锻造过程中变形均匀性控制及模具优化设计

以直接设计预成形模具形状为目标 ,提出并建立了一种控制变形均匀性的灵敏度分析理论和模具形状优化设计方法 .以任意单元的等效应变与所有单元的平均等效应变的差值的平方和作为目标函数 ,B样条曲线表示模具形状 ,B样条曲线控制点坐标作为优化设计变量 ,优化设计的目标就是通过设计预成形模具形状使目标函数最小 ,即使整个工件的变形尽可能均匀 .给出了目标函数的表达形式 ,详细推导了目标函数、单元等效应变率和单元各应变率分量对优化设计变量的灵敏度 .并对圆柱体平面镦粗工艺进行预成形优化设计 ,给出了相应的优化设计结果 .     

Optimal Preform die Design through Controlling Deformation Uniformity in Metal Forging

A finite element based sensitivity analysis method for preform die shape design in metal forging is developed.The optimization goal is to obtain more uniform deformation within the final forging by controlling the deformation uniformity.The objective function expressed by the effective strain is constructed.The sensitivity equations of the objective function,elemental volume,elemental effective strain rate and the elemental strain rate with respect to the design variables are constituted.The preform die shapes of an H-shaped forging process in axisymmetric deformation are designed using this method.     

Shape optimization of preform tools in forging of aerofoil using a metamodel-assisted multi-island genetic algorithm

Preform design plays an important role in improving the material flow, mechanical properties and reducing defects for forgings with complex shapes. In this paper, a study on shape optimization of preform tools in forging of an airfoil is carried out based on a multi-island genetic algorithm combined with a metamodel technique. An optimal Latin hypercube sampling technique is employed for sampling with the expected coverage of parameter space. Finite element (FE) simulations of multistep forging processes are implemented to obtain the objective function values for evaluating the forging qualities. For facilitating the optimization process, a radial basis function surrogate model is established to predict the responses of the hot forging process to the variation of the preform tool shape. In consideration of the compromise between different optimal objectives, a set of Pareto-optimal solutions are identified by the suggested genetic algorithm to provide more selections. Finally, according to the proposed fitness function, the best solution of multi-objective optimization on the Pareto front is confirmed and the corresponding preform tool shape proves optimal performances with substantially improved forging qualities via FE validation.     

基于有限元灵敏度分析的预锻模具形状优化设计

在控制锻件几何形状的前提下 ,采用有限元灵敏度分析方法 ,对预锻模具形状进行优化设计 .针对下模速度为零时 ,速度灵敏度边界条件为零 ,其形状在优化迭代过程中得不到优化的情况 ,对速度灵敏度边界条件提出改进措施 ,使上下模具形状同时能够得到优化 .最后给出了优化设计实例 ,验证该方法的可靠性 .     

A continuum sensitivity method for finite thermo‐inelastic deformations with applications to the design of hot forming processes

A computational framework is presented to evaluate the shape as well as non‐shape (parameter) sensitivity of finite thermo‐inelastic deformations using the continuum sensitivity method (CSM). Weak sensitivity equations are developed for the large thermo‐mechanical deformation of hyperelastic thermo‐viscoplastic materials that are consistent with the kinematic, constitutive, contact and thermal analyses used in the solution of the direct deformation problem. The sensitivities are defined in a rigorous sense and the sensitivity analysis is performed in an infinite‐dimensional continuum framework. The effects of perturbation in the preform, die surface, or other process parameters are carefully considered in the CSM development for the computation of the die temperature sensitivity fields. The direct deformation and sensitivity deformation problems are solved using the finite element method. The results of the continuum sensitivity analysis are validated extensively by a comparison with those obtained by finite difference approximations (i.e. using the solution of a deformation problem with perturbed design variables). The effectiveness of the method is demonstrated with a number of applications in the design optimization of metal forming processes. Copyright © 2002 John Wiley & Sons, Ltd.     

PREFORM DIE SHAPE DESIGN IN METAL FORMING USING AN OPTIMIZATION METHOD

An optimization algorithm for preform die shape design in metal-forming processes is developed in this paper. The preform die shapes are represented by cubic B-spline curves. The control points of the B-spline are used as the design variables. The optimization objective is to reduce the difference between the realized and desired final forging shapes. The sensitivities of the objective function with respect to the design variables are developed in detail. The numerical examples show that the optimization method and the sensitivity analysis developed in this paper are very useful and the design results are satisfactory. Importantly, the preform die shapes designed by this method are easily manufacturable and can be implemented in practical metal-forming operations. This optimization method and the sensitivity analysis can also be applied in the preform design of complex industrial metal-forming problems. © 1997 by John Wiley & Sons, Ltd.     

