Showing papers in "Structural and Multidisciplinary Optimization in 2008"
TL;DR: A nondominated sorting genetic algorithm is employed to search for Pareto solution to a full-scale vehicle design problem that undergoes both the full frontal and 40% offset-frontal crashes, demonstrating the capability and potential of this procedure in solving the crashworthiness design of vehicles.
Abstract: In automotive industry, structural optimization for crashworthiness criteria is of special importance. Due to the high nonlinearities, however, there exists substantial difficulty to obtain accurate continuum or discrete sensitivities. For this reason, metamodel or surrogate model methods have been extensively employed in vehicle design with industry interest. This paper presents a multiobjective optimization procedure for the vehicle design, where the weight, acceleration characteristics and toe-board intrusion are considered as the design objectives. The response surface method with linear and quadratic basis functions is employed to formulate these objectives, in which optimal Latin hypercube sampling and stepwise regression techniques are implemented. In this study, a nondominated sorting genetic algorithm is employed to search for Pareto solution to a full-scale vehicle design problem that undergoes both the full frontal and 40% offset-frontal crashes. The results demonstrate the capability and potential of this procedure in solving the crashworthiness design of vehicles.
288 citations
TL;DR: The aim of the work is to analyze the performances of an alternative methodology to the ε-relaxation introduced in Cheng and Guo (Struct Optim 13:258–266, 1997), which handles the well-known stress singularity problem.
Abstract: The paper deals with the imposition of local stress constraints in topology optimization. The aim of the work is to analyze the performances of an alternative methodology to the e-relaxation introduced in Cheng and Guo (Struct Optim 13:258–266, 1997), which handles the well-known stress singularity problem. The proposed methodology consists in introducing, in the SIMP law used to apply stress constraints, suitable penalty exponents that are different from those that interpolate stiffness parameters. The approach is similar to the classical one because its main effect is to produce a relaxation of the stress constraints, but it is different in terms of convergence features. The technique is compared with the classical one in the context of stress-constrained minimum-weight topology optimization. Firstly, the problem is studied in a modified truss design framework, where the arising of the singularity phenomenon can be easily shown analytically. Afterwards, the analysis is extended to its natural context of topology bidimensional problems.
285 citations
TL;DR: In this paper, a meta-heuristic search method based on the analogy between the performance process of natural music and searching for solutions to optimization problems was developed for optimum design of steel frames.
Abstract: In this article, harmony search algorithm was developed for optimum design of steel frames. Harmony search is a meta-heuristic search method that has been developed recently. It bases on the analogy between the performance process of natural music and searching for solutions to optimization problems. The objective of the design algorithm is to obtain minimum weight frames by selecting suitable sections from a standard set of steel sections such as American Institute of Steel Construction (AISC) wide-flange (W) shapes. Strength constraints of AISC load and resistance factor design specification and displacement constraints were imposed on frames. The effectiveness and robustness of harmony search algorithm, in comparison with genetic algorithm and ant colony optimization-based methods, were verified using three steel frames. The comparisons showed that the harmony search algorithm yielded lighter designs.
230 citations
TL;DR: In this article, an extension of the hierarchical model for topology optimisation to three-dimensional structures is presented, which covers the simultaneous characterisation of the optimal topology of the structure and the optimal design of the cellular material used in its construction.
Abstract: This paper presents an extension of the hierarchical model for topology optimisation to three-dimensional structures. The problem addressed covers the simultaneous characterisation of the optimal topology of the structure and the optimal design of the cellular material used in its construction. In this study, hierarchical suggests that the optimisation model works at two interconnected levels, the global and local levels identified, respectively, with the structure and its material. The class of cellular materials, defining the material microstructure, is restricted to single scale cellular materials, with the cell geometry locally optimised for the given objective function and constraints. The model uses the asymptotic homogenisation model to obtain the equivalent material properties for the specific local microstructures designed using a SIMP based approach. The necessary optimality conditions for the hierarchical optimal design problem are discussed and approximated numerically by a proper finite element discretisation of the global and local analysis and design problems. Examples to explore and demonstrate the model developed are presented.
