SoftFEM: The Soft Finite Element Method

While the finite element method (FEM) has now reached full maturity both in academy and industry, its use in optimization pipelines remains either computationally intensive or cumbersome. In particular, currently used optimization schemes leveraging FEM still require the choice of dedicated optimization algorithms for a specific design problem, and a “black box” approach to FEM-based optimization remains elusive. To this end, we propose here an integrated finite element-soft computing method, ie, the soft FEM (SoftFEM), which integrates a finite element solver within a metaheuristic search wrapper. To illustrate this general method, we focus here on solid mechanics problems. For these problems, SoftFEM is able to optimize geometry changes and mechanistic measures based on geometry constraints and material properties inputs. From the optimization perspective, the use of a fitness function based on finite element calculation imposes a series of challenges. To bypass the limitations in search capabilities of the usual optimization techniques (local search and gradient-based methods), we propose, instead a hybrid self adaptive search technique, the multiple offspring sampling (MOS), combining two metaheuristics methods: one population-based differential evolution method and a local search optimizer. The formulation coupling FEM to the optimization wrapper is presented in detail and its flexibility is illustrated with three representative solid mechanics problems. More particularly, we propose here the MOS as the most versatile search algorithm for SoftFEM. A new method for the identification of nonfully determined parameters is also proposed.     

Limited-Memory BFGS Diagonal Preconditioners for a Data Assimilation Problem in Meteorology

This paper uses two simple variational data assimilation problems with the 1D viscous Burgers' equation on a periodic domain to investigate the impact of various diagonal-preconditioner update and scaling strategies, both on the limited-memory BFGS (Broyden, Fletcher, Goldfarb and Shanno) inverse Hessian approximation and on the minimization performance. These simple problems share some characteristics with the large-scale variational data assimilation problems commonly dealt with in meteorology and oceanography.The update formulae studied are those proposed by Gilbert and Lemaréchal (Math. Prog., vol. 45, pp. 407–435, 1989) and the quasi-Cauchy formula of Zhu et al. (SIAM J. Optim., vol. 9, pp. 1192–1204, 1999). Which information should be used for updating the diagonal preconditioner, the one to be forgotten or the most recent one, is considered first. Then, following the former authors, a scaling of the diagonal preconditioner is introduced for the corresponding formulae in order to improve the minimization performance. The large negative impact of such a scaling on the quality of the L-BFGS inverse Hessian approximation led us to propose an alternate updating and scaling strategy, that provides a good inverse Hessian approximation and gives the best minimization performance for the problems considered. With this approach the quality of the inverse Hessian approximation improves steadily during the minimization process. Moreover, this quality and the L-BFGS minimization performance improves when the amount of stored information is increased.     

A general optimization procedure for stereotactic small-beam multi-isocentric radiotherapy

Clinical experience in single-dose stereotactic radiotherapy of irregular complex lesions has shown that new developments in optimization procedures were necessary to improve dose distribution, making this conformal technique more efficient. We propose a conformal procedure for stereotactic radiotherapy of complex lesions treated with multiple isocenters based on the associated targets methodology. The successive steps of this conformal procedure are: (a) the determination of the number of subvolumes; (b) the choice of collimator diameters using a dosimetric data basis; (c) the search by the inverse SVD optimizer algorithm for the optimal irradiation space and the minibeam weights for each subvolume using singular value decomposition; (d) the basis of quantitative evaluation criteria for the choice of satisfactory solutions on dose-volume histograms and clinical considerations. The efficiency of the SVD optimizer to planify multi-isocentric treatments was examined in the case of an irregular lesion planified with two isocenters. The condition number of this dual isocentric configuration showed a more ill-conditioned problem than in the monoisocentric case. Examining different reconstructed weighting vectors, we observed that optimal solutions are obtained with the first singular components and an important healthy tissue overdosage occurs when the number of singular components used in the SVD expansion increases. Our SVD optimization approach is a general procedure which can be applied to different radiotherapy techniques.     

A parallel code for a self-consistent charge density functional based tight binding method: Total energy calculations for extended systems

We present a parallel version of a selfconsistent-charge density-functional based tight-binding (SCC-DFTB) method for total energy calculations and geometry optimizations of clusters and periodic structures. On single processor machines the SCC-DFTB method has been successfully applied to systems up to several hundred atoms with an accuracy comparable to sophisticated selfconsistent field density-functional theory (SCF-DFT) methods. The new parallel code allows to treat systems which are larger by an order of magnitude in reasonable time. The freely available ScaLAPACK and PBLAS libraries are used for linear algebra operations. We tested the scaling of our program for a realistic system (III–V semiconductor surface) with different sizes and give a short outlook on current applications.     

