Large-scale Structural Optimization Research Articles

GPU-based parallel programming for FEM analysis in the optimization of steel frames

ABSTRACT Optimization of large-scale frame structures consumes a vast amount of time since the analysis of such complex systems contains several iterative processes. Mitigating computational burden and reducing this time to a reasonable level is possible by running GPU (Graphical Processing Unit) processors, which can be found on standard computers. This study presents an algorithm for the acceleration of size optimization of steel frames by using the BBO (Biogeography-Based Optimization) method that is suitable for GPU architecture. The GPU-based parallel algorithm, designed for FEM (Finite Element Method) analysis, is applied to three hypothetical steel-frame case structures with different numbers of members and nodes; and processed on four different computers which are available on the market. The presented case studies revealed that the proposed solution’s efficiency increases as the number of members increases and confirmed the ability of the acceleration algorithm for optimization of large-scale frame structures and provided time efficiency.

Large-scale sandwich structures optimization using Bayesian method

Benefiting from advanced features like high stiffness-to-weight ratios, sandwich structures are widely used in aerospace for primary and secondary structures. As tasks grow more complex and structures increase in scale, high-dimensional design spaces inevitably arise. Optimizing large-scale sandwich structures efficiently and intelligently presents certain challenges. Additionally, functional requirements and constraints, such as thermal deformation, should be fully considered in the design of practical structures like solar arrays, which involve a large expensive analysis and make the problem more complicated. This paper proposes to use efficient Bayesian optimization with the active subspace (AC) method to address this type of problem. The active subspace method, combined with the adaptive kriging and global sensitivity analysis (GSA), is employed for dimension reduction and reconstructing the design space. Then the structure is optimized within the reconstructed design space using the efficient Bayesian method. The proposed optimization strategy is applied to a case study of solar array with sandwich panels, demonstrating that the developed framework is feasible and effective for structural optimization of large-scale sandwich structures.

Effective thermal conductivity estimation using a convolutional neural network and its application in topology optimization

Topology optimization of heterogeneous structures can find significant use in a wide range of applications, and its fabrication has been made possible by recent advances in additive manufacturing. However, the optimization procedure is computationally expensive, as each structural update requires the re-evaluation of the properties. The computational time is the major limiting factor in large-scale and complex structural optimization. In this study, a convolutional neural network (CNN) model for predicting effective thermal conductivity inspired by the VGG networks is proposed. Trained using 130,000 unique binary images, the model achieves high predictive accuracy. Specifically, it shows a mean absolute percent error (MAPE) of 0.35% in testing when the thermal conductivity of the solid is ten times larger than the fluid, and when the thermal conductivities assigned are that of aluminum and water, the MAPE is 2.35%. The prediction time is 15 ms for a single image with 128 × 128 pixels, which is 3 to 5 orders of magnitude faster than a finite volume simulation. When employed in topology optimization, the CNN retains a MAPE between 0.67% and 11.8% for different cases. The CNN model correctly predicts trends in effective thermal conductivity and improves the structure to close proximity of a theoretical maximum in all cases.

A Problem-Independent Machine Learning (PIML) enhanced substructure-based approach for large-scale structural analysis and topology optimization of linear elastic structures

Structural analysis for structures composed of highly heterogeneous materials which often involves the solution of large-scale linear algebraic equations is very time-consuming even within the linear elastic regime. Furthermore, the tremendous computational cost of iterative large-scale finite element analysis also prevents the widespread use of topology optimization as a powerful design tool especially when the desired design resolution is very high. In order to break the bottleneck hindering the efficient solutions of large-scale structural analysis and design optimization problems, in the present article, a general machine learning (ML) enhanced substructure-based framework is proposed. The essential idea is resorting to the classical substructure-based finite element analysis approach and establishing an implicit mapping between the parameters characterizing the material distribution within a substructure and the corresponding condensed stiffness matrix/numerical shape functions through offline trained deep neural networks. In contrast to most of the existing ML enhanced approaches, the proposed framework is truly independent of the forms of structural geometry, boundary condition and external load, and can be applied to solve various boundary value problems governing by the same type of partial differential equation once the offline training is completed. Armed with modern artificial intelligence techniques, the proposed approach in some sense revives the classical substructure approach in finite element analysis. Compared with the traditional paradigm, it can achieve 104–105 times solution efficiency for tested large-scale examples with satisfactory accuracy. The effectiveness of the approach was also validated for the non-adjoint topology optimization problems, i.e., 3D compliant mechanism design. Finally, to demonstrate the proposed approach’s capability in dealing with extremely large-scale three-dimensional problems, a three-dimensional topology optimization problem with about 109 design variables and 3×109 degrees of freedom (Dofs) is solved on a laptop without resorting to any parallel computing techniques.

