Optimal topology selection of continuum structures with displacement constraints

This paper deals with the optimal topology selection of continuum structures subject to displacement constraints by using the performance-based design concept. The optimal topology of a continuum structure is generated by gradually eliminating underutilized elements from the discretized design domain. A performance index is developed for monitoring the optimization process and is used as a termination criterion in the optimization algorithm so that the global optimum can be selected from the optimization history. Maximizing the performance index in the design space is proposed as the performance-based optimization criterion. The performance index can be utilized to compare the efficiency of structural topologies produced by different continuum topology optimization methods. Several examples are provided to demonstrate the capabilities of the performance-based optimization approach in selecting the best configuration for the minimum-weight design of continuum structures with maximum stiffness.     

Non-probabilistic reliability-based topology optimization with multidimensional parallelepiped convex model

In this paper, a new non-probabilistic reliability-based topology optimization (NRBTO) method is proposed to account for interval uncertainties considering parametric correlations. Firstly, a reliability index is defined based on a newly developed multidimensional parallelepiped (MP) convex model, and the reliability-based topology optimization problem is formulated to optimize the topology of the structure, to minimize material volume under displacement constraints. Secondly, an efficient decoupling scheme is applied to transform the double-loop NRBTO into a sequential optimization process, using the sequential optimization & reliability assessment (SORA) method associated with the performance measurement approach (PMA). Thirdly, the adjoint variable method is used to obtain the sensitivity information for both uncertain and design variables, and a gradient-based algorithm is employed to solve the optimization problem. Finally, typical numerical examples are used to demonstrate the effectiveness of the proposed topology optimization method.     

Single-loop system reliability-based topology optimization considering statistical dependence between limit-states

This paper presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization (TO) problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method, which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The paper provides numerical examples of two- and three-dimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared with one another to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.     

Maximum energy dissipation for elasto-plastic plates via isogeometric shape optimization

In this research, optimum shape of plate structures is sought to maximize the energy dissipation via structural shape optimization. To achieve this, isogeometric analysis (IGA) is utilized for structural analysis of plates considering elasto-plastic behavior of materials. The von Mises material model is employed for this purpose. Non-uniform rational B-splines basis functions are used for both geometry definition and approximating the unknown deformation field. The optimization problem is to maximize the structural dissipated energy until a prescribed displacement is reached and a fixed amount of material is considered in the design domain. A direct shape sensitivity analysis is performed and a mathematical based approach is employed for the optimization process. To demonstrate the efficiency of the proposed algorithm three examples are illustrated. Using the IGA prevents adjusting analysis model during the optimization process, which is time-consuming especially when iterative nonlinear analysis is performed. The results also show that large geometry modifications can be properly managed by the proposed algorithm.

     

Structural topology design optimization using Genetic Algorithms with a bit-array representation

In this paper, a bit-array representation method for structural topology optimization using the Genetic Algorithm (GA) is implemented. The importance of structural connectivity in a design is further emphasized by considering the total number of connected objects of each individual explicitly in an equality constraint function. To evaluate the constrained objective function, Deb’s constraint handling approach is further developed to ensure that feasible individuals are always better than infeasible ones in the population to improve the efficiency of the GA. A violation penalty method is proposed to drive the GA search towards the topologies with higher structural performance, less unusable material and fewer separate objects in the design domain. An identical initialization method is also proposed to improve the GA performance in dealing with problems with long narrow design domains. Numerical results of structural topology optimization problems of minimum weight and minimum compliance designs show the success of this bit-array representation method and suggest that the GA performance can be significantly improved by handling the design connectivity properly.     

A novel displacement constrained optimization approach for black and white structural topology designs under multiple load cases

The gray problem of displacement constrained topology volume minimization under multiple load cases still is an opening topic of research. A series of topologies with clear profiles generated from an optimization process are very beneficial to method engineering applications. In this paper, a novel displacement constrained optimization approach for black and white structural topology designs under multiple load cases, is proposed to obtain a series of topologies with clear profiles. Firstly, a distribution feature of constraint displacement derivatives is investigated. Secondly, an adaptive adjusting approach of design variable bounds is proposed, and an improved approximate model with varied constraint limits and a volume penalty objective function are constructed. Thirdly, an improved density-based optimization method is proposed for the displacement constrained topology volume minimization under multiple load cases. Finally, several examples are given to demonstrate that the results obtained by the proposed method provide a series of topologies with clear profiles during an optimization process. It is concluded from examples that the proposed method is effective and robust for generating an optimal topology.     