Multiobjective design of actively controlled structures using a hybrid optimization method

A multiobjective approach to the combined structure and control optimization problem for flexible space structures is presented. The proposed formulation addresses robustness considerations for controller design, as well as a simultaneous determination of optimum actuator locations. The structural weight, controlled system energy, stability robustness index and damping augmentation provided by the active controller are considered as objective functions of the multiobjective problem which is solved using a cooperative game-theoretic approach. The actuator locations and the cross-sectional areas of structural members are treated as design variables. Since the actuator locations are spatially discrete, whereas the cross-sectional areas are continuous, the optimization problem has mixed discrete-continuous design variables. A solution approach to this problem based on a hybrid optimization scheme is presented. The hybrid optimizer is a synergetic blend of artificial genetic search and gradient-based search techniques. The computational procedure is demonstrated through the design of an ACOSS-FOUR space structure. The optimum solutions obtained using the hybrid optimizer are shown to outperform the optimum results obtained using gradient-based search techniques.     

A stochastically evolving non-local search and solutions to inverse problems with sparse data

Building on a martingale approach to global optimization, a powerful stochastic search scheme for the global optimum of cost functions is proposed using change of measures on the states that evolve as diffusion processes and splitting of the state-space along the lines of a Bayesian game. To begin with, the efficacy of the optimizer, when contrasted with one of the most efficient existing schemes, is assessed against a family of N_p-hard benchmark problems. Then, using both simulated and experimental data, potentialities of the new proposal are further explored in the context of an inverse problem of significance in photoacoustic imaging, wherein the superior reconstruction features of a global search vis-à-vis the commonly adopted local or quasi-local schemes are brought into relief.     

A generic framework for handling constraints with agent-based optimization algorithms and application to aerodynamic design

A generic constraint handling framework for use with any swarm-based optimization algorithm is presented. For swarm optimizers to solve constrained optimization problems effectively modifications have to be made to the optimizers to handle the constraints, however, these constraint handling frameworks are often not universally applicable to all swarm algorithms. A constraint handling framework is therefore presented in this paper that is compatible with any swarm optimizer, such that a user can wrap it around a chosen swarm algorithm and perform constrained optimization. The method, called separation-sub-swarm, works by dividing the population based on the feasibility of individual agents. This allows all feasible agents to move by existing swarm optimizer algorithms, hence promoting good performance and convergence characteristics of individual swarm algorithms. The framework is tested on a suite of analytical test function and a number of engineering benchmark problems, and compared to other generic constraint handling frameworks using four different swarm optimizers; particle swarm, gravitational search, a hybrid algorithm and differential evolution. It is shown that the new framework produces superior results compared to the established frameworks for all four swarm algorithms tested. Finally, the framework is applied to an aerodynamic shape optimization design problem where a shock-free solution is obtained.     

Shape and size optimization of trusses with multiple frequency constraints using harmony search and ray optimizer for enhancing the particle swarm optimization algorithm

In this paper, size and shape optimization of truss structures is performed using an efficient hybrid method. This algorithm uses a particle swarm strategy and ray optimizer, and utilizes additional harmony search for a better exploitation. Here, multiple frequency constraints are considered making the optimization a highly nonlinear problem. Some basic benchmark problems are solved by this hybrid method, and the numerical results demonstrate the efficiency and robustness of this method compared to other mathematical and heuristic algorithms.     

A geometry projection method for the topology optimization of curved plate structures with placement bounds

Single‐curvature plates are commonly encountered in mechanical and civil structures. In this paper, we introduce a topology optimization method for the stiffness‐based design of structures made of curved plates with fixed thickness. The geometry of each curved plate is analytically and explicitly represented by its location, orientation, dimension, and curvature radius, and therefore, our method renders designs that are distinctly made of curved plates. To perform the primal and sensitivity analyses, we use the geometry projection method, which smoothly maps the analytical geometry of the curved plates onto a continuous density field defined over a fixed uniform finite element grid. A size variable is ascribed to each plate and penalized in the spirit of solid isotropic material with penalization, which allows the optimizer to remove a plate from the design. We also introduce in our method a constraint that ensures that no portion of a plate lies outside the design envelope. This prevents designs that would otherwise require cuts to the plates that may be very difficult to manufacture. We present numerical examples to demonstrate the validity and applicability of the proposed method.     