Acceleration of structural topology optimization using symmetric element-by-element strategy for unstructured meshes on GPU

PurposeStructural topology optimization is computationally expensive due to the involvement of high-resolution mesh and repetitive use of finite element analysis (FEA) for computing the structural response. Since FEA consumes most of the computational time in each optimization iteration, a novel GPU-based parallel strategy for FEA is presented and applied to the large-scale structural topology optimization of 3D continuum structures.Design/methodology/approachA matrix-free solver based on preconditioned conjugate gradient (PCG) method is proposed to minimize the computational time associated with solution of linear system of equations in FEA. The proposed solver uses an innovative strategy to utilize only symmetric half of elemental stiffness matrices for implementation of the element-by-element matrix-free solver on GPU.FindingsUsing solid isotropic material with penalization (SIMP) method, the proposed matrix-free solver is tested over three 3D structural optimization problems that are discretized using all hexahedral structured and unstructured meshes. Results show that the proposed strategy demonstrates 3.1× –3.3× speedup for the FEA solver stage and overall speedup of 2.9× –3.3× over the standard element-by-element strategy on the GPU. Moreover, the proposed strategy requires almost 1.8× less GPU memory than the standard element-by-element strategy.Originality/valueThe proposed GPU-based matrix-free element-by-element solver takes a more general approach to the symmetry concept than previous works. It stores only symmetric half of the elemental matrices in memory and performs matrix-free sparse matrix-vector multiplication (SpMV) without any inter-thread communication. A customized data storage format is also proposed to store and access only symmetric half of elemental stiffness matrices for coalesced read and write operations on GPU over the unstructured mesh.

Optimal Design of Large-scale Dome Truss Structures with Multiple Frequency Constraints Using Success-history Based Adaptive Differential Evolution Algorithm

The success-history based adaptive differential evolution (SHADE) algorithm is an efficient modified version of the differential evolution (DE) algorithm, and it has been successfully applied to solve some real-world optimization problems. However, to the best of our knowledge, it has been rarely applied in the field of structural optimization. The optimal design of structures with frequency constraints is well known as a highly nonlinear and non-convex optimization problem with many local optima. In this paper, the SHADE algorithm is examined in the context of size optimization of large-scale truss structures with multiple frequency constraints. In SHADE, a historical memory of successful control parameter settings is used to guide the generation of new control parameters. In order to demonstrate the effectiveness and efficiency of SHADE, three truss optimization problems with multiple frequency constraints are presented. The three examples considered in this paper include a 600-bar single-layer dome-shaped truss, a 1180-bar single-layer dome-shaped truss, and a 1410-bar double-layer dome-shaped truss. The results obtained by the SHADE algorithm are presented and compared with the best-known results reported in the literature. Numerical results indicate the effectiveness and superior performance of SHADE over other algorithms in terms of solution accuracy and robustness. It is worth mentioning that in all the three cases considered, the optimal designs obtained by SHADE are the best ones reported in the literature so far. However, SHADE often requires fewer structural analyses than those required by the other algorithms.

A Dimension Reduction Technique for Large-Scale Structured Sparse Optimization Problems with Application to Convex Clustering

In this paper, we propose a novel adaptive sieving (AS) technique and an enhanced AS (EAS) technique, which are solver independent and could accelerate optimization algorithms for solving large scale convex optimization problems with intrinsic structured sparsity. We establish the finite convergence property of the AS technique and the EAS technique with inexact solutions of the reduced subproblems. As an important application, we apply the AS technique and the EAS technique on the convex clustering model, which could accelerate the state-of-the-art algorithm SSNAL by more than 7 times and the algorithm ADMM by more than 14 times.

