for quasi‐state one way coupled thermo‐elasto‐plastic systems is presented. The formulation is suitable for modeling material processes such as welding and laser surfacing. In an Eulerian frame, the solution field of a quasi‐state process becomes steady state for the heat transfer problem and static for the stress problem. A mixed small deformation displacement elasto‐plastic formulation is proposed. The formulation accounts for temperature dependent material properties and exhibits a robust convergence. Streamline upwind Petrov–Galerkin (SUPG) is used to remove spurious oscillations. Smoothing functions are introduced to relax the non‐differentiable evolution equations and allow for the use of gradient (stiffness) solution scheme via the Newton–Raphson method. A 3‐dimensional simulation of a laser surfacing process is presented to exemplify the formulation. Results from the Eulerian formulation are in good agreement with results from the conventional Lagrangian formulation. However, the Eulerian formulation is approximately 15 times faster than the Lagrangian. Copyright © 2001 John Wiley & Sons, Ltd.     

Sensitivity computation and shape optimization for a non-linear arch model with limit-points instabilities

A shape optimization method for geometrically non-linear structural mechanics based on a sensitivity gradient is proposed. This gradient is computed by means of an adjoint state equation and the structure is analysed with a total Lagrangian formulation. This classical method is well understood for regular cases, but standard equations have to be modified for limit points and simple bifurcation points. These modifications introduce numerical problems which occur at limit points. Numerical systems are very stiff and the quadratic convergence of Newton–Raphson algorithm vanishes, then higher-order derivatives have to be computed with respect to state variables. A geometrically non-linear curved arch is implemented with a finite element method via a formal calculus approach. Thickness and/or shape for differentiable costs under linear and non-linear constraints are optimized. Numerical results are given for linear and non-linear examples and are compared with analytic solutions. © 1998 John Wiley & Sons, Ltd.     

Assessment of methods for computing the closest point projection,penetration, and gap functions in contact searching problems

In computational contact mechanics problems, local searching requires calculation of the closest point projection of a contactor point onto a given target segment. It is generally supposed that the contact boundary is locally described by a convex region. However, because this assumption is not valid for a general curved segment of a three‐dimensional quadratic serendipity element, an iterative numerical procedure may not converge to the nearest local minimum. To this end, several unconstrained optimization methods are tested: the Newton–Raphson method, the least square projection, the sphere and torus approximation method, the steepest descent method, the Broyden method, the Broyden–Fletcher–Goldfarb–Shanno method, and the simplex method. The effectiveness and robustness of these methods are tested by means of a proposed benchmark problem. It is concluded that the Newton–Raphson method in conjunction with the simplex method significantly increases the robustness of the local contact search procedure of pure penalty contact methods, whereas the torus approximation method can be recommended for contact searching algorithms, which employ the Lagrange method or the augmented Lagrangian method. Copyright © 2015 John Wiley & Sons, Ltd.     

An alternative formulation for quasi-static frictional and cohesive contact problems

It is known by Engineering practitioners that quasi-static contact problems with friction and cohesive laws often present convergence difficulties in Newton iteration. These are commonly attributed to the non-smoothness of the equilibrium system. However, non-uniqueness of solutions is often an obstacle for convergence. We discuss these conditions in detail and present a general algorithm for 3D which is shown to have quadratic convergence in the Newton–Raphson iteration even for parts of the domain where multiple solutions exist. Chen–Mangasarian replacement functions remove the non-smoothness corresponding to both the stick-slip and normal complementarity conditions. Contrasting with Augmented Lagrangian methods, second-order updating is performed for all degrees-of-freedom. Stick condition is automatically selected by the algorithm for regions with multiple solutions. The resulting Jacobian determinant is independent of the friction coefficient, at the expense of an increased number of nodal degrees-of-freedom. Aspects such as a dedicated pivoting for constrained problems are also of crucial importance for a successful solution finding. The resulting 3D mixed formulation, with 7 degrees-of-freedom in each node (displacement components, friction multiplier, friction force components and normal force) is tested with representative numerical examples (both contact with friction and cohesive force), which show remarkable robustness and generality.     

