On the enhancement of computation and exploration of discretization approaches for meshless shape design sensitivity analysis

A new implementation of Reproducing Kernel Particle Method (RKPM) is proposed to enhance the process of shape design sensitivity analysis (DSA). The acceleration process is accomplished by expressing RKPM shape functions and their derivatives explicitly in terms of kernel function moments. In addition, two different discretization approaches are explored elaborately, which emanate from discretizing design sensitivity equation using the direct differentiation method. Comparison of these two approaches is made, and the equivalence of these two superficially different approaches is demonstrated through two elastostatics problems. The effectiveness of the enhanced RKPM is also verified by comparison of consumption of computer time between the classical method and the improved method.     

Structural optimization using global stress-deviation objective function via the level-set method

The paper deals with minimum stress design using a novel stress-related objective function based on the global stress-deviation measure. The shape derivative, representing the shape sensitivity analysis of the structure domain, is determined for the generalized form of the global stress-related objective function. The optimization procedure is based on the domain boundary evolution via the level-set method. The elasticity equations are, instead of using the usual ersatz material approach, solved by the extended finite element method. The Hamilton-Jacobi equation is solved using the streamline diffusion finite element method. The use of finite element based methods allows a unified numerical approach with only one numerical framework for the mechanical problem as also for the boundary evolution stage. The numerical examples for the L-beam benchmark and the notched beam are given. The results of the structural optimization problem, in terms of maximum von Mises stress corresponding to the obtained optimal shapes, are compared for the commonly used global stress measure and the novel global stress-deviation measure, used as the stress-related objective functions.     

Design optimization of stiffened storage tank for spacecraft

Stiffened storage tank is an important structural component in spacecraft. Its structural weight is one of the key criterions in the design phase. This paper focuses on the design optimization of the structure by using finite element method, structural sensitivity analysis techniques, and sequential linear/quadratic programming aimed to reduce the structural weight. Design variables include the numbers of stiffeners, stiffeners’ section dimensions, and shell thickness distribution. Detailed finite element modeling processes are presented, which are the ways to construct the stiffener (beam orientation and offset) and shell elements and the ways to determine the analysis model and structural boundary conditions. A brief introduction to sensitivity analysis and optimization solution algorithm is also given. Main attention is paid to the studies of design optimization of the tank structure, including the selection of design cases, evaluation, and comparison of the optimal results. There are six design cases considered in the design procedures. Numerical results show that by using the above computational techniques, the structural weight is effectively reduced. In this work, MSC.Patran/Nastran is employed to construct the Finite Element Model (FEM), and JIFEX, which is developed in our group, is used to conduct the structural design optimization. JIFEX is a structural analysis and optimization software package developed by Gu and colleagues in the Dalian University of Technology Department of Engineering Mechanics. Among its many functions is the ability to analyze and optimize piezoelectric smart structures.     

Design of distributed compliant micromechanisms with an implicit free boundary representation

In this paper, a parameterization approach is presented for structural shape and topology optimization of compliant mechanisms using a moving boundary representation. A level set model is developed to implicitly describe the structural boundary by embedding into a scalar function of higher dimension as zero level set. The compactly supported radial basis function of favorable smoothness and accuracy is used to interpolate the level set function. Thus, the temporal and spatial initial value problem is now converted into a time-separable parameterization problem. Accordingly, the more difficult shape and topology optimization of the Hamilton–Jacobi equation is then transferred into a relatively easy size optimization with the expansion coefficients as design variables. The design boundary is therefore advanced by applying the optimality criteria method to iteratively evaluate the size optimization so as to update the level set function in accordance with expansion coefficients of the interpolation. The optimization problem of the compliant mechanism is established by including both the mechanical efficiency as the objective function and the prescribed material usage as the constraint. The design sensitivity analysis is performed by utilizing the shape derivative. It is noted that the present method is not only capable of simultaneously addressing shape fidelity and topology changes with a smooth structural boundary but also able to avoid some of the unfavorable numerical issues such as the Courant–Friedrich–Levy condition, the velocity extension algorithm, and the reinitialization procedure in the conventional level set method. In particular, the present method can generate new holes inside the material domain, which makes the final design less insensitive to the initial guess. The compliant inverter is applied to demonstrate the availability of the present method in the framework of the implicit free boundary representation.     

