scholarly journals Revisiting non-convexity in topology optimization of compliance minimization problems

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mohamed Abdelhamid ◽  
Aleksander Czekanski
Keyword(s):  
Topology Optimization ◽  
Optimization Problems ◽  
Global Optimum ◽  
Continuation Methods ◽  
Comprehensive Treatment ◽  
Mathematical Treatment ◽  
Content Type ◽  
Compliance Minimization ◽  
Intermediate Density ◽  
Filtering Techniques

PurposeThis is an attempt to better bridge the gap between the mathematical and the engineering/physical aspects of the topic. The authors trace the different sources of non-convexification in the context of topology optimization problems starting from domain discretization, passing through penalization for discreteness and effects of filtering methods, and end with a note on continuation methods.Design/methodology/approachStarting from the global optimum of the compliance minimization problem, the authors employ analytical tools to investigate how intermediate density penalization affects the convexity of the problem, the potential penalization-like effects of various filtering techniques, how continuation methods can be used to approach the global optimum and how the initial guess has some weight in determining the final optimum.FindingsThe non-convexification effects of the penalization of intermediate density elements simply overshadows any other type of non-convexification introduced into the problem, mainly due to its severity and locality. Continuation methods are strongly recommended to overcome the problem of local minima, albeit its step and convergence criteria are left to the user depending on the type of application.Originality/valueIn this article, the authors present a comprehensive treatment of the sources of non-convexity in density-based topology optimization problems, with a focus on linear elastic compliance minimization. The authors put special emphasis on the potential penalization-like effects of various filtering techniques through a detailed mathematical treatment.

A novel approach of reliability-based topology optimization for continuum structures under interval uncertainties

2019 ◽  
Vol 25 (9) ◽  
pp. 1455-1474 ◽  
Author(s):  
Lei Wang ◽  
Haijun Xia ◽  
Yaowen Yang ◽  
Yiru Cai ◽  
Zhiping Qiu
Keyword(s):  
Topology Optimization ◽  
Reliability Index ◽  
Large Scale ◽  
Optimization Problems ◽  
Uncertainty Propagation ◽  
Content Type ◽  
Continuum Structures ◽  
Novel Approach ◽  
Design Variables ◽  
Interval Uncertainties

Purpose The purpose of this paper is to propose a novel non-probabilistic reliability-based topology optimization (NRBTO) method for continuum structural design under interval uncertainties of load and material parameters based on the technology of 3D printing or additive manufacturing. Design/methodology/approach First, the uncertainty quantification analysis is accomplished by interval Taylor extension to determine boundary rules of concerned displacement responses. Based on the interval interference theory, a novel reliability index, named as the optimization feature distance, is then introduced to construct non-probabilistic reliability constraints. To circumvent convergence difficulties in solving large-scale variable optimization problems, the gradient-based method of moving asymptotes is also used, in which the sensitivity expressions of the present reliability measurements with respect to design variables are deduced by combination of the adjoint vector scheme and interval mathematics. Findings The main findings of this paper should lie in that new non-probabilistic reliability index, i.e. the optimization feature distance which is defined and further incorporated in continuum topology optimization issues. Besides, a novel concurrent design strategy under consideration of macro-micro integration is presented by using the developed RBTO methodology. Originality/value Uncertainty propagation analysis based on the interval Taylor extension method is conducted. Novel reliability index of the optimization feature distance is defined. Expressions of the adjoint vectors between interval bounds of displacement responses and the relative density are deduced. New NRBTO method subjected to continuum structures is developed and further solved by MMA algorithms.

scholarly journals Self-Directed Online Machine Learning for Topology Optimization

2021 ◽  
Author(s):  
Changyu Deng ◽  
Yizhou Wang ◽  
Can Qin ◽  
Wei Lu
Keyword(s):  
Topology Optimization ◽  
Optimization Problems ◽  
State Of The Art ◽  
Computational Cost ◽  
Region Of Interest ◽  
Global Optimum ◽  
Training Data ◽  
Computational Time ◽  
Minimization Problems ◽  
Design Variables

Abstract Topology optimization by optimally distributing materials in a given domain requires gradient-free optimizers to solve highly complicated problems. However, with hundreds of design variables or more involved, solving such problems would require millions of Finite Element Method (FEM) calculations whose computational cost is huge and impractical. Here we report a Self-directed Online Learning Optimization (SOLO) which integrates Deep Neural Network (DNN) with FEM calculations. A DNN learns and substitutes the objective as a function of design variables. A small number of training data is generated dynamically based on the DNN's prediction of the global optimum. The DNN adapts to the new training data and gives better prediction in the region of interest until convergence. Our algorithm was tested by compliance minimization problems and fluid-structure optimization problems. It reduced the computational time by 2 ~ 5 orders of magnitude compared with directly using heuristic methods, and outperformed all state-of-the-art algorithms tested in our experiments. This approach enables solving large multi-dimensional optimization problems.