Optimization of Large-Scale Frame Structures Using Fuzzy Adaptive Quantum Inspired Charged System Search

Optimization of Large-Scale Frame Structures Using Fuzzy Adaptive Quantum Inspired Charged System Search

A partition and microstructure based method applicable to large-scale topology optimization

Topology optimization (TO) of large-scale structures is a computationally demanding process that challenges its widespread adoption in industrial design. We present a new partition based TO framework applicable to the design of large-scale problems and apply it to mechanical stiffness optimization problems. The method employs, first, a data-driven physical partitioning of strain energy contours to partition the design domain, and then a partition based TO. In contrast to conventional topology optimization in which the number of design variables is typically equal to the number of elements in the discretized domain, the proposed method assigns density design variables to each spatial partition leading to significant computational cost reduction and convergence acceleration. The constitutive matrix required for finite element analysis is iteratively determined according to the Hashin-Shtirkman upper bounds based on partition densities. Once the optimized partition densities are achieved, the manufacturable binary structure is realized by mapping a set of generated high-performance isotropic microstructure cells onto partition elements. To validate the capability, effectiveness, and efficiency of the proposed method, several numerical examples are provided. The optimized structures using macrostructural analysis exhibit comparable performance to the conventional SIMP method for small-size problems. Besides, the TO results for the large-scale problems suggest significant computational cost efficiency.

CPU parallel-based adaptive parameterized level set method for large-scale structural topology optimization

CPU parallel-based adaptive parameterized level set method for large-scale structural topology optimization

Improved multi-objective Hybrid Genetic Algorithm for Shape and Size Optimization of Free-form latticed structures

Industrial and public buildings generally adopt spatial latticed structures as roofs, where the number of members is too large for the traditional genetic algorithm (GA) to conduct size optimization. In this paper, an improved multi-objective hybrid genetic algorithm incorporating the fully stressed design (FSD), called NSGA-II-FSD, is proposed for the shape and size optimization of large-scale free-form spatial latticed structures. The proposed method uses the B-spline theory, which makes it easier to generate free-form surfaces that meet the aesthetic requirements. The total mass and strain energy of the structure are selected as the objective functions, and the strength, as well as the stability of the member, are taken as the constraints. By incorporating the FSD, one can also set the stress ratio limit and get an optimal set of cross-sections of the members under the desired margin of safety. Besides, a significant mutation strategy is proposed in order to expand the searching boundary, which can avoid premature convergence. Through numerical examples of a cylindrical latticed shell and a free-form space truss , it is proved that the proposed method is effective, and engineers can choose the solution according to their preferences in practical design. • Method for shape and size optimization of latticed structures is proposed. • Proposed method can handle multi-objective optimization problems. • A significant mutation strategy is proposed to expand the searching boundary. • The improved FSD is effective to solve the problems with multiple load cases. • All individuals in Pareto front can be directly adopted by designers.

High-dimensional optimization of large-scale steel truss structures using guided stochastic search

Despite a plethora of truss optimization algorithms devised in the recent literature of structural optimization, still high-dimensional large-scale truss optimization problems have not been properly tackled basically due to the excessive computational effort required to handle the foregoing instances. In this study, application of a recently developed design-driven heuristic, namely guided stochastic search (GSS), is extended to a more challenging class of truss optimization problems having thousands of design variables. Two variants of the algorithm, namely GSSA and GSSB, have been employed for sizing optimization of four high-dimensional examples of steel trusses, i.e., a 2075-member single-layer onion dome, a 2688-member double-layer open dome, a 6000-member double-layer scallop dome, and a 15048-member double-layer grid as per AISC-LRFD specification. The numerical results obtained indicate the efficiency of GSSA and GSSB in handling high-dimensional instances of large-scale steel trusses with up to 15048 discrete design variables.