196 citations
TL;DR: In this article, the eigenvector dimension reduction (EDR) method was proposed for probability analysis that makes a significant improvement based on univariate dimension reduction method for estimating statistical moments of mildly nonlinear system responses in engineering applications.
Abstract: This paper presents the eigenvector dimension reduction (EDR) method for probability analysis that makes a significant improvement based on univariate dimension reduction (DR) method. It has been acknowledged that the DR method is accurate and efficient for assessing statistical moments of mildly nonlinear system responses in engineering applications. However, the recent investigation on the DR method has found difficulties of instability and inaccuracy for highly nonlinear system responses while maintaining reasonable efficiency. The EDR method integrates the DR method with three new technical components: (1) eigenvector sampling, (2) one-dimensional response approximation, and (3) a stabilized Pearson system. First, 2N+1 and 4N+1 eigenvector sampling schemes are proposed to resolve correlated and asymmetric random input variables. The eigenvector samples are chosen along the eigenvectors of the covariance matrix of random parameters. Second, the stepwise moving least squares (SMLS) method is proposed to accurately construct approximate system responses along the eigenvectors with the response values at the eigenvector samples. Then, statistical moments of the responses are estimated through recursive numerical integrations. Third, the stabilized Pearson system is proposed to predict probability density functions (PDFs) of the responses while eliminating singular behavior of the original Pearson system. Results for some numerical and engineering examples indicate that the EDR method is a very accurate, efficient, and stable probability analysis method in estimating PDFs, component reliabilities, and qualities of system responses.
172 citations
TL;DR: In this paper, a methodology of sequential optimization and reliability assessment for MDO is proposed to improve the efficiency of reliability-based MDO, which decouple the reliability analysis from MDO with sequential cycles of reliability analysis and deterministic MDO.
Abstract: With higher reliability and safety requirements, reliability-based design has been increasingly applied in multidisciplinary design optimization (MDO). A direct integration of reliability-based design and MDO may present tremendous implementation and numerical difficulties. In this work, a methodology of sequential optimization and reliability assessment for MDO is proposed to improve the efficiency of reliability-based MDO. The central idea is to decouple the reliability analysis from MDO with sequential cycles of reliability analysis and deterministic MDO. The reliability analysis is based on the first-order reliability method (FORM). In the proposed method, the reliability analysis and the deterministic MDO use two MDO strategies, the multidisciplinary feasible approach and the individual disciplinary feasible approach. The effectiveness of the proposed method is illustrated with two example problems.
169 citations
TL;DR: Multidisciplinary design optimization (MDO) is an emerging optimization method that considers a design environment with multiple disciplines and seven methods have been proposed for MDO.
Abstract: Recently, engineering systems are quite large and complicated. The design requirements are fairly complex and it is not easy to satisfy them by considering only one discipline. Therefore, a design methodology that can consider various disciplines is needed. Multidisciplinary design optimization (MDO) is an emerging optimization method that considers a design environment with multiple disciplines. Seven methods have been proposed for MDO. They are Multiple-discipline-feasible (MDF), Individual-discipline-feasible (IDF), All-at-once (AAO), Concurrent subspace optimization (CSSO), Collaborative optimization (CO), Bi-level integrated system synthesis (BLISS), and Multidisciplinary design optimization based on independent subspaces (MDOIS). Through several mathematical examples, the performances of the methods are evaluated and compared. Specific requirements are defined for comparison and new types of mathematical problems are defined based on the requirements. All the methods are coded and the performances of the methods are compared qualitatively and quantitatively.
147 citations
TL;DR: In this paper, the authors evaluated the performance of simulated annealing, genetic and evolutionary algorithms with respect to a single crash case and linear statics and dynamics, and a lateral impact problem for multi-criteria optimization.
