STRUCTURAL OPTIMIZATION USING ADAPTIVE FINITE ELEMENT ANALYSIS AND DESIGN SENSITIVITY ERROR CONTROL C. A. de Carvalho Silva and M. L. Bittencourt* Departamento de Projeto Mecanico Faculdade de Engenharia Mecanica Universidade Estadual de Campinas Campinas, SP, Brazil, 13083-970 *Corresponding author, Email: mlb@fem.unicamp.br A structural optimization environment has been implemented using mathematical programming, finite element method, continuum sensitivity analysis, adaptive finite element analysis, sensitivity analysis of the error estimator, and error control of the sensitivity of the structural performance functionals. In this work aspects of adaptive finite element procedures applied to the optimization process are discussed. The application of structural optimization techniques has become standard in the design of components and systems. This has raised issues about the reliability of the optimum solutions obtained which in general may be very sensitive to the discretization adopted. This fact can lead to suboptimum or infeasible solutions and cause convergence problems in the mathematical programming algorithm due to the inaccuracy of the gradient evaluation. One way to overcome those difficulties is to use adaptive finite element procedures for the solution of the state equation. Therefore, it is possible to obtain a discretization-independent optimization process. In addition, the sensitivity of the constraint functionals are evaluated more precisely due to the better accuracy of the calculated displacement field. However, adaptive analysis becomes very expensive for complex problems which decreases the efficiency of the iterative optimization procedure. The optimization process can be improved using results of a previous design to accelerate the convergence of the current analysis. This can be done by evaluating of the sensitivity of the estimated error to the design variables on the current design and then estimating the error distribution in the new design. In this way, the next step of the adaptive analysis will start with an approximately refined mesh. The global efficiency and reliability of the optimization process can be also improved by the application of a sensitivity analysis error estimator to control the gradient evaluation. The efficiency and reliability of the proposed methodology are verified by means of several examples SENSITIVITY ANALYSIS IN A PASSIVE VIBRATION ISOLATION SYSTEM A. Coronado M. *, F. A. Rochinha ** and R. Sampaio * Mechanical Engineering Department * Pontific Catholic University of Rio de Janeiro, ** Federal University of Rio de Janeiro Corresponding author, e-mail : betocm@mec.puc-rio.br Evaluation of the sensitivity to parameter variations is a basic aspect in the analysis of mechanical systems [1,2]. In a complex system whose dynamical behavior is governed by several parameters it is important to know which ones are the most influent, if we want to change only a few parameters. In this work we study a passive vibration isolation system whose efficacy is measured by the power transmitted to the base of the system. This quantity has been chosen because it considers the generalized forces and velocities in a single number eliminating any possible ambiguity and allowing to use a multidirectional excitation. We use sub-structuring to analyze the passive system [3], it is composed of a rigid mass as a source, two flexible isolators and a rigid or flexible base as a receiver. The source has 3 degrees of freedom (two translational and one rotational) and in this case it is considered its mobility matrix, the two isolators are modeled with analytical expressions of their impedance matrix. The mobility matrix of the receiver has two possibilities to being modeled, the first is as a rigid mass supported by two springs and the second is as a flexible beam with two simple supports, since this last case has complex boundary conditions, the mobilities were computed using the finite element method and a dynamic condensation technique. We have noted that the configuration (place of isolators, for example) is very important in the design of passive isolation systems. Moreover, it will be show that each parameter studied has a great dependence of the kind of excitation and of the specific frequency (or frequency range) analyzed, this demonstrates that without the tool of sensitivity analysis would be very difficult to chose which parameters must be modified. We have also studied the influence of another parameters like, mass, rigidity, inclination angle, loss factor, etc. References [1] Frank Paul M. `Introduction to system sensitivity theory' Academic Press (1978) [2] Nalecz A.G. and Brooks P.C. `Sensitivities of frequency response functions of linear dynamic systems to variations in design parameters values' Journal of Sound and Vibration (1988) 120(3), 517-526 [3] Gardonio P., Elliot S.J. and Pinnington