Multipoint Approximation Research Articles

An integrated optimization method for the damper placements, shape and size of truss structures

This work extends the application of the approximation concept in dynamic topology optimization for truss structures. The corresponding problem contains both continuous and discrete variables, encompassing damper placements, truss shape and cross-sectional area. First, the utilization of a branched multipoint approximation (BMA) function establishes sequential approximation problems that also involve such mixed variables, marking its innovative application in dynamic optimization. Then, a modified genetic algorithm (GA) is used to optimize discrete variables. To enhance convergence stationarity, the concept of ‘adjacent individuals’ is proposed to control the degree of difference in topology configuration between initial population members and the current optimal. Additionally, adjustments to the sequence-based crossover process ensure compliance with damper quantity constraints. Examples demonstrate that the introduction of adjacent individuals is beneficial to convergence stationarity, and the shape change of the truss increases the mode damping ratio by more than 20%. The created optimization platform can also tackle other mixed variables optimization problems.

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.

Technology Focus: Formation Evaluation (February 2022)

This marks the first of two Formation Evaluation features for 2022. The COVID-19 pandemic naturally has affected SPE meetings, causing many to be rescheduled or postponed indefinitely, but SPE papers continue to be a crucial source of technical knowledge. The selected papers explore simple and complex innovative approaches toward reservoir characterization to work around the absence of certain data. Paper SPE 203903 addresses issues related to safe and optimal operation of geoenergy applications related to induced seismicity as a result of pore-pressure fluctuations. The authors developed a finite-volume method with a multipoint approximation of fluxes for geomechanics and poromechanics that will cope with discontinuities in displacements, as occur in faults, on the level of discretization. The next paper demonstrates local calibration of a simple rock physics model that can provide valuable insights into the elastic properties, which then can be combined with seismic data for quantitative reservoir characterization away from the wellbore. The key, according to the author of paper OTC 30359, is the use of the Hashin-Shtrikman effective medium model with absolute upper and lower bounds for compressional and shear velocities of media that are composed of multiple constituents. The third paper explores the challenges of identifying and quantifying hydrocarbon sections because of little resistivity contrast between reservoirs and water zones in the absence of radioactive tools. The authors of paper SPE 202420 develop a work flow that integrates nuclear magnetic resonance with azimuthal resistivity to overcome the uncertainty of fluid typing and optimal wellbore placement. Despite the challenges our industry continues to face, authors of SPE conference papers continue to provide an important source of knowledge and experience. I hope you enjoy these papers and encourage you to visit OnePetro to find similar works that may deepen your interests and enhance your skills. Recommended additional reading at OnePetro: www.onepetro.org. SPE 203130 - Managing Sanding Risk in Sandstone Reservoir Through a New Constitutive Model by Surej Kumar Subbiah, Universiti Teknologi Malaysia and Schlumberger, et al. SPE 201672 - Estimating Reservoir and Fracture Properties From Stage-by-Stage Pressure-Decline Analysis in Horizontal Wells by HanYi Wang, The University of Texas at Austin, et al. SPE 202248 A Digital Core Laboratory Characterizing Oil/Water Transition Zones by Graeme Morrison, Woodside Energy

An improved Hoeffding’s inequality for sum of independent random variables

Using a multipoint approximation to the moment-generating function, this paper presents an improved version of Hoeffding’s inequality which can be evaluated easily. The optimal multipoint degree is selected by a change-point detection method. Some numerical comparisons are also demonstrated.

Large-Scale Truss Topology and Sizing Optimization by an Improved Genetic Algorithm with Multipoint Approximation

Truss size and topology optimization problems have recently been solved mainly by many different metaheuristic methods, and these methods usually require a large number of structural analyses due to their mechanism of population evolution. A branched multipoint approximation technique has been introduced to decrease the number of structural analyses by establishing approximate functions instead of the structural analyses in Genetic Algorithm (GA) when GA addresses continuous size variables and discrete topology variables. For large-scale trusses with a large number of design variables, an enormous change in topology variables in the GA causes a loss of approximation accuracy and then makes optimization convergence difficult. In this paper, a technique named the label–clip–splice method is proposed to improve the above hybrid method in regard to the above problem. It reduces the current search domain of GA gradually by clipping and splicing the labeled variables from chromosomes and optimizes the mixed-variables model efficiently with an approximation technique for large-scale trusses. Structural analysis of the proposed method is extremely reduced compared with these single metaheuristic methods. Numerical examples are presented to verify the efficacy and advantages of the proposed technique.