Neutron Production from Structured Targets Irradiated By an Ultrashort Laser Pulse

We study the laser-driven generation of thermonuclear neutrons from targets with a microstructured surface in the form of deuterated microwires, using three-dimensional numerical simulation with previously obtained results of large-scale structural optimization of the target, which provides its best heating by femtosecond laser pulses of moderate intensity. We show that, for modern laser technologies, femtosecond lasers of low (several mJ) energy are even better for creating a neutron source than more powerful (∼1 J) lasers because the mode of high (∼1 kHz) pulse repetition rate is practically available. Microlayers (relief) and cylindrical microholes on the irradiated side are considered as alternative microstructured targets. For the latter, we demonstrate accumulation of ions on the axis of the holes, which leads to increase in the ion density above the initial value and consequently to a possible increase in the yield of thermonuclear neutrons.

Frequency-constrained optimization of large-scale dome-shaped trusses using chaotic water strider algorithm

The frequency constrained design of large-scale structures is a complex optimization problem with many local optima. In order to tackle this problem, in this paper, a chaotic version of a newly-established metaheuristic algorithm, the so-called Water Strider Algorithm (WSA), is introduced. This version repetitively diversifies solutions and then focuses on promising areas during the course of iterations. The chaotic algorithm involves two types of succession states, namely explorative and exploitative states, and it probabilistically switches between them using the Circle map with a quasi-cyclic behavior. To examine the effectiveness of this algorithm, it is applied to structural optimization of large-scale structures with multiple frequency constraints such as 600-, 1180- and 1410-bar dome trusses. The results show that the new version outperforms its basic version as well as some other well-established algorithms in the literature. The statistical measures reveal that it is reasonably capable of providing a satisfactory performance in terms of both obtaining lighter designs and convergence speed.

Large-Scale Truss-Sizing Optimization with Enhanced Hybrid HS Algorithm

Metaheuristic algorithms currently represent the standard approach to engineering optimization. A very challenging field is large-scale structural optimization, entailing hundreds of design variables and thousands of nonlinear constraints on element stresses and nodal displacements. However, very few studies documented the use of metaheuristic algorithms in large-scale structural optimization. In order to fill this gap, an enhanced hybrid harmony search (HS) algorithm for weight minimization of large-scale truss structures is presented in this study. The new algorithm, Large-Scale Structural Optimization–Hybrid Harmony Search JAYA (LSSO-HHSJA), developed here, combines a well-established method like HS with a very recent method like JAYA, which has the simplest and inherently most powerful search engine amongst metaheuristic optimizers. All stages of LSSO-HHSJA are aimed at reducing the number of structural analyses required in large-scale structural optimization. The basic idea is to move along descent directions to generate new trial designs, directly through the use of gradient information in the HS phase, indirectly by correcting trial designs with JA-based operators that push search towards the best design currently stored in the population or the best design included in a local neighborhood of the currently analyzed trial design. The proposed algorithm is tested in three large-scale weight minimization problems of truss structures. Optimization results obtained for the three benchmark examples, with up to 280 sizing variables and 37,374 nonlinear constraints, prove the efficiency of the proposed LSSO-HHSJA algorithm, which is very competitive with other HS and JAYA variants as well as with commercial gradient-based optimizers.

Accelerating Large-scale Topology Optimization: State-of-the-Art and Challenges

Large-scale structural topology optimization has always suffered from prohibitively high computational costs that have till date hindered its widespread use in industrial design. The first and major contributor to this problem is the cost of solving the Finite Element equations during each iteration of the optimization loop. This is compounded by the frequently very fine 3D models needed to accurately simulate mechanical or multi-physical performance. The second issue stems from the requirement to embed the high-fidelity simulation within the iterative design procedure in order to obtain the optimal design. The prohibitive number of calculations needed as a result of both these issues, is often beyond the capacities of existing industrial computers and software. To alleviate these issues, the last decade has opened promising pathways into accelerating the topology optimization procedure for large-scale industrial sized problems, using a variety of techniques, including re-analysis, multi-grid solvers, model reduction, machine learning and high-performance computing, and their combinations. This paper attempts to give a comprehensive review of the research activities in all of these areas, so as to give the engineer both an understanding as well as a critical appreciation for each of these developments.

