Dangers inherited in shells optimized with linear assumptions

In this paper, continuum-based sizing and configuration design sensitivity analysis (DSA) and optimization methods using the established transient dynamic analysis code DYNA3D are developed. For transient dynamic analysis, the explicit central difference method is used. For the structural domain, the updated Lagrangian formulation is used since it is appropriate for the path dependent problem. The first-order variations of the kinetic energy form, strain energy form, load form, and kinematic and structural responses with respect to the sizing and configuration design variables are derived. For the updated Lagrangian formulation, the design velocity field that defines the mapping between the initial and perturbed designs is updated at each updated configuration. The direct differentiation method is used in this paper for DSA. Numerical implementation of sizing DSA, configuration DSA, and optimization of a vehicle frame and A-pillar models has been carried out using the nonlinear finite element analysis (FEA) code DYNA3D.

In this paper is described a procedure to synthesize the optimal topology, shape, and size of compliant continua for a given nonlinear output path. The path is prescribed using a finite number of distinct precision points much in accordance with the synthesis for path generation in traditional kinematics. Geometrically nonlinear analysis is employed to model large displacements of the constituent members. It is also essential to employ nonlinear analysis to allow the output port to negotiate the prescribed path accurately. The topology synthesis problem is addressed in its original binary form in that the corresponding design variables are only allowed to assume values of “0” for no material and “1” for the material present at a site in the design region. Shape and size design variables are modeled using continuous functions. Owing to the discrete nature of topology design variables, since gradient based optimization methods cannot be employed, a genetic algorithm is used that utilizes only the objective values to approach an optimum solution. A notable advantage of a genetic algorithm over its gradient based counterparts is the implicit circumvention of nonconvergence in the large displacement analysis, which is another reason why a genetic algorithm is chosen for optimization. The least squared objective is used to compare the design and desired output responses. To allow a user to specify preference for a precision point, individual multiple least squared objectives, same in number as the precision points are used. The multiple objectives are solved using Nondominated Sorting in Genetic Algorithm (NSGA-II) to yield a set of pareto optimal solutions. Thus, multiple solutions for compliant mechanisms can be obtained such that a mechanism can traverse one or some precision points among those specified more precisely. To traverse the entire path, a solution that minimizes the sum of individual least square objectives may be chosen. Synthesis examples are presented to demonstrate the usefulness of the proposed method that is capable of generating a solution that can be manufactured as is without requiring any interpretation.

A shape design sensitivity analysis and optimization procedure is proposed using a meshfree method. A CAD tool connection is established to facilitate the seamless integration of structural modeling, simulation, and design optimization. A new CAD-based method for design velocity computation is proposed, applicable to both the meshfree method and the finite element method. A shape design variable can be chosen among the geometric parameters defined in the CAD model in order to support a concurrent engineering environment. A new CAD-based method for updating the discrete model is proposed, such that the meshfree model is automatically updated in a continuous and consistent way during design optimization. The proposed method is applied to 2D and 3D linear elastic problems, as well as to a large deformation hyperelastic problem.

Several new formulations for structural design sensitivity analysis (DSA) of elastic solids are presented that consistently account for steady and transient thermal loading, along with centrifugal and gravity loading. The approach employs implicit differentiation of the discretized boundary integral equations and facilitates the production of accurate and efficient computational procedures that obviate the need for matrix factorizations in the DSA step. For centrifugal, gravity and general thermal loading, a new DSA concept employing a particular integral technique is developed. For centrifugal, gravity and steady thermal effects, a surface integration approach is also presented, and the two methods compared. Alternative strategies for modelling the behaviour of temperature distributions of evolving shapes undergoing shape optimization and the ramifications of these strategies in the DSA process are also discussed. It is shown that, for certain classes of thermoelastic problems, shape optimization can be performed without the need to incorporate thermal analysis or DSA into the overall process. A thermoelastic surface stress sensitivity analysis formulation is presented that does not require additional numerical integration. A new thermoelastic sharp corner traction component consistent sensitivity analysis formulation is also presented. This technique is shown to work for any initial stress distribution. Numerical sensitivity results are presented for many problems with analytical solutions.

