Sensitivity analysis and shape optimization for preform design in thermo‐mechanical coupled analysis

In complex forging processes, it is essential to find the optimal deformation path and the optimal preform shape that will lead to the desired final shape and service properties. A sensitivity analysis and optimization for preform billet shape in thermo‐mechanical coupled simulation is developed in this work. Non‐linear sensitivity analysis of temperatures, flow‐stresses, strains and strain‐rates are presented with respect to design variables. Both analytical and finite‐difference gradients are employed to validate the effectiveness of sensitivity analysis developed in this work. Numerous iterations of coupled thermo‐mechanical analysis are performed to determine an optimum preform shape based on a given criterion of minimizing the objective function on effective strain variance within the final forging. The design constraints are imposed on die underfill, material scrap, folding defects and temperatures. In addition, a method for material data processing is given in order to determine the flow stress and its derivatives. The shape optimization scheme is demonstrated with the preform designs of an axisymmetric disk and a plane strain problem. Copyright © 1999 John Wiley & Sons, Ltd.     

An updated Lagrangian finite element sensitivity analysis of large deformations using quadrilateral elements

A continuum parameter and shape sensitivity analysis is presented for metal forming processes using the finite element method. The sensitivity problem is posed in a novel updated Lagrangian framework as suitable for very large deformations when remeshing operations are performed during the analysis. In addition to exploring the issue of transfer of variables between meshes for finite deformation analysis, the complex problem of transfer of design sensitivities (derivatives) between meshes for large deformation inelastic analyses is also discussed. A method is proposed that is shown to give accurate estimates of design sensitivities when remeshing operations are performed during the analysis. Sensitivity analysis for the consistent finite element treatment of near incompressibility within the context of the assumed strain methods is also proposed. In particular, the performance of four‐noded quadrilateral elements for the sensitivity analysis of large deformations is studied. The results of the continuum sensitivity analysis are validated by a comparison with those obtained by a finite difference approximation (i.e. using the solution of a perturbed deformation problem). The effectiveness of the method is demonstrated by applications in the design optimization of metal forming processes. Copyright © 2001 John Wiley & Sons, Ltd.     

Die shape design optimization of sheet metal stamping process using meshfree method

A die shape design sensitivity analysis (DSA) and optimization for a sheet metal stamping process is proposed based on a Lagrangian formulation. A hyperelasticity‐based elastoplastic material model is used for the constitutive relation that includes a large deformation effect. The contact condition between a workpiece and a rigid die is imposed through the penalty method with a modified Coulomb friction model. The domain of the workpiece is discretized by a meshfree method. A continuum‐based DSA with respect to the rigid die shape parameter is formulated using a design velocity concept. The die shape perturbation has an effect on structural performance through the contact variational form. The effect of the deformation‐dependent pressure load to the design sensitivity is discussed. It is shown that the design sensitivity equation uses the same tangent stiffness matrix as the response analysis. The linear design sensitivity equation is solved at each converged load step without the need of iteration, which is quite efficient in computation. The accuracy of sensitivity information is compared to that of the finite difference method with an excellent agreement. A die shape design optimization problem is solved to obtain the desired shape of the workpiece to minimize spring‐back effect and to show the feasibility of the proposed method. Copyright © 2001 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.     

Design sensitivity analysis and optimization for minimizing springback of sheet‐formed part

The springback is a manufacturing defect in the stamping process and causes difficulty in product assembly. An impediment to the use of lighter‐weight, higher‐strength materials in manufacturing is relative lack of understanding about how these materials respond to complex forming processes. The springback can be reduced by using an optimized combination of die, punch, and blank holder shapes together with friction and blank‐holding force. An optimized process can be determined using a gradient‐based optimization to minimize the springback. For an effective optimization of the stamping process, development of an efficient design sensitivity analysis (DSA) for the springback with respect to these process parameters is crucial. A continuum‐based shape and configuration DSA method for the stamping process has been developed using a non‐linear shell model. The material derivative is used to develop the continuum‐based design sensitivity. The design sensitivity equation is solved without iteration at each converged load step in the finite deformation elastoplastic non‐linear analysis with frictional contact, which makes sensitivity calculation very efficient. Numerical implementation of the proposed shape and configuration DSA method is performed using the meshfree method. The accuracy and efficiency of the proposed method are illustrated by minimizing the springback in a benchmark S‐rail problem. Copyright © 2007 John Wiley & Sons, Ltd.     

Continuum‐based design sensitivity analysis and optimization of nonlinear shell structures using meshfree method

A continuum‐based shape and configuration design sensitivity analysis (DSA) method for a finite deformation elastoplastic shell structure has been developed. Shell elastoplasticity is treated using the projection method that performs the return mapping on the subspace defined by the zero‐normal stress condition. An incrementally objective integration scheme is used in the context of finite deformation shell analysis, wherein the stress objectivity is preserved for finite rotation increments. The material derivative concept is used to develop a continuum‐based shape and configuration DSA method. Significant computational efficiency is obtained by solving the design sensitivity equation without iteration at each converged load step using the same consistent tangent stiffness matrix. Numerical implementation of the proposed shape and configuration DSA is carried out using the meshfree method. The accuracy and efficiency of the proposed method is illustrated using numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

挤压成形过程灵敏度分析及模具型腔参数优化设计

以高温下的挤压成形加工过程优化设计问题为背景,针对应变率相关粘性固体本构关系的材料稳态非弹性变形过程提出了形状灵敏度分析方法,并对算法的效率进行了讨论,在此基础上研究了挤压成形模具的形状优化设计问题的数学模型和数值求解算法。给出了挤压成形模具的二维形状优化设计算例,数值结果验证了所给出的灵敏度分析算法和优化设计模型的正确性和可用性。     

Numerical simulation of forging a ball from a cylindrical billet

Abstract

Rigid‐viscoplastic finite element equations are used for the analysis of metal forming process. With the view of solution accuracy and computation efficiency, the possible overconstraint of the incompressibility condition is avoided by modifying the penalty method in the variational formulation; and both the direct iterative method and Newton‐Raphson iterative method are combined to solve the finite element equations. The forging process of a ball from a cylindrical workpiece is completely simulated by a remesh procedure. The computed results agree well with the experimental measurements. It is shown, during the early stage of plastic deformation, the effect of friction is small, but gradually increases with further plastic deformation. In