Structural shape optimization integrated with CAD environment

The research work presented here is based on the concept of the integration of optimization techniques and numerical analysis with the finite element method (FEM) and computer-aided design (CAD). A microcomputer aided optimum design system, MCADS, has been developed for general structures. Certain techniques to be discussed in the paper, e.g. the semi-analytical method for design sensitivity analysis, optimization analysis modelling for shape design, application oriented user interfaces and the coupling of automated optimization and user intervention have rendered MCADS pratical and versatile in applications for engineering structures. The above techniques and an application are presented in this paper.     

Structural shape optimization using Cartesian grids and automatic <Emphasis Type="Italic">h</Emphasis>-adaptive mesh projection

We present a novel approach to 3D structural shape optimization that leans on an Immersed Boundary Method. A boundary tracking strategy based on evaluating the intersections between a fixed Cartesian grid and the evolving geometry sorts elements as internal, external and intersected. The integration procedure used by the NURBS-Enhanced Finite Element Method accurately accounts for the nonconformity between the fixed embedding discretization and the evolving structural shape, avoiding the creation of a boundary-fitted mesh for each design iteration, yielding in very efficient mesh generation process. A Cartesian hierarchical data structure improves the efficiency of the analyzes, allowing for trivial data sharing between similar entities or for an optimal reordering of the matrices for the solution of the system of equations, among other benefits. Shape optimization requires the sufficiently accurate structural analysis of a large number of different designs, presenting the computational cost for each design as a critical issue. The information required to create 3D Cartesian h-adapted mesh for new geometries is projected from previously analyzed geometries using shape sensitivity results. Then, the refinement criterion permits one to directly build h-adapted mesh on the new designs with a specified and controlled error level. Several examples are presented to show how the techniques here proposed considerably improve the computational efficiency of the optimization process.     

Structural shape optimization—A survey

This paper is a survey of structural shape optimization with an emphasis on techniques dealing with shape optimization of the boundaries of two- and three-dimensional bodies. Attention is focused on the special problems of structural shape optimization which are due to a finite element model which must change during the optimization process. These problems include the requirement for sophisticated automated mesh generation techniques and careful choice of design variables. They also include special problems in obtaining sufficiently accurate sensitivity derivatives.     

An error analysis and mesh adaptation method for shape design of structural components

A finite element error analysis and mesh adaptation method that can be used for improving analysis accuracy in carrying out shape design of structural components is presented in this paper. The simple error estimator developed by Zienkiewicz is adopted in this study for finite element error analysis, using only post-processing finite element data. The mesh adaptation algorithm implemented in ANSYS is investigated and the difficulties found are discussed. An improved algorithm that utilizes ANSYS POST1 capabilities is proposed and found to be more efficient than the ANSYS algorithm. An example is given to show the efficiency. An interactive mesh adaptation method that utilizes PATRAN meshing and result-displaying capabilities is proposed. This proposed method displays error distribution and stress contour of analysis results using color plots, to help the designer in identifying the critical regions for mesh refinement. Also, it provides guidance for mesh refinement by computing and displaying the desired element size information, based on error estimate and a mesh refinement criterion defined by the designer. This method is more efficient and effective than the semi-automatic algorithm implemented in ANSYS, and is suitable for structural shape design. This method can be applied not only to set-up a finite element mesh of the structure at initial design but to ensure analysis accuracy in the design process. Examples are given to demonstrate feasibility of the proposed method.     

Analysis and optimal design of plates and shells under dynamic loads – I: finite element and sensitivity analysis

In this paper we consider the development, integration, and application of reliable and efficient computational tools for the geometry modeling, mesh generation, structural analysis, and sensitivity analysis of variable-thickness plates and free-form shells under dynamic loads. A flexible shape-definition tool for surface modeling using Coons patches is considered to represent the shape and the thickness distribution of the structure, followed by an automatic mesh generator for structured meshes on the shell surface. Nine-node quadrilateral Mindlin–Reissner shell elements degenerated from 3D elements and with an assumed strain field, the so-called Huang–Hinton elements, are used for the FE discretization of the structure. The Newmark direct integration algorithm is used for the time discretization of the dynamic equilibrium equations for both the structural analysis and the semi-analytical (SA) sensitivity analysis. Alternatively, the sensitivities are computed by using the global finite difference (FD) method. Several examples are considered. In a companion paper, the tools presented here are combined with mathematical programming algorithms to form a robust and reliable structural optimization process to achieve better dynamic performance on the shell designs.     

