Multipoint Approximation Method Research Articles

Application of approximate concept and graph theory to beam and plate topology optimization

Approximate concept and graph theory are developed and applied to solve the integrated optimization problem of engineering structure containing both beam and plate discrete variables. Firstly, both structural responses and explicit requirements for process manufacturing are considered as constraints, which makes an optimization system based on the two-level multi-point approximation method more engineering oriented. Secondly, a practical evaluation method of topology rationality is proposed based on graph theory in order to avoid invalid structural analysis and improve optimization efficiency. The finite element model corresponding to the binary point-line relation in graph theory and judgements of force nodes, lump masses, structural integrity, etc. are unified. An engineering example shows that the two-level multi-point approximation method is still efficient in solving topology optimization problems with participating plates and beams, and the proposed topological rationality evaluation method can effectively eliminate about 15% of unreasonable configurations.

Wing jig shape optimisation with gradient-assisted metamodel building in a trust-region optimisation framework

Significant computational resources are required to obtain an optimised wing jig shape by solving a high-fidelity large-scale aero-structural design optimisation problem. Gradient-based methods are efficient; however, some of the features of real-life engineering problems including numerical noise that pollutes the function values and occurrences of failed evaluations in the optimisation may limit their performance. To address these issues, this paper presents the latest developments in the multipoint approximation method (MAM) based on a gradient-assisted metamodel assembly technique within a trust-region optimisation framework. The proposed method is tested by a benchmark case first, and then, an aircraft wing jig shape optimisation problem is offered to demonstrate its performance. The gradient-based optimisation is used as a benchmark case, and the metamodel-based optimisation utilises the latest developments in MAM to solve the same problem. The results show that the proposed method can achieve the same design goal as the gradient-based method but with enhanced robustness and efficient performance. In the wing jig shape optimisation, the difference in the design objective, the global equivalent drag coefficient, between the two aforementioned optimisation approaches is 0.20 counts, whose relative difference is approximately 0.10%. Three approximate sub-optimisations have been conducted in every iteration of the metamodel-based optimisation to reduce the possibility of local optimality, while the overall elapsed time of the metamodel-based optimisation is approximately 1.98 times that of one gradient-based optimisation, which confirms the competitiveness of the proposed method bearing in mind the added safeguards for numerical noise, failed evaluations and possible local optimality.

Computationally Efficient Approximations Using Adaptive Weighting Coefficients for Solving Structural Optimization Problems

With rapid development of advanced manufacturing technologies and high demands for innovative lightweight constructions to mitigate the environmental and economic impacts, design optimization has attracted increasing attention in many engineering subjects, such as civil, structural, aerospace, automotive, and energy engineering. For nonconvex nonlinear constrained optimization problems with continuous variables, evaluations of the fitness and constraint functions by means of finite element simulations can be extremely expensive. To address this problem by algorithms with sufficient accuracy as well as less computational cost, an extended multipoint approximation method (EMAM) and an adaptive weighting-coefficient strategy are proposed to efficiently seek the optimum by the integration of metamodels with sequential quadratic programming (SQP). The developed EMAM stems from the principle of the polynomial approximation and assimilates the advantages of Taylor’s expansion for improving the suboptimal continuous solution. Results demonstrate the superiority of the proposed EMAM over other evolutionary algorithms (e.g., particle swarm optimization technique, firefly algorithm, genetic algorithm, metaheuristic methods, and other metamodeling techniques) in terms of the computational efficiency and accuracy by four well-established engineering problems. The developed EMAM reduces the number of simulations during the design phase and provides wealth of information for designers to effectively tailor the parameters for optimal solutions with computational efficiency in the simulation-based engineering optimization problems.