R.J. `Active isolation of structural vibration on multiple-degree-of-freedom system, Part I : The dynamics of the system' Journal of Sound and Vibration (1997) 207(1), 61-9 Optimization of precipitate evolution in quench processes Huang, Sobh, Yin, Haber, Hyland, Tortorell OPTIMIZATION OF THE MATERIAL QUALITY DURING THE BRIGDMAN CASTING PROCESS G. Laschet*, D. Ma and M. Schallmo ACCESS e.V., Intzestrasse 5, D-52056 Aachen, Germany Phone ++49-241-80-5890, Fax ++49-241-38578, Email g.laschet@access.rwth-aachen.de Turbine components of modern aircraft engines and power plants are commonly produced by the investment casting process. As their mechanical properties are strongly related to their grain structures, an increasing world wide demand for Directional Solidified (DS) and Single Crystal (SC) turbine blades exists. These blades are made of Ni-base superalloy and produced by directional solidification in a Bridgman furnace. Its apparently simple principle of generating a directional heat flow by withdrawing the shell mould out of a heating zone into a cooling zone constitutes in fact a complex optimization problem for real blade geometries [1]. Technically relevant casting parameters, such as heater tempe-rature and withdrawal velocity, are currently determined by series of expensive experiments. Therefore, based on a validated casting design tool [1], an optimization strategy will be presented which is able to optimize the casting quality with respect to achieving low process time and costs. After specifying the main features and design parameters of the Bridgman process, a multiobjective function, defined over the whole domain, is introduced. This function is built on a set of violation terms characterizing the superalloy quality. For example, to produce a high quality DS turbine blade, following features are the most relevant : - the probability of freckle formation, which is governed by the cooling rate at Tliq ; - the degree of curvature of the solidification front; the critical G/v ratio at the solidification front which describes the transition from columnar dendritic growth to equiaxed grain structure. In order to formulate a constrained optimization problem, more suitable for the implemented optimi-zation algorithms, mean local qualities are also evaluated on selected subdomains of the blade. These values allow to define a constraint min-max optimization problem. In this study only the withdrawal velocity vw , described by a polyline of N parameters is chosen as design variable. A convex lineariza-tion algorithm, CONLIN [2], and a globally convergent version of MMA, GCM [3], are used as optimizer. This procedure is then applied to the optimization of the withdrawal profile for the DS casting process of a complex 3-D dummy turbine blade with two thin platforms. Interesting optimization results, obtained either with CONLIN or CGM as optimizer, are then discussed and compared with previous results [1]. During this investigation, the influence of the sensitivity analyses on the results is also pointed out. In order to validate these optimization results, the temperature fields obtained with the initial and optimized withdrawal profile are introduced in 3-D CAFE model [4]. In this post-processor phase, cellular automata nucleation and growth algorithms are used to predict the grain structure formation. These analyses show clearly the better quality of the predicted DS macro-grain structure with the optimized withdrawal profile. Finally, two pairs of casting trials are processed respectively with the initial and optimized withdrawal velocity in order to confirm the optimization and CAFE model results. [1] G.Laschet, M. Schallmo & N. Hofmann: "Optimization tools for Bridgman casting process", Proc. of 7th Conf. on Casting, Welding & advanced Solidification, Ed. B. Thomas & C. Beckermann, TMS Editions, San Diego, June 98, pp 1095-1102 [2] C. Fleury: "First and second order convex approximation strategies in structural optimization", Structural Optimization, vol. 1, pp 3-10, 1989. [3] K. Svanberg. "A globally convergent version of MMA without linesearch", Proc. of WCSO-1, Goslar, 1995, Ed. Olhoff, pp 9-17. [4] J.-L. Desbiolles, Ch.