Shape design sensitivity analysis and optimization of two-dimensional periodic thermal problems using BEM

 A general procedure to perform shape design sensitivity analysis for two-dimensional periodic thermal diffusion problems is developed using boundary integral equation formulation. The material derivative concept to describe shape variation is used. The temperature is decomposed into a steady state component and a perturbation component. The adjoint variable method is used by utilizing integral identities for each component. The primal and adjoint systems are solved by boundary element method. The sensitivity results compared with those by finite difference show good accuracy. The shape optimal design problem of a plunger model for the panel of a television bulb, which operates periodically, is solved as an example. Different objectives and amounts of heat flux allowed are studied. Corresponding optimum shapes of the cooling boundary of the plunger are obtained and discussed. Received 15 August 2001 / Accepted 28 February 2002     

Multiobjective optimization of multi-cell sections for the crashworthiness design

Plastic deformation of structures absorbs substantial kinetic energy when impact occurs. For this reason, energy-absorbing components have been extensively used in the structural design of vehicles to intentionally absorb a large portion of crash energy to reduce the severe injury of occupants. On the other hand, high peak crushing force may to a certain extent indicate the risk of structural integrity and biomechanical damage of occupants. For this reason, it is of great significance to maximize the energy absorption and minimize the peak force by seeking for optimal design of these components. This paper aims to design the multi-cell cross-sectional thin-walled columns with these two crashworthiness criteria. An explicit finite element analysis (FEA) is used to derive higher-order response surfaces for these two objectives. Both the single-objective and multi-objective optimizations are performed for the single, double, triple and quadruple cell sectional columns under longitudinal impact loading. A comparative analysis is consequently given to explore the relationship between these two design criteria with the different optimization formulations.     

Adaptive improved response surface method for reliability-based design optimization

Reliability-based design optimization (RBDO) requires the evaluation of probabilistic constraints (or reliability), which can be very time consuming. Therefore, a practical solution for efficient reliability analysis is needed. The response surface method (RSM) and dimension reduction (DR) are two well-known approximation methods that construct the probabilistic limit state functions for reliability analysis. This article proposes a new RSM-based approximation approach, named the adaptive improved response surface method (AIRSM), which uses the moving least-squares method in conjunction with a new weight function. AIRSM is tested with two simplified designs of experiments: saturated design and central composite design. Its performance on reliability analysis is compared with DR in terms of efficiency and accuracy in multiple RBDO test problems.     

Response surface approximations for structural optimization

Response surface methodology can be used to construct global and midrange approximations to functions in structural optimization. Since structural optimization requires expensive function evaluations, it is important to construct accurate function approximations so that rapid convergence may be achieved. In this paper techniques to find the region of interest containing the optimal design, and techniques for finding more accurate approximations are reviewed and investigated. Aspects considered are experimental design techniques, the selection of the ‘best’ regression equation, intermediate response functions and the location and size of the region of interest. Standard examples in structural optimization are used to show that the accuracy is largely dependent on the choice of the approximating function with its associated subregion size, while the selection of a larger number of points is not necessarily cost-effective. In a further attempt to improve efficiency, different regression models were investigated. The results indicate that the use of the two methods investigated does not significantly improve the results. Finding an accurate global approximation is challenging, and sufficient accuracy could only be achieved in the example problems by considering a smaller region of the design space. © 1998 John Wiley & Sons, Ltd.     

Thermal optimization in transient thermoelasticity using response surface approximations

Response surface methodology is used to construct approximations to temperature and stress in transient thermoelastic analysis of non-linear systems. The analysis forms the core component of a heating/cooling rate maximization problem in which the ordinates of the ambient temperature at equally spaced time intervals are chosen as the design variables. Polynomials or cubic splines are fitted through the ordinates to describe the ambient temperature profile required for the convective heat transfer analysis. An experimental design method based on D-optimality and a genetic algorithm was used to select the design points used to create the approximations. Linear response surfaces were found to be sufficiently accurate, thereby minimizing the number of finite element analyses. Two examples of which one is a thick-walled pressure vessel are used to illustrate the methodology. © 1998 John Wiley & Sons, Ltd.     