Integrated optimal topology design and shape optimization using neural networks

In this paper, neural network- and feature-based approaches are introduced to overcome current shortcomings in the automated integration of topology design and shape optimization. The topology optimization results are reconstructed in terms of features, which consist of attributes required for automation and integration in subsequent applications. Features are defined as cost-efficient simple shapes for manufacturing. A neural network-based image-processing technique is presented to match the arbitrarily shaped holes inside the structure with predefined features. The effectiveness of the proposed approach in integrating topology design and shape optimization is demonstrated with several experimental examples.     

Automated interrogation and adaptive subdivision of shape using medial axis transform

We present a method for the generation of coarse and fine finite element meshes on multiply connected surfaces. Our method is based on the medial axis transform (MAT) which is employed to decompose a complex shape into topologically simple subdomains. One important property of our approach is that MAT is effectively employed to automatically extract some important shape characteristics and their length scales. Using this technique, we can create a coarse subdivision of a complex surface and select local element size to generate fine triangular meshes within individual subregions. The MAT allows us to carry out these processes in an automated manner. Thus, our approach can lead to integration of automated finite element (FE) mesh generation schemes into existing FE preprocessing systems. We also briefly discuss several design and analysis applications, which include adaptive surface approximations and adaptive h- and p-version finite element analysis (FEA) processes, in order to demonstrate our method.     

Second-order shape design sensitivity using P-version finite element analysis

Thep-version finite element analysis (FEA) approach is attractive for design sensitivity analysis (DSA) and optimization due to its high accuracy of analysis results, even with coarse mesh; insensitivity to finite element mesh distortion and aspect ratio; and tolerance for large shape design changes during design iterations. A continuum second-order shape DSA formulation is derived and implemented usingp-version FEA. The second-order shape design sensitivity can be used for reliability based analysis and design optimization by incorporating it with the second-order reliability analysis method (SORM). Both the second-order shape DSA formulations with respect to the single and mixed shape design parameters are derived for elastic solids using the material derivative concept. Both the direct differentiation and hybrid methods are presented in this paper. A shape DSA is implemented by using an establishedp-version FEA code, STRESS CHECK. Two numerical examples, a connecting rod and bracket, are presented to demonstrate the feasibility and accuracy of the proposed seond-order shape DSA approach.     

Design Sensitivity Analysis and Optimization tool (DSO) for shape design applications

The Design Sensitivity Analysis and Optimization (DSO) tool, developed initially for sizing design application, has been extended to support shape design applications of structural components. The new capabilities including shape design parameterization, error analysis and mesh adaptation, design velocity field computation, shape design sensitivity analysis, and interactive design steps, are discussed. These capabilities are integrated on the top of the DSO framework that includes databases, user interface, foundation class and remote module. The DSO allows the design engineer to easily create geometric, design, and analysis models; define performance measures; perform design sensitivity analysis (DSA); and carry out a four-step interactive design process that includes visual display of design sensitivity, what-if study, trade-off analysis, and interactive design optimization. Additionally, a 3-D tracked vehicle clevis is presented in this paper to demonstrate the new capabilities.     

Improved image interpreting and modeling technique for automated structural optimization system

The automated structural optimization system (ASOS) proposed in the previous work incorporates the image-preprocessing techniques, the image-interpreting technique, and the automated shape-optimization modeling techniques to successfully obtain an autonomously integrated topology and shape optimization. However, the characteristic value-based image-interpreting technique used in ASOS is unable to accurately interpret complicated hole shapes, necessitating the use of the hole shape-adjusting strategy in addition to the hole-expanding strategy and the interference analysis in the automated shape-optimization modeling techniques to obtain a viable initial design and side constraints of design variables. In order to solve the above-mentioned problem in ASOS, this paper proposes the improved automated structural optimization system (IASOS) and uses the polygonal image-interpreting technique to replace the characteristic value-based image-interpreting technique used in ASOS. This alteration significantly increases the accuracy of image interpretation. Moreover, it can simplify the process of automated shape-optimization modeling techniques, reduce the design duration, and produce better results.     