Lightweight flatbed trailer design by using topology and thickness optimization

A new design for a lightweight flatbed trailer with high bending stiffness and torsional frequency is presented. The design procedure consists of two main steps: topology optimization and thickness optimization. During topology optimization, a creative frame layout different from existing ladder-type frames can be obtained by searching the best layout out of all possible layouts of a simplified design domain model. After approximating the result of topology optimization as a thin-walled structure, the approximated thicknesses of the plates are optimized to minimize the mass of a trailer. The bending stiffness and torsional frequency obtained by topology optimization are set as design constraints for thickness optimization. Due to the closed cross-section, the optimized trailer can efficiently increase the stiffness-to-mass ratio to a large extent. Discrete thicknesses are employed as design variables for thickness optimization so that the thicknesses of the plates of a trailer can be included in those of commercially available high-strength steel products. The final model has a 29% reduction in total mass, a 21% decrease in mean compliance with a uniform bending load, and a 169% increase in torsional frequency.     

A new multi-objective programming scheme for topology optimization of compliant mechanisms

This paper presents an alternative method in implementing multi-objective optimization of compliant mechanisms in the field of continuum-type topology optimization. The method is designated as “SIMP-PP” and it achieves multi-objective topology optimization by merging what is already a mature topology optimization method—solid isotropic material with penalization (SIMP) with a variation of the robust multi-objective optimization method—physical programming (PP). By taking advantages of both sides, the combination causes minimal variation in computation algorithm and numerical scheme, yet yields improvements in the multi-objective handling capability of topology optimization. The SIMP-PP multi-objective scheme is introduced into the systematic design of compliant mechanisms. The final optimization problem is formulated mathematically using the aggregate objective function which is derived from the original individual design objectives with PP, subjected to the specified constraints. A sequential convex programming method, the method of moving asymptotes (MMA) is then utilized to process the optimization evolvement based on the design sensitivity analysis. The main findings in this study include distinct advantages of the SIMP-PP method in various aspects such as computation efficiency, adaptability in convex and non-convex multi-criteria environment, and flexibility in problem formulation. Observations are made regarding its performance and the effect of multi-objective optimization on the final topologies. In general, the proposed SIMP-PP method is an appealing multi-objective topology optimization scheme suitable for “real world” problems, and it bridges the gap between standard topological design and multi-criteria optimization. The feasibility of the proposed topology optimization method is exhibited by benchmark examples.     

Bridge topology optimisation with stress, displacement and frequency constraints

The principal stress based evolutionary structural optimisation method is presented herein for topology optimisation of arch, tied arch, cable-stayed and suspension bridges with both stress and displacement constraints. Two performance index formulas are developed to determine the efficiency of the topology design. A refined mesh scheme is proposed to improve the details of the final topology without resorting to the complete analysis of a finer mesh. Furthermore, cable-supported bridges are optimised with frequency constraint incorporating the “nibbling” technique. The applicability, simplicity and effectiveness of the method are demonstrated through the topology optimisation of the four types of bridges.     

Evolutionary topology optimization of continuum structures with a global displacement control

The conventional compliance minimization of load-carrying structures does not directly deal with displacements that are of practical importance. In this paper, a global displacement control is realized through topology optimization with a global constraint that sets a displacement limit on the whole structure or certain sub-domains. A volume minimization problem is solved by an extended evolutionary topology optimization approach. The local displacement sensitivities are derived following a power-law penalization material model. The global control of displacement is realized through multiple local displacement constraints on dynamically located critical nodes. Algorithms are proposed to secure the stability and convergence of the optimization process. Through numerical examples and by comparing with conventional stiffness designs, it is demonstrated that the proposed approach is capable of effectively finding optimal solutions which satisfy the global displacement control. Such solutions are of particular importance for structural designs whose deformed shapes must comply with functioning requirements such as aerodynamic performances.     