Reliability based design optimization using response surfaces in application to multidisciplinary systems

Traditionally, reliability based design optimization (RBDO) is formulated as a nested optimization problem. For these problems the objective is to minimize a cost function while satisfying the reliability constraints. The reliability constraints are usually formulated as constraints on the probability of failure corresponding to each of the failure modes or a single constraint on the system probability of failure. The probability of failure is usually estimated by performing a reliability analysis. The difficulty in evaluating reliability constraints comes from the fact that modern reliability analysis methods are themselves formulated as an optimization problem. Solving such nested optimization problems is extremely expensive for large scale multidisciplinary systems which are likewise computationally intensive. In this research, a framework for performing reliability based multidisciplinary design optimization using approximations is developed. Response surface approximations (RSA) of the limit state functions are used to estimate the probability of failure. An outer loop is incorporated to ensure that the approximate RBDO converges to the actual most probable point of failure. The framework is compared with the exact RBDO procedure. In the proposed methodology, RSAs are employed to significantly reduce the computational expense associated with traditional RBDO. The proposed approach is implemented in application to multidisciplinary test problems, and the computational savings and benefits are discussed.     

Multi-objective optimization of tire carcass contours using a systematic aspiration-level adjustment procedure

 While forming a basic tire configuration and supporting most static and dynamic loads of automobiles, tire carcass influences major tire performances according to its contour. Among significant tire performances, we in this study intend to improve the automobile maneuverability and the tire durability by optimizing the sidewall carcass contour. In order to effectively maximize these multi-objectives, we refine the conventional satisficing trade-off methods (STOM) which were proposed originally for the multi-objective structural optimization, by introducing a systematic aspiration-level adjustment procedure. According to the systematic procedure, we perform the sidewall contour optimization that ideally distributes the sidewall carcass tension and minimizes strain-energy density at the belt edge. Since the tire analysis is highly nonlinear problem we employ an incremental analysis scheme, together with the finite-difference sensitivity scheme. Through the numerical experiment, we confirmed that the refined multi-objective optimization technique systematically leads to a final optimum sidewall contour, together with the stable and rapid convergence. Received: 20 August 2001 / Accepted: 29 July 2002 This work was supported by Kumho Industries Co., Ltd in Korea and by the Ministry of Science and Technology under the NRL program (M10203000017-02J0000-00910).     

Shape optimization of periodic structures

This paper describes a numerical approach to the optimization of effective properties of periodic perforations in an infinite body, in the frameworks of heat conduction and of linear elasticity. We implement a special finite element mesh in order to deal with the periodic nature of the problem. We compute the gradient of the functional to be minimized. We describe the process of mesh deformation and mesh regeneration. We give several numerical examples, some of them having practical relevance.     

Evolutionary optimization in thermoelastic problems using the boundary element method

 The paper is devoted to application of evolutionary algorithms and the boundary element method to shape optimization of structures for various thermomechanical criteria, inverse problems of finding an optimal distribution of temperature on the boundary and identification of unknown boundary. Design variables are specified by Bezier curves. Several numerical examples of evolutionary computation are presented. Received 6 November 2000     

Configuration design sensitivity analysis and optimization of beam structures

 A general method for configuration design sensitivity analysis over a three-dimensional beam structure is developed based on a variational formulation of the classical beam in linear elasticity. A sensitivity formula is derived based on a variational equation for the beam structure using the material derivative concept and adjoint variable method. The formulation considers not only the shape variation in a three dimensional direction, which includes translational as well as rotational change of the beam but also the orientation angle variation of the beam's cross section. The sensitivity formula can be evaluated with generality and ease even by employing a piecewise linear design velocity field despite the fact that the bending model is a fourth order differential equation. The design sensitivity analysis is implemented using the post-processing data of a commercial code ANSYS. Several numerical examples are given to show the excellent accuracy of the method. Optimization is carried out for a tilted arch bridge and an archgrid structure to show the method's applicability. Received 29 September 2001 / Accepted 20 March 2002     

Technologic parameter optimization of gas quenching process using response surface method