A CAD-based design parameterization for shape optimization of elastic solids

In this paper a CAD-based design sensitivity analysis (DSA) and optimization method using Pro/ENGINEER for shape design of structural components is presented. The CAD-based design model is critically important for multidisciplinary shape design optimization. Only when each discipline can compute the design sensitivity coefficients of the CAD-based design model, can a true multidisciplinary what-if study, trade-off analysis, and design optimization be carried out. The proposed method will allow the design engineer to compute design sensitivity coefficients of structural performance measures such. as stress and displacement, evaluated using existing finite element analysis (FEA) tools, both h- and p-versions, with respect to design variables defined in the parameterized CAD model. The proposed method consists of (i) a CAD-based design parameterization technique that ties the structural DSA and optimization to a CAD tool; (ii) a design velocity field computation that defines material point movement due to design change in CAD geometry, satisfies linearity and regularity requirements, and supports both hand p-version FEA meshed using existing mesh generators; and (iii) a design optimization method that supports structural geometric and finite element model updates in Pro/ENGINEER during the optimization process.     

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.     

Skeleton-Sectional Structural Analysis for 3D Printing

3D printing has become popular and has been widely used in various applications in recent years. More and more home users have motivation to design their own models and then fabricate them using 3D printers. However, the printed objects may have some structural or stress defects as the users may be lack of knowledge on stress analysis on 3D models. In this paper, we present an approach to help users analyze a model’s structural strength while designing its shape. We adopt sectional structural analysis instead of conventional FEM (Finite Element Method) analysis which is computationally expensive. Based on sectional structural analysis, our approach imports skeletons to assist in integrating mesh designing, strength computing and mesh correction well. Skeletons can also guide sections building and load calculation for analysis. For weak regions with high stress over a threshold value in the model from analysis result, our system corrects them by scaling the corresponding bones of skeleton so as to make these regions stiff enough. A number of experiments have demonstrated the applicability and practicability of our approach.     

High-fidelity aerostructural optimization with integrated geometry parameterization and mesh movement

This paper extends an integrated geometry parameterization and mesh movement strategy for aerodynamic shape optimization to high-fidelity aerostructural optimization based on steady analysis. This approach provides an analytical geometry representation while enabling efficient mesh movement even for very large shape changes, thus facilitating efficient and robust aerostructural optimization. The geometry parameterization methodology uses B-spline surface patches to describe the undeflected design and flying shapes with a compact yet flexible set of parameters. The geometries represented are therefore independent of the mesh used for the flow analysis, which is an important advantage to this approach. The geometry parameterization is integrated with an efficient and robust grid movement algorithm which operates on a set of B-spline volumes that parameterize and control the flow grid. A simple technique is introduced to translate the shape changes described by the geometry parameterization to the internal structure. A three-field formulation of the discrete aerostructural residual is adopted, coupling the mesh movement equations with the discretized three-dimensional inviscid flow equations, as well as a linear structural analysis. Gradients needed for optimization are computed with a three-field coupled adjoint approach. Capabilities of the framework are demonstrated via a number of applications involving substantial geometric changes.     

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.     

Velocity fields using NURBS with distortion control for structural shape optimization

The paper presents methods for the calculation of design velocity fields and mesh updating in the context of shape optimization. Velocity fields have a fundamental role in the integration of the main conceptual and software components in shape design optimization. Nonuniform rational B-splines are used to parameterize the domain boundary. A Newton/Raphson procedure is used to calculate the curve and surface internal parameters. A preconditioning iterative conjugate gradient method with low precision is used to improve the solution performance of the auxiliar problem in the calculation of the velocity fields. The velocity fields are also used to perturb the finite element mesh and element distortion measures are introduced. Finally, examples of two- and three-dimensional elastic problems are presented to illustrate the application of the algorithms.     

Structural optimization system based on trabecular bone surface adaptation

In the paper the structural optimization system based on trabecular bone surface adaptation is presented. The basis of the algorithm formulation was the phenomenon of bone adaptation to mechanical stimulation. This process, called remodeling, leads to the optimization of the trabecular network in the bone. The simulation system, as well as the finite element mesh generation, decision criteria for structural adaptation, and the finite element analysis in a parallel environment are described. The possibility of applying the system in mechanical design is discussed. Some computation results using the developed system are presented, including the comparison to the topology optimization method.