Sensitivity Analysis Based Multiple Objective Preform Die Shape Optimal Design in Metal Forging

The multiple objective preform design optimization was put forward. The final forging＇s shape and deformation uniformity were considered in the multiple objective. The objective is to optimize the shape and the deformation uniformity of the final forging at the same time so that a more high integrate quality of the final forging can be obtained. The total objective was assembled by the shape and uniformity objective using the weight adding method. The preform die shape is presented by cubic B-spline curves. The control points of B-spline curves are used as the design variables. The forms of the total objective function, shape and uniformity sub-objective function are given. The sensitivities of the total objective function and the sub-objective functions with respect to the design variables are developed. Using this method, the preform die shape of an H-shaped forging process is optimally designed. The optimization results are very satisfactory.     

Analysis of center burst during hot forging

Carbon steel axle forgings were rejected due to internal cracks observed during final machining. To determine the cause of the cracks, the preforms of the forging were analyzed in detail at each stage of the forging. The analysis revealed a large central burst in the intermediate stage of the forging preform, which subsequently increased in the final stage. A high upset strain during forging, especially in the final stage, accentuated the center burst by high lateral flow of the metal. It was concluded that the center burst of the axle forging resulted from a high concentration of nonmetallic inclusions in the central portion of the raw bar stock rather than the usual problem of improper forging temperature. Strict control over the inclusion content in the raw material by changing the vendor eliminated the problem.     

Experimental characterisation of a CuAg alloy for thermo‐mechanical applications. Part 2: Design strain‐life curves estimated via statistical analysis

Strain‐life fatigue data on copper alloys, especially type CuAg, are seldom available in the literature. This work fills this gap by estimating the strain‐life curves of a CuAg alloy used for thermo‐mechanical applications, from isothermal low‐cycle fatigue tests at 3 temperatures (room temperature, 250°C, 300°C). Regression analysis is used to estimate the median fatigue curves at 50% survival probability. The comparison of median curves with the Universal Slopes Equation model, calibrated on monotonic tensile properties, shows a fairly good agreement. Design strain‐life curves with a lower failure probability and given confidence are estimated by several approximate statistical methods (“Equivalent Prediction Interval,” univariate tolerance interval, Owen's tolerance interval for regression). When higher survival probabilities are considered, the results show a marked decrease in the allowable design strain at a prescribed fatigue life. The suggested procedure thus improves the durability analysis of components loaded thermo‐mechanically.     

Microstructure optimization design methods of the forging process and applications

A microstructure optimization design method of the forging process is proposed. The optimization goal is the fine grain size and homogeneous grain distribution. The optimization object is the forging process parameters and the shape of the preform die. The grain size sub-objective function, the forgings shape sub-objective function and the whole objective function including the shape and the grain size are established, respectively. The detailed optimization steps are given. The microstructure optimization program is developed using the micro-genetic algorithm and the finite element method. Then, the upsetting process of the cylindrical billet is analyzed using a self-developed program. The forging parameters and the shape of preform die of the upsetting process are optimized respectively. The fine size and homogenous distribution of the grain can be achieved by controlling the shape of the preform die and improving the friction condition.     

Modelling of reinforced polymer composites subject to thermo‐mechanical loading

A coupled finite element model is developed to analyse the thermo‐mechanical behaviour of a widely used polymer composite panel subject to high temperatures at service conditions. Thermo‐chemical and thermo‐mechanical models of previous researchers have been extended to study the thermo‐chemical decomposition, internal heat and mass transfer, deformation and the stress state of the material. The phenomena of heat and mass transfer and thermo‐mechanical deformation are simulated using three sets of governing equations, i.e. energy, gas mass diffusion and deformation equations. These equations are then assembled into a coupled matrix equation using the Bubnov–Galerkin finite element formulation and then solved simultaneously at each time interval. An experimentally tested 1.09 cm thick glass‐fibre woven‐roving/polyester resin composite panel is analysed using the numerical model. Results are presented in the form of temperature, pore pressure, deformation, strain and stress profiles and discussed. The maximum normal stress failure criterion is used in order to establish the load‐bearing capability of the composite panel. Significant pore gas pressure build‐ups (to 0.8 MPa and higher) have been perceived at high thermo‐chemical decomposition rates where the material experiences a complex expansion/contraction phenomenon. It is found that the composite panel experiences structural instability at elevated temperatures up to 300°C but retains its integrity even under moderate external loading. Copyright © 2005 John Wiley & Sons, Ltd.     