A step function model with time is presented in the paper, and an axisymmetric component is regarded as the study objective in this model. The heat transfer coefficient during the gas quenching process is described as a function of time in this model, and five design variables are selected to do the design of Box–Behnken experiment with five factors and three levels. The levels of design variables that attain from the result of Box–Behnken experiment design are regard as the technical parameters of gas quenching to simulate the gas quenching process using the FEM software developed in the paper. Some mathematical models of response surface are gained by the mixed regression method and response surface method. These mathematical models show the dependencies of distortion, surface average equivalent residual stress, standard deviation of equivalent residual stress, average surface hardness and standard deviation of surface hardness with respect to the design variables. The optimization model is presented with the distortion as the optimization objective, and the model is optimized with an upper limit, a lower limit and the constraint function by the non-linear method and the Lagrange multiplier method.     

Iterative mesh partitioning optimization for parallel nonlinear dynamic finite element analysis with direct substructuring

 This work presents a novel iterative approach for mesh partitioning optimization to promote the efficiency of parallel nonlinear dynamic finite element analysis with the direct substructure method, which involves static condensation of substructures' internal degrees of freedom. The proposed approach includes four major phases – initial partitioning, substructure workload prediction, element weights tuning, and partitioning results adjustment. The final three phases are performed iteratively until the workloads among the substructures are balanced reasonably. A substructure workload predictor that considers the sparsity and ordering of the substructure matrix is used in the proposed approach. Several numerical experiments conducted herein reveal that the proposed iterative mesh partitioning optimization often results in a superior workload balance among substructures and reduces the total elapsed time of the corresponding parallel nonlinear dynamic finite element analysis. Received 22 August 2001 / Accepted 20 January 2002     

响应表面法在薄壁构件耐撞性优化设计中的应用研究

汽车结构的耐撞性及碰撞吸能优化是现代汽车工业重要的研究内容。耐撞性的优化涉及材料与结构的众多参数。传统的设计、碰撞仿真及试验往往只能在一定程度上改善结构的碰撞性能而无法达到限定条件下的最优状态。利用国际上近年来新发展起来的一种优化理论方法--响应表面法,结合传统的优化手段以及非线性有限元程序对薄壁构件的耐撞性问题进行了优化研究。耐撞性优化的结果表明,该方法具有较高的精确性和有效性。     

基于复合优化方法立式数控加工中心的多目标优化设计

为实现加工中心动静态性能不低于优化前性能,达到整机重量最轻的要求,本文提出了一种复合优化方法来研究多变量、多约束和多目标的数控加工中心优化设计。采用有限元分析和实验模态测试方法分析各大件动态性能,并验证了有限元模型的精确性。然后以该有限元模型为基础进行静态分析,得出各大件的最大变形及应力等。以柔度为目标,采用变密度法拓扑优化设计立柱结构的外形框架;以固有频率为目标,基于元结构的可适应性动态优化方法设计加工中心的筋板结构;以固有频率和质量为目标,基于响应面法的尺寸优化确定各结构的最优尺寸。最后将优化后的各大件进行整机装配,分析校核整机动静态性能。分析结果表明,优化后的整机在保证加工中心动静态性能的条件下,整机质量从12749kg减少到12127kg,减重达到4.9%,达到了整机的优化设计要求,说明该方法具有较高的精度和较强的工程实用性。     

Simulation of airbag inflation processes using a coupled fluid structure approach

 This paper explores simulation techniques for airbag inflation problems using a coupled fluid structure approach. It is to be seen as an initial study on the phenomena occurring in an airbag during a so called out of position occupant impact. The problem studied in this paper is an airbag which is set to impact a head form. The head form is positioned at a very short distance from the airbag. A multi material arbitrary Lagrangian Eulerian technique in the explicit finite element code LS-DYNA is used for the fluid and it is coupled to the structure using a penalty based fluid structure contact algorithm. The results for the head form acceleration and velocity show a good agreement to experimentally obtained values. At the early stages of the inflation process a high pressure zone is found to develop between the gas inlet and the head form. Consequently the pressure difference between the inlet and the high pressure zone is too low for an a priori assumption of sonic flow at the inlet, which is a common requirement in the control volume models used in the industry today. Received: 8 February 2002 / Accepted: 4 June 2002 This work is funded by Autoliv Research and the Swedish Agency for Innovation Systems (VINOVA). The authors would like to thank Autoliv Research for test data and fruitful discussions, Dr. John Hallquist for allowing us to use LS-DYNA in this project and Dr. Lars Olovsson for invaluable help on the fluid structure coupling. In addition, we would like to thank Mr. Claes-Fredrik Lindh for helping us carry out the experiments in this investigation.     