-A. Gandin, J-F Joyeux, M. Rappaz & Ph. Thevoz: "A 3-D CAFE model for the prediction of solidification grain structures", Proc. of 7th Conf. on Casting, Welding & advanced Solidification, Ed. B. Thomas & C. Beckermann, TMS Editions, San Diego, June 98, pp 433-440. FINITE ELEMENT SIMULATION OF SMART STRUCTURES USING AN OPTIMAL OUTPUT FEEDBACK CONTROLLER FOR VIBRATION AND NOISE CONTROL Young-Hun Lim, G.V. Senthil, Vasundara V. Varadan* and Vijay K. Varadan * Corresponding author. E-mail: vvvesm@engr.psu.edu Research Center for the Engineering of Electronic and Acoustic Materials, Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA 16802 ABSTRACT This numerical study presents a detail optimal control design based on general finite element approach for the integrated design of a structure and its control system. Linear quadratic (LQ) theory with output feedback is considered on the basis of the state space modal of system. Three-dimensional finite elements are used to model the smart structure containing discrete piezoelectric sensors and actuators by the use of combination of solid, transition, and shell elements. Since several discrete piezoelectric patches are spatially distributed in the structure to effectively observe and control the vibration of a structure, the system model is thus utilized to design multi-input-multi-output (MIMO) controller. A modal analysis is performed to transform the coupled finite element equations of motion into the state space model of system in the modal coordinates. The output feedback controller is then employed to emulate the optimal controller by solving the Riccati equations from modal space model. An optimal controller design for the vibration suppression of a clamped plate is presented for both the steady state and the transient case. Numerical simulation is also used to predict the reduction in the sound pressure level inside an enclosure radiated from this optimally controlled plate. AN HP-ADAPTIVE TOPOLOGY OPTIMIZATION USING FEM AND MESHLESS METHODS Tadeusz J. Liszka*, C. Armando Duarte, Osama Hamzeh The Computational Mechanics Company *Corresponding author, e-mail: liszka@comco.com The paper presents results of the feasibility tests of merging three technologies: - adaptive computational methods, - meshless computational methods, and - topological and size and shape optimization, into a homogeneous design/analysis tool. The approach tested herein encapsulates the preliminary design, topology optimization, and size and shape optimization into a single design step requiring minimal user intervention and significantly reducing the overall development time and prototype costs. The hp-Cloud method, a particular variant of the meshless technology used in the project, does not rely on the mesh of elements like the one required in the finite element method. Instead of this the approximation of the solution is built around a collection of nodes sprinkled arbitrarily within the domain on which the problem is formulated. There is no fixed connectivity between these nodes, thus the absence of the mesh makes the definition of the problem, and later implementation of h- and p- adaptivity techniques very straightforward. The use of the h-adaptivity during the shape resolution, both in the context of the finite element method and in the meshless formulation produced dramatic improvement in the quality of the results at the same time limiting the cost of the solution. hp-adaptive meshless formulation allows more flexible adaptation than that in the finite elements, thus allowing to resolve final shapes with fewer adaptations and lower cost. This flexible adaptation is especially important in the topology optimization, where the initial discretization cannot be properly aligned with the final structure. Use of p-adaptation (higher order approximation of the solution) allows for very good resolution of stresses, but also improved results of topology optimization by eliminating the instabilities in the form of checkerboard pattern and by increasing the size of the smallest details created in the structure (thus increasing manufacturability of the designs). During the project, the computer code (named PHLEX-topo) has been designed, coded and tested. The code combines hp-adaptive meshless formulation and hp-adaptive finite element formulation with the topology optimization technique, to deliver mechanical designs with high quality shape resolution and good estimation of stresses. Using the PHLEX-topo code, several design problems have been solved. Comparison of results from the meshless formulation with those from the hp-adaptive FEM as well as those from the existing commercial codes was used to validate results and to evaluate potential advantages of the proposed formulation. [1]. Bendsoe, M P, and Kikuchi, N: Generating optimal topologies in structural design using a homogenization method, CMAME 71:197-224, 1988 [2]. Duarte, C A M, and Oden, J T: hp-clouds an hp meshless method. NMPDE, 12:673-705, 1996 Supported by NSF SBIR grant 966160 OPTIMIZATION OF THERMO-MECHANICAL PROCESSES USING AN EULERIAN FORMULATION AND APPLICATION IN WELDING S. M. Rajadhyaksha*, P. Michaleris Department of Mechanical and Nuclear Engineering, Pennsylvania State University A systematic design approach has been developed for solving quasi- state thermo-mechanical material processes using finite element methods, sensitivity analysis and optimization. The thermo-mechanical processes considered are governed by one-way coupled thermo-elasto-plastic response. The quasi-state thermal problem is solved first using an Eulerian