Adaptive Multi-Level Search for Global Optimization: An Integrated Swarm Intelligence-Metamodelling Technique

Over the last decade, metaheuristic algorithms have emerged as a powerful paradigm for global optimization of multimodal functions formulated by nonlinear problems arising from various engineering subjects. However, numerical analyses of many complex engineering design problems may be performed using finite element method (FEM) or computational fluid dynamics (CFD), by which function evaluations of population-based algorithms are repetitively computed to seek a global optimum. It is noted that these simulations become computationally prohibitive for design optimization of complex structures. To efficiently and effectively address this class of problems, an adaptively integrated swarm intelligence-metamodelling (ASIM) technique enabling multi-level search and model management for the optimal solution is proposed in this paper. The developed technique comprises two steps: in the first step, a global-level exploration for near optimal solution is performed by adaptive swarm-intelligence algorithm, and in the second step, a local-level exploitation for the fine optimal solution is studied on adaptive metamodels, which are constructed by the multipoint approximation method (MAM). To demonstrate the superiority of the proposed technique over other methods, such as conventional MAM, particle swarm optimization, hybrid cuckoo search, and water cycle algorithm in terms of computational expense associated with solving complex optimization problems, one benchmark mathematical example and two real-world complex design problems are examined. In particular, the key factors responsible for the balance between exploration and exploitation are discussed as well.

Multi-Disciplinary Design Optimisation of the Cooled Squealer Tip for High Pressure Turbines

The turbine tip geometry can significantly alter the performance of the turbine stage; its design represents a challenge for a variety of reasons. Multiple disciplines are involved in its design and their requirements limit the creativity of the designer. Multi-Disciplinary Design Optimisation (MDO) offers the capability to improve the performance whilst satisfying all the design constraints. This paper presents a novel design of a turbine tip achieved via MDO techniques. A fully parametrised Computer-Aided Design (CAD) model of the turbine rotor is used to create the squealer geometry and to control the location of the cooling and dust holes. A Conjugate Heat Transfer Computational Fluid Dynamics (CFD) analysis is performed for evaluating the aerothermal performance of the component and the temperature the turbine operates at. A Finite Element (FE) analysis is then performed to find the stress level that the turbine has to withstand. A bi-objective optimisation reduces simultaneously the aerodynamic loss and the stress level. The Multipoint Approximation Method (MAM) recently enhanced for multi-objective problems is chosen to solve this optimisation problem. The paper presents its logic in detail. The novel geometry offers a significant improvement in the aerodynamic performance whilst reducing the maximum stress. The flow associated with the new geometry is analysed in detail to understand the source of the improvement.

Efficient hybrid algorithms to solve mixed discrete-continuous optimization problems

PurposeIn real-world cases, it is common to encounter mixed discrete-continuous problems where some or all of the variables may take only discrete values. To solve these non-linear optimization problems, the use of finite element methods is very time-consuming. The purpose of this study is to investigate the efficiency of the proposed hybrid algorithms for the mixed discrete-continuous optimization and compare it with the performance of genetic algorithms (GAs).Design/methodology/approachIn this paper, the enhanced multipoint approximation method (MAM) is used to reduce the original nonlinear optimization problem to a sequence of approximations. Then, the sequential quadratic programing technique is applied to find the continuous solution. Following that, the implementation of discrete capability into the MAM is developed to solve the mixed discrete-continuous optimization problems.FindingsThe efficiency and rate of convergence of the developed hybrid algorithms outperforming GA are examined by six detailed case studies in the ten-bar planar truss problem, and the superiority of the Hooke–Jeeves assisted MAM algorithm over the other two hybrid algorithms and GAs is concluded.Originality/valueThe authors propose three efficient hybrid algorithms, the rounding-off, the coordinate search and the Hooke–Jeeves search-assisted MAMs, to solve nonlinear mixed discrete-continuous optimization problems. Implementations include the development of new procedures for sampling discrete points, the modification of the trust region adaptation strategy and strategies for solving mix optimization problems. To improve the efficiency and effectiveness of metamodel construction, regressors f defined in this paper can have the form in common with the empirical formulation of the problems in many engineering subjects.