Computational method for optimal machine scheduling problem with maintenance and production

This paper considers an optimal scheduling problem of maintenance and production for a machine. Firstly, the problem is formulated as a stochastic switched impulsive optimal control problem. However, there exists the stochastic disturbance in this model. Thus, it is difficult to solve the problem by conventional optimisation techniques. To overcome this difficulty, the stochastic switched impulsive optimal control problem is transformed into a deterministic switched impulsive optimal control problem with continuous state inequality constraints. Then, by combining a time-scaling transformation, a second-order smoothing technique and a penalty function method, an improved Newton algorithm is developed for solving this problem. Convergence results indicate that the algorithm is globally convergent with quadratic rate. Finally, two numerical examples are provided to illustrate the effectiveness of the developed algorithm.     

Trajectory Optimization for a Multi-Stage Launch Vehicle Using Time Finite Element and Direct Collocation Methods

An indirect time finite element method is applied to solve the trajectory optimization problem for a multi-stage launch vehicle, and the numerical results are compared with the numerical solutions obtained by using a direct collocation and nonlinear program-ming method. A nonlinear programming problem is solved by the sequential quadratic programming algorithm with an augmented Lagrangian merit function, and the converged Lagrange multiplier is used for estimating the costate variables of the optimal trajectory. As a numerical example, a multi-stage launch vehicle trajectory optimization problem with a control variable constraint is solved, and the results are compared.     

On sequential approximate simultaneous analysis and design in classical topology optimization

We study the simultaneous analysis and design (SAND) formulation of the ‘classical’ topology optimization problem subject to linear constraints on material density variables. Based on a dual method in theory, and a primal‐dual method in practice, we propose a separable and strictly convex quadratic Lagrange–Newton subproblem for use in sequential approximate optimization of the SAND‐formulated classical topology design problem. The SAND problem is characterized by a large number of nonlinear equality constraints (the equations of equilibrium) that are linearized in the approximate convex subproblems. The availability of cheap second‐order information is exploited in a Lagrange–Newton sequential quadratic programming‐like framework. In the spirit of efficient structural optimization methods, the quadratic terms are restricted to the diagonal of the Hessian matrix; the subproblems have minimal storage requirements, are easy to solve, and positive definiteness of the diagonal Hessian matrix is trivially enforced. Theoretical considerations reveal that the dual statement of the proposed subproblem for SAND minimum compliance design agrees with the ever‐popular optimality criterion method – which is a nested analysis and design formulation. This relates, in turn, to the known equivalence between rudimentary dual sequential approximate optimization algorithms based on reciprocal (and exponential) intervening variables and the optimality criterion method. Copyright © 2016 John Wiley & Sons, Ltd.     

Topology optimization of continuum structures with stress constraints and uncertainties in loading

Topology optimization using stress constraints and considering uncertainties is a serious challenge, since a reliability problem has to be solved for each stress constraint, for each element in the mesh. In this paper, an alternative way of solving this problem is used, where uncertainty quantification is performed through the first‐order perturbation approach, with proper validation by Monte Carlo simulation. Uncertainties are considered in the loading magnitude and direction. The minimum volume problem subjected to local stress constraints is formulated as a robust problem, where the stress constraints are written as a weighted average between their expected value and standard deviation. The augmented Lagrangian method is used for handling the large set of local stress constraints, whereas a gradient‐based algorithm is used for handling the bounding constraints. It is shown that even in the presence of small uncertainties in loading direction, different topologies are obtained when compared to a deterministic approach. The effect of correlation between uncertainties in loading magnitude and direction on optimal topologies is also studied, where the main observed result is loss of symmetry in optimal topologies.     