Mesh refinement for shape optimization

The numerical solution of shape optimization problems is considered. The algorithm of successive optimization based on finite element techniques and design sensitivity analysis is applied. Mesh refinement is used to improve the quality of finite element analysis and the computed numerical solution. The norm of the variation of the Lagrange augmented functional with respect to boundary variation (residuals in necessary optimality conditions) is taken as an a posteriori error estimator for optimality conditions and the Zienkiewicz—Zhu error estimator is used to improve the quality of structural analysis. The examples presented show meaningful effects obtained by means of mesh refinement with a new error estimator.     

Numerical method for shape optimization using T-spline based isogeometric method

Numerical methods for shape design sensitivity analysis and optimization have been developed for several decades. However, the finite-element-based shape design sensitivity analysis and optimization have experienced some bottleneck problems such as design parameterization and design remodeling during optimization. In this paper, as a remedy for these problems, an isogeometric-based shape design sensitivity analysis and optimization methods are developed incorporating with T-spline basis. In the shape design sensitivity analysis and optimization procedure using a standard finite element approach, the design boundary should be parameterized for the smooth variation of the boundary using a separate geometric modeler, such as a CAD system. Otherwise, the optimal design usually tends to fall into an undesirable irregular shape. In an isogeometric approach, the NURBS basis function that is used in representing the geometric model in the CAD system is directly used in the response analysis, and the design boundary is expressed by the same NURBS function as used in the analysis. Moreover, the smoothness of the NURBS can allow the large perturbation of the design boundary without a severe mesh distortion. Thus, the isogeometric shape design sensitivity analysis is free from remeshing during the optimization process. In addition, the use of T-spline basis instead of NURBS can reduce the number of degrees of freedom, so that the optimal solution can be obtained more efficiently while yielding the same optimum design shape.     

A study on X-FEM in continuum structural optimization using a level set model

In this paper, we implement the extended finite element method (X-FEM) combined with the level set method to solve structural shape and topology optimization problems. Numerical comparisons with the conventional finite element method in a fixed grid show that the X-FEM leads to more accurate results without increasing the mesh density and the degrees of freedom. Furthermore, the mesh in X-FEM is independent of the physical boundary of the design, so there is no need for remeshing during the optimization process. Numerical examples of mean compliance minimization in 2D are studied in regard to efficiency, convergence and accuracy. The results suggest that combining the X-FEM for structural analysis with the level set based boundary representation is a promising approach for continuum structural optimization.     

Structural application of a shape optimization method based on a genetic algorithm

This paper presents a structural application of a shape optimization method based on a Genetic Algorithm (GA). The method produces a sequence of fixed-distance step-wise movements of the boundary nodes of a finite element model to derive optimal shapes from an arbitrary initial design space. The GA is used to find the optimal or near-optimal combination of boundary nodes to be moved for a given step movement. The GA uses both basic and advanced operators. For illustrative purposes, the method has been applied to structural shape-optimization. The shape-optimization methodology presented allows local optimization, where only crucial parts of a structure are optimized as well as global shape-optimization which involves finding the optimal shape of the structure as a whole for a given environment as described by its loading and freedom conditions. Material can be removed or added to reach the optimal shape. Two examples of structural shape optimization are included showing local and global optimization through material removal and addition. Received October 14, 1999     

Numerical method for shape optimization using meshfree method

A numerical method for continuum-based shape design sensitivity analysis and optimization using the meshfree method is proposed. The reproducing kernel particle method is used for domain discretization in conjunction with the Gauss integration method. Special features of the meshfree method from a sensitivity analysis viewpoint are discussed, including the treatment of essential boundary conditions, and the dependence of the shape function on the design variation. It is shown that the mesh distortion that exists in the finite element-based design approach is effectively resolved for large shape changing design problems through 2-D and 3-D numerical examples. The number of design iterations is reduced because of the accurate sensitivity information.     