Abstract: Rising complexity of industrial development in the automotive industry is leading to a higher degree of interdisciplinarity, which is especially true in the virtual design area. New methods and solution procedures have to be evaluated and integrated in the overall process. For example, in car body design process, a new topic emerged recently: the multidisciplinary optimization of car bodies with respect to crash and NVH (noise, vibration, and harshness). Because rigorous evaluation of appropriate numerical algorithms is still missing, an intense study was realized at the research center of BMW. The results are summarized in this article. Four benchmarks have been studied: (a) a full vehicle model for NVH analysis, (b) a simplified multidisciplinary problem with a single crash case and linear statics and dynamics, (c) a lateral impact problem for multi-criteria optimization, and finally, (d) a small shape optimization problem was included to demonstrate the potential of transferring the results to the more complex problem of optimizations based on real changes in the shape of the structures. Because response surface methods have already been discussed in the literature and because of their failure in certain industrial cases, the focus was set on the evaluation of stochastic algorithms: simulated annealing, genetic and evolutionary algorithms were tested. Finally, a complete industrial multidisciplinary example from the current development process was studied for the validation of the results.
135 citations
TL;DR: This note considers topology optimization of large scale 2D and 3D Stokes flow problems using parallel computations on a shared memory computer consisting of Sun UltraSparc IV CPUs.
Abstract: This note considers topology optimization of large scale 2D and 3D Stokes flow problems using parallel computations. We solve problems with up to 1,125,000 elements in 2D and 128,000 elements in 3D on a shared memory computer consisting of Sun UltraSparc IV CPUs.
132 citations
TL;DR: In this paper, a method for topology optimization of periodic structures using the bi-directional evolutionary structural optimization (BESO) technique is presented, where the designable domain is divided into a certain number of identical unit cells.
Abstract: This paper presents a method for topology optimization of periodic structures using the bi-directional evolutionary structural optimization (BESO) technique. To satisfy the periodic constraint, the designable domain is divided into a certain number of identical unit cells. The optimal topology of the unit cell is determined by gradually removing and adding material based on a sensitivity analysis. Sensitivity numbers that consider the periodic constraint for the repetitive elements are developed. To demonstrate the capability and effectiveness of the proposed approach, topology design problems of 2D and 3D periodic structures are investigated. The results indicate that the optimal topology depends, to a great extent, on the defined unit cells and on the relative strength of other non-designable part, such as the skins of sandwich structures.
114 citations
TL;DR: In this paper, the eigenvector dimension reduction (EDR) method is proposed for Bayesian reliability analysis, which takes a sensitivity-free approach for reliability analysis so that it is very efficient and accurate compared with other reliability methods such as FORM/SORM.
Abstract: In practical engineering design, most data sets for system uncertainties are insufficiently sampled from unknown statistical distributions, known as epistemic uncertainty. Existing methods in uncertainty-based design optimization have difficulty in handling both aleatory and epistemic uncertainties. To tackle design problems engaging both epistemic and aleatory uncertainties, reliability-based design optimization (RBDO) is integrated with Bayes theorem. It is referred to as Bayesian RBDO. However, Bayesian RBDO becomes extremely expensive when employing the first- or second-order reliability method (FORM/SORM) for reliability predictions. Thus, this paper proposes development of Bayesian RBDO methodology and its integration to a numerical solver, the eigenvector dimension reduction (EDR) method, for Bayesian reliability analysis. The EDR method takes a sensitivity-free approach for reliability analysis so that it is very efficient and accurate compared with other reliability methods such as FORM/SORM. Efficiency and accuracy of the Bayesian RBDO process are substantially improved after this integration.
TL;DR: In this article, a topology optimization method for synthesis of such structures, employing a time domain formulation, is developed and extended to synthesis of pulse converting structures with possible applications in optical communication.
Abstract: Structures exhibiting band gap properties, ie, having frequency ranges for which the structure attenuates propagating waves, have applications in damping of acoustic and elastic wave propagation and in optical communication A topology optimization method for synthesis of such structures, employing a time domain formulation, is developed The method is extended to synthesis of pulse converting structures with possible applications in optical communication
TL;DR: In this article, a coupled shape and topology optimization (CSTO) technique is proposed to study the layout design of the components and their supporting structures in a finite packing space.