formulation, where the heat flux is fixed in space and the material flows through a control volume. For constant velocity and heat flux distribution, the Eulerian formulation reduces to a steady state problem, whereas the Lagrangian formulation remains transient. Streamline Upwinding Petrov Galerkin stabilization is employed to suppress The spurious oscillations occurring due to the nature of the Eulerian formulation for the thermal problem. The quasi-state non-linear mechanical problem is then solved using an Eulerian formulation importing the results of the thermal problem. Streamlines are used to determine the temperature history. The plasticity evolution is then determined by integrating along the streamline path. The design sensitivities are computed using the direct differentiation approach. The sensitivities are then employed to determine the optimal objective function. The developed approach is applied to a mechanical problem, where a moving weld torch passes over the material to be welded. The optimization objective is to minimize the distortion after the welding process, varying the thermal tensioning heater parameters, which are the design variables 3D SHAPE OPTIMIZATION FOR INDUSTRIAL CONFIGURATIONS USING INCOMPLETE SENSITIVITIES Bijan Mohammadi Univ. Montpellier and INRIA - France e-mail: Bijan.Mohammadi@Inria.fr We show how to use incomplete sensitivities evaluations together with gradient based methods for shape optimization. The major ingredients are an automatic differentiation tool for sensitivity analysis, incomplete gradients and an adaptive CAD-Free framework for shape and unstructured mesh deformations. The AD tool we use is developed at Inria Sophia Antipolis. We show how to use AD as a tool for analysis rather than solution. The effects of different operators involved in the design loop is analysed and their contribution to the sensitivities identified. Hence, we discoverd some interesting features of these sensitivities for cost functions lying on the shape through boundary integrals in shape optimization problems. We show the impact of this approximation on the sensitivities. The other important ingredient is the CAD-Free framework which enables for optimization without CAD. The interface with CAD is only made before (through the surfacic mesh) and after optimization. This former point is to be created for each new application. We noticed that in fact CAD parameters are not necessarly suitable for optimization and that we need to develop a different parametrization for optimization. Therefore, we perfered to use the richest one available (i.e. the surfacic mesh). In that sense, this approach is close to topological optimization. We show how this plateform has been adapted to various 3D industrial applications coming from aeronautics. We also used this plateform for unsteady applications and also together with mesh adaptation. If time, we will also give some examples of them An evaluation of Partition Based Solution Approach for Topology Optimization Problems. D. Natekar and G. Subbarayan * Mechanical Engineering Department University of Colorado, Boulder *Corresponding author, e-mail : [ ganesh@colorado.edu ] Topology and size optimization problems are aimed at determining the optimal connectivity and material distribution within a structure. These problems are inherently expensive as they involve discretization of a continuum. Partitioned approaches have received much attention in the recent years with their promise of handling large and computationally explosive problems by decomposing the problem into smaller subsystems, thus reducing the problem size, and enhancing computational efficiency through parallel processing of the subsystems. It is therefore natural that the topology optimization problem be solved using a partition solution technique. However, there does not appear to have been any attempt at application of a general decomposition approach to topology optimization problems. In this paper we apply a general decomposition based optimization technique for solving topology optimization problems. The technique also involves the outlining and implementation of a definite coordination and solution strategy. The paper further explores the use of the partitioned approach as a means to achieve improved efficiency in solving topological optimization problems. Both the partitioning and the optimization schemes use analytically calculated variational sensitivity values obtained using a program developed by the authors. Topology optimization problems studied include the classical Mitchell cantilever beam problem and a simple plate problem. The optimization problems are solved using a nonlinear constrained optimization program. It is shown that in these classes of problems, partitioning may not always lead to computational efficiency. Furthermore, partitioned inputs to optimization solvers act as different initial