Multiparametric shape optimal design of disks

The external boundary of a structure described by a set of design parameters undergoes shape modification. Arbitrary stress, strain and displacement functionals are defined within the domain of the structure and its first- and second-order sensitivities with respect to varying structural shape are discussed. The optimal shape design problem is then formulated and solved using the first- and second-order sensitivity information. The iterative analysis-redesign algorithm is formulated using the finite element method. Some illustrative examples are included.     

Free vibration analysis and shape optimization of variable thickness plates and shells—II. Sensitivity analysis and shape optimization

In this paper an automated approach is used to carry out sensitivity analysis and to obtain optimum shapes for plates and shells in which the natural frequencies are maximized. The free vibration analysis is carried out with the nine-noded, degenerated, Huang-Hinton shell element implemented and tested in Part I of this paper. Design variables that specify either the shape or thickness distribution of the structures are considered. Special attention is focused on the sensitivity calculations and problems connected with their accuracy and performance are highlighted when the semi-analytical and finite difference methods are used. Advantages and disadvantages of each method are discussed. The optimal solution is found by the use of a structural optimization algorithm which integrates the finite element module (Part I), sensitivity analysis and a mathematical programming method: sequential quadratic programming (SQP). Optimal forms are then obtained for a set of benchmark examples using the two sensitivity analysis techniques and their results are compared. The results obtained for optimum solutions in the present paper justify the usage of the semi-analytical method for sensitivities calculations for structural shape optimization purposes.     

Structural optimization of the cross-beam of a gantry machine tool based on grey relational analysis

Crossbeam structural design of gantry machine tool is a multi-level, multi-index and multi-scheme decision-making problem. In order to solve the above problem, the optimum seeking model of crossbeam structure was built through using the grey relational analysis and Analytic Hierarchy Process. The finite element analysis of the static and dynamic performance parameters for four kinds of crossbeam structural schemes designed had been done, and the optimal design scheme was selected by using the optimum seeking model. After conducting sensitivity analysis for the optimal crossbeam selected, the reasonable design variables were obtained, and the dynamic optimization design model of crossbeam was established. Six groups of non-inferior solutions were obtained after solving the optimization design model. The optimal solution was selected from the non-inferior solution set through using the crossbeam structural optimization method based on grey relational analysis again, which makes the crossbeam’s dynamic performance improving greatly. The dynamic experiments on the crossbeams before and after optimization design were conducted, then the experimental results show that the first four order natural frequencies of the crossbeam increase 17.56 %, 19.36 %, 17.04 % and 19.58 % respectively, which proves that the structural optimization design method based on grey relational analysis proposed in this paper is reasonable and practicable.     

Eulerian shape design sensitivity analysis and optimization with a fixed grid

Conventional shape optimization based on the finite element method uses Lagrangian representation in which the finite element mesh moves according to shape change, while modern topology optimization uses Eulerian representation. In this paper, an approach to shape optimization using Eulerian representation such that the mesh distortion problem in the conventional approach can be resolved is proposed. A continuum geometric model is defined on the fixed grid of finite elements. An active set of finite elements that defines the discrete domain is determined using a procedure similar to topology optimization, in which each element has a unique shape density. The shape design parameter that is defined on the geometric model is transformed into the corresponding shape density variation of the boundary elements. Using this transformation, it has been shown that the shape design problem can be treated as a parameter design problem, which is a much easier method than the former. A detailed derivation of how the shape design velocity field can be converted into the shape density variation is presented along with sensitivity calculation. Very efficient sensitivity coefficients are calculated by integrating only those elements that belong to the structural boundary. The accuracy of the sensitivity information is compared with that derived by the finite difference method with excellent agreement. Two design optimization problems are presented to show the feasibility of the proposed design approach.     

