Efficient structural optimization using reanalysis and sensitivity reanalysis

Optimization of large-scale structures using conventional formulations often involves much computational effort. Repeated solutions of the analysis and sensitivity analysis equations usually require most of this effort. The computational cost may become prohibitive in large-scale structures having complex analysis models. To alleviate this difficulty, various procedures are integrated in this study into a general optimization approach. The approach is suitable for different classes of response types and optimization methods, including linear and non-linear response; static and dynamic response; direct and gradient optimization methods. Combined approximations are used for reanalysis and repeated sensitivity analysis. The advantage is that the efficiency of local approximations and the improved quality of global approximations are combined to obtain effective solution procedures. Approximate reanalysis and finite-difference sensitivity reanalysis are considered for each intermediate design during the solution process. Reductions in the computational effort may reach several orders of magnitude. Typical numerical examples show that the results achieved by the approach presented are similar to those obtained by exact reanalysis and sensitivity analysis.     

A systematic approach on computational analysis and optimization design: for a nonlinear coupling shock absorber

The aim of this article is to provide a systematic approach to perform computational simulation and optimization design of parameters matching selection for a nonlinear coupling shock absorber. A theoretical mathematical model with nonlinear coupling for shock absorber is induced based on relative literature. The model considers the coupling of quadratic damping, viscosity damping, coulomb damping and nonlinear spring. Approximate computational solution is deduced by introducing harmonic balance method and Fourier transform method. These approximate theoretical solutions include output response of the system, absolute acceleration transmissibility in vibration or impact, and the maximum relative displacement in impact process, etc. The approximate computational results are compared with those obtained by numerical integration to confirm the validity of the mathematical model. In the meantime, an optimization design model for parameters is built. The design example is illustrated to confirm the validity of the modeling method and the theoretical solution.     

Exact and Accurate Reanalysis of Structures for Geometrical Changes

A reanalysis approach for geometrical changes in structural systems is presented. The solution procedure is based on the combined approximations method, where the binomial series terms are used as basis vectors in reduced basis approximations. The calculations are based on results of a single exact analysis, calculation of derivatives is not required, and each reanalysis involves a small computational effort. The method is easy to implement, and can be used with general finite element programs. Exact solutions are obtained efficiently for low-rank modifications in the geometry. Accurate solutions are achieved in cases where the basis vectors come close to being linearly dependent. Such solutions are also achieved for nearly scaled geometries, when the angle between the two vectors representing the initial design and modified design is small. Numerical examples demonstrate the high accuracy achieved with a small number of basis vectors     

A hybrid OC–GA approach for fast and global truss optimization with frequency constraints

The truss optimization constrained with vibration frequencies is a highly nonlinear and more computational cost problem. To speed up the convergence and obtain the global solution of this problem, a hybrid optimality criterion (OC) and genetic algorithm (GA) method for truss optimization is presented in this paper. Firstly, the OC method is developed for multiple frequency constraints. Then, the most efficient variables are identified by sensitivity analysis and modified as iteration scheme. Finally, OC method, serving as a local search operator, is integrated with GA. The numerical results verify that the hybrid method provides powerful ability in searching for more optimal solution and reducing computational effort.     

Topology optimization in OpenMDAO

Recently, topology optimization has drawn interest from both industry and academia as the ideal design method for additive manufacturing. Topology optimization, however, has a high entry barrier as it requires substantial expertise and development effort. The typical numerical methods for topology optimization are tightly coupled with the corresponding computational mechanics method such as a finite element method and the algorithms are intrusive, requiring an extensive understanding. This paper presents a modular paradigm for topology optimization using OpenMDAO, an open-source computational framework for multidisciplinary design optimization. This provides more accessible topology optimization algorithms that can be non-intrusively modified and easily understood, making them suitable as educational and research tools. This also opens up further opportunities to explore topology optimization for multidisciplinary design problems. Two widely used topology optimization methods—the density-based and level-set methods—are formulated in this modular paradigm. It is demonstrated that the modular paradigm enhances the flexibility of the architecture, which is essential for extensibility.