Extended Multipoint Approximation Method

Stemming from polynomial metamodels, multipoint approximation method (MAM) and moving least square method (MLSM) focus on the development of metamodels for the objective and constraint functions in solving a mid-range optimization problem with a trust region. Although both of these methods could solve problems successfully, there is still some room for improvement on the computational effort and search capability. To address this problem, the extended multipoint approximation method is proposed to seek the optimal solution in this paper. The developed method assimilating the advantage of Taylor’s expansion used in MLSM demonstrates its superiority over other methods in terms of the computational efficiency and accuracy by some well-established benchmark problems.

Enhancing Efficiency of Two Level Multipoint Approximation Method by an Efficient Move Limit Approach

This paper presents an improvement to the efficiency of Two Level Multipoint Approximation Method (TMA) for structure optimization by means of move limits definition. A move limit strategy, which was initially presented by Bloebaum, C. L., can efficiently enhance robustness of the method especially for the achievement of gauge (least) values of design variables in truss structures. Method’s capability has been exploited for assigning a different move limit to each design variable based upon statistical distribution of effectiveness coefficients. This paper suggests an improved version of TMA as well as an improved definition of highest value of move limit for least heuristic definition of move limits.

Structural optimization for multiple structure cases and multiple payload cases with a two-level multipoint approximation method

This paper is to address structural optimization problems where multiple structure cases or multiple payload cases can be considered simultaneously. Both types of optimization problems involve multiple finite element models at each iteration step, which draws high demands in optimization methods. Considering the common characteristic for these two types of problems, which is that the design domain keeps the same no matter what the structure cases or payload cases are, both problems can be formulated into the unified expressions. A two-level multipoint approximation (TMA) method is firstly improved with the use of analytical sensitivity analysis for structural mass, and then this improved method is utilized to tackle these two types of problems. Based on the commercial finite element software MSC.Patran/Nastran, an optimization system for multiple structure cases and multiple payload cases is developed. Numerical examples are conducted to verify its feasibility and efficiency, and the necessity for the simultaneous optimizations of multiple structure cases and multiple payload cases are illustrated as well.

Robust optimisation of railway crossing geometry

ABSTRACTThis paper presents a methodology for improving the crossing (frog) geometry through the robust optimisation approach, wherein the variability of the design parameters within a prescribed tolerance is included in the optimisation problem. Here, the crossing geometry is defined by parameterising the B-spline represented cross-sectional shape and the longitudinal height profile of the nose rail. The dynamic performance of the crossing is evaluated considering the variation of wheel profiles and track alignment. A multipoint approximation method (MAM) is applied in solving the optimisation problem of minimising the contact pressure during the wheel–rail contact and constraining the location of wheel transition at the crossing. To clarify the difference between the robust optimisation and the normal deterministic optimisation approaches, the optimisation problems are solved in both approaches. The results show that the deterministic optimum fails under slight change of the design variables; the robust optimum, however, has improved and robust performance.

Implementation of Discrete Capability into the Enhanced Multipoint Approximation Method for Solving Mixed Integer-Continuous Optimization Problems

Multipoint approximation method (MAM) focuses on the development of metamodels for the objective and constraint functions in solving a mid-range optimization problem within a trust region. To develop an optimization technique applicable to mixed integer-continuous design optimization problems in which the objective and constraint functions are computationally expensive and could be impossible to evaluate at some combinations of design variables, a simple and efficient algorithm, coordinate search, is implemented in the MAM. This discrete optimization capability is examined by the well established benchmark problem and its effectiveness is also evaluated as the discreteness interval for discrete design variables is increased from 0.2 to 1. Furthermore, an application to the optimization of a lattice composite fuselage structure where one of design variables (number of helical ribs) is integer is also presented to demonstrate the efficiency of this capability.