Three methods for design sensitivity such as numerical differentiation, analytical method and semianalytical method have been developed for the last three decades. Although analytical design sensitivity analysis is exact, it is hard to implement for practical design problems. Therefore, numerical method such as finite difference method is widely used to simply obtain the design sensitivity in most cases. The numerical differentiation is sufficiently accurate and reliable for most linear problems. However, it turns out that the numerical differentiation is inefficient and inaccurate because its computational cost depends on the number of design variables and large numerical errors can be included especially in nonlinear design sensitivity analysis. Thus semi-analytical method is more suitable for complicated design problems. Moreover semianalytical method is easy to be performed in design procedure, which can be coupled with an analysis solver such as commercial finite element package. In this paper, implementation procedure for the semi-analytical design sensitivity analysis outside of the commercial finite element package is studied and computational technique is proposed, which evaluates the pseudo-load for design sensitivity analysis easily by using the design variation of corresponding internal nodal forces. Errors in semi-analytical design sensitivity analysis are examined and numerical examples are illustrated to confirm the reduction of numerical error considerably.

Free vibration analysis and shape optimization of box-girder bridges in straight and curved planform

It is important to establish an integrated CAD system for mechanical engineering. The geometric model is constructed with primitives, such as wire frame, surface and solid on this system. The finite element model is induced from the substructure, which is a component of the geometric model. After the extraction of the substructure from the model, the mesh for the finite element analyses is automatically generated against the substructure. An approach to automatic mesh generation is discussed on this paper, based upon the techniques of interpolation with the blending function, originally developed for geometric modeling. But, the blending function may produce unexpected results when the mesh is generated, due to the nature of the algorithm. In order to conclude with expected results for automatic mesh generation, and accuracy-fitting method is introduced.

The shapes of waveguide structures are optimized for the uniform electromagnetic field illumination. The vector finite element method is used as an analysis method. As the design algorithm, we employ the shape optimization with a design sensitivity analysis. Performance measure function representing the field uniformity is defined in this paper and used for the calculation of the design sensitivity. The design procedure is applied to the H-plane shape optimization in the rectangular cavity structure with a rectangular waveguide feeder and the E-plane shape optimization in the two plates waveguide.

Multi-objective free-form optimization for shape and thickness of shell structures with composite materials

This paper deals with structural shape optimization of vibrating prismatic shells and folded plates. The finite strip method is used to determine the natural frequencies and modal shapes based on Mindlin‐Reissner shell theory which allows for transverse shear deformation and rotatory inertia effects. An automated optimization procedure is adopted which integrates finite strip analysis, parametric cubic spline geometry definition, automatic mesh generation, sensitivity analysis and mathematical programming methods. The objective is to maximize the fundamental frequency by changing thickness and shape design variables defining the cross‐section of the structure, with a constraint that the total volume of the structure remains constant. A series of examples is presented to highlight various features of the optimization procedure as well as the accuracy and efficiency of finite strip method.

A fuzzy logic knowledge-based approach, FUZZYMESH, for finite element mesh generation and analysis is presented. The proposed approach initiates the adaptive process with a high quality initial mesh that is more refined around the critical points/regions in the problem domain. In order to create high quality initial meshes, the heuristic knowledge, past experience, common sense, and ad hoc methods of finite element specialists are incorporated into the knowledge base of the fuzzy system. Using the linguistic variable concept and approximate reasoning techniques, the fuzzy system makes expert decisions about the initial mesh design by considering the geometric information, as well as the boundary and loading conditions. The decision process includes the determination of priority of critical points/regions and the prediction of mesh sizes for them. According to the mesh size information, a near-optimal initial mesh is created with an automatic mesh generator that is based on the advancing front mesh generation technique. The performance of the proposed approach was measured and evaluated in terms of efficiency and accuracy. The evaluation included comparison between the results of a code based on the proposed fuzzy logic knowledge-based approach, FUZZYMESH, and the conventional approach, which starts the finite element analysis with different meshes, by solving various problems. The global as well as local errors of different solutions were examined and compared. The CPU times for different approaches to achieve a particular accuracy were also measured and compared. The results showed that due to better quality of initial meshes, FUZZYMESH results in lower levels and more accurate error estimates. In turn, the proposed approach is able to solve the problem with a more accurate solution at less cost.