An implicit boundary element solution with consistent linearization for free surface flows and non‐linear fluid–structure interaction of floating bodies

In this work, a new comprehensive method has been developed which enables the solution of large, non‐linear motions of rigid bodies in a fluid with a free surface. The application of the modern Eulerian–Lagrangian approach has been translated into an implicit time‐integration formulation, a development which enables the use of larger time steps (where accuracy requirements allow it). Novel features of this project include: (1) an implicit formulation of the rigid‐body motion in a fluid with a free surface valid for both two or three dimensions and several moving bodies; (2) a complete formulation and solution of the initial conditions; (3) a fully consistent (exact) linearization for free surface flows valid for any boundary elements such that optimal convergence properties are obtained when using a Newton–Raphson solver. The proposed framework has been completed with details on implementation issues referring mainly to the computation of the complete initial conditions and the consistent linearization of the formulation for free surface flows. The second part of the paper demonstrates the mathematical and numerical formulation through numerical results simulating large free surface flows and non‐linear fluid structure interaction. The implicit formulation using a fully consistent linearization based on the boundary element method and the generalized trapezoidal rule has been applied to the solution of free surface flows for the evolution of a triangular wave, the generation of tsunamis and the propagation of a wave up to overturning. Fluid–structure interaction examples include the free and forced motion of a circular cylinder and the sway, heave and roll motion of a U‐shaped body in a tank with a flap wave generator. The presented examples demonstrate the applicability and performance of the implicit scheme with consistent linearization. Copyright © 2001 John Wiley & Sons. Ltd.     

Algorithms for non-linear contact constraints with application to stability problems of rods and shells

In this paper a class of non-linear problems is discussed where stability as well as post-buckling behaviour is coupled with contact constraints. The contact conditions are introduced via a perturbed Lagrangian formulation. From this formulation the penalty and Lagrangian multiplier method are derived. Both algorithms are investigated together with an algorithm based on an augmented Lagrangian method. The resulting finite element formulation is applied to structural problems of beams and shells undergoing finite elastic deflections and rotations. For the examination of the post-buckling behaviour the arc-length method is used. The performance of the element formulation and a comparison of the different contact algorithms are demonstrated by numerical examples.     

Frictionless geometrically non-linear contact using quadratic programming

The successive quadratic programming (SQP) method is used with the finite element method (FEM) to solve frictionless geometrically non-linear contact problems involving large deformations of the elastica in the presence of flat rigid walls. To formulate the SQP problems, the potential energy (PE) is expanded in a Taylor series of second order in displacement increments about a configuration near a contact solution. The SQP problems consist of minimizing the Taylor expansion of the PE subject to the inequality constraints which represent contact. The quadratic programming (QP) method is made part of a Newton–Raphson (NR) search in which the QP corrections are made when a NR step does not satisfy the constraints. A revised simplex method developed by Rusin is used to solve the QP problems. The elastica is modelled with a total Lagrangian FEM developed by Fried. Solutions are obtained for the end loaded buckled elastica in point contact with a rigid wall and for a uniformly loaded elastica in regional contact with a rigid wall. The problems are also solved using a penalty method. The results obtained for the point contact problem are compared to an analytical solution. Calculations were made to obtain numerical information on maximum load step size and the number of inverse operations required for each load step. Cases in which the elastica stiffened substantially as a result of the initiation of contact are also discussed.     

An implicit integration algorithm with a projection method for viscoplastic constitutive equations

Robinson's viscoplastic model, a representative of the so-called overstress models, is integrated by use of the generalized midpoint rule. The solution of the non-linear system of algebraic equations arising from time discretization of the constitutive equations is determined using a projection method in combination with Newton's method. Consistent tangent moduli are calculated and the quadratic convergence of the global Newton equilibrium iteration is shown. The time increment size is controlled by the convergence behaviour of the equilibrium iteration and the accuracy of the numerical integration. Various numerical examples are considered to demonstrate the efficiency of the methods.     

Local stress‐constrained and slope‐constrained SAND topology optimisation

We study the alternative ‘simultaneous analysis and design’ (SAND) formulation of the local stress‐constrained and slope‐constrained topology design problem. It is demonstrated that a standard trust‐region Lagrange–Newton sequential quadratic programming‐type algorithm—based, in this case, on strictly convex and separable approximate subproblems—may converge to singular optima of the local stress‐constrained problem without having to resort to relaxation or perturbation techniques. Moreover, because of the negation of the sensitivity analyses—in SAND, the density and displacement variables are independent—and the immense sparsity of the SAND problem, solutions to large‐scale problem instances may be obtained in a reasonable amount of computation time. Copyright © 2016 John Wiley & Sons, Ltd.