Level set based robust shape and topology optimization under random field uncertainties

A robust shape and topology optimization (RSTO) approach with consideration of random field uncertainty in loading and material properties is developed in this work. The proposed approach integrates the state-of-the-art level set methods for shape and topology optimization and the latest research development in design under uncertainty. To characterize the high-dimensional random-field uncertainty with a reduced set of random variables, the Karhunen–Loeve expansion is employed. The univariate dimension-reduction (UDR) method combined with Gauss-type quadrature sampling is then employed for calculating statistical moments of the design response. The combination of the above techniques greatly reduces the computational cost in evaluating the statistical moments and enables a semi-analytical approach that evaluates the shape sensitivity of the statistical moments using shape sensitivity at each quadrature node. The applications of our approach to structure and compliant mechanism designs show that the proposed RSTO method can lead to designs with completely different topologies and superior robustness.     

Structural design for desired eigenfrequencies and mode shapes using topology optimization

A technique is proposed for determining the material distribution of a structure to obtain desired eigenmode shapes for problems of maximizing the fundamental eigenfrequency. The design objective is achieved using the solid isotropic method with penalization (SIMP) for topology optimization. Weighted constraints added in bound formulation are proposed to maximize the fundamental natural frequency, which provides an easy and straightforward way to prevent mode switching in the optimization process. Aside from maximizing the fundamental frequency, a method to modify existing eigenmodes to continuously evolve and assume the same shapes as the desired modes within the optimization process is proposed. The topology layout of a structure with desired eigenmodes is obtained by adding the modal assurance criterion (MAC) as additional constraints in the bound formulation optimization. Examples are presented to illustrate the proposed method, and a potential application of the proposed technique in decoupling a mechanical system is demonstrated.     

Artificial neural network based hole image interpretation techniques for integrated topology and shape optimization

Homogenization based and density based topology optimization seeks the best conceptual structural configuration on a predefined design domain with specific boundary and loading conditions. Such structural configuration is most often the minimum-compliance design under a fixed material usage constraint. Shape optimization must be subsequently executed so as to ensure the satisfaction of other practical design constraints such as stress and displacement, and attain the detailed definition of the structure configuration with a smooth circumference and interior hole contours. Complicated procedures involved in connection between topology and shape optimization are major obstacle for most design engineers to overcome. A fully automated configuration optimization system was developed [C.Y. Lin, L.S. Chao, Automated image interpretation for integrated topology and shape optimization, Structural and Multidisciplinary Optimization 20(2) (2000) 125–137] to execute the entire configuration design process automatically with room of improvements in the hole representation templates and hole interpretation reliabilities. In response, this paper proposes two-stage artificial neural networks based hole image interpretation techniques with improved template variety and recognition reliability.     

Topology optimization of planar and plate structures using conformal mapping

This paper proposes a new method to determine the optimum topology of planar and plate structures using conformal mapping. First it is proved that two invariants of stresses, which are the sum and difference of principal stresses, satisfy the Laplace equation. Additionally, it is proved that two invariants of bending moments are the sum and difference of principal bending moments, and is proved that they satisfy the Laplace equation under certain condition. Finally, we show that corresponding relationships between fluid mechanics and electromagnetism are valid to the theory of planar elasticity and plate bending. From these considerations, a convenient design method to optimize topologies of planar and plate structures for complicate design domains is proposed using conformal mapping. Several numerical examples of optimum topologies are treated by the proposed method. Through numerical results the effectiveness and validity of proposed method are discussed quantitatively and qualitatively.     

Developable polynomial surface approximation to smooth surfaces for fabrication parameters of a large curved shell plate by Differential Evolution

This paper presents a simple and efficient method to approximate a developable surface to a compound design surface by a polynomial. It is required to predict a final shape of roll bending in the fabrication of a curved shell plate. The roll bending process usually makes the cylindrical or conical curvature from an initial flat plate. It means that the final shape is developable or the surface representation has zero Gaussian curvature. The fabrication shape is important in order to estimate process parameters of roller bending.An optimization problem is formulated to determine the polynomial surface which is in the closest proximity to the design surface or the given shell plate, which is subjected to developability. The results and the efficiency of this algorithm are verified and evaluated by applying it to some shell plates which are obtained from a real ship model. The predicted bending shape becomes fundamental information in determining more process parameters for the fabrication of a compound curved shell plate.     