In spite of recent advancements in computer aided design and engineering (CAD/CAE), such as parametric geometry and automatic mesh generation, a large gap exists between computer based design model and analysis model. In this paper, a new product specific approach to integrate design and analysis integration is proposed. The primary objective of this approach is to capture designer’s intent from the limited interaction with the designer in a language that is familiar to designer and to provide the feed back on the suitability of the design based on the finite element analysis (FEA) so that the design is automatically and iteratively modified. Hence, the proposed modeling paradigm provides capabilities of analysis to the designer and acts as a black box without requiring the designer to possess relevant knowledge of FEA. Priority has been given to the class of problems termed as routine analysis – the regularly used, established analysis model in the product design. A representative case study pertaining to Belleville spring is discussed in the context of the proposed methodology to demonstrate that it enables highly automated routine analysis through FEA for the model.

The goal of this work is to develop a systematic method for optimizing the structural design of a segmented wheel concept to improve its operating performance. In this study, a wheel concept is parameterized into a set of size and shape design variables, and a finite element model of the wheel component is created. A multi-objective optimization problem is formulated to optimize its directional compliance and reduce stress concentrations, which has a direct affect on the efficiency, traction, rider comfort, maneuverability, and reliability of the wheel. To solve the optimization problem, a Matlab-FE simulation loop is built and a multi-objective genetic algorithm is used to find the Pareto front of optimal solutions. A trade-off design is selected which demonstrates an improvement from the original concept. Finally, recommendations will be made to apply the structural optimization framework to alternative wheel conceptual designs.

A constitutive theory, finite element formulation and topology optimization for anti-vibration rubber are presented. Many vibration isolators made of rubbers are operating under small oscillatory load superimposed on large static deformation. A~viscoelastic constitutive equation for rubber is proposed considering the influence of large static pre-deformation on the dynamic properties. The proposed model is derived through linearization of Simo's viscoelastic constitutive model and introduction of static deformation correction factor. And then the model is implemented in a finite element code to analyze the behavior of rubber elements under general loading conditions. Dynamic tests are performed in order to verify the model under multi-axial deformation. The computed results by the FEA code are compared with the experimental results and the suggested constitutive equation with static deformation correction factor shows good agreement with the test values. For the stability and low transmissibility of isolation systems, both static and dynamic performance must be concurrently considered in the design process. The continuum-based design sensitivity analyses (DSA) of both the static hyperelastic model and dynamic viscoelastic model are developed. And then the topology optimization methodology is used in order to generate the system layouts considering both the static and dynamic performance.

Bellows is a component in piping systems which absorbs mechanical deformation with flexibility. Its geometry is an axial symmetric shell which consists of two toroidal shells and one annular plate or conical shell. In order to analyze bellows, this study presents the finite element analysis using a conical frustum shell element. A finite element analysis is developed to analyze various bellows. The validity of the developed program is verified by the experimental results for axial and lateral stiffness. The formula for calculating the natural frequency of bellows is made by the simple beam theory. The formula for fatigue life is also derived by experiments. The shape optimal design problem is formulated using multiple objective optimization. The multiple objective functions are transformed to a scalar function by weighting factors. The stiffness, strength and specified stiffness are considered as the multiple objective function. The formulation has inequality constraints imposed on the fatigue limit, the natural frequencies, and the manufacturing conditions. Geometric parameters of bellows are the design variables. The recursive quadratic programming algorithm is selected to solve the problem. The results are compared to existing bellows, and the characteristics of bellows is investigated through optimal design process. The optimized shape of bellows is expected to give quite a good guideline to practical design.