Reliability-based design optimization with progressive surrogate models

Reliability-based design optimization (RBDO) has traditionally been solved as a nested (bilevel) optimization problem, which is a computationally expensive approach. Unilevel and decoupled approaches for solving the RBDO problem have also been suggested in the past to improve the computational efficiency. However, these approaches also require a large number of response evaluations during optimization. To alleviate the computational burden, surrogate models have been used for reliability evaluation. These approaches involve construction of surrogate models for the reliability computation at each point visited by the optimizer in the design variable space. In this article, a novel approach to solving the RBDO problem is proposed based on a progressive sensitivity surrogate model. The sensitivity surrogate models are built in the design variable space outside the optimization loop using the kriging method or the moving least squares (MLS) method based on sample points generated from low-discrepancy sampling (LDS) to estimate the most probable point of failure (MPP). During the iterative deterministic optimization, the MPP is estimated from the surrogate model for each design point visited by the optimizer. The surrogate sensitivity model is also progressively updated for each new iteration of deterministic optimization by adding new points and their responses. Four example problems are presented showing the relative merits of the kriging and MLS approaches and the overall accuracy and improved efficiency of the proposed approach.     

A multi-objective variable-fidelity optimization method for genetic algorithms

A novel variable-fidelity optimization (VFO) scheme is presented for multi-objective genetic algorithms. The technique uses a low- and high-fidelity version of the objective function with a Kriging scaling model to interpolate between them. The Kriging model is constructed online through a fixed updating schedule. Results for three standard genetic algorithm test cases and a two-objective stiffened panel optimization problem are presented. For the stiffened panel problem, statistical analysis of four performance metrics are used to compare the Pareto fronts between the VFO method, full high-fidelity optimizer runs, and Pareto fronts developed by enumeration. The fixed updating approach is shown to reduce the number of high-fidelity calls significantly while approximating the Pareto front in an efficient manner.     

State-transition-matrix optimization for reconfiguration manoeuvres of formation-flying satellites

The on-orbit reconfiguration of a pair of formation-flying satellites in low Earth orbit is studied in the presence of J₂–J₆ gravitational perturbations. A methodology for determining a robust and accurate impulsive thrusting scheme is developed with the aim of minimizing reconfiguration overshoot errors and fuel expenditure (Δ V). The method uses a state transition matrix based on the Hill–Clohessy–Wiltshire linear equations of relative motion and the analytical solution to the state-space model to solve for a pair of impulsive thrusts. The manoeuvre is then propagated through a fully nonlinear orbital simulator with the thrusts implemented non-impulsively. A Sequential Quadratic Programming optimizer adjusts the inputs to the linear state transition matrix to produce impulses that, when applied in the high-fidelity orbital propagator, mitigates the Δ V of the manoeuvre while maintaining acceptable overshoot errors.     

Particle swarm optimization approach for multi-objective composite box-beam design

This paper presents a multi-agent search technique to design an optimal composite box-beam helicopter rotor blade. The search technique is called particle swarm optimization (‘inspired by the choreography of a bird flock’). The continuous geometry parameters (cross-sectional dimensions) and discrete ply angles of the box-beams are considered as design variables. The objective of the design problem is to achieve (a) specified stiffness value and (b) maximum elastic coupling. The presence of maximum elastic coupling in the composite box-beam increases the aero-elastic stability of the helicopter rotor blade. The multi-objective design problem is formulated as a combinatorial optimization problem and solved collectively using particle swarm optimization technique. The optimal geometry and ply angles are obtained for a composite box-beam design with ply angle discretizations of 10°, 15° and 45°. The performance and computational efficiency of the proposed particle swarm optimization approach is compared with various genetic algorithm based design approaches. The simulation results clearly show that the particle swarm optimization algorithm provides better solutions in terms of performance and computational time than the genetic algorithm based approaches.     

Fuzzy entropy‐based MR brain image segmentation using modified particle swarm optimization

This article presents an image segmentation technique based on fuzzy entropy, which is applied to magnetic resonance (MR) brain images in order to detect brain tumors. The proposed method performs image segmentation based on adaptive thresholding of the input MR images. The image is classified into two membership functions (MFs) of the fuzzy region: Z‐function and S‐function. The optimal parameters of these fuzzy MFs are obtained using modified particle swarm optimization (MPSO) algorithm. The objective function for obtaining the optimal fuzzy MF parameters is considered to be the maximum fuzzy entropy. Through a number of examples, The performance is compared with existing entropy based object segmentation approaches and the superiority of the proposed method is demonstrated. The experimental results are compared with the exhaustive search method and Otsu's segmentation technique. The result shows the proposed fuzzy entropy‐based segmentation method optimized using MPSO achieves maximum entropy with proper segmentation of infected areas and with minimum computational time. © 2013 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 23, 281–288, 2013     

An improved weighting method with multibounds formulation and convex programming for multicriteria structural optimization

This paper presents an improved weighting method for multicriteria structural optimization. By introducing artificial design variables, here called as multibounds formulation (MBF), we demonstrate mathematically that the weighting combination of criteria can be transformed into a simplified problem with a linear objective function. This is a unified formulation for one criterion and multicriteria problems. Due to the uncoupling of involved criteria after the transformation, the extension and the adaptation of monotonic approximation‐based convex programming methods such as the convex linearization (CONLIN) or the method of moving asymptotes (MMA) are made possible to solve multicriteria problems as efficiently as for one criterion problems. In this work, a multicriteria optimization tool is developed by integrating the multibounds formulation with the CONLIN optimizer and the ABAQUS finite element analysis system. Some numerical examples are taken into account to show the efficiency of this approach. Copyright © 2001 John Wiley & Sons, Ltd.     