Quasi-Rational Analytic Approximation for the Modified Bessel Function I1(x) with High Accuracy

A new simple and accurate expression to approximate the modified Bessel function of the first kind I1(x) is presented in this work. This new approximation is obtained as an improvement of the multi-point quasi-rational approximation technique, MPQA. This method uses the power series of the Bessel function, its asymptotic expansion, and a process of optimization to fit the parameters of a fitting function. The fitting expression is formed by elementary functions combined with rational ones. In the present work, a sum of hyperbolic functions was selected as elementary functions to capture the first two terms of the asymptotic expansion of I1(x), which represents an important improvement with respect to previous research, where just the leading term of the asymptotic series was captured. The new approximation function presents a remarkable agreement with the analytical solution I1(x), decreasing the maximum relative error in more than one order of magnitude with respect to previous similar expressions. Concretely, the relative error was reduced from 10−2 to 4×10−4, opening the possibility of applying the new improved method to other Bessel functions. It is also remarkable that the new approximation is valid for all positive and negative values of the argument.

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.

Improved genetic algorithm with two-level multipoint approximation for complex frame structural optimization

In this paper, an improved structural topology and sizing optimization method is developed for the fast and efficient engineering design of complex frame structures where beam elements are mainly used in the structures. Discrete and continuous variables are included that the elimination or existence of beam elements are treated as discrete variables (0,1), and the continuous sizing variables of beam cross sections are considered to be continuous variables. To solve the mixed variable problem, the paper introduces a two-level multipoint approximation strategy (TMA). The first-level approximate problem is established by using the branched multipoint approximate function, which includes both two types of variables. Genetic algorithm (GA) is used to determine the absence or presence of beam members. The second-level approximate problem that only involving retained continuous size variables is made on this basis, which uses Taylor expansion and dual methods to solve the inner layer continuous optimization problem. Meanwhile, a strategy of adding a new complementary design point is adopted to expend the search scopes and improve the precision. Temporal deletion techniques are used to temporarily remove redundant constraints and local vibration modes processing techniques are used for continuum topology optimization under frequency constraints. Several representative examples are investigated to validate the effectiveness of the improved method.

A Combined Discrete-and-Continuous Multipoint Model of a Countercurrent Heat Exchanger

The article addresses matters concerned with developing the linear theory of constructing multipoint models intended for synthesizing automatic closed loop control systems of thermal power facilities. The multipoint approximation models are oriented at solving practical problems to an extent greater than the models with distributed parameters. The transfer functions of multipoint linear models can be obtained using the signal graph method; however, this method is not always suitable for being used without additional simplifications, in particular, the assumption about “independent heating.” In this study, for obtaining the dynamic characteristics of countercurrent heat exchangers involving longitudinal motion of coolants, it is proposed to use the discrete-continuous Laplace transform method. The method is developed taking the boiler economizer as an example. The article shows the way in which a shift can be made to the process circuit with longitudinal countercurrent motion by using the method of decomposing the economizer surface with cross flow of coolants into segments. Analytical expressions for the model’s transfer functions of an arbitrary order are obtained. Complex frequency responses for eight economizer channels are calculated. The characteristics of multipoint models and models with distributed parameters only for water and for both coolants are compared with each other. Recommendations for selecting the minimally admissible order depending on the input disturbance kind are given. The possibility of using a model with lumped parameters for the external coolant is revealed. It is shown that simplification of the external coolant model down to the first order has hardly any effect on the characteristics of only the internal coolant’s two temperature channels connected with its own input actions in terms of flowrate and temperature. However, such simplification cannot be applied for the remaining six channels.

