Optimal design of compliant mechanisms using functionally graded materials

This research applies topology optimization to create feasible functionally graded compliant mechanism designs with the aim of improving structural performance compared to traditional homogeneous compliant mechanism designs. Converged functionally graded designs will also be compared with two-material compliant mechanism designs. Structural performance is assessed with respect to mechanical/geometric advantage and stress distributions. Two design problems are presented – a gripper and a mechanical inverter. A novel modified solid isotropic material with penalization (SIMP) method is introduced for representing local element material properties in functionally graded structures. The method of moving asymptotes (MMA) is used in conjunction with adjoint sensitivity analysis to find the optimal distribution of material properties. Geometric non-linear analysis is used to solve the mechanics problem based on the Neo-Hookean model for hyperelastic materials. Functionally graded materials (FGMs) have material properties that vary based on spatial position. Here, FGMs are implemented using two different resource constraints – one on the mechanism’s volume and the other on the integral of the Young’s modulus distribution throughout the design domain. Tensile tests are performed to obtain the material properties used in the analysis. Results suggest that FGMs can achieve the desired improvements in mechanical/geometric advantage when compared to both homogeneous and two-material mechanisms.     

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.     

Shape design sensitivity analysis of nonlinear structural systems

A unified approach is presented for shape design sensitivity analysis of nonlinear structural systems that include trusses and beams. Both geometric and material nonlinearities are considered. Design variables that specify the shape of components of built-up structures are treated, using the continuum equilibrium equations and the material derivative concept. To best utilize the basic character of the finite element method, shape design sensitivity information is expressed as domain integrals. For numerical evaluation of shape design sensitivity expressions, two alternative methods are presented: the adjoint variable and direct differentiation methods. Advantages and disadvantages of each method are discussed. Using the domain formulation of shape design sensitivity analysis, and the adjoint variable and direct differentiation methods, design sensitivity expressions are derived in the continuous setting in terms of shape design variations. A numerical method to implement the shape design sensitivity analysis, using established finite element codes, is discussed. Unlike conventional methods, the current approach does not require differentiation of finite element stiffness and mass matrices.     

T-stress,mixed-mode stress intensity factors,and crack initiation angles in functionally graded materials: a unified approach using the interaction integral method

For linear elastic functionally graded materials (FGMs), the fracture parameters describing the crack tip fields include not only stress intensity factors (SIFs) but also T-stress (nonsingular stress). These two fracture parameters are important for determining the crack initiation angle under mixed-mode loading conditions in brittle FGMs (e.g. ceramic/ceramic such as TiC/SiC). In this paper, the mixed-mode SIFs and T-stress are evaluated by means of the interaction integral, in the form of an equivalent domain integral, in combination with the finite element method. In order to predict the crack initiation angle in brittle FGMs, this paper makes use of a fracture criterion which incorporates the T-stress effect. This type of criterion involves the mixed-mode SIFs, the T-stress, and a physical length scale r_c (representative of the fracture process zone size). Various types of material gradations are considered such as continuum models (e.g. exponentially graded material) and micromechanics models (e.g. self-consistent model). Several examples are given to show the accuracy and efficiency of the interaction integral scheme for evaluating mixed-mode SIFs, T-stress, and crack initiation angle. The techniques developed provide a basic framework for quasi-static crack propagation in FGMs.     

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.     

Efficient design sensitivity derivatives for multi-load case structures as an integrated part of finite element analysis

In most structural optimization problems the accurate calculation of design sensitivity derivatives is required many times during the optimization process. For large structures with multi-load cases the computational costs are sometimes prohibitive. In this paper an approach for incorporating design sensitivity calculations into the finite element analysis of multi-load case structures is presented. A formulation designed to minimize the computational time for the assembled stiffness matrix derivatives is discussed for different element types. The formulation depends on the implicit differentiation method and requires few additional calculations to obtain the design sensitivity derivatives. The approach is developed and implemented to calculate the design sensitivities for continuum and structural isoparametric elements. To demonstrate the accuracy and efficiency of the developed approach some test cases using different structural and continuum element types are presented.     