Using the hierarchical Kriging model to optimize the structural dynamics of rocket engines

Traditional fine models are often inefficient in optimizing the dynamics of complex structural systems such as large liquid rocket engines. To enhance the design efficiency, this paper proposes a fast optimization method based on finite element modeling, central Latin hypercube experimental design with minimax criteria, hierarchical Kriging surrogate model, and combined multi-island genetic algorithm + nonlinear sequential quadratic programming approach. First, a low-frequency dynamic fine finite element model of the entire engine structure was established, and the sampling points were obtained by considering the experimental design and a hierarchical Kriging model that describes the relationship between the design variables of the engine structure parameters and the modal frequency characteristics. Next, the height of the frame, size of the gimbal mount, and stiffness and installation angle of the servo mechanism were considered as design variables. To satisfy the design requirements in terms of the first modal frequency of the engine structure and realize the optimal design of the modal characteristics, the global optimization of the high-precision Kriging model was performed in the design space by using the hybrid optimization algorithm. Finally, the optimization results were substituted into the finite element model to perform the modal analysis, and modal tests were conducted to verify the effectiveness of the optimization method in realizing large-scale structural system optimization. The results indicated that the first modal frequency of the engine structure was increased from 7.53 Hz to 11.50 Hz and the optimization results could satisfy the requirements for the dynamics design. The proposed optimization design method can considerably improve the optimization efficiency by ensuring high accuracy of the results. The presented modeling strategies and optimization concepts can be applied to several engineering applications and provide novel solutions for the dynamic optimization problems of large and complex structures.

An ANSYS APDL code for topology optimization of structures with multi-constraints using the BESO method with dynamic evolution rate (DER-BESO)

This paper presents a 390-line code written in ANSYS Parametric Design Language (APDL) for topology optimization of structures with multi-constraints. It adopts the bi-directional evolutionary structural optimization method with the proposed dynamic evolution rate strategy (DER-BESO) to accelerate the iteration convergence. The complete APDL program includes the modules of finite element modeling, element sensitivity calculation, Lagrange multiplier updating, and the element updating module using DER-BESO method. It allows users to conduct the finite element analysis and optimization iterations just in one platform without having to switch repeatedly between the finite element analysis software and the optimization program. Through a cantilever example, the evolution procedure of DER-BSEO is compared to the primary BESO method to show its better performance in convergence speed. Different constraint cases are also considered to examine the robustness of the DER_BESO program. Example extensions for 3D structures and periodic structures with geometric restraints are also presented and discussed. Since ANSYS is a powerful finite element analysis platform, the given code has a perspective of extending to the optimization of large-scale structures or more complicated optimization problems that consider nonlinear or buckling effects.

A bi-level methodology for solving large-scale mixed categorical structural optimization

In this work, large-scale structural optimization problems involving non-ordinal categorical design variables and continuous variables are investigated. The aim is to minimize the weight of a structure with respect to cross-section areas, with materials and stiffening principles selection. First, the problem is formulated using a bi-level decomposition involving master and slave problems. The master problem is given by a first-order-like approximation that helps to drastically reduce the combinatorial explosion raised by the categorical variables. Continuous variables are handled in a slave problem solved using a gradient-based approach, where the categorical variables are driven by the master problem. The proposed algorithm is tested on three different structural optimization test cases. A comparison to state-of-the-art algorithms emphasize efficiency of the proposed algorithm in terms of the optimum quality, the computation cost, and the scaling with respect to the problem dimension.

Large-scale structural optimization using a fuzzy reinforced swarm intelligence algorithm

In contrast with conventional structural optimization benchmark problems real size structures mostly contain a large number of members and their optimal design provide a serious challenging area for optimization methods. In this regard, current study deals with assessing the search performance of the recently developed fuzzy reinforced metaheuristic technique so called Interactive Fuzzy Search Algorithm on the optimization of large-scale structures. This method is a self-adaptive and parameter-free algorithm which applies a dual-module fuzzy decision mechanism to adjust its search behavior during the optimization process. This mechanism employs two nine rule fuzzy modules which permanently monitor the agents updating process and based on the governing conditions of the problem emphasize their exploration or exploitation search behavior. Attained results show that proposed method can adopt itself with the extensive search space of the studied problems. Form both accuracy and stability aspects Interactive Fuzzy Search Algorithm provides promising results on solving large-scale structural optimization problems.