Characterization of Hidden Chirality: Two-Fold Helicity in β-Strands

A β-strand is a component of a β-sheet and is an important structural motif in biomolecules. An α-helix has clear helicity, while chirality of a β-strand had been discussed on the basis of molecular twists generated by forming hydrogen bonds in parallel or non-parallel β-sheets. Herein we describe handedness determination of two-fold helicity in a zig-zag β-strand structure. Left- (M) and right-handedness (P) of the two-fold helicity was defined by application of two concepts: tilt-chirality and multi-point approximation. We call the two-fold helicity in a β-strand, whose handedness has been unrecognized and unclarified, as hidden chirality. Such hidden chirality enables us to clarify precise chiral characteristics of biopolymers. It is also noteworthy that characterization of chirality of high dimensional structures like a β-strand and α-helix, referred to as high dimensional chirality (HDC) in the present study, will contribute to elucidation of the possible origins of chirality and homochirality in nature because such HDC originates from not only asymmetric centers but also conformations in a polypeptide chain.

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.

Multipoint Approximation of Statistical Descriptors of Local Strain and Stress Fields in Heterogeneous Media Using Integral Equation Method

This paper is devoted to derivation of analytic expressions for statistical descriptors of stress and strain fields in heterogeneous media. Multipoint approximations of solutions of stochastic elastic boundary value problems for representative volume elements are investigated. The stress and strain fields are represented in the form of random coordinate functions, for which analytical expressions for the first- and second-order statistical central moments are obtained. Such moments characterize distribution of fields under prescribed loading of a representative volume element and depend on the geometry features and location of components within a volume. The information of the internal geometrical structure is taken into account by means of multipoint correlation functions. Within the framework of the second approximation of the boundary value problem, the correlation functions up to the fifth order are required to calculate the statistical characteristics. Using the method of Green’s functions, analytical expressions for the moments in distinct phases of the microstructure are obtained explicitly in a form of integral equations. Their analysis and comparison with previously obtained results are performed.

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.

Analysis of Deterministic Control and Its Improvements for an Inventory Problem with Multiproduct Batch Differentiation

We consider a multiperiod planning problem faced by a biopharmaceutical firm that must coordinate the production and allocation of batches of intermediate products to end products for multiple markets. This is a challenging problem to solve optimally, so we derive a theoretical bound on the performance of a deterministic control (DC) in which all random variables are replaced by their expected values and the corresponding deterministic optimization problem is solved. This is a variant of an approach that is widely used in practice. We show that while DC can perform poorly in some instances, simple reoptimizations of DC (i.e., certainty equivalent control (CEC)) improve the performance of DC. That said, the benefit of reoptimization decreases as the magnitude of demand variation increases. To address this, and also to provide guidance for heuristic design, we derive performance bounds for two additional heuristic controls—(1) open-loop feedback control (OLFC), which reoptimizes a sequence of stochastic programs with a complete demand distribution information, and (2) multipoint approximation control (MPAC), which solves a dynamic program using an approximate demand distribution that incorporates slightly more distributional information beyond simple expected values. We show that although OLFC outperforms CEC, its improvement can be limited. On the other hand, with carefully chosen approximate demand distributions, MPAC can significantly outperform OLFC. This suggests that, in our setting, fully capturing decision dynamics coupled with a small amount of additional distributional information beyond expected values is more useful than fully capturing the demand distribution but coupling that with insufficient modeling of decision dynamics. The online appendix is available at https://doi.org/10.1287/opre.2017.1647 .

Topology and Sizing Optimization for Frame Structures with a Two-Level Approximation Method

This paper discusses the integrated topology and sizing optimization of frame structures where beam elements are used in the structural analysis model. Based on the ground structure approach, discrete variables (0, 1) representing the absence or presence of beam members and continuous sizing variables for beam cross sections are involved in the problem. A two-level multipoint approximation strategy is introduced to address this mixed-variable problem. A first-level approximate problem is first constructed using the branched multipoint approximate function involving both discrete and continuous variables. For solving this problem, discrete variables are optimized through a genetic algorithm, and when calculating the fitness in the genetic algorithm, a second-level approximate problem solved by the dual method is established to optimize the cross sections for the retained beam members. Temporal deletion techniques are used for displacement and stress constraints. Meanwhile, by considering local vibration modes in frequency-constrained problems, material density is assigned with an extremely small value for the removed beams to avoid these local modes. Thence, the singularities encountered in stress, displacement, and frequency constraints are overcome. Typical examples and a satellite application are given to verify the feasibility and efficiency of this method in handling the integrated topology and sizing optimization for frame structures.

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 Topology Optimization using Integrated Multi-Point Approximation

Structural Topology Optimization using Integrated Multi-Point Approximation