Nonlinear Mathematical Programming Techniques Research Articles

Assessment of ship manoeuvrability by using a coupling between a nonlinear transient manoeuvring model and mathematical programming techniques

In this paper, a numerical method based on a coupling between a mathematical model of nonlinear transient ship manoeuvring motion in the horizontal plane and Mathematical Programming (MP) techniques is proposed. The aim of the proposed procedure is an efficient estimation of optimal ship hydrodynamic parameters in a dynamic model at the early design stage. The proposed procedure has been validated through turning circle and zigzag manoeuvres based on experimental data of sea trials of the 190 000dwt oil tanker. Comparisons between experimental and computed data show a good agreement of overall tendency in manoeuvring trajectories.

Offshore wind farm layout optimization using mathematical programming techniques

Offshore wind power is a renewable energy of growing relevance in current electric energy systems, presenting favorable wind conditions in comparison with the sites on land. However, the higher energy yield has to compensate the increment in installation and maintenance costs, thus the importance of optimizing resources. One relevant aspect to increase profitability is the wind farm layout. The aim of this paper is to propose a new method to maximize the expected power production of offshore wind farms by setting the appropriate layout, i.e. minimizing the wake effects. The method uses a sequential procedure for global optimization consisting of two steps: i) an heuristic method to set an initial random layout configuration, and ii) the use of nonlinear mathematical programming techniques for local optimization, which use the random layout as an initial solution. The method takes full advantage of the most up-to-date mathematical programming techniques while performing a global optimization approach, which can be easily parallelized. The performance of the proposed procedure is tested using the German offshore wind farm Alpha Ventus, located in the North Sea, yielding an increment of expected annual power production of 3.52% with respect to the actual configuration. According to current electricity prices in Germany, this constitutes an expected profit increment of almost 1 M€ per year.

A Hybrid Differential Dynamic Programming Algorithm for Constrained Optimal Control Problems. Part 1: Theory

A new algorithm is presented to solve constrained nonlinear optimal control problems, with an emphasis on highly nonlinear dynamical systems. The algorithm, called HDDP, is a hybrid variant of differential dynamic programming, a proven second-order technique that relies on Bellman's Principle of Optimality and successive minimization of quadratic approximations. The new hybrid method incorporates nonlinear mathematical programming techniques to increase efficiency: quadratic programming subproblems are solved via trust region and range-space active set methods, an augmented Lagrangian cost function is utilized, and a multiphase structure is implemented. In addition, the algorithm decouples the optimization from the dynamics using first- and second-order state transition matrices. A comprehensive theoretical description of the algorithm is provided in this first part of the two paper series. Practical implementation and numerical evaluation of the algorithm is presented in Part 2.

Identification of elastic properties of composite plate

Composite laminates are used extensively in the aerospace industry, especially for the fabrication of high-performance structures. The determination of stiffness parameters for complex materials, such as fibre-reinforced composites, is much more complicated than for isotropic materials. A conventional way is testing the coupon specimens, which are manufactured by technology similar to that used for the real, large structures. When such a method is used, the question arises of whether the material properties obtained from the coupon tests are the same as those in the large structure. Therefore, the determination of actual material properties for composite laminates using non-destructive evaluation techniques has been widely investigated. A number of various non-destructive evaluation techniques have been proposed for determining the material properties of composite laminates. In the present study, attention is focused on the identification of the elastic properties of laminated plate using vibration test data. The problem associated with vibration testing is converting the measured modal frequencies to elastic constants. A standard method for solving this problem is the use of a numerical-experimental model and optimization techniques. The identification functional represents the gap between the numerical model response and the experimental one. This gap should be minimized, taking into account the side constraints on the design variables (elastic constants). The minimization problem is solved by using non-linear mathematical programming techniques and sensitivity analysis. The results obtained were verified by comparing the experimentally measured eigenfrequencies with the numerical ones obtained by FEM at the point of optima

Second-order nonsmooth optimization for H ∞ synthesis

The standard way to compute H ∞ feedback controllers uses algebraic Riccati equations and is therefore of limited applicability. Here we present a new approach to the H ∞ output feedback control design problem, which is based on nonlinear and nonsmooth mathematical programming techniques. Our approach avoids the use of Lyapunov variables, and is therefore flexible in many practical situations.