Shape design sensitivity analysis and optimization of spatially rotating objects

Shape design sensitivity analysis (DSA) and optimization of spatially rotating objects is presented in this paper. Design sensitivity expressions are derived using a continuum DSA method for spatial objects rotating with angular velocity and angular acceleration, based on three definitions of the finite element mass matrix: consistent, lumped, and diagonalized. The design sensitivity expression derived using a diagonalized element mass matrix, which is consistent with the finite element analysis (FEA) method used in ANSYS, is implemented, although the method can work with other FEA codes, such as MSC/NASTRAN or ABAQUS. Since the continuum DSA method is used, sensitivity information can be computed outside the FEA codes by postprocessing finite element data. Rotating block and turbine blade examples are presented to validate the proposed DSA method. The turbine blade example is optimized using an integrated optimization module of the Design Sensitivity Analysis and Optimization (DSO) tool developed at the University of Iowa. The integrated module consists of ANSYS, MSC/NASTRAN, or ABAQUS for FEA; Design Optimization Tool (DOT) for nonlinear programming; and DSA and design model update programs developed in DSO.     

Evolutionary topology optimization of continuum structures with smooth boundary representation

This paper develops an extended bi-directional evolutionary structural optimization (BESO) method for topology optimization of continuum structures with smoothed boundary representation. In contrast to conventional zigzag BESO designs and removal/addition of elements, the newly proposed evolutionary topology optimization (ETO) method, determines implicitly the smooth structural topology by a level-set function (LSF) constructed by nodal sensitivity numbers. The projection relationship between the design model and the finite element analysis (FEA) model is established. The analysis of the design model is replaced by the FEA model with various elemental volume fractions, which are determined by the auxiliary LSF. The introduction of sensitivity LSF results in intermediate volume elements along the solid-void interface of the FEA model, thus contributing to the better convergence of the optimized topology for the design model. The effectiveness and robustness of the proposed method are verified by a series of 2D and 3D topology optimization design problems including compliance minimization and natural frequency maximization. It has been shown that the developed ETO method is capable of generating a clear and smooth boundary representation; meanwhile the resultant designs are less dependent on the initial guess design and the finite element mesh resolution.     

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.     

An effective multilayered sandwich beam-link finite element for solution of the electro-thermo-structural problems

The first part of this contribution deals with deriving the effective matrices of the multilayered sandwich 2D-link/beam finite element made of FGM (functionally graded material) with variation of material properties and rectangular cross-section for solution of weak-coupled electric–thermal–structural problems. The variation of effective material properties is caused by both continuous longitudinal and layer-wise symmetric transversal variation of the constituent’s volume fractions and constituent’s material properties. The second part of this contribution is completed by numerical validation, which documents the high accuracy and effectiveness of our proposed electro-thermo-structural composite (FGMs) link/beam finite element.     

Accuracy test of sensitivity analysis in the semi-analytic method with respect to configuration variables

Configuration optimization is a structural optimization method where the geometrical shape of the structures can be changed during the optimization process. Sensitivity informations are required in the general optimization and quite costly. Especially, they are extemely expensive in the structural optimization where the finite element analysis is utilized. Since the nodal coordinates are regarded as design variables in the configuration optimization, the sensitivities according to the nodal coordinates must be calculated. The characteristics of the configuration optimization is that the transformation matrix in the finite element analysis is a function of design variables. Thus the sensitivity calculation in the configuration optimization is even more complicated. For the efficient sensitivity calculations, various methods have been proposed. They are the analytic method (AM), overall finite difference method (OFD), and semi-analytic method (SM). The semi-analytic method consists of the forward and central difference approximation. This study has been conducted to choose an appropriate method by comparison based on the mathematical and numerical aspects. Some standard structural problems are selected for the evaluations.     

Analysis of optimality criteria and gradient projection methods for optimal structural design

A general mathematical model for optimal design of structural and mechanical systems is defined. The finite element techniques for analysis of these systems are incorporated into the model. Optimality conditions for the model are derived. It is shown that an analytical solution of the optimality conditions is impossible except for trivial design problems. Numerical methods for solving these optimality conditions are presented. A direct method based on the gradient projection concept is derived. Numerical aspects for the method are discussed. These include step size selection, calculation of Langrange multipliers, identification of dependent design sensitivity vectors of constraint functions, and convergence criterion. Both direct and indirect approaches require total derivatives of cost and various constraint functions with respect to design variables. This is accomplished by integrating into both the approaches the state space design sensitivity analysis that has proven to be very general and efficient. It is shown that other major calculations of the two approaches are essentially the same. Thus there is potential for continuous transition between various algorithms for structural optimization. Optimal solutions for two design examples are obtained by hybrid methods to show the potential for further development of these methods.     