BLISS/S: a new method for two-level structural optimization

The paper describes a two-level method for structural optimization for a minimum weight under the local strength and displacement constraints. The method divides the optimization task into separate optimizations of the individual substructures (in the extreme, the individual components) coordinated by the assembled structure optimization. The substructure optimizations use local cross-sections as design variables and satisfy the highly nonlinear local constraints of strength and buckling. The design variables in the assembled structure optimization govern the structure overall shape and handle the displacement constraints. The assembled structure objective function is the objective in each of the above optimizations. The substructure optimizations are linked to the assembled structure optimization by the sensitivity derivatives. The method was derived from a previously reported two-level optimization method for engineering systems, e.g. aerospace vehicles, that comprise interacting modules to be optimized independently, coordination provided by a system-level optimization. This scheme was adapted to structural optimization by treating each substructure as a module in a system, and using the standard finite element analysis as the system analysis. A numerical example, a hub structure framework, is provided to show the new method agreement with a standard, no-decomposition optimization. The new method advantage lies primarily in the autonomy of the individual substructure optimization that enables concurrency of execution to compress the overall task elapsed time. The advantage increases with the magnitude of that task. Received December 5, 1999?Revised mansucript received April 26, 2000     

Shape optimization of structures under earthquake loadings

The optimum design of structures under static loads is well-known in the design world; however, structural optimization under dynamic loading faces many challenges in real applications. Issues such as the time-dependent behavior of constraints, changing the design space in the time domain, and the cost of sensitivities could be mentioned. Therefore, optimum design under dynamic loadings is a challenging task. In order to perform efficient structural shape optimization under earthquake loadings, the finite element-based approximation method for the transformation of earthquake loading into the equivalent static loads (ESLs) is proposed. The loads calculated using this method are more accurate and reliable than those obtained using the building regulations. The shape optimization of the structures is then carried out using the obtained ESLs. The proposed methodologies are transformed into user-friendly computer code, and their capabilities are demonstrated using numerical examples.     

Solution to boundary shape optimization problem of linear elastic continua with prescribed natural vibration mode shapes

This paper presents a practical method of numerical analysis for boundary shape optimization problems of linear elastic continua in which natural vibration modes approach prescribed modes on specified sub-boundaries. The shape gradient for the boundary shape optimization problem is evaluated with optimality conditions obtained by the adjoint variable method, the Lagrange multiplier method, and the formula for the material derivative. Reshaping is accomplished by the traction method, which has been proposed as a solution to boundary shape optimization problems of domains in which boundary value problems of partial differential equations are defined. The validity of the presented method is confirmed by numerical results of three-dimensional beam-like and plate-like continua.     

A level set method for structural shape and topology optimization using radial basis functions

This paper presents an alternative level set method for shape and topology optimization of continuum structures. An implicit free boundary representation model is established by embedding structural boundary into the zero level set of a higher-dimensional level set function. An explicit parameterization scheme for the level set surface is proposed by using radial basis functions with compact support. In doing so, the originally more difficult shape and topology optimization, driven by the temporal and spatial Hamilton–Jacobi partial differential equation (PDE), is transformed into a relatively easier size optimization of the expansion coefficients of the basis functions. The design optimization is converted to an iterative numerical process that combines the parameterization with a derivation of the shape sensitivity of the design functions, so as to allow using mathematical programming algorithms to solve the level set-based design problem and avoid directly solving the Hamilton–Jacobi PDE. Furthermore, a numerically more stable and efficient volume integration scheme is proposed to implement calculations of the shape derivatives, leading to the creation of new holes which are generated initially along the boundary and then propagated to the interior of the design domain. Two widely studied examples are used to demonstrate the effectiveness of the proposed optimization method.