     

Efficient reanalysis for topological optimization

An efficient reanalysis method for the topological optimization of structures is presented. The method is based on combining the computed terms of a series expansion, used as high quality basis vectors, and coefficients of a reduced basis expression. The advantage is that the efficiency of local approximations and the improved quality of global approximations are combined to obtain an effective solution procedure.The method is based on results of a single exact analysis and can be used with a general finite element system. It is suitable for different types of structures, such as trusses, frames, grillages, etc. Calculations of derivatives is not required, and the errors involved in the approximations can readily be evaluated.Several numerical examples illustrate the effectiveness of the solution procedure. It is shown that excellent results can be achieved with small computational effort for very large changes in the cross-sections and in the topology of the structure.     

A unified reanalysis approach for structural analysis,design, and optimization

This study presents a unified reanalysis approach for structural analysis, design, and optimization that is based on the Combined Approximations (CA) method. The method is suitable for various analysis models (linear, nonlinear, elastic, plastic, static, dynamic), different types of structures (trusses, frames, grillages, continuum structures), and all types of design variables (cross-sectional, material, geometrical, topological). The calculations are based on results of a single exact analysis. The computational effort is usually much smaller than that needed to carry out a complete analysis of modified designs. Accurate results are achieved by low-order approximations for significant changes in the design. It is possible to improve the accuracy by considering higher-order terms, and exact solutions can be achieved in certain cases. The solution steps are straightforward, and the computational procedures presented can readily be used with general finite element systems. Typical results are demonstrated by numerical examples.     

Metaheuristic algorithms for approximate solution to ordinary differential equations of longitudinal fins having various profiles

Differential equations play a noticeable role in engineering, physics, economics, and other disciplines. Approximate approaches have been utilized when obtaining analytical (exact) solutions requires substantial computational effort and often is not an attainable task. Hence, the importance of approximation methods, particularly, metaheuristic algorithms are understood. In this paper, a novel approach is suggested for solving engineering ordinary differential equations (ODEs). With the aid of certain fundamental concepts of mathematics, Fourier series expansion, and metaheuristic methods, ODEs can be represented as an optimization problem. The target is to minimize the weighted residual function (error function) of the ODEs. The boundary and initial values of ODEs are considered as constraints for the optimization model. Generational distance and inverted generational distance metrics are used for evaluation and assessment of the approximate solutions versus the exact (numerical) solutions. Longitudinal fins having rectangular, trapezoidal, and concave parabolic profiles are considered as studied ODEs. The optimization task is carried out using three different optimizers, including the genetic algorithm, the particle swarm optimization, and the harmony search. The approximate solutions obtained are compared with the differential transformation method (DTM) and exact (numerical) solutions. The optimization results obtained show that the suggested approach can be successfully applied for approximate solving of engineering ODEs. Providing acceptable accuracy of the proposed technique is considered as its important advantage against other approximate methods and may be an alternative approach for approximate solving of ODEs.     

Vibration reanalysis using frequency-shift combined approximations

In the structural dynamic optimization procedure, many repeated analyses are conducted to evaluate vibration performance of successively modified structural designs. A new procedure for structural vibration (or eigenproblem) reanalysis is developed based on iteration and inverse iteration method with frequency-shift and linear combination acceleration to reduce the high computational cost of structure reanalysis. With a suitable frequency-shift factor, the Frequency-Shift Combined Approximations (FSCA) method allows to calculate higher modes accurately. Three numerical examples are presented to demonstrate the accuracy of the proposed method. Excellent results can be obtained in cases where large modifications are made and higher modes are needed.     

Stress-based topology optimization

Previous research on topology optimization focussed primarily on global structural behaviour such as stiffness and frequencies. However, to obtain a true optimum design of a vehicle structure, stresses must be considered. The major difficulties in stress based topology optimization problems are two-fold. First, a large number of constraints must be considered, since unlike stiffness, stress is a local quantity. This problem increases the computational complexity of both the optimization and sensitivity analysis associated with the conventional topology optimization problem. The other difficulty is that since stress is highly nonlinear with respect to design variables, the move limit is essential for convergence in the optimization process. In this research, global stress functions are used to approximate local stresses. The density method is employed for solving the topology optimization problems. Three numerical examples are used for this investigation. The results show that a minimum stress design can be achieved and that a maximum stiffness design is not necessarily equivalent to a minimum stress design.     