Modeling optimal dietary pigmentation strategies in farmed Atlantic salmon: Application of mixed-integer non-linear mathematical programming techniques

The most important visual quality characteristic of Atlantic salmon is the red/pink flesh color. The primary source of this coloration in salmon is caused by deposition of relatively large amounts of pigments, such as astaxanthin, obtained from their diet. Astaxanthin is expensive, and in commercial farming practice, dietary color pigments comprises about 15–20% of the total feed cost. One important operational process in commercial fish farms is therefore to minimize pigment costs. Based on recent models on the effects of dietary pigment concentration and fish size on visual color perception of Atlantic salmon, this study has built a mathematical programming model designed to optimize dietary astaxanthin concentrations throughout the grow-out period that results in well-pigmented fish at minimum cost. We have applied a mixed-integer non-linear programming algorithm to solve this problem. Various managerial implications of applying optimization models in product quality management of farmed salmon are discussed.

Limit analysis of composite materials based on an ellipsoid yield criterion

Employing repeating unit cell (RUC) to represent the microstructure of periodic composite materials, this paper develops a numerical technique to calculate the plastic limit loads and failure modes of composites by means of homogenization technique and limit analysis in conjunction with the displacement-based finite element method. With the aid of homogenization theory, the classical kinematic limit theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. Based on nonlinear mathematical programming techniques, the finite element modelling of kinematic limit analysis is then developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the limit load of a composite is then obtained. The nonlinear formulation has a very small number of constraints and requires much less computational effort than a linear formulation. An effective, direct iterative algorithm is proposed to solve the resulting nonlinear programming problem. The effectiveness and efficiency of the proposed method have been validated by several numerical examples. The proposed method can provide theoretical foundation and serve as a powerful numerical tool for the engineering design of composite materials.

A nonlinear programming approach to kinematic shakedown analysis of frictional materials

This paper develops a novel nonlinear numerical method to perform shakedown analysis of structures subjected to variable loads by means of nonlinear programming techniques and the displacement-based finite element method. The analysis is based on a general yield function which can take the form of most soil yield criteria (e.g. the Mohr–Coulomb or Drucker–Prager criterion). Using an associated flow rule, a general yield criterion can be directly introduced into the kinematic theorem of shakedown analysis without linearization. The plastic dissipation power can then be expressed in terms of the kinematically admissible velocity and a nonlinear formulation is obtained. By means of nonlinear mathematical programming techniques and the finite element method, a numerical model for kinematic shakedown analysis is developed as a nonlinear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load can be calculated. An effective, direct iterative algorithm is then proposed to solve the resulting nonlinear programming problem. The calculation is based on the kinematically admissible velocity with one-step calculation of the elastic stress field. Only a small number of equality constraints are introduced and the computational effort is very modest. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples.

Tool design optimization in extrusion processes

Tool design optimization in steady-state extrusion processes, based on the finite element discretization and non-linear mathematical programming techniques, is considered in this paper. In order to perform the optimization the die geometry is adequately parameterized, according to the polynomial representation. By imposing real technological and geometry constraints the optimal tool design is effected by extremizing the actual optimization objectives. In particular, the minimization of the forming energy consumption and the maximization of the possible area reduction are considered. The results, obtained numerically by assuming the Lagrange incremental elastic–plastic finite element formulation in modelling the material flow and the considered optimization approach, are compared to the known theoretical solutions, in cases where such solutions are available. It is demonstrated that the discussed approach is quite effective.