Development of a Hybrid Algorithm for Efficiently Solving Mixed Integer-Continuous Optimization Problems

Problems with mixed integer-continuous design variables are a class of complicated optimization problems that commonly exist in practical engineering design work. In this paper, a hybrid algorithm combining metamodel-based Multipoint Approximation Method (MAM) and Hooke-Jeeves direct search technique is presented to efficiently seek the optimum solutions for mixed integer-continuous optimization problems. First, optimal continuous values are obtained by the Sequential Quadratic Programming method (SQP) on the approximated functions in a current trust region. Then, continuous values are rounded to the nearest integer values for discrete variables. Utilizing integer values as a starting point, the Hooke-Jeeves assisted MAM is applied to search for the discrete optimal solution in the sub-space of discrete variables as well as accordingly update the sub-optimal values for continuous design variables by SQP. The proposed hybrid algorithm is examined by the well established benchmark example and the obtained results demonstrate the superiority of the developed algorithm over GA in terms of computational cost and the quality of solutions.

Generation of Supramolecular Chirality around Twofold Rotational or Helical Axes in Crystalline Assemblies of Achiral Components

A multi-point approximation method clarifies supramolecular chirality of twofold rotational or helical assemblies as well as bundles of the one-dimensional (1D) assemblies. While one-point approximation of materials claims no chirality generation of such assemblies, multi-point approximations do claim possible generation in the 1D assemblies of bars and plates. Such chirality derives from deformations toward three-axial directions around the helical axes. The chiral columns are bundled in chiral ways through symmetry operations. The preferable right- or left-handed columns are bundled together to yield chiral crystals with right- or left-handedness, respectively, indicating that twofold helix symmetry operations cause chiral crystals composed of achiral components via a three-stepwise and three-directional process.

Actuator placement optimization for adaptive trusses using a two-level multipoint approximation method

Parameters in adaptive trusses considering placements of actuators, control gains as well as structure cross-sectional sizes, have been simultaneously optimized to suppress vibrations and to maintain stringent shape specifications. Since the placements of actuators are treated as (0,1) variables but the control gains and structural member sizes are continuous, the studied problem then becomes mixed (0,1)-continuous optimization, which is difficult to be tackled with conventional optimization methods since they are often encountered with local optimal solutions or numerous computational costs. A two-level multipoint approximation strategy combined with genetic algorithm, which proved to be applicable in truss topology-size optimizations involving discrete-continuous design variables, has been developed to address the current actuator placement problem. To enhance the efficiency of this method, a branched multi-point approximation function was introduced to construct the first-level approximation problem. Besides, a new fitness function as well as adaptive crossover and mutation operators were adopted to prevent the optimization process from too early converging to local optimal solutions. Numerical results were presented to illustrate the efficacy of this strategy in dealing with actuator placement problems.

Improvement of vehicle–turnout interaction by optimising the shape of crossing nose

Proper rail geometry in the crossing part is essential for reducing damage on the nose rail. To improve the dynamic behaviour of turnout crossings, a numerical optimisation approach to minimise rolling contact fatigue (RCF) damage and wear in the crossing panel by varying the nose rail shape is presented in the paper. The rail geometry is parameterised by defining several control cross-sections along the crossing. The dynamic vehicle–turnout interaction as a function of crossing geometry is analysed using the VI-Rail package. In formulation of the optimisation problem a combined weighted objective function is used consisting of the normal contact pressure and the energy dissipation along the crossing responsible for RCF and wear, respectively. The multi-objective optimisation problem is solved by adapting the multipoint approximation method and a number of compromised solutions have been found for various sets of weight coefficients. Dynamic behaviour of the crossing has been significantly improved after optimisations. Comparing with the reference design, the heights of the nose rail are notably increased in the beginning of the crossing; the nominal thicknesses of the nose rail are also changed. All the optimum designs work well under different track conditions.