Diploidy in evolutionary algorithms for dynamic optimization problems

2015 ◽  
Vol 8 (4) ◽  
pp. 312-329 ◽  
Author(s):  
Boris Shabash ◽  
Kay C. Wiese
Keyword(s):  
Evolutionary Algorithms ◽  
Dynamic Optimization ◽  
Genetic Information ◽  
Optimization Problems ◽  
Fitness Function ◽  
Search Space ◽  
Global Optimum ◽  
Dynamic Optimization Problems ◽  
Content Type ◽  
Optimization Function

Purpose – In this work, the authors show the performance of the proposed diploid scheme (a representation where each individual contains two genotypes) with respect to two dynamic optimization problems, while addressing drawbacks the authors have identified in previous works which compare diploid evolutionary algorithms (EAs) to standard EAs. The paper aims to discuss this issue. Design/methodology/approach – In the proposed diploid representation of EA, each individual possesses two copies of the genotype. In order to convert this pair of genotypes to a single phenotype, each genotype is individually evaluated in relation to the fitness function and the best genotype is presented as the phenotype. In order to provide a fair and objective comparison, the authors make sure to compare populations which contain the same amount of genetic information, where the only difference is the arrangement and interpretation of the information. The two representations are compared using two shifting fitness functions which change at regular intervals to displace the global optimum to a new position. Findings – For small fitness landscapes the haploid (standard) and diploid algorithms perform comparably and are able to find the global optimum very quickly. However, as the search space increases, rediscovering the global optimum becomes more difficult and the diploid algorithm outperforms the haploid algorithm with respect to how fast it relocates the new optimum. Since both algorithms use the same amount of genetic information, it is only fair to conclude it is the unique arrangement of the diploid algorithm that allows it to explore the search space better. Originality/value – The diploid representation presented here is novel in that instead of adopting a dominance scheme for each allele (value) in the vector of values that is the genotype, dominance is adopted across the entire genotype in relation to its homologue. As a result, this representation can be extended across any alphabet, for any optimization function.

h-adaptive topology optimization considering variations of material properties and energy error density recovery

2020 ◽  
Vol 37 (9) ◽  
pp. 3209-3241
Author(s):  
Jéderson da Silva ◽  
Jucélio Tomás Pereira ◽  
Diego Amadeu F. Torres
Keyword(s):  
Topology Optimization ◽  
Material Properties ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Computational Cost ◽  
Error Estimator ◽  
Content Type ◽  
Energy Error ◽  
Error Density ◽  
Discretization Errors

Purpose The purpose of this paper is to propose a new scheme for obtaining acceptable solutions for problems of continuum topology optimization of structures, regarding the distribution and limitation of discretization errors by considering h-adaptivity. Design/methodology/approach The new scheme encompasses, simultaneously, the solution of the optimization problem considering a solid isotropic microstructure with penalization (SIMP) and the application of the h-adaptive finite element method. An analysis of discretization errors is carried out using an a posteriori error estimator based on both the recovery and the abrupt variation of material properties. The estimate of new element sizes is computed by a new h-adaptive technique named “Isotropic Error Density Recovery”, which is based on the construction of the strain energy error density function together with the analytical solution of an optimization problem at the element level. Findings Two-dimensional numerical examples, regarding minimization of the structure compliance and constraint over the material volume, demonstrate the capacity of the methodology in controlling and equidistributing discretization errors, as well as obtaining a great definition of the void–material interface, thanks to the h-adaptivity, when compared with results obtained by other methods based on microstructure. Originality/value This paper presents a new technique to design a mesh made with isotropic triangular finite elements. Furthermore, this technique is applied to continuum topology optimization problems using a new iterative scheme to obtain solutions with controlled discretization errors, measured in terms of the energy norm, and a great resolution of the material boundary. Regarding the computational cost in terms of degrees of freedom, the present scheme provides approximations with considerable less error if compared to the optimization process on fixed meshes.