A comparison of several reanalysis methods for structural layout modifications with added degrees of freedom

The goal of this paper is to compare several reanalysis methods for structural layout modifications with added degrees of freedom. These methods include the modified initial analysis methods, the modified initial analyses method with a scalar multiplier, and the preconditioned conjugate gradient method. These reanalysis methods are compared in terms of condition numbers of the preconditioned system and their computational accuracy. It is concluded in this paper that the preconditioner of the first method is not positive definite in general, while for the second method a positive definite preconditioner for a small scalar multiplier can be obtained, but the condition number of the preconditioned system is still large. For the third method, the condition numbers of the preconditioned system are significantly reduced, and highly accurate approximate solutions can be obtained in a few iterations.     

Dynamic response topology optimization in the time domain using model reduction method

The dynamic response topology optimization problems are usually computationally expensive, so it is necessary to employ the model reduction methods to reduce computational cost. This work will investigate the effectiveness of the mode displacement method(MDM) and mode acceleration method(MAM) for time-domain response problems within the framework of density-based topology optimization. Three objective functions, the mean dynamic compliance, mean strain energy and mean squared displacement are considered. It is found that, in general cases, MDM is not suitable for time-domain response topology optimization problems due to its low accuracy of approximation, while MAM works well for problems of a wide range, and when there are many time steps, the MAM based topology optimization approach is more efficient than the direct integration based approach. So for practical applications, when the problem needs many time steps, the MAM based approach is preferred and otherwise, the direct integration based approach is suggested.     

Reanalysis and sensitivity reanalysis by combined approximations

One of the main obstacles in the solution of structural optimization problems is the need to repeat solutions of the analysis and sensitivity analysis equations. In large-scale structures, having complex analysis models, the computational effort may become prohibitive. To alleviate this difficulty a general approach for repeated analysis and repeated sensitivity analysis, called combined approximations, was developed during the last 15 years. The solution is based on the integration of several algorithms and methods. As a result, accurate results can be achieved efficiently. In previous studies, solution procedures for various particular problems were developed. This article summarizes the various formulations and solution procedures for reanalysis and sensitivity reanalysis of linear, nonlinear, static and dynamic systems. It is shown that the various solution procedures are based on applications of similar basic algorithms. Numerical examples demonstrate the efficiency of the calculations and the accuracy of the results.     

Fast structural optimization with frequency constraints by genetic algorithm using adaptive eigenvalue reanalysis methods

Structural optimization with frequency constraints is highly nonlinear dynamic optimization problems. Genetic algorithm (GA) has greater advantage in global optimization for nonlinear problem than optimality criteria and mathematical programming methods, but it needs more computational time and numerous eigenvalue reanalysis. To speed up the design process, an adaptive eigenvalue reanalysis method for GA-based structural optimization is presented. This reanalysis technique is derived primarily on the Kirsch’s combined approximations method, which is also highly accurate for case of repeated eigenvalues problem. The required number of basis vectors at every generation is adaptively determined and the rules for selecting initial number of basis vectors are given. Numerical examples of truss design are presented to validate the reanalysis-based frequency optimization. The results demonstrate that the adaptive eigenvalue reanalysis affects very slightly the accuracy of the optimal solutions and significantly reduces the computational time involved in the design process of large-scale structures.     

A modified stochastic perturbation algorithm for closely-spaced eigenvalues problems based on surrogate model