Modified fully utilized design (MFUD) method for stress and displacement constraints

The traditional fully stressed method performs satisfactorily for stress-limited structural design. When this method is extended to include displacement limitations in addition to stress constraints, it is known as the Fully Utilized Design (FUD). Typically, the FUD produces an overdesign, which is the primary limitation of this otherwise elegant method. We have modified FUD in an attempt to alleviate the limitation. This new method, called the Modified Fully Utilized Design (MFUD) method, has been tested successfully on a number of problems that were subjected to multiple loads and had both stress and displacement constraints. The solutions obtained with MFUD compare favourably with the optimum results that can be generated by using non-linear mathematical programming techniques. The MFUD method appears to have alleviated the overdesign condition and offers the simplicity of a direct, fully stressed type of design method that is distinctly different from optimization and optimality criteria formulations. The MFUD method is being developed for practicing engineers who favour traditional design methods rather than methods based on advanced calculus and non-linear mathematical programming techniques. The Integrated Force Method (IFM) was found to be the appropriate analysis tool in the development of the MFUD method. In this paper, the MFUD method and its optimality are examined along with a number of illustrative examples. © 1998 This paper was produced under the auspices of the U.S. Government and it is therefore not subject to copyright in the U.S.

Modified fully utilized design (MFUD) method for stress and displacement constraints

The traditional fully stressed method performs satisfactorily for stress-limited structural design. When this method is extended to include displacement limitations in addition to stress constraints, it is known as the Fully Utilized Design (FUD). Typically, the FUD produces an overdesign, which is the primary limitation of this otherwise elegant method. We have modified FUD in an attempt to alleviate the limitation. This new method, called the Modified Fully Utilized Design (MFUD) method, has been tested successfully on a number of problems that were subjected to multiple loads and had both stress and displacement constraints. The solutions obtained with MFUD compare favourably with the optimum results that can be generated by using non-linear mathematical programming techniques. The MFUD method appears to have alleviated the overdesign condition and offers the simplicity of a direct, fully stressed type of design method that is distinctly different from optimization and optimality criteria formulations. The MFUD method is being developed for practicing engineers who favour traditional design methods rather than methods based on advanced calculus and non-linear mathematical programming techniques. The Integrated Force Method (IFM) was found to be the appropriate analysis tool in the development of the MFUD method. In this paper, the MFUD method and its optimality are examined along with a number of illustrative examples. © 1998 This paper was produced under the auspices of the U.S. Government and it is therefore not subject to copyright in the U.S.

Optimization of single and multistep wire drawing processes with respect to minimization of the forming energy

Optimization of single and multistep wire drawing processes, based on the finite element analysis and nonlinear mathematical programming techniques, is considered. Since the investigated problem involves the optimal selection of the extrusion die angles and the correspondent reduction rate sequence, it can be considered as one belonging to a class of optimum shape design problems. Characteristic to the considered problem is that the domain boundary, the shape of which is to be optimized, is subject to the shape dependent forming loads. The minimization of the objective function, which is related to the process energy consumption, is sought by taking several technological and geometrical constraints into account. The numerically evaluated examples demonstrate that by using the proposed optimization technique not only a substantial cost reduction, but also a considerable improvement in the homogeneity of plastic deformation through the cross-section of the drawn wire can be achieved.

Modelling and design of conveyor belt dryers

A mathematical model suitable for the design of conveyor belt dryers was developed. The objective of design was the evaluation of optimum flowsheet structure, construction characteristics, and operational conditions. The methodology followed was based on constructing a superstructure which involves a large number of minor structures, and optimizing it by means of non-linear mathematical programming techniques. The optimization problem can be stated in an equivalent way so that the computational effort involved in its solution is greatly reduced. This decomposition strategy transforms the original problem into an optimization problem of a major and a minor stage.

