Discrete Variable Optimization Research Articles

Discrete variable topology optimization for maximizing single/multiple natural frequencies and frequency gaps considering the topological constraint

AbstractFinding optimized structural topology design for maximizing natural frequencies and frequency gaps of continuum structures is crucial for engineering applications. However, two significant numerical issues must be addressed: non‐smoothness caused by multiple frequencies and Artificial Localized Rigid Motion (ALRM) modes due to the violation of the topological constraint related to isolated islands and point‐connections. The above two issues are solved by employing the discrete variable topology optimization method based on Sequential Approximate Integer Programming (SAIP). First, the directional differentiability and multiple frequencies preservation constraints are formulated as the linear integer constraints. And then, the integer programming with these linear integer constraints is established and solved by the discrete variable linear or quadratic programming solver with multi‐constraint Canonical Relaxation Algorithm (CRA). We also prove that the popular Average Modal Frequencies (AMF) strategy, like the Kreisselmeier‐Steinhauser (KS) function, cannot rigorously tackle this non‐smoothness caused by multiple frequencies. Furthermore, to eliminate the ALRM modes and concentrate on the real structural global modes, the burning method is employed to impose the topological constraint of the first Betti number that represents the number of isolated islands and point‐connections. Numerical examples, including 2D and 3D, two‐fold and three‐fold multiple frequencies, natural frequencies and frequency gaps, are presented to show the effectiveness of the proposed method.

Optimized topology design of finned ducts for internal flow thermal performance by discrete variable eigenvalue-related optimization

Optimized topology design of finned ducts for internal flow thermal performance by discrete variable eigenvalue-related optimization

Multi-material structural discrete variable topology optimization with minimum length scale control under mass constraint

Topology optimization of multi-material structure has a larger design space compared with single material optimization, and it also requires efficient material selection methods to provide references for designers. The multi-material additive manufacturing ensures the manufacturability of topology design of multi-material structures, but also puts forward new manufacturability requirements, such as the minimum size of multi-material and the connection interface. This paper focuses on the development of a topology optimization algorithm for multi-material structures under the total mass constraint based on discrete variables. Recursive multiphase materials interpolation (RMMI) scheme is utilized to establish the relationship between discrete design variables and element elastic modulus and density, and sequential approximate integer programming (SAIP) and canonical relaxation algorithms are constructed to update the discrete design variables. It is shown that this method can obtain multi-material topology design with clear interface display, and has the ability of free material coexistence and degradation, which indicates that the discrete variables have more advantages than the continuous variables in the RMMI scheme. Based on this, the material selection under mass constraints is discussed in depth, and it is pointed out that the traditional determination of material coexistence or degradation in design based on specific stiffness has limitations. Thus, the present work expands the necessary conditions of material coexistence. Then, a minimum size control method of multi-material design based on post-processing is proposed to solve the minimum size geometry constraint problem by SAIP. Numerical examples show that this method can not only precisely control the minimum size of each material phase (including void phase), but also resolve the problem of interface and boundary diffusion of the multi-material phase, and significantly improve the manufacturability of multi-material design. Finally, it is applied to optimization problems with complex design domain and irregular meshes which are closer to engineering design.

Efficient two-phase approach to reliability-based discrete variable topology optimization of continuum structures with multimodal distributions

In practical applications, some random variables follow multimodal distributions. However, conventional reliability analysis methods in reliability-based topology optimization (RBTO), such as the first order reliability method, often result in considerable computational errors when dealing with problems involving multimodal distributions. Consequently, the corresponding RBTO design is incredible. Moreover, the RBTO problem also faces the challenge of high computational cost. To this end, this paper proposes an efficient two-phase approach for RBTO of continuum structures with multimodal distributions combining sequential approximate integer programming with trust region (SAIP-TR) and direct probability integral method (DPIM). Firstly, DPIM is advanced to address the difficult problems of failure probability estimation and efficient sensitivity analysis under multimodal distributions. Secondly, a reliability-based discrete variable topology optimization framework based on SAIP-TR and DPIM is established, which yields clear topology configurations and facilitates engineering manufacturing. Owing to the merit of SAIP-TR, the original RBTO process is divided into two phases: the first phase performs deterministic topology optimization, and the second phase focuses on RBTO. Moreover, an adaptive selection strategy of representative points, considering structural compliance as performance function, is devised to further enhance computational efficiency. Finally, several examples illustrate high efficiency and accuracy of the proposed approach. The multimodal random variable and the random field are employed separately to describe global and local uncertainties in materials. In contrast, the optimized result considering global material uncertainty is more suitable for additive manufacturing. The proposed approach also presents potential in handling complex RBTO problems with random fields.