On material forces and finite element discretizations

 The idea of using material forces also termed configurational forces in a computational setting is presented. The theory of material forces is briefly recast in the terms of a non-linear elastic solid. It is shown, how in a computational setting with finite elements (FE) the discrete configurational forces are calculated once the classical field equations are solved. This post-process calculation is performed in a way, which is consistent with the approximation of the classical field equations. Possible physical meanings of this configurational forces are discussed. A purely computational aspect of material forces is pointed out, where material forces act as an indicator to obtain softer discretizations. Received 12 December 2001 / Accepted 18 March 2002     

The measurement and analysis of micro bonding force for electroplated CBN grinding wheels based on response surface methodology

The superabrasive (e.g. CBN or diamond) grain dislodgement occurrence on the wheel surface due to insufficient bonding force is the major failure phenomena in the grinding process with electroplated grinding tools. This failure leads to the abrupt increase of load on the immediate grains, accelerating more grain dislodgement on wheel surface. Ultimately, the aggregated grain dislodgement causes the workpiece profile accuracy degradation and catastrophic wheel sharpness loss. Therefore, the provision of sufficient and uniform micro bonding force all through the wheel surface is the critical task in electroplated superabrasive grinding wheel design. Considering the complexity in the micro bonding force enabling factors, e.g. the grain shape, dimensional size, spatial orientation, and bond layer thickness, it is vital to establish the quantitative and comprehensive relationship between these factors with the micro bonding force for optimal electroplated grinding wheel design. In this paper, an inclined micro-thread turning test is developed to measure the single grain micro bonding force. In addition, the finite element model of single CBN grain bonding force is established and validated to simulate the grain dislodgement. Finally, the response surface methodology (RSM) is applied to build the comprehensive correlation of the bonding force with its dimensional size, spatial orientation, and bond layer thickness. Therefore, the optimal bonding condition through regressed prediction model is identified to provide the quantitative basis for the electroplated CBN grinding wheels design, which indicates that the bonding force can be predicted for specific wheel manufacturing parameters and improved by related variable adjustment.     

Application of the material force method to isotropic continuum damage

 The objective of this work is the exploitation of the notion of material forces in computational continuum damage mechanics. To this end we consider the framework of isotropic geometrically non–linear continuum damage and investigate the spatial and material settings that lead to either spatial or material forces, respectively. Thereby material forces essentially represent the tendency of material defects to move relative to the ambient material. In this work we combine an internal variable approach towards damage mechanics with the material force method. Thus the appearance of distributed material volume forces that are conjugated to the damage field necessitates the discretization of the damage variable as an independent field in addition to the deformation field. Consequently we propose a monolithic solution strategy for the corresponding coupled problem. The underlying kinematics, strong and weak forms of the coupled problem will be presented and implemented within a standard Galerkin finite element procedure. As a result in particular global discrete nodal quantities, the so–called material node point (surface) forces, are obtained and are studied for a number of computational examples. Received: 19 August 2002 / Accepted: 16 October 2002     

A mixed least squares method for solving problems in linear elasticity: theoretical study

 In a previous paper we proposed a mixed least squares method for solving problems in linear elasticity. The solution to the equations of linear elasticity was obtained via minimization of a least squares functional depending on displacements and stresses. The performance of the method was tested numerically for low order elements for classical examples with well known analytical solutions. In this paper we derive a condition for the existence and uniqueness of the solution of the discrete problem for both compressible and incompressible cases, and verify the uniqueness of the solution analytically for two low order piece-wise polynomial FEM spaces. Received: 20 January 2001 / Accepted: 14 June 2002 The authors gratefully acknowledge the financial support provided by NASA George C. Marshall Space Flight Centre under contract number NAS8-38779.