Improved proportional topology optimization algorithm for minimum volume problem with stress constraints

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Wenming Cheng ◽  
Hui Wang ◽  
Min Zhang ◽  
Run Du
Keyword(s):  
Topology Optimization ◽  
Optimization Problem ◽  
Optimization Problems ◽  
Weighting Function ◽  
Target Material ◽  
Stress Constraints ◽  
Convergence Speed ◽  
Minimum Volume ◽  
Content Type ◽  
Density Filter

Purpose The purpose of this paper is to propose an improved proportional topology optimization (IPTO) algorithm for tackling the stress-constrained minimum volume optimization problem, which can meet the requirements that are to get rid of the problems of numerical derivation and sensitivity calculation involved in the process of obtaining sensitivity information and overcome the drawbacks of the original proportional topology optimization (PTO) algorithm. Design/methodology/approach The IPTO algorithm is designed by using the new target material volume update scheme and the new density variable update scheme and by introducing the improved density filter (considering the weighting function based on the Gaussian distribution) and Heaviside-type projection operator on the basis of the PTO algorithm. The effectiveness of the IPTO algorithm is demonstrated by solving the stress-constrained minimum volume optimization problems for two numerical examples and being compared with the PTO algorithm. Findings The results of this paper show that the uses of the proposed strategies contribute to improving the optimized results and the performance (such as the ability to obtain accurate solutions, robustness and convergence speed) of the IPTO algorithm. Compared with the PTO algorithm, the IPTO algorithm has the advantages of fast convergence speed, enhancing the ability to obtain accurate solutions and improving the optimized results. Originality/value This paper achieved the author’s intended purpose and provided a new idea for solving the stress-constrained optimization problem under the premise of avoiding obtaining sensitivity information.

scholarly journals Global optimization based on active preference learning with radial basis functions

2020 ◽  
Author(s):  
Alberto Bemporad ◽  
Dario Piga
Keyword(s):  
Global Optimization ◽  
Objective Function ◽  
Decision Maker ◽  
Optimization Problems ◽  
Global Optimum ◽  
Bayesian Optimization ◽  
Preference Learning ◽  
Global Optimizer ◽  
Radial Basis ◽  
Decision Vector

AbstractThis paper proposes a method for solving optimization problems in which the decision-maker cannot evaluate the objective function, but rather can only express a preference such as “this is better than that” between two candidate decision vectors. The algorithm described in this paper aims at reaching the global optimizer by iteratively proposing the decision maker a new comparison to make, based on actively learning a surrogate of the latent (unknown and perhaps unquantifiable) objective function from past sampled decision vectors and pairwise preferences. A radial-basis function surrogate is fit via linear or quadratic programming, satisfying if possible the preferences expressed by the decision maker on existing samples. The surrogate is used to propose a new sample of the decision vector for comparison with the current best candidate based on two possible criteria: minimize a combination of the surrogate and an inverse weighting distance function to balance between exploitation of the surrogate and exploration of the decision space, or maximize a function related to the probability that the new candidate will be preferred. Compared to active preference learning based on Bayesian optimization, we show that our approach is competitive in that, within the same number of comparisons, it usually approaches the global optimum more closely and is computationally lighter. Applications of the proposed algorithm to solve a set of benchmark global optimization problems, for multi-objective optimization, and for optimal tuning of a cost-sensitive neural network classifier for object recognition from images are described in the paper. MATLAB and a Python implementations of the algorithms described in the paper are available at http://cse.lab.imtlucca.it/~bemporad/glis.

Numerical Experiments in Using Voronoi Cell Finite Elements for Topology Optimization

2009 ◽  
Author(s):  
James M. Gibert ◽  
Georges M. Fadel
Keyword(s):  
Topology Optimization ◽  
Isotropic Material ◽  
Numerical Experiments ◽  
Voronoi Cell ◽  
Material Model ◽  
Topological Optimization ◽  
Advantages And Disadvantages ◽  
Compliance Minimization ◽  
Design Variables ◽  
Standard Linear

This paper provides two separate methodologies for implementing the Voronoi Cell Finite Element Method (VCFEM) in topological optimization. Both exploit two characteristics of VCFEM. The first approach utilizes the property that a hole or inclusion can be placed in the element: the design variables for the topology optimization are sizes of the hole. In the second approach, we note that VCFEM may mesh the design domain as n sided polygons. We restrict our attention to hexagonal meshes of the domain while applying Solid Isotropic Material Penalization (SIMP) material model. Researchers have shown that hexagonal meshes are not subject to the checker boarding problem commonly associated with standard linear quad and triangle elements. We present several examples to illustrate the efficacy of the methods in compliance minimization as well as discuss the advantages and disadvantages of each method.