ISOCOMP: Unified geometric and material composition for optimal topology design

In this paper, a unified strategy is developed to simultaneously insert inclusions or holes of regular shape as well as redistribute the material to effect optimal topologies of solids. We demonstrate the unified optimal design strategy through three possible choices of design variables: (1) purely geometrical, (2) purely material, and (3) geometrical-material. We couple the geometrical approach with the topological derivative of the objective function and a condition derived for optimally inserting an infinitesimal ellipsoidal heterogeneity (hole or inclusion) into the structure. The approximations of the geometry, material and behavioral fields are isoparametric (or “isogeometric”) and are composed consistent with the Hierarchical Partition of Unity Field Compositions (HPFC) theory (Rayasam et al., Int J Numer Methods Eng 72(12):1452–1489, 2007). Specifically, analogous to the constructive solid geometry procedure of CAD, the complex material as well as the behavioral field is modeled hierarchically through a series of pair-wise compositions of primitive fields defined on the primitive geometrical domains. The geometrical, material and behavioral approximations are made using Non-Uniform Rational B-Splines (NURBS) basis functions. Thus, the proposed approach seamlessly unifies the explicit representation of boundary shapes with the implicit representations of boundaries arising out of material redistribution, and is termed ISOCOMP, or isoparametric compositions for topology optimization. The methodology is demonstrated first on a set of example problems that increase in complexity of design variable choice culminating in simultaneous optimization of hole location, hole shape and material distribution within the domain. This is followed by a detailed case study involving topology optimization of a bicycle “dropout.”     

Continuum topology optimization with non-probabilistic reliability constraints based on multi-ellipsoid convex model

Using a quantified measure for non-probab ilistic reliability based on the multi-ellipsoid convex model, the topology optimization of continuum structures in presence of uncertain-but-bounded parameters is investigated. The problem is formulated as a double-loop optimization one. The inner loop handles evaluation of the non-probabilistic reliability index, and the outer loop treats the optimum material distribution using the results from the inner loop for checking feasibility of the reliability constraints. For circumventing the numerical difficulties arising from its nested nature, the topology optimization problem with reliability constraints is reformulated into an equivalent one with constraints on the concerned performance. In this context, the adjoint variable schemes for sensitivity analysis with respect to uncertain variables as well as design variables are discussed. The structural optimization problem is then solved by a gradient-based algorithm using the obtained sensitivity. In the present formulation, the uncertain-but bounded uncertain variations of material properties, geometrical dimensions and loading conditions can be realistically accounted for. Numerical investigations illustrate the applicability and the validity of the present problem statement as well as the proposed numerical techniques. The computational results also reveal that non-probabilistic reliability-based topology optimization may yield more reasonable material layouts than conventional deterministic approaches. The proposed method can be regarded as an attractive supplement to the stochastic reliability-based topology optimization.     

Optimum material design of minimum structural compliance under seepage constraint

In filtration and chemical engineering industry the load carrying capacity and seepage performances are very important for a successful filter design. We study a two-scale structural design optimization problem to minimize structural compliance under given seepage flow rate and material porosity constraints. Structural size, shape and topology are given because of other functional requirements. Structural material used is macro homogeneous porous material with periodic microstructure and is to be designed. Since structural compliance and seepage performances in macro-scale are implicit functions of material microstructural topology, it becomes a two-scale design optimization problem. The cross scale sensitivities are derived by the adjoint method. A new volume preserving nonlinear density filter is proposed which makes the process of optimization iteration more stable. The optimization problem is solved by GCMMA. Examples under the equality constraints of different seepage flow rate are presented to illustrate the effectiveness of two-scale design optimization formulation and solution approach.     

Mixed elements in the optimal design of plates

The theory of design sensitivity analysis of structures, based on mixed finite element models, is developed for static, dynamic and stability constraints. The theory is applied to the optimal design of plates with minimum weight, subject to displacement, stress, natural frequencies and buckling stresses constraints. The finite element model is based on an eight node mixed isoparametric quadratic plate element, whose degrees of freedom are the transversal displacement and three moments per node. The corresponding nonlinear programming problem is solved using the commercially available ADS (Automated Design Synthesis) program. The sensitivities are calculated by analytical, semi-analytical and finite difference techniques. The advantages and disadvantages of mixed elements in design optimization of plates are discussed with reference to applications.