Optimisation of the elastic track properties of turnout crossings

Rail pads and under sleeper pads (USPs) are resilient elements inserted between the rail and the sleeper, and between the sleeper and the ballast, respectively. They improve the elastic properties of the track’s superstructure. In this paper, the approach of estimating the performance of a turnout by using the dynamic forces acting on the crossing as indicators of the extent of crossing nose damage is improved by tuning the stiffness and damping of the rail pads and USPs using a numerical optimisation method. In the optimisation problem, the dynamic forces acting on rails, sleepers and the ballast bed, which should be minimised, are considered in the objective function. Constraints are imposed on the displacements of the structural elements of the turnout crossing. The combined multi-objective optimisation problem is solved using the multipoint approximation method. The results of the optimisation show that application of softer rail pads combined with USPs can significantly reduce the dynamic forces acting on the rails, sleepers and ballast. Moreover, the track elasticity should be varied along the crossing.

Optimization of the dynamic properties of the ladder track system to control rail vibration using the multipoint approximation method

The combination model of the vehicle and ladder track system that accounts for the wheel–rail interface has been developed to analyze the wheelset and ladder track vibration characteristics in the frequency and time domains. In the wheel–rail interface, the vertical contact is represented by contact stiffness derived from the Hertz theory of normal elastic contact, and lateral contact is idealized by linear contact used Kalker linear rolling contact theory. The considerable noise and vibration frequency of the ladder track is determined by experimental investigation results. Then the effect of the most significant ladder track parameters on the track vibration is analyzed in time domain using the vehicle–track coupling model. Based on the results of the parametric study, an optimization of the mechanical properties of the ladder track to reduce the track vibrations is performed using the multipoint approximation method. The results of the optimization are presented and discussed.

Aerodynamic Optimization of High-Pressure Turbines for Lean-Burn Combustion System

Modern lean-burn combustors make use of high flow swirl to maintain flame stability. The swirling flow can persist downstream of the turbine first vane, changing the loading on the rotor, leading to a reduction in efficiency. This paper presents the results of an automatic optimization study carried out to mitigate the effect of high swirling flow on a high pressure turbine stage. A high-fidelity computational fluid dynamics (CFD)-based design optimization using a multipoint approximation (response surface) method is carried out to produce a new vane and a new rotor configuration with a significantly improved aerodynamic performance. It is demonstrated that the novel optimization methodology can cope well with a number of near equality constraints needed for a practical design.

多点近似による二回らせん集合体とその束集合体の左右決定

Two-fold helical assemblies of points, lines and faces were analyzed by the multipoint approximation method on the basis of three-dimensional space geometry. The analysis has led to the new concept that assemblies generated from a rotational operation by 180 degrees exhibit supramolecular chirality with distinguishable right- or left-handedness, termed as supramolecular tilt chirality, in contrast with the conventional theory in crystallography. This concept has enabled us to determine handedness of two-fold helical assemblies of various organic molecules and macromolecules. Such helical assemblies with three-axial chirality are bundled in various manners through three-stepwise and three-direction processes by using weak interactions. The bundling manners explain formation of chiral crystals starting from achiral molecules, guiding us toward an interpretation of chirality recognition mechanism in host-guest systems. This methodology can unify various kinds of chirality in molecules, supramolecules, macromolecules, and daily life materials on the basis of four-point chirality recognition in three-dimensional space geometry.

Mid-range metamodel assembly building based on linear regression for large scale optimization problems

In this work an approach to building a high accuracy approximation valid in a larger range of design variables is investigated. The approach is based on an assembly of multiple surrogates into a single surrogate using linear regression. The coefficients of the model assembly are not weights of the individual models but tuning parameters determined by the least squares method. The approach was utilized in the Multipoint Approximation Method (MAM) method within the mid-range approximation framework. The developed technique has been tested on several benchmark problems with up to 1000 design variables and constraints. The obtained results show a high degree of accuracy of the built approximations and the efficiency of the technique when applied to large-scale optimization problems.