Interactive use of finite element programs—or—how to get more out of your finite element model

The paper outlines the application of sensitivity methods to explore the effect of element changes, or to modify the finite element model of a structure. The sensitivity algorithms for structural applications, initially studied at AEO under ESA contract, are partly implemented at AEO for interactive use in conjunction with the finite element analysis of structures. The algorithms are computationally very efficient and hence suited to interactive use in the design and development of structures. Because of the greater insight into structural behaviour that it can provide, sensitivity analysis might open a new, more efficient and cost effective era in the field of structural design and analysis.     

Efficient sensitivity analysis and optimization of shell structures by the ABAQUS code

In this paper, a new efficient sensitivity analysis procedure is presented for the optimization of shell structures without access to the finite element source code. It is devised as a general interface tool to extend existing finite element systems from pure structural analysis to design capability. The implementation is performed based on the ABAQUS code. Kirchhoff flat shell elements are taken into account in the study with the element thickness as design variables. To ensure the performance and the validity of the proposed procedure, satisfactory sensitivity and optimization results are illustrated for numerical examples.     

A global sensitivity analysis in structural mechanics

A methodology is developed to evaluate the response sensitivity of structural systems to variations in their design parameters. The sensitivity is evaluated by considering the global behavior of the system response when the parameters vary within a bounded region. The design parameters are characterized by means of baseline values plus fluctuating components, and the sensitivity of the system is measured in terms of the global variability of the response with respect to its baseline response. The methodology is then extended into the context of optimum redesign analysis of structural systems. Application of the method is made to a structural system defined by two-dimensional beam-column elements and to a system defined by plate elements. The numerical implementation of the global sensitivity approach is made by means of the finite element method. Several analyses are performed and the results are discussed. Finally, some extensions of the present work are presented.     

基于结构拓扑优化方法的发动机支架轻量化设计

为实现某发动机支架轻量化设计,在支架总成整体结构有限元分析的基础上,运用等效刚度原则建立支架局部有限元模型,得到不同工况下的支架位移分布结果.基于TOSCA软件,通过提取结构分析结果对支架结构拓扑优化模型进行敏度分析并形成优化模型列式.优化求解后,在每轮循环迭代中将新的单元密度值重新赋予结构模型,直至满足预先给定的收敛判定条件;设置过滤半径和各类制造加工约束,消除结构拓扑优化中的数值不稳定性问题,改善优化结果的可加工性.对优化前后的计算结果进行对比分析.     

Shape design sensitivity analysis and optimization of general shape arches

A method is presented for the shape design sensitivity analysis as applied to general arches whose shapes cannot be mapped on one plane. The shape design sensitivity formulation with respect to the perturbation in the direction normal to the middle surface of the shallow arch curve is derived using the material derivative and the adjoint variable method with the system variational equation expressed in a Cartesian coordinate system. A general shape arch is subdivided into segments each of which can be considered as a shallow arch. On each subdivision of the arch, a Cartesian coordinate system is installed and the shape design sensitivity for the shallow arch is applied. For numerical implementation, the finite element method is adopted and each finite element can be considered as such a subdivision. Numerical examples for sensitivity analysis and optimization are presented to illustrate the finite element sensitivity method proposed.     

Optimum structural design with parallel finite element analysis

Structural analysis is an important part of the optimum structural design process. Therefore, extra effort should be devoted to make this part as efficient as possible. Since finite element analysis is the most powerful and widely used tool in the structural analysis field, in this paper a new method for structural optimization by parallel finite element method is presented. This method divides the original structure into several substructures and assigns each substructure to one processor. Each processor handles its finite element calculation independently with limited communication between processors. Some numerical examples on the Cray X-MP multiprocessor system with their obtained speedups are presented.