guesses and result in different, though physically viable, solutions. This increases the uncertainty in the solution HOMOGENIZATION DESIGN METHOD APPLIED TO THE DESIGN OF FLEXTENSIONAL TRANSDUCERS Emilio Carlos Nelli Silva*, Shinji Nishiwaki, and Noboru Kikuchi Escola Politecnica da Universidade de Sao Paulo Sao Paulo, SP 05508-900 Brazil *Corresponding author, email: ecnsilva@usp.br Flextensional transducers consist of a piezoceramic connected to a flexible mechanical structure that converts and amplifies the output displacement of the piezoceramic. Among the applications, they can be used as actuators, sonar, and hydrophones. Flextensional transducers have been developed by using simple analytical models and experimental techniques [1], and the finite element method (FEM) [2]. However, the design is limited to the optimization of some dimension of a specific topology chosen for the coupling structure. These studies showed that the performance and resonance frequency depend on the distribution of mass, stiffness and flexibility in the coupling structure domain, which is related to the coupling structure topology. By designing other types of coupling structures connected to the piezoceramic, we can obtain novel types of flextensional transducers with enhanced performance for the desired application. In this work, we propose a method for designing flextensional transducers using topology optimization technique based on the homogenization design method developed by Bendsoe and Kikuchi [3]. Essentially, the method consists of finding the optimal material distribution in a perforated design domain with infinite microscale voids. The material in each element can vary from void to full material, also assuming intermediate materials. Since complex topologies are expected, the finite element method is used for transducer modeling. The problem is posed as the design of a flexible structure coupled to the piezoceramic that produces high output displacements in a specified point of the domain and direction, in a specified frequency. The distribution of stiffness and flexibility in the coupling structure is obtained by defining two quantities: mean transduction and mean compliance. The mean transduction was obtained by extending the Betti's theorem to the piezoelectric medium. The formulation of the mean transduction takes into account the inertia effect in the case periodic loads are applied to the medium. Its maximization is related to the maximization of the output displacement of the transducer in a specified region of the domain. However, the maximization of the mean transduction generates a very weak topology, and therefore, the second quantity, mean compliance, must be minimized to provide some stiffness in the final design. The optimization problem is constructed by defining a multi-objective function that combines both quantities, mean transduction and mean compliance. The homogenization method is used for the relaxation of the design domain. As a result, designs of flextensional transducers are presented. References: [1] Q.C.Xu, S.Yoshikawa, J.R.Belsick, and R.E.Newnham, ``Piezoelectric Composites with High Sensitivity and High Capacitance for Use at High Pressures,'' IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, vol.38, no.6, pp.634-639 November, 1991. [2] T.Inoue, T.Nada, and K.Sugiuchi, ``Low-Frequency Flextensional Piezoelectric Underwater Transmitter with Displacement Amplifier,'' IEEE Transactions of the IEICE, vol.E 72, no.12, pp.1410-1416, December, 1989. [3] M.P.Bendsoe and N.Kikuchi, ``Generating Optimal Topologies in Structural Design Using a Homogenization Method,'' Computer Methods in Applied Mechanics and Engineering, vol.71, pp.197-224, 1988. EVOLUTIONARY ALGORITHMS APPLIED TO STRUCTURAL OPRIMIZATION PROBLEMS M. Papadrakakis*, N. Lagaros and G. Kokossalakis Institute of Structural Analysis and Seismic Research National Technical University of Athens Athens, 15773, Greece *Corresponding author, e-mail: mpapadra@central.ntua.gr The objective of this study is to investigate the efficiency of various Evolutionary Algorithms (EAs), such as Evolution Strategies (ESs) and Genetic Algorithms (GAs), when applied to large-scale sizing optimization problems. Both ESs and GAs imitate biological evolution in nature and have three characteristics that differ from other conventional optimization algorithms: (i) In place of the usual deterministic operators, they use randomized operators: mutation, selection and recombination. (ii) Instead of a single design point, they work simultaneously with a population of design points in the space of design variables. (iii) They can handle with minor modifications continuous, discrete or mixed optimization problems. The most time-consuming part of a gradient-based Mathematical Programming (MP) approach is related to the sensitivity analysis phase, which is an important ingredient