Abstract: The purpose of this paper was to study the layout design of the components and their supporting structures in a finite packing space. A coupled shape and topology optimization (CSTO) technique is proposed. On one hand, by defining the location and orientation of each component as geometric design variables, shape optimization is carried out to find the optimal layout of these components and a finite-circle method (FCM) is used to avoid the overlap between the components. On the other hand, the material configuration of the supporting structures that interconnect components is optimized simultaneously based on topology optimization method. As the FE mesh discretizing the packing space, i.e., design domain, has to be updated itertively to accommodate the layout variation of involved components, topology design variables, i.e., density variables assigned to density points that are distributed regularly in the entire design domain will be introduced in this paper instead of using traditional pseudo-density variables associated with finite elements as in standard topology optimization procedures. These points will thus dominate the pseudo-densities of the surrounding elements. Besides, in the CSTO, the technique of embedded mesh is used to save the computing time of the remeshing procedure, and design sensitivities are calculated w.r.t both geometric variables and density variables. In this paper, several design problems maximizing structural stiffness are considered subject to the material volume constraint. Reasonable designs of components layout and supporting structures are obtained numerically.
TL;DR: A novel approach is proposed, combining proper orthogonal decomposition to decrease the amount of data exchanged between disciplines, with surrogate models based on moving least squares to reduce disciplines in an original 2D wing demonstrator involving two disciplines.
Abstract: Multidisciplinary optimization (MDO) is a growing field in engineering, with various applications in aerospace, aeronautics, car industry, etc. However, the presence of multiple disciplines leads to specific issues, which prevent MDO to be fully integrated in industrial design methodology. In practice, the key issues in MDO lie in the management of the interconnections between disciplines, along with the high number of simulations required to find a feasible multidisciplinary (optimal) solution. Therefore, in this paper, a novel approach is proposed, combining proper orthogonal decomposition to decrease the amount of data exchanged between disciplines, with surrogate models based on moving least squares to reduce disciplines. This method is applied to an original 2D wing demonstrator involving two disciplines (fluid and structure). The numerical results obtained for an optimization task show its benefits in diminishing both the interfaces between disciplines and the overall computational time.
TL;DR: It is shown how reliable structural design can be obtained using the proposed techniques based on the interplay of convex optimization and randomization, which permit to solve efficiently some suitable probabilistic relaxation of the indicated problems.
Abstract: Many real-world engineering design prob- lems are naturally cast in the form of optimization programs with uncertainty-contaminated data. In this context, a reliable design must be able to cope in some way with the presence of uncertainty. In this paper, we consider two standard philosophies for finding optimal solutions for uncertain convex opti- mization problems. In the first approach, classical in the stochastic optimization literature, the optimal de- sign should minimize the expected value of the ob- jective function with respect to uncertainty (average approach), while in the second one it should mini- mize the worst-case objective (worst-case or min-max approach). Both approaches are briefly reviewed in this paper and are shown to lead to exact and nu- merically efficient solution schemes when the uncer- tainty enters the data in simple form. For general uncertainty dependence however, these problems are numerically hard. In this paper, we present two tech- niques based on uncertainty randomization that permit to solve efficiently some suitable probabilistic relax- ation of the indicated problems, with full generality with respect to the way in which the uncertainty en- ters the problem data. In the specific context of truss topology design, uncertainty in the problem arises,
TL;DR: In this paper, a new utopia hyperplane is proposed to improve the original normalized normal constraint method using two approaches: a redefinition of the anchor points and an exact linear transformation between the design objectives space and the normalized space.