Optimization design of titanium alloy connecting rod based on structural topology optimization

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Bin Zheng ◽  
Yi Cai ◽  
Kelun Tang
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Topology Optimization ◽  
Optimization Design ◽  
Equivalent Stress ◽  
Connecting Rod ◽  
Element Analysis ◽  
Content Type ◽  
The Finite Element Analysis ◽  
Maximum Equivalent Stress

Purpose The purpose of this paper is to realize the lightweight of connecting rod and meet the requirements of low energy consumption and vibration. Based on the structural design of the original connecting rod, the finite element analysis was conducted to reduce the weight and increase the natural frequencies, so as to reduce materials consumption and improve the energy efficiency of internal combustion engine. Design/methodology/approach The finite element analysis, structural optimization design and topology optimization of the connecting rod are applied. Efficient hybrid method is deployed: static and modal analysis; and structure re-design of the connecting rod based on topology optimization. Findings After the optimization of the connecting rod, the weight is reduced from 1.7907 to 1.4875 kg, with a reduction of 16.93%. The maximum equivalent stress of the optimized connecting rod is 183.97 MPa and that of the original structure is 217.18 MPa, with the reduction of 15.62%. The first, second and third natural frequencies of the optimized connecting rod are increased by 8.89%, 8.85% and 11.09%, respectively. Through the finite element analysis and based on the lightweight, the maximum equivalent stress is reduced and the low-order natural frequency is increased. Originality/value This paper presents an optimization method on the connecting rod structure. Based on the statics and modal analysis of the connecting rod and combined with the topology optimization, the size of the connecting rod is improved, and the static and dynamic characteristics of the optimized connecting rod are improved.

Solving Mono- and Multi-Objective Problems Using Hybrid Evolutionary Algorithms and Nelder-Mead Method

2021 ◽  
Vol 12 (4) ◽  
pp. 98-116
Author(s):  
Noureddine Boukhari ◽  
Fatima Debbat ◽  
Nicolas Monmarché ◽  
Mohamed Slimane
Keyword(s):  
Convergence Rate ◽  
Optimization Problem ◽  
Optimization Problems ◽  
State Of The Art ◽  
Hybrid Approach ◽  
Optimization Methods ◽  
Global Optimum ◽  
Stochastic Methods ◽  
Local Optima ◽  
Portfolio Optimization Problem

Evolution strategies (ES) are a family of strong stochastic methods for global optimization and have proved their capability in avoiding local optima more than other optimization methods. Many researchers have investigated different versions of the original evolution strategy with good results in a variety of optimization problems. However, the convergence rate of the algorithm to the global optimum stays asymptotic. In order to accelerate the convergence rate, a hybrid approach is proposed using the nonlinear simplex method (Nelder-Mead) and an adaptive scheme to control the local search application, and the authors demonstrate that such combination yields significantly better convergence. The new proposed method has been tested on 15 complex benchmark functions and applied to the bi-objective portfolio optimization problem and compared with other state-of-the-art techniques. Experimental results show that the performance is improved by this hybridization in terms of solution eminence and strong convergence.

PSO and GA tuned conventional and fractional order PID controllers for quadrotor control

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Hasan Saribas ◽  
Sinem Kahvecioglu
Keyword(s):  
Fractional Order ◽  
Pid Controller ◽  
Controller Design ◽  
Optimization Problems ◽  
Parameter Tuning ◽  
Absolute Error ◽  
Cost Functions ◽  
Pid Controllers ◽  
Content Type ◽  
Quadrotor Control

Purpose This study aims to compare the performance of the conventional and fractional order proportional-integral-derivative (PID and FOPID) controllers tuned with a particle swarm optimization (PSO) and genetic algorithm (GA) for quadrotor control. Design/methodology/approach In this study, the gains of the controllers were tuned using PSO and GA, which are included in the heuristic optimization methods. The tuning processes of the controller’s gains were formulated as optimization problems. While generating the objective functions (cost functions), four different decision criteria were considered separately: integrated summation error (ISE), integrated absolute error, integrated time absolute error and integrated time summation error (ITSE). Findings According to the simulation results and comparison tables that were created, FOPID controllers tuned with PSO performed better performances than PID controllers. In addition, the ITSE criterion returned better results in control of all axes except for altitude control when compared to the other cost functions. In the control of altitude with the PID controller, the ISE criterion showed better performance. Originality/value While a conventional PID controller has three parameters (Kp, Ki, Kd) that need to be tuned, FOPID controllers have two additional parameters (µ). The inclusion of these two extra parameters means more flexibility in the controller design but much more complexity for parameter tuning. This study reveals the potential and effectiveness of PSO and GA in tuning the controller despite the increased number of parameters and complexity.

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