3D preform shape optimization in forging using reduced basis techniques

Three-dimensional preform shape optimization of complex forgings with a weighted summation of multiple basis shapes is presented in this article. Currently, 2D preform shape optimization is well developed; however, in cases in which the parts are neither axisymmetric nor plane strain, 2D assumptions do not hold well. The number of design variables required to define the 3D preform shape is high, making most iterative design methods impractical for shape optimization. The goal here is to make design optimization practical and efficient by developing reduced-order modeling techniques for 3D preform shape optimization. The preform shape is treated as a linear combination of various billet shapes, called basis shapes, with the weights for each basis shape used as design variables, thereby reducing the number of design variables. It is very difficult to obtain the necessary gradient information for 3D forging simulations, so a non-gradient method is used to build the surrogate model on which optimization is performed. The optimization problem is formulated to minimize strain variance while placing constraints on underfill. Representative problems are used to demonstrate the effectiveness of the approach.     

A new three-dimensional finite element model for the simulation of powder forging processes: application to hot forming of P/M connecting rod

In powder metallurgy (P/M) the forming of industrial artifacts requires consolidation of loose powder into dense material leading to near-to-net shape components. In order to realize the economic advantages of the near-to-net shape formation, it is essential to understand the mechanical behaviour of powder deforming domain. The conventional P/M forming process consists of different stages such as closed cold die compaction, sintering and hot/cold forging. In the present study a finite element based computational model has been formulated to study the hot forging stage with particular reference to forging of P/M connecting rods. In order to achieve this purpose, a new finite element formulation has been developed to model the powder deformation under a given thermo-mechanical loading. Essentially, the computational model is formulated based on a visco-plastic Green type material model considering an independent idealization of strain rates into a total deformation part and a dilatational part, and the yield criterion takes into account the pressure sensitivity. The model is set in a perturbed Lagrangian functional, leading to a three field mixed formulation with velocity, Lagrangian multiplier/pressure and volumetric strain rates as three basic unknowns in the finite element domain. The developed finite element model can be used right from the compressible domain to the incompressible domain with the Lagrangian multiplier becoming then the pressure. A relative density-dependent visco-plastic type of friction law is used for characterizing the friction behaviour between the powder preform and the die. The required various material parameters are determined from experiments on aluminium powder preforms. In order to facilitate the non-isothermal deformation study, a powder-based transient thermal analysis has been developed. A computational scheme has been used to couple the mechanical and thermal calculations. Using the developed three-dimensional code, hot forging of automotive components can be simulated and which in the present study is exemplified by simulation of hot forging of a P/M connecting rod. © 1997 John Wiley & Sons, Ltd.     

A continuum Lagrangian sensitivity analysis for metal forming processes with applications to die design problems

A continuum sensitivity analysis is presented for large inelastic deformations and metal forming processes. The formulation is based on the differentiation of the governing field equations of the direct problem and development of weak forms for the corresponding field sensitivity equations. Special attention is given to modelling of the sensitivity boundary conditions that result due to frictional contact between the die and the workpiece. The contact problem in the direct deformation analysis is modelled using an augmented Lagrangian formulation. To avoid issues of non‐differentiability of the contact conditions, appropriate regularizing assumptions are introduced for the calculation of the sensitivity of the contact tractions. The proposed analysis is used for the calculation of sensitivity fields with respect to various process parameters including the die surface. The accuracy and effectiveness of the proposed method are demonstrated with a number of representative example problems. In the die design applications, a Bézier representation of the die curve is introduced. The control points of the Bézier curve are used as the design parameters. Comparison of the computed sensitivity results with those obtained using the direct analysis for two nearby dies and a finite difference approximation indicate a very high accuracy of the proposed analysis. The method is applied to the design of extrusion dies that minimize the standard deviation of the material state in the final product or minimize the required extrusion force for a given reduction ratio. An open‐forging die is also designed which for a specified stroke and initial workpiece produces a final product of desired shape. Copyright © 2000 John Wiley & Sons, Ltd.     