Abstract: In industrial applications, several objectives are often managed simultaneously (e.g., minimizing the cost and the weight of a mechanical structure satisfying some constraints). Although lots of optimization studies deal with only one objective, this approach is often not realistic for engineering optimization. Therefore, improvements in multiobjective optimization methods are required. This paper presents the formulation of a new utopia hyperplane that improves the proposal of the original normalized normal constraint method using two approaches: a redefinition of the anchor points and an exact linear transformation between the design objectives space and the normalized space. Both approaches always produce a normalized space with equal scales that improves the even distribution of the solutions over the Pareto frontier. Examples of the method proposed are presented related with mechanical engineering and structure design including a challenging non-convex Pareto frontier.
TL;DR: In this paper, the effectiveness of surrogate modeling of helicopter vibrations, and the use of the surrogates for minimization of helicopter rotor vibrations are studied, and accuracies of kriging, radial basis function interpolation, and polynomial regression surrogates are compared.
Abstract: The effectiveness of surrogate modeling of helicopter vibrations, and the use of the surrogates for minimization of helicopter rotor vibrations are studied. The accuracies of kriging, radial basis function interpolation, and polynomial regression surrogates are compared. In addition, the surrogates are used to generate an objective function which is employed in an optimization study. The design variables consist of the cross-sectional dimensions of the structural member of the blade and non-structural masses. The optimized blade is compared with a baseline rotor blade which resembles an MBB BO-105 blade. Results indicate that: (a) kriging surrogates best approximate vibratory hub loads over the entire design space and (b) the surrogates can be used effectively in helicopter rotor vibration reduction studies.
TL;DR: Issues and difficulties arising when a state-of-the-art parallel linear solver is applied to topology optimization problems and attempts to improve it by applying additional scaling and/or preconditioning strategies are discussed.
Abstract: Parallel computing is an integral part of many scientific disciplines In this paper, we discuss issues and difficulties arising when a state-of-the-art parallel linear solver is applied to topology optimization problems Within the topology optimization framework, we cannot readjust domain decomposition to align with material decomposition, which leads to the deterioration of performance of the substructuring solver We illustrate the difficulties with detailed condition number estimates and numerical studies We also report the practical performances of finite element tearing and interconnection/dual–primal solver for topology optimization problems and our attempts to improve it by applying additional scaling and/or preconditioning strategies The performance of the method is finally illustrated with large-scale topology optimization problems coming from different optimal design fields: compliance minimization, design of compliant mechanisms, and design of elastic surface wave-guides
TL;DR: In this paper, the authors combine the advanced multidisciplinary design optimization (MDO) methodology to optimize vehicle structure by using uniform Latin hypercube sampling and subset selection regression methods to construct the response surface models for the highly nonlinear impact and seatbelt pull responses.
Abstract: Tailor rolled blank (TRB) is an emerging steel rolling process to produce lightweight vehicle components. It allows continuous metal thickness changes, and as a result, it offers opportunities for automotive design in weight reduction, part complexity reduction, reduced capital investment, yet, maintains equal to or better strength characteristics. The objective of this research is to take advantages of the TRB manufacturing technology and combine with the advanced multidisciplinary design optimization (MDO) methodology to optimize vehicle structure. The process begins with noise vibration and harshness (NVH) optimization. The outputs of the optimal NVH response sensitivities are employed to build the first order response surface models. Uniform Latin Hypercube sampling and subset selection regression methods are used to construct the response surface models for the highly nonlinear impact and seatbelt pull responses. The optimal NVH design is then used as the starting point for MDO to obtain the optimal thickness profiles for the TRB parts. A vehicle application considering multiple impact modes, seatbelt pulls, and NVH, is used to demonstrate the proposed process for vehicle underbody TRB design. Results of this MDO TRB study is presented and discussed.
TL;DR: The virtual distortion method has proved to be a versatile reanalysis tool in various applications, including structures and truss-like systems, both in statics and dynamics, in linear and nonlinear analysis.
Abstract: For 20 years of development, the virtual distortion method (VDM) has proved to be a versatile reanalysis tool in various applications, including structures and truss-like systems. This article presents a summary of principal achievements, demonstrating the capabilities of the VDM both in statics and dynamics, in linear and nonlinear analysis. The major advantage of VDM is its exactness and no need for matrix inversion in the reanalysis algorithm. The influence matrix—numerical core of the VDM—contains the whole mechanical knowledge about a structure, by looking at all global responses due to local disturbances. The strength of the method is demonstrated for truss structures.