An extension of the directed search domain algorithm to bilevel optimization

A method is developed for generating a well-distributed Pareto set for the upper level in bilevel multiobjective optimization. The approach is based on the Directed Search Domain (DSD) algorithm, which is a classical approach for generation of a quasi-evenly distributed Pareto set in multiobjective optimization. The approach contains a double-layer optimizer designed in a specific way under the framework of the DSD method. The double-layer optimizer is based on bilevel single-objective optimization and aims to find a unique optimal Pareto solution rather than generate the whole Pareto frontier on the lower level in order to improve the optimization efficiency. The proposed bilevel DSD approach is verified on several test cases, and a relevant comparison against another classical approach is made. It is shown that the approach can generate a quasi-evenly distributed Pareto set for the upper level with relatively low time consumption.     

A constrained‐optimization methodology for the detection phase in contact mechanics simulations

The detection phase in computational contact mechanics can be subdivided into a global search and a local detection. When potential contact is detected by the former, a rigorous local detection determines which surface elements come or may come in contact in the current increment. We first introduce a rigorous definition of the closest point for non‐differentiable lower‐dimensional manifolds. We then simplify the detection by formulating an optimization problem subject to inequality constraints. The formulation is then solved using different techniques from the field of mathematical optimization, for both linear and quadratic finite element meshes. The resulting general and robust detection scheme is tested on a set of problems and compared with other techniques commonly used in computational geometry. Copyright © 2013 John Wiley & Sons, Ltd.     

Continuous families of Lax-Wendroff schemes

Families of difference schemes are developed for both the linear and nonlinear inviscid Burger's equation and a system of shallow water equations. Single-step and two-step (Lax–Wendroff type) families are described and comparisons are made with other difference schemes. The schemes are developed using a very limited portion of a space–time continua, coupled with Galerkin's method applied over a few triangular elements. The elements are geometrically flexible as a function of a geometry factor α, creating the idea of difference families. Stability is commented on, including the continuous relationship between stability and the families of schemes as a function of α. Numerical results are presented and commented on for the system of shallow water equations and the inviscid Burger's equation. Several areas for future research are described.     

MULTIOBJECTIVE DESIGN OF A MARINE VEHICLE WITH AID OF DESIGN KNOWLEDGE

To generate the Pareto optimal set efficiently in multiobjective optimization, a hybrid optimizer is developed by coupling the genetic algorithm and the direct search method. This method determines a candidate region around the global optimum point by using the genetic algorithm, then searches the global optimum point by the direct search method concentrating in this region, thus reducing calculation time and increasing search efficiency. Although the hybrid optimizer provides cost-effectiveness, the design optimization process involves a number of tasks which require human expertise and experience. Therefore, methods of optimization and associated programs have been used mostly by experts in the real design world. Hence, this hybrid optimizer incorporates a knowledge-based system with heuristic and analytic knowledge, thereby narrowing the feasible space of the objective function. Some domain knowledge is retrieved from database and design experts. The obtained knowledge is stored in the knowledge base. The results of this paper, through application to marine vehicle design with multiobjective optimization, show that the hybrid optimizer with aid of design knowledge can be a useful tool for multiobjective optimum design. © 1997 John Wiley & Sons, Ltd.     

Power plant maintenance scheduling using ant colony optimization: an improved formulation

It is common practice in the hydropower industry to either shorten the maintenance duration or to postpone maintenance tasks in a hydropower system when there is expected unserved energy based on current water storage levels and forecast storage inflows. It is therefore essential that a maintenance scheduling optimizer can incorporate the options of shortening the maintenance duration and/or deferring maintenance tasks in the search for practical maintenance schedules. In this article, an improved ant colony optimization-power plant maintenance scheduling optimization (ACO-PPMSO) formulation that considers such options in the optimization process is introduced. As a result, both the optimum commencement time and the optimum outage duration are determined for each of the maintenance tasks that need to be scheduled. In addition, a local search strategy is presented in this article to boost the robustness of the algorithm. When tested on a five-station hydropower system problem, the improved formulation is shown to be capable of allowing shortening of maintenance duration in the event of expected demand shortfalls. In addition, the new local search strategy is also shown to have significantly improved the optimization ability of the ACO-PPMSO algorithm.