A data-driven hybrid interval reactive power optimization based on the security limits method and improved particle swarm optimization

The integration of renewable power generation introduces randomness and uncertainties in power systems, and the reactive power optimization with interval uncertainty (RPOIU) problem has been constructed to acquire the voltage control strategy. However, the large amount of uncertain data and the coexistence of discrete and continuous control variables increase the difficulty of solving the RPOIU problem. This paper proposes a data-driven hybrid interval reactive power optimization based on the security limits method (SLM) and the improved particle swarm optimization (IPSO) to solve the RPOIU problem. In this method, the large amount of historical uncertain data is processed by data-driven to obtain the boundary of optimal uncertainty set. The control variable optimization is decomposed into continuous variable optimization and discrete variable optimization. The continuous variables are optimized by applying the SLM with the discrete variables fixed, and the discrete variables are optimized by the IPSO with the continuous variables fixed. The two processes are applied alternately, and the values of the control variables obtained by each method are used as the fixed variables of the other method. Based on simulations carried out for the IEEE 30-bus system with three optimization methods, we verified that the voltage control strategy obtained by the data-driven hybrid optimization could ensure that the state variable intervals satisfied the constraints. Meanwhile, the values of the real power losses obtained by the proposed method were smaller than those obtained by the SLM and IPSO. The simulation results demonstrated the effectiveness and value of the proposed method.

Multi-Patch Black-White Topology Optimization in Isogeometric Analysis

Topological optimization plays a guiding role in the conceptual design process. This paper conducts research on structural topology optimization algorithm within the framework of isogeometric analysis. For multi-component structures, the Nitsche’s method is used to glue different meshes to perform isogeometric multi-patch analysis. The discrete variable topology optimization algorithm based on integer programming is adopted in order to obtain clear boundaries for topology optimization. The sensitivity filtering method based on the Helmholtz equation is employed for averaging of curved elements' sensitivities. In addition, a simple averaging method along coupling interfaces is proposed in order to ensure the material distribution across coupling areas is reasonably smooth. Finally, the performance of the algorithm is demonstrated by numerical examples, and the effectiveness of the algorithm is verified by comparing it with the results obtained by single-patch and ABAQUS cases.

Inequality constraint on the maximum genus for 3D structural compliance topology optimization

Structural topology constraints in topology optimization are an important research topic. The structural topology is characterized by the topological invariance of the number of holes. The holes of a structure in 3D space can be classified as internally enclosed holes and external through-holes (or tunnels). The genus is the number of tunnels. This article proposes the quotient set design variable method (QSDV) to implement the inequality constraint on the maximum genus allowed in an optimized structure for 3D structural topology optimization. The principle of the QSDV is to classify the changing design variables according to the connectivity of the elements in a structure to obtain the quotient set and update the corresponding elements in the quotient set to meet the topological constraint. Based on the standard relaxation algorithm discrete variable topology optimization method (DVTOCRA), the effectiveness of the QSDV is illustrated in numerical examples of a 3D cantilever beam.