TL;DR: In this paper, an ant algorithm consisting of the Ant System and API (after “apicalis” in Pachycondylaapsicalis) was proposed to find optimal truss structures to achieve minimum weight objective under stress, deflection, and kinematic stability constraints.
Abstract: An ant algorithm, consisting of the Ant System and API (after “apicalis” in Pachycondylaapicalis) algorithms, was proposed in this study to find optimal truss structures to achieve minimum weight objective under stress, deflection, and kinematic stability constraints. A two-stage approach was adopted in this study; first, the topology of the truss structure was optimized from a given ground structure employing the Ant System algorithm due to its discrete characteristic, and then the size and/or shape of member was optimized utilizing the API algorithm. The effectiveness of the proposed ant algorithm was evaluated through numerous different 2-D and 3-D truss-structure problems. The proposed algorithm was observed to find truss structures better than those reported in the literature. Moreover, multiple different truss topologies with almost equal overall weights can be found simultaneously.
TL;DR: In this article, a new univariate decomposition method for design sensitivity analysis and reliability-based design optimization of mechanical systems subject to uncertain performance functions in constraints is presented, which involves a novel univariate approximation of a general multivariate function in the rotated Gaussian space for reliability analysis.
Abstract: This paper presents a new univariate decomposition method for design sensitivity analysis and reliability-based design optimization of mechanical systems subject to uncertain performance functions in constraints. The method involves a novel univariate approximation of a general multivariate function in the rotated Gaussian space for reliability analysis, analytical sensitivity of failure probability with respect to design variables, and standard gradient-based optimization algorithms. In both reliability and sensitivity analyses, the proposed effort has been reduced to performing multiple one-dimensional integrations. The evaluation of these one-dimensional integrations requires calculating only conditional responses at selected deterministic input determined by sample points and Gauss–Hermite integration points. Numerical results indicate that the proposed method provides accurate and computationally efficient estimates of the sensitivity of failure probability, which leads to accurate design optimization of uncertain mechanical systems.
TL;DR: In this paper, the authors proposed a new mixed-variable design optimization (MVDO) method using the performance measure approach (PMA) for structural design problems with sufficient and insufficient data.
Abstract: If the statistical data for the input uncertainties are sufficient to construct the distribution function, the input uncertainties can be treated as random variables to use the reliability-based design optimization (RBDO) method; otherwise, the input uncertainties can be treated as fuzzy variables to use the possibility-based design optimization (PBDO) method. However, many structural design problems include both input uncertainties with sufficient and insufficient data. This paper proposes a new mixed-variable design optimization (MVDO) method using the performance measure approach (PMA) for such design problems. For the inverse analysis, this paper proposes a new most probable/possible point (MPPP) search method called maximal failure search (MFS), which is an integration of the enhanced hybrid mean value method (HMV+) and maximal possibility search (MPS) method. This paper also improves the HMV+ method using an angle-based interpolation. Mathematical and physical examples are used to demonstrate the proposed inverse analysis method and MVDO method.
TL;DR: A new algorithm called the ranking selection-based PSO (RSPSO), which is an effective and widely applicable optimizer for optimization problems in engineering design in comparison with the state-of-the-art algorithms in the area.
Abstract: Particle swarm optimization (PSO) algorithms have been proposed to solve optimization problems in engineering design, which are usually constrained (possibly highly constrained) and may require the use of mixed variables such as continuous, integer, and discrete variables. In this paper, a new algorithm called the ranking selection-based PSO (RSPSO) is developed. In RSPSO, the objective function and constraints are handled separately. For discrete variables, they are partitioned into ordinary discrete and categorical ones, and the latter is managed and searched directly without the concept of velocity in the standard PSO. In addition, a new ranking selection scheme is incorporated into PSO to elaborately control the search behavior of a swarm in different search phases and on categorical variables. RSPSO is relatively simple and easy to implement. Experiments on five engineering problems and a benchmark function with equality constraints were conducted. The results indicate that RSPSO is an effective and widely applicable optimizer for optimization problems in engineering design in comparison with the state-of-the-art algorithms in the area.