of all MP optimization methods and may consume a large part of the total computational effort. On the other hand the application of evolutionary algorithms that are based on probabilistic searching, such as ESs and GAs, do not need gradient information and therefore avoid to perform the computationally expensive sensitivity analysis step. MP methods make use of local curvature information derived from linearization of the original functions by using their derivatives with respect to the design variables. These methods present a satisfactory local rate of convergence, but they cannot assure that the global optimum can be found. On the other hand, evolutionary algorithms are in general more robust and present a better global behaviour than the MP methods. They may suffer, however, from a slow rate of convergence towards the global optimum. In this work the efficiency of various evolutionary algorithms in structural sizing optimization problems is investigated. Furthermore, in order to benefit from the advantages of both methodologies a combination of MP and EAs is also examined in an attempt to increase the robustness as well as the computational efficiency of the optimization procedure. The numerical tests presented demonstrate the computational advantages of the proposed approach, which become more pronounced in large-scale and computationally intensive optimization problems. REFERENCES 1. M. Papadrakakis, and N.D. Lagaros, G. Thierauf, and J. Cai; Advanced Solution Methods in Structural Optimization Based on Evolution Strategies, Engineering Computations, vol. 15(1), pp.12-34, 1998. 2. M. Papadrakakis, Y. Tsompanakis, N.D. Lagaros, E. Hinton, J. Sienz, G. Thierauf, and J. Cai; Innovative Computational Methods for Structural Optimization, In Papadrakakis, M., and Topping, B.H.V. (eds), Innovative Computational Methods for Structural Mechanics, Saxe-Coburg Publications,1999. STRUCTURAL OPTIMIZATION INCLUDING NONLINEAR STRUCTURAL RESPONSE Ekkehard Ramm*, Stefan Schwarz and Roman Kemmler Institute of Structural Mechanics, University of Stuttgart 70550 Stuttgart, Germany e-mail:eramm@statik.uni-stattgaart The value of optimization results in the design of structures strongly depends on the quality of the underlying mechanical model. The more the mechanicalmodel is simplified, for example neglecting the in general nonlinear structural response or by approximating a 3-dimensional stress state by 1- or 2-dimensional models, the less meaningful the optimization results may be. Optimization is known to be the generator of sensitivities, this also holds for structures optimized with a reduced mechanical model which often turn out to be prone to the neglected effects. Maximizing linear elastic buckling loads may lead to clustering of failure modes and an extreme sensitivity with respect to geometrical imperfections. Elastic structures optimized for maximum stiffness often yield brittle failure. To remedy these defects either the anticipated sensitivities have to be included in the problem formulation or the model has to be extended to include these phenomena. The present contribution addresses this aspect and extends the formulation into the nonlinear structural range. These may be either large deformation problems including instability phenomena or materially nonlinear problems, here in particular classical small strain metal plasticity. The formulations are embedded in a general, so called adaptive optimization concept for both, shape and topology optimization and their combination. Here adaptivity can be understood in the geometrical sense, i.e. for the design model, and in its usual mechanical meaning as finite element adaptivity based on error control. An important point is to include not only the discretization error of the state variables but also the error of their sensitivities. This extension not only improves the optimization result but also the numerical efficiency. The crucial point of an optimization procedure is the determination of the derivatives with respect to the optimization variables, i.e. the sensitivity analysis. According to the mechanical model different formulations for the sensitivity analysis are applied. For path-dependent problems, like elastoplasticity, a variational approach turned out to be suitable while for problems with geometrically nonlinear behavior, e.g. buckling, a discrete approach is favourable. The presented optimization procedures are verified by several problems for plane stress, plate and shell structures. References Kleiber, M.; Antunez, H.; Hien, T.D.; Kowalczyk, P.: Parameter Sensitivity in Nonlinear Mechanics. Wiley (1997). Maute, K.: Topologie- und Formoptimierung von dunnwandigen Tragwerken. Ph.D. Thesis, University of Stuttgart (1998). Maute, K.; Schwarz, S. ; Ramm, E.: Adaptive topology optimization of elastoplastic structures. Structural Optimization (1998) 81-91. Reitinger, R.; Ramm, E.: Buckling and imperfection sensitivity in the optimization of shell structures. Thin-walled Structures (1995) 159-177. Ryu, Y.S.; Haririan, M.; Wu, C.C.; Arora, J.S.: Structural design sensitivity analysis of nonlinear response. Computers \& Structures (1988) 245-255. Vidal, C.A.; Haber, R.B.: Design sensitivity analysis for rate-independent elastoplasticity. Comp. Meth. Appl. Mech. Eng. (1993) 391-431. Sensitivity Analysis for Large Deformation Plasticity-Contact- Friction Problems: Formulation and Metal Forming Applications M. J. Saran* and M. Kleiber** *Case Western Reserve University, Cleveland, OH **Institute of Fundamental Technological Research, Warsaw, Poland *Corresponding author, e-mail: [mjs7@po.cwru.edu] A formulation and an algorithm have been developed for sensitivity analysis of large deformation plasticity-contact-friction problems. The general concept of the theory of systems sensitivity is applied to be able to perform automatic calculation of quantitative information on problem sensitivity in response to variation of selected system parameters. The approach is targeted, among others, for industrial metal forming applications characterized by quasi-static loading, irregularly shaped contact surfaces with rigid tools, large relative movement between material and tools, and finally, unknown and changing domain of contact. The mechanical problem is defined as follows. The updated Lagrangian description of motion is used. The material is characterized by the elastic plastic model with plastic yielding behavior described by Hill's nonquadratic yield condition. A nonlinear hardening law is assumed and the normal anisotropy is taken into account. The friction conditions are described by the modified Coulomb friction model. The CFS solution algorithm is used with a consistent tangent matrix, insuring a quadratic rate of convergence and considerably increasing the numerical stability of the solution. The sensitivity equations have been derived. The direct differentiation method has been applied to the considered path dependent problems. The most significant material, process, and tool dimension parameters have been established and are used as the design variables. Sensitivities of the primary degrees of freedom, and of the secondary variables, can be considered, such as sensitivities of the strain, stress, and the springback. Selected results of calculations will be presented and discussed. A Gate Parameterization Scheme for Sensitivity Analysis and Optimization of the Polymer Injection Molding Process Douglas E. Smith Division of Engineering Colorado School of Mines Golden, CO email: dsmith@mines.edu ABSTRACT Plastics have an ever-increasing presence in modern design because they offer numerous combinations of engineering properties and enjoy many versatile manufacturing methods unparalleled in other materials. To support plastic product development, computer aided engineering tools are now used extensively to evaluate the performance of plastic components in their service environment and the manufacturing process used in their production. Developments in process modeling over the past two decades have made it possible to analyze the complex flow of polymer melt in arbitrary mold geometries. The application of optimization methods to make these simulation tools more effective in process design, however, has received much less attention. The cost and complexity of the manufacturing process for making plastic products such as automotive instrument panels, for example, demands the development of optimization methods for the plastics injection molding process. The current work enhances the usefulness of polymer processing analyses by including mold-filling simulations within an optimization methodology to design, not just evaluate, the injection molding process. In this study, the molding process is parameterized by the injection pressure and the location of the edge gate or gates (i.e., the location where polymer melt enters the mold cavity). The location of each gate is specified using a B-Spline surface to represent the part where one of the parametric coordinates becomes the design variable. New gate locations and the injection pressure profiles are computed with gradient-based optimization algorithms to achieve better mold filling conditions. The numerical simulations used to measure the performance of the mold filling process employs the Hele-Shaw approximation for viscous flow in thin cavities. The flow behavior of the non-Newtonian polymer melt is computed with nonlinear transient finite element simulations and the moving boundary at the polymer melt / air interface is tracked using the Volume of Fluid method. This is a coupled problem since the position of the melt / air interface defines the computational domain in the fluid flow analysis and the flow front motion is a function of melt pressure and velocity. Design