Flexible Interconnected Distribution Network with Embedded DC System and Its Dynamic Reconfiguration

Flexible interconnection transformation of a distribution network using all-control power electronics technology will help to overcome the technical bottleneck of distributed generation (DG) penetration and high-quality power supply in the traditional distribution network. However, the flexible interconnection form that uses back-to-back power electronic converters to replace tie switches has disadvantages such as high cost and large footprint, so its mass promotion and application are bound to be limited. Therefore, a new topology of flexible interconnected distribution network with embedded DC system (FDN+EDC) with a better economy is proposed. Firstly, the topology of FDN+EDC is introduced to improve the controllability level of the distribution network under high percentage DG penetration, and is compared with the conventional flexible interconnection method. Secondly, in order to realize the coordinated optimal economic operation of source, network and load in the FDN+EDC, a dynamic reconfiguration model of switches status and continuous adjustment of flexible interconnection device (FID) are proposed. Then, to achieve a fast and globally optimal solution of the model, a hierarchical coordinated optimization strategy is designed based on an improved particle swarm algorithm for the optimization of discrete and continuous variables. Finally, the validity of the algorithm is verified, and an 88-node FDN+EDC is constructed to verify the advantages of the proposed embedded DC flexible interconnection topology and realize the joint optimal dispatch of network-source-load.

Sensitivity analysis of discrete variable topology optimization

Sensitivity analysis of discrete variable topology optimization

Optimal Placement of Vibration Sensors for Industrial Robots Based on Bayesian Theory

This paper presents an optimal sensor placement method for vibration signal acquisition in the field of industrial robot health monitoring and fault diagnosis. Based on the general formula of Bayes and relative entropy, the evaluation function of sensor placement is deduced, and the modal confidence matrix is used to express the redundancy of sensor placement. The optimal placement of vibration sensors is described as a discrete variable optimization problem, which is defined as whether the existing sensor layout can obtain joint state information efficiently. The initial layout of the sensor was obtained from the structural simulation results of the industrial robots, and the initial layout was optimized by the derived objective function. The efficiency of the optimized layout in capturing joint state information is proven by the validation experiment with a simulation model. The problem of popularizing the optimization method in engineering is solved by a verification experiment without a simulation model. The optimal sensor placement method provides a theoretical basis for industrial robots to acquire vibration data effectively.

A Discrete-Steepest Descent Framework for the Simultaneous Process and Control Design of Multigrade Reactive Distillation Columns

The simultaneous optimization of continuous and discrete design variables, operating conditions, and controller's tuning parameters of reactive distillation (RD) columns is investigated in this work. For this purpose, the capabilities of a recently proposed modular economic optimization strategy based on a Discrete-Steepest Descent (D-SDA) framework are investigated. The D-SDA is a decomposition method that aims to improve an initial design by systematically modifying its discrete decisions, e.g., number of stages, until a design that optimizes the process economics while meeting the desired specifications is found. A case study involving the production of ethyl tert-butyl-ether (ETBE) in a RD unit was considered. The simultaneous design and control of the RD column was solved under two scenarios, i.e., product changeovers between four different grades and the production of a single grade of ETBE under a step disturbance in the feed composition. The results show that the modular strategy can specify economic design and control schemes in reasonable computational times.

Explicit control of 2D and 3D structural complexity by discrete variable topology optimization method

The structural complexity (the number of holes) of the 2D or 3D continuum structures can be measured by their topology invariants (i.e., Euler and Betti numbers). Controlling the 2D and 3D structural complexity is significant in topology optimization design because of the various consideration, including manufacturability and necessary structural redundancy, but remains a challenging subject. In this paper, we propose a programmable Euler–Poincaré formula to efficiently calculate the Euler and Betti numbers for the 0–1 pixel-based structures. This programmable Euler–Poincaré​ formula only relates to the nodal density and nodal characteristic vector that represents the nodal neighbor relation so that it avoids manually counting the information of the vertices, edges, and planes on the surfaces of the structure. As a result, the explicit formulations between the structural complexity (the number of holes) and the discrete density design variables for 2D and 3D continuum structures can be efficiently constructed. Furthermore, the discrete variable sensitivity of the structural complexity is calculated through the programmable Euler–Poincaré formula so that the structural complexity control problem can be efficiently and mathematically solved by Sequential Approximate Integer Programming and Canonical relaxation algorithm Various 2D and complicated 3D numerical examples are presented to demonstrate the effectiveness of the method. We further believe that this study bridges the gap between structural topology optimization and mathematical topology analysis, which is much expected in the structural optimization community.