TL;DR: In this paper, the authors proposed a general applicable optimisation strategy that makes use of FEM simulations of metal forming processes, which can be applied to a hydroforming process in general and more specific to a specific metal forming problem.
Abstract: Product improvement and cost reduction have always been important goals in the metal forming industry. The rise of finite element (FEM) simulations for processes has contributed to these goals in a major way. More recently, coupling FEM simulations to mathematical optimisation techniques has shown the potential to make a further giant contribution to product improvement and cost reduction. Much research on the optimisation of metal forming processes has been published during the last couple of years. Although the results are impressive, the optimisation techniques are generally only applicable to specific optimisation problems for specific products and specific metal forming processes. As a consequence, applying optimisation techniques to other metal forming problems requires a lot of optimisation expertise, which forms a barrier for more general industrial application of these techniques. In this paper, we overcome this barrier by proposing a generally applicable optimisation strategy that makes use of FEM simulations of metal forming processes. It consists of a structured methodology for modelling optimisation problems related to metal forming. Subsequently, screening is applied to reduce the size of the optimisation problem by selecting only the most important design variables. Finally, the reduced optimisation problem is solved by an efficient optimisation algorithm. The strategy is generally applicable in a sense that it is not constrained to a certain type of metal forming problem, product or process. Also, any FEM code may be included in the strategy. Furthermore, the structured approach for modelling and solving optimisation problems should enable non-optimisation specialists to apply optimisation techniques to improve their products and processes. The optimisation strategy has been successfully applied to a hydroforming process, which demonstrates the potential of the optimisation of metal forming processes in general and more specific the proposed optimisation strategy.
TL;DR: In this article, the design objective is the maximization of the buckling load, and the design variable is considered as the fiber orientation, where the first-order shear deformation theory is used for the finite element analysis and the modified feasible direction method is used to solve the optimization problems.
Abstract: This paper addresses optimal design of simply supported symmetrically laminated composite plates with central circular holes. The design objective is the maximization of the buckling load, and the design variable is considered as the fiber orientation. The first-order shear deformation theory is used for the finite element analysis. The study is complicated because the effects of bending–twisting coupling are also included for the buckling optimization. The modified feasible direction method is used to solve the optimization problems. Finally, the effect of different number of layers, boundary conditions, width-to-thickness ratio, plate aspect ratios, hole daimeter-to-width ratio, and load ratios on the results is investigated.
TL;DR: The present paper further elaborates the SAP for PMA and presents error analysis and shows that in the ɛ-vicinity of optimum design and corresponding MPTP, the difference between the Taylor expansion of PPM and the linear expansion of approximate PPM is of higher order of ɔ.
Abstract: Compared to the traditional deterministic optimization based on safety factors, the probabilistic structural design optimization (PSDO) is considered to be a more rational design philosophy because of reasonable account of uncertainties in material properties, loading, boundary condition and geometry, and even mathematical representation of the system model. However, it is well known that the computation for PSDO can be prohibitive when the associated function evaluation is expensive. As a result, many approximate PSDO methods have been developed in recent literatures. In previous works, we developed two sequential approximate programming (SAP) strategies for PSDO based on reliability index approach (RIA) and performance measure approach (PMA). In PMA with SAP, a sequence of approximate programming of PSDO was formulated and solved before the final optimum was located. In each subprogramming, rather than relying on direct linear Taylor expansion of the probabilistic performance measure (PPM), we developed a formulation for approximate PPM at the current design point and used its linearization instead. The approximate PPM and its sensitivity were obtained by approximating the optimality conditions in the vicinity of the minimum performance target point (MPTP). The present paper further elaborates the SAP for PMA. In addition to detailed description of the algorithm, we present error analysis and show that in the ɛ-vicinity of optimum design and corresponding MPTP, the difference between the Taylor expansion of PPM and the linear expansion of approximate PPM is of higher order of ɛ. Four examples are optimized by six algorithms appearing in recent literatures for efficiency comparison. The effect of target reliability index and statistical distribution of random variables on the comparison is discussed. The third example shows that PMA with SAP performs well even for the problem for which reliability index calculation by first order reliability method (FORM) fails. Finally, the fourth example with 144 probabilistic constraints is shown to demonstrate the effectiveness of PMA with SAP. All example results illustrate that with the algorithm PMA with SAP, we get concurrent convergence of both design optimization and probabilistic performance measure calculation, which agrees well with the error analysis.