sensitivities are computed with a transient coupled direct differentiation method which incorporates a general gate parameterization and various process performance measures. A shape design sensitivity analysis method is developed to compute the change in the performance measures with respect the parametric coordinates that define each gate location. The decomposed tangent stiffness matrix from the converged Newton Raphson iteration at each time step is used to efficiently and accurately compute the design sensitivities. Example problems illustrates the design methodology where the fill time is minimized subject to constraints on mold clamp force, injection pressure and flow rate, and flow-front velocity. Additionally, results show that it is possible to evaluate the influence of the design variables on weld line position by viewing contour plots of the design sensitivities. Structural Optimization in Magnetic Fields Using the Homogenization Design Method: Maximizing Magnetic Mean Compliance Jeonghoon Yoo*, Noboru Kikuchi Computational Mechanics Laboratory Dept. of Mechanical Engineering and Applied Mechanics University of Michigan Ann Arbor, MI 48105 * Corresponding author, e-mail:yoojh@engin.umich.edu In the field of topology optimization, The homogenization design method has been successfully applied to elastic problems [1,2]. The homogenization design method is based on the homogenization theory[3] and the finite element method. In the shape optimization magnetic fields, gradient based optimization method combined with the finite element method has been used as a conventional design method [4]. Although this method is effective to some problems, this method can decide only the outer shape of a design domain and dfficult to obtain the optimal shape considering the magnetic energy of the whole design domain. To overcome these disadvantages, the homogenization design method is introdueced to obtain an optimal shape of a structure in magnetic fields. The homogenized magnetic permeability is computed using the homogenization theory. The topology optimization process is developed to get an optimal shape to maximize the magnetic mean compliance. References [1] Bendsoe, M. P., and N. Kikuchi. "Generating optimal topologies in structural design using a homogenization method." Computer Methods in Applied Mechanics and Engineering 71 (1988): 197-224. [2] Suzuki, K., and N. Kikuchi. "A homogenization method for shape and topology optimization." Computer Methods in Applied Mechanics and Engineering 93 (1991): 291-318. [3] Sanchez-Palencia, E. Non-Homogeneous Media and Vibration Theory. Berlin: Springer, 1980. [4] Nakata, T., and N. Takahashi. "New design method of permanent magnets by using the finite element method." IEEE Transactions on Magnetics 19.6 (1983):2494-2497. A LAGRANGIAN SENSITIVITY ANALYSIS FOR FINITE INELASTIC DEFORMATIONS AND METAL FORMING PROCESSES Nicholas Zabaras*, Akkaram Srikanth and Yangang Bao Sibley School of Mechanical and Aerospace Engineering Cornell University Ithaca, NY 14853--3801 *Corresponding author: E-mail: zabaras@cornell.edu A direct differentiation based computational model is presented for the sensitivity analysis of finite deformations. In particular, of interest is the calculation of the sensitivities of material fields (such as stress, state variables, etc.) with respect to design parameters. In metal forming, such design parameters may include the die shape and the preform geometry. The obtained sensitivities are important in the design of processes and preforms that produce a desired geometry and material properties in the final product under certain constraints. In this paper, a sensitivity analysis is introduced for a rigorous mathematical formulation and solution of die and preform design forming problems. An updated Lagrangian finite element method is used to model the large hyperelastic viscoplastic deformations during forming. A reference configuration, that is not related to the deformation of the workpiece, is introduced for a proper definition of the shape derivatives of the deformation gradient and material state fields. Appropriate weak sensitivity problems are defined and special attention is given to the modeling of the sensitivity oundary conditions that result due to the frictional contact between the die and the workpiece. To avoid issues of non-differentiability of the contact conditions, a number of regularizing assumptions are introduced for the calculation of the sensitivity of the contact tractions. Various examples of preform and die design problems will be presented to demonstrate the effectiveness and accuracy of the method developed. Finally, the obtained sensitivity fields will be used to design extrusion and forging processes that result in a final product of desired geometry and properties with minimum work (or force) effort. --------------4EEDF617E25DB0782365E45C--