Correction to: Discrete variable topology optimization for compliant mechanism design via Sequential Approximate Integer Programming with Trust Region (SAIP-TR)

Correction to: Discrete variable topology optimization for compliant mechanism design via Sequential Approximate Integer Programming with Trust Region (SAIP-TR)

Design of curved composite panels for maximum buckling load using sequential permutation search algorithm

New buckling optimal solutions for curved composite panels are presented in terms of principle of virtual work and sequential permutation search (SPS) algorithm. By utilizing the principle of virtual displacements, the buckling load is calculated and the accuracy of the derived buckling solutions is verified by comparison with experimental data and numerical results. SPS utilizes the inherent mechanical properties of composite laminates in which the bending stiffness of the outer layers is more sensitive than those close to the mid-surface. Curved composite panels exhibit a variety of length-to-arc ratios and radius-to-length ratios, under axial compression, pure shear, and combined loadings, are optimized. Some of the results are compared with the global optima obtained by enumeration method, and SPS shows good robustness in locating the global optima. In addition, the buckling load carrying capacity of optimized curved composite panels can be improved by as much as twice over cross-ply laminates. Low computational costs of SPS algorithm imply its potential for solving high-dimensional discrete variable optimization problems of composite structures.

Peloton Dynamics Optimization: Algorithm for Discrete Structural Optimization

A discrete variable optimization algorithm is presented based on peloton dynamics that occur during bicycle racing. Peloton dynamics are mainly attributable to the physical capacity of cyclists, energy saving by the coupling effects of drafting, and the capacity for cyclists to pass others. It also includes cooperating with other cyclists by changing positions inside the peloton, competitors’ positions, and their relative energy levels. The algorithm’s performance is tested on nine discrete benchmark truss structures subjected to multiple loading conditions. The performance of the peloton dynamics optimization (PDO) algorithm is compared with other optimization algorithms based on the success rate (ability to find the best solution) and computational effort (number of structural analyses required). Other metaheuristic methods generally consist of different components, and these components have several parameters that need to be set manually. As a result, it is not uncommon that these parameters need to be tuned to just the right values to obtain better results and are often specific to the problem to be solved. For the present algorithm, only two parameters are needed: the number of cyclists (solutions) and the maximum number of iterations. Results demonstrate that the performance of the PDO algorithm in terms of finding near optimum solutions and convergence behavior is comparable or better than various alternative optimization methods but with less user-specific parameter settings.

Multi-strategy co-evolutionary differential evolution for mixed-variable optimization

In real world, engineering design problems have been regarded as common and complex problems which always include non-linear standard functions, many complex constraints and decision variables containing both continuous and discrete variables. Problems with these features also can be named as mixed-variable optimization problems (MVOPs). However, because the decision variables of MVOPs present different spatial distribution features, how to design an effective algorithm to solve MVOPs is still a challenge. In this paper, a new variation of differential evolution (DE), named multi-strategy co-evolutionary differential evolution (MCDEmv) is proposed to solve MVOPs. A mixed-variable co-evolutionary scheme simultaneously considering continuous and discrete variables in MVOPs is utilized. Based on this scheme, a multi-strategy co-evolutionary approach considering a dynamic adaptive selection mechanism and the combination of different characteristic mutation strategies and crossover operators is proposed for adapting to the all-inclusive situations in MVOPs. Furthermore, a statistics-based local search (SBA) specially designed for the optimization of discrete variables in MVOPs is proposed for enhancing the efficiency and flexibility of the MCDEmv. 28 artificial benchmark functions are adopted to verify the effectiveness and efficiency of the proposed algorithm. The experimental results show that our proposed algorithm is more competitive and efficient than other compared similar algorithms, i.e., AEDA, ACOMV, DEMV and PSOmv. Comparing with other similar algorithms in coping with two mixed-variable engineering design problems, MCDEmv also shows higher efficiency and performance.