TL;DR: In this article, an element-based search scheme is introduced to identify the load surface in a topology optimization of continuum structures with design-dependent loads, where the load surfaces are formed by the connection of the real boundary of elements and the pressures are transferred directly to corresponding element nodes.
Abstract: The identification of the load surface is a key problem in solving topology optimization of continuum structures with design-dependent loads. In this paper, an element-based search scheme is introduced to identify the load surface. The load surfaces are formed by the connection of the real boundary of elements and the pressures are transferred directly to corresponding element nodes. The search scheme is very convenient to apply and is found to be efficient and effective in identifying the load surfaces. Only slight modifications to the load codes in the routine procedure are required and there is no need to calculate the sensitivities of the load with respect to the material density changes. Numerical examples are presented to demonstrate the efficiency of the boundary search scheme.
TL;DR: In this article, the implicit topology description function is integrated into Reproducing Kernel Particle Method and presents a new implementation of topology optimization of continua, which is carried out by using the meshless reproducing kernel approximations.
Abstract: The implicit topology description function method is integrated into Reproducing Kernel Particle Method and presents a new implementation of topology optimization of continua. The structural response analysis and the sensitivity analysis are carried out by using the meshless reproducing kernel approximations. Compared with mesh-based methods, the construction of an explicit mesh and the definition of nodal connectivity are avoided. The differences between the finite element method and the meshless method for topology optimization problems are highlighted. Formulations for imposition of concentrated forces and analysis of sensitivity in meshless method are derived in details. Several two-dimensional linear elastic topology optimization problems are solved successfully by the proposed method. The method is found robust and no checkerboarding is found in our numerical examples. Without any worry of mesh-entanglement, the method is expected to be further developed for the topology optimization of nonlinear structures with large deformations.
TL;DR: In this article, an adaptive inner-front level set method is proposed to solve compliance minimization problems of linear elastic structures, where the size, position, and number of new innerfronts during the optimization process can be globally and consistently identified.
Abstract: A new topology optimization using adaptive inner-front level set method is presented. In the conventional level set-based topology optimization, the optimum topology strongly depends on the initial level set due to the incapability of inner-front creation during the optimization process. In the present work, in this regard, an algorithm for inner-front creation is proposed in which the sizes, the positions, and the number of new inner-fronts during the optimization process can be globally and consistently identified. In the algorithm, the criterion of inner-front creation for compliance minimization problems of linear elastic structures is chosen as the strain energy density along with volumetric constraint. To facilitate the inner-front creation process, the inner-front creation map is constructed and used to define new level set function. In the implementation of inner-front creation algorithm, to suppress the numerical oscillation of solutions due to the sharp edges in the level set function, domain regularization is carried out by solving the edge smoothing partial differential equation (smoothing PDE). To update the level set function during the optimization process, the least-squares finite element method (LSFEM) is adopted. Through the LSFEM, a symmetric positive definite system matrix is constructed, and non-diffused and non-oscillatory solution for the hyperbolic PDE such as level set equation can be obtained. As applications, three-dimensional topology optimization of shell structures is treated. From the numerical examples, it is shown that the present method brings in much needed flexibility in topologies during the level set-based topology optimization process.