Aiming at uncertainty propagation and dynamic reanalysis of closely-spaced eigenvalues, with consideration of uncertainties in design variables, a modified stochastic perturbation method is proposed. Concerning quasi-symmetric or partial-symmetric structures that frequently appear, one of their primary features is closely-distributed natural frequencies. For structure with closely-spaced eigenvalues, due to its instability and sensitivity to the changes of design variables and its excessively concentrated adjacent eigenvalues, conventional uncertainty analysis or dynamic reanalysis methods for distinct eigenvalue are no longer available. Initially, the spectral decompositions of stiffness and mass matrices are provided; by transfer technique, the eigen-problem of closely-spaced eigenvalues is converted to that of repeated eigenvalues with two perturbation parts appended; then the perturbed closely-spaced eigenvalue is rewritten as the sum of original closely-spaced eigenvalues’ mean value and surrogate model which approximates the first-order perturbation term by polynomial chaos expansions. According to this method, statistical quantities of perturbed closely-spaced eigenvalues are calculated directly and accurately, which contributes to its uncertainty analysis and dynamic reanalysis. Furthermore, the capability of proposed method in dealing with relatively large uncertainties and complex engineering structure is demonstrated. The accuracy and efficiency of proposed method have been verified sufficiently by numerical examples.     

Approximate structural reanalysis based on series expansion

In most optimal design procedures the analysis of the structure must be repeated many times. This operation, which involves much computational effort, is one of the main difficulties in applying optimization methods to large systems. This study deals with approximate reanalysis methods based on series expansion. Both design variables and inverse variables formulations are presented. It is shown that a Taylor series expansion of the nodal displacements or the redundant forces is equivalent to a series obtained from a simple iteration procedure. The series coefficients can readily be computed, providing efficient and high-degree polynomial approximations.To further improve the quality of the approximations, a modified nonpolynomial series is proposed. To reduce the amount of calculations, the possibility of reanalysis along a given line in the variables space is demonstrated. All the proposed procedures require a single exact analysis to obtain an explicit behaviour model along a line.The relationship between the various methods is discussed and numerical examples demonstrate applications. The results obtained are encouraging and indicate that the proposed methods provide efficient and high quality approximations for the structural behavior. This may lead to a wider use of optimization methods in the design of large structural systems.     

On simultaneous shape and material layout optimization of shell structures

This work presents a computational method for integrated shape and topology optimization of shell structures. Most research in the last decades considered both optimization techniques separately, seeking an initial optimal topology and refining the shape of the solution later. The method implemented in this work uses a combined approach, were the shape of the shell structure and material distribution are optimized simultaneously. This formulation involves a variable ground structure for topology optimization, since the shape of the shell mid-plane is modified in the course of the process. It was considered a simple type of design problem, where the optimization goal is to minimize the compliance with respect to the variables that control the shape, material fraction and orientation, subjected to a constraint on the total volume of material. The topology design problem has been formulated introducing a second rank layered microestructure, where material properties are computed by a “smear-out” procedure. The method has been implemented into a general optimization software called ODESSY, developed at the Institute of Mechanical Engineering in Aalborg. The computational model was tested in several numerical applications to illustrate and validate the approach.     

Sensitivity analysis of optimum solutions by updating active constraints: application in aircraft structural design

Often the parameters considered as constants in an optimization problem have some uncertainty and it is interesting to know how the optimum solution is modified when these values are changed. The only way to continue having the optimal solution is to perform a new optimization loop, but this may require a high computational effort if the optimization problem is large. However, there are several procedures to obtain the new optimal design, based on getting the sensitivities of design variables and objective function with respect to a fixed parameter. Most of these methods require obtaining second derivatives which has a significant computational cost. This paper uses the feasible direction-based technique updating the active constraints to obtain the approximate optimum design. This procedure only requires the first derivatives and it is noted that the updating set of active constraints improves the result, making possible a greater fixed parameter variation. This methodology is applied to an example of very common structural optimization problems in technical literature and to a real aircraft structure.     

A comparative study of dynamic analysis methods for structural topology optimization under harmonic force excitations

This work is focused on the topology optimization related to harmonic responses for large-scale problems. A comparative study is made among mode displacement method (MDM), mode acceleration method (MAM) and full method (FM) to highlight their effectiveness. It is found that the MDM results in the unsatisfactory convergence due to the low accuracy of harmonic responses, while MAM and FM have a good accuracy and evidently favor the optimization convergence. Especially, the FM is of superiority in both accuracy and efficiency under the excitation at one specific frequency; MAM is preferable due to its balance between the computing efficiency and accuracy when multiple excitation frequencies are taken into account.