Discrete variable topology optimization for simplified convective heat transfer via sequential approximate integer programming with trust‐region

AbstractThis article presents a discrete variable topology optimization method to solve the simplified convective heat transfer (SCHT) design optimization modeled by Newton's law of cooling. The discrete variable topology optimization is based on the proposed sequential approximate integer programming with trust‐region. Due to the discrete variables, identifying the convective boundary, and implementing this design‐dependent convective boundary condition can be precisely undertaken. As a result, the consistent precise temperature field compared with the commercial software is captured. Besides, the interpolation scheme of the convective coefficient is unnecessary to analyze this SCHT problem. Furthermore, an analytical sensitivity formulation that can simultaneously incorporate the conductive and convective effect is also deduced. Finally, several 2D and 3D valid thermal designs are presented to illustrate the effectiveness of the method. Based on the optimized designs, we find that favorable configurations for a simplified convective problem may be hollowed structure or the dense needle‐like structure. Further, the checkerboard pattern should be interpreted as a convection oscillatory feature but not the discretization error because it cannot be eliminated by using higher‐order elements.

Research on Pattern Synthesis of Time Modulated Sparse Array Based on Discrete Variable Convex Optimization

Effective resource utilization is an important problem in the application of array, especially for the new time modulated array. Considering the problem of full utilization of array elements in time modulated array, a sparse optimization algorithm based on discrete variable convex optimization is proposed in this paper. The pattern optimization of equal excitation time modulation array is realized in two stages: In the first stage, the number of working array elements is as low as possible under the condition of suppressing the sidelobe of the central frequency. In the second stage, the sideband is suppressed by iterative convex optimization. The numerical simulation results are compared with other methods to verify the effectiveness of the proposed method in pattern optimization of equal excitation time modulation array. Finally, the optimization performance of the algorithm with different array parameters is verified.

Discrete variable optimization of structures subjected to dynamic loads using equivalent static loads and metaheuristic algorithms

This paper presents a new computational procedure for optimization of structures subjected to dynamic loads. The optimization problem is formulated with discrete design variables that represent the members from a table of commercially available members. Also, the requirements in the American Institute of Steel Construction (AISC) manual are formulated as constraints. This results in a nondifferentiable optimization problem. In the new procedure, the dynamic load is transformed into equivalent static loads (ESLs). Then the static response optimization problem having discrete design variables is solved using a metaheuristic optimization algorithm. Three methods to calculate the ESLs are investigated. It is found that the ESL cycles cannot converge to the final design. Therefore after a few ESL cycles, the original dynamic loads need to be used in the optimization process. Four example problems are solved to analyze the procedure. Based on this analysis, it is concluded that the new procedure is more efficient compared to a procedure that does not use the ESL cycles because it reduces the total CPU effort to obtain the final design. Also, better final designs are found. The reason is that many more designs are analyzed very efficiently with the ESL procedure.

Mixed-integer second-order cone optimization for composite discrete ply-angle and thickness topology optimization problems

Discrete variable topology optimization problems are usually solved by using solid isotropic material with penalization (SIMP), genetic algorithms (GA), or mixed-integer nonlinear optimization (MINLO). In this paper, we propose formulating discrete ply-angle and thickness topology optimization problems as a mixed-integer second-order cone optimization (MISOCO) problem. Unlike SIMP and GA methods, MISOCO efficiently finds the problem’s globally optimal solution. Furthermore, in contrast with existing MISOCO formulations of discrete ply-angle optimization problems, our reformulations allow the structure to change topology, consider the more realistic Tsai–Wu stress yield criteria constraint, and eliminate checkerboard patterns using simple linear constraints. We address two types of discrete ply-angle and thickness problems: a structural mass minimization problem and a compliance optimization problem where the objective is to maximize the structural stiffness. For each element, one first chooses if the element is present or not in the structure. One can then choose the element’s ply-angle and thickness from a finite set of possibilities for the former case. The discrete design space for ply-angle and thickness is a result of manufacturing limitations. To improve the problem’s MISOCO solution approach, we develop valid inequality constraints to tighten the continuous relaxation of the MISOCO reformulation. We compare the performance of various MISOCO solvers: Gurobi, CPLEX, and MOSEK to solve the MISOCO reformulation. We also use BARON to solve the original MINLO formulations of the problems. Our results show that solving the MISOCO problem’s formulation using MOSEK is the most efficient solution approach.