On the limit analysis of cylindrical shells with a single cutout

Approximate lower bounds to the limit load and torque of cylindrical shells with a single cutout are obtained by means of non-linear mathematical programming techniques. The bounds are approximate since some form of approximation is inherent with the application of such techniques to non-symmetric problems. General cylindrical shell equations are used and the yield surface is one proposed by Illyushin obeying the Von Mises criteria. The results are presented in non-dimensional form in terms of a shell parameter, α = √( Rt) . The tolerance factor used in the minimization process is ‘calibrated’ using a limit load solution obtained from a finite element analysis.

Estimating freshwater inflow needs for Texas estuaries by mathematical programming

As mandated by the Texas State Legislature, the Texas Department of Water Resources conducted studies of the effect of freshwater inflows upon the bays and estuaries of Texas. Developed as part of these studies, a mathematical programming model is described for computing estimates of the monthly and seasonal freshwater inflows necessary to meet specified environmental conditions in each of the major estuaries of the Texas Gulf Coast. The optimization model relates freshwater inflow to the key estuarine indicators of salinity, marsh inundation, and commercial fisheries harvests. Three management proposals are formulated for each estuary, corresponding to ecosystem subsistence, maintenance of fisheries harvests, and fisheries harvest enhancement. Linear and nonlinear mathematical programming techniques are used to determine the optimal inflows for each of these three management alternatives in all but one of the seven major estuaries, where only one of the management proposals could be solved.

Stress constraints and screening for optimality criteria design

Procedures that emphasize the processing of stress constraints within optimality criteria designs for low structural weight with stress and compliance constraints are described. Prescreening criteria are used to partition stress constraints into either potentially active primary sets or passive secondary sets that require minimal processing. Side-constraint boundaries for passive constraints are derived by projections from design histories to modify conventional stress-ratio boundaries. Other procedures described apply partial structural modification reanalysis to correct stress constraint violations of unfeasible designs. Sample problem results show effective design convergence and, in particular, advantages for reanalysis in obtaining lower feasible design weights. HERE have been many significant developments1 during the past two decades in the structural-design optimization procedures that select member-size design variables to achieve low structure weight subject to constraints on displacement compliances and member stresses. For this type of problem, the difficulties of satisfying the stress constraints are usually more severe than those associated with the compliance constraints. Stress constraints are more troublesome because they are direct functions of strains; displacement constraints, however, are integral functions that benefit from the smoothing of local strain variations. This paper concentrates on the processing of stress constraints to expedite com- putations and to produce effective designs within optimality criteria methods. These are indirect design methods for which the solution space is equal to the number of active constraints and is typically smaller than the number of design variables. On the other hand, the solution space for direct design methods is equal in size to the number of design variables. Consequently, optimality criteria methods have the ad- vantages of larger problem-size potential and smaller com- putational effort. Optimality criteria solution methods determine Lagrange multipliers for the active constraints and then use these within the criteria to determine the design variables. The procedures to determine the Lagrange multipliers employ sensitivity (or flexibility) coefficients that are assumed not to depend upon the design variables. Nevertheless, the extent of structural redundancy weakens this assumption, and for the typically redundant structure it is necessary to develop the design through a number of iterative cycles. In each cycle: 1) the structure is reanalyzed to update the coefficients so that they are consistent with the current stage of design; 2) candidate active constraints are screened; 3) design space algorithms are applied to construct the active set of constraints and to determine the associated multipliers; and 4) the optimality criteria are applied to determine the design variables for the next cycle. Nonlinear mathematical programming techniques are used to determine the Lagrange multipliers2 for the set of active constraints. Methods for constructing this set apply hier- archical strategies in which constraints from a candidate set are assembled, added, or deleted. The determination of the multipliers and the construction of the sets of active con-

Optimality Criterion Techniques Applied to Mechanical Design

Presented are two optimality criterion techniques for the minimum weight design of mechanical and structural systems subject to limitations on stresses and natural frequencies. The results, based on their application to several examples, are compared with those obtained by other researchers and with standard nonlinear mathematical programming techniques, as well as with a special linear approach. It is shown in all cases that the optimality criterion methods are capable of locating the optimal design and are far superior to other methods in terms of computational time.