Optimization of thermo‐elasto‐plastic processes using Eulerian sensitivity analysis

A computational scheme for the analysis and optimization of quasi‐static thermo‐mechanical processes is presented in this paper. In order to obtain desirable mechanical transformations in a workpiece using a thermal treatment process, the optimal control parameters need to be determined. The problem is addressed by posing the process as a decoupled thermo‐mechanical finite element problem and performing an optimization using gradient methods. The forward problem is solved using the Eulerian formulation since it is computationally more efficient compared to an equivalent Lagrangian formulation. The design sensitivities required for the optimization are developed analytically using direct differentiation. This systematic design approach is applied to optimize a laser forming process. The objective is to maximize the angular distortion of a specimen subject to the constraint that the phase transition temperature is not exceeded at any point in the model. Copyright © 2003 John Wiley & Sons, Ltd.     

Variational approach to the analysis of steady‐state thermo‐elastic wear regimes

A transient wear process on frictional interface of two thermo‐elastic bodies in a relative steady sliding motion induces shape evolution of contact interface and tends to a steady state for which the wear process occurs at fixed contact stress and strain distribution. The temperature field generated by frictional and wear dissipation on the contact surface is assumed to reach a steady state. This state is assumed to correspond to minimum of the wear dissipation power and the temperature field corresponds to maximum of the heat entropy production. The stationarity conditions of the response functionals provide the contact pressure distribution and the corresponding temperature field. The present approach extends the authors previous analyses of optimal or steady‐state contact shapes by accounting for coupled wear and thermal distortion effects. The wear rule is assumed as a non‐linear relation of wear rate to shear stress and relative sliding velocity. The analysis of disk and drum brakes is presented with account for thermal distortion effect. It is shown that the contact shape in a steady thermo‐elastic state essentially differs from that specified for mechanical loading with neglect of thermal effects. The thermal instability regimes are not considered in the paper. Copyright © 2009 John Wiley & Sons, Ltd.     

Shape optimization of axisymmetric preform tools in forging using a direct differentiation method

A method for computing shape sensitivity in the frame of non‐linear and non‐steady‐state forging is presented. Derivatives of tool geometry, velocity and state variables with respect to the shape parameters are calculated by a direct differentiation of discrete equations. Because of the important part played by the accuracy of finite element calculations, an efficient transfer method is used between meshes during remeshings and the contact algorithms are carefully differentiated. The resulting inverse design procedure is successfully applied to two industrial examples of forging of automobile parts, with fold‐over and piping defects occurring during the intermediate designs. It makes it possible to suggest reasonable preform shapes, with or without any available knowledge of the forging process. Copyright © 2001 John Wiley & Sons, Ltd.     

On variational sensitivity analysis and configurational mechanics

This contribution is concerned with the application of variational design sensitivity analysis in the context of structural optimization and configurational mechanics. In both disciplines we consider variations of the material configuration and we use techniques from variational sensitivity analysis in order to solve these problems. We derive the physical and material residual problem in one step by using standard optimization procedures. Furthermore, we investigate the sensitivity of the physical as well as the material residual problem and obtain the coupled saddle point problem based on these sensitivities. Both problems are coupled by the pseudo load operator, which plays an important role by the solution of structural optimization problems. By means of computational examples from mesh optimization and shape optimization, we demonstrate the capability of the proposed theoretical framework.     

OPTIMAL DESIGN FOR NON-STEADY-STATE METAL FORMING PROCESSES—I. SHAPE OPTIMIZATION METHOD

We suggest a shape optimization method for a non-linear and non-steady-state metal forming problem. It consists in optimizing the initial shape of the part as well as the shape of the preform tool during a two-step forging operation, for which the shape of the second operation is known. Shapes are described using spline functions and optimal parameter values of the splines are searched in order to produce, at the end of the forging sequence, a part with a prescribed geometric accuracy, optimal metallurgical properties and for a minimal production cost. The finite element method, including numerous remeshing operations, is used for the simulation of the process. We suggest using a least-squares-type algorithm for the unconstrained optimization method (based on external penalty) for which we describe the calculation of the derivatives of the objective function. We show that it can reduce to calculations which are equivalent to the derivative calculations of steady-state processes and to evolution equations. Therefore, the computational cost of such an optimization is quite reasonable, even for complex forging processes. Lastly, in order to reduce the errors due to the numerous remeshings during the simulation, we introduce error estimation and adaptive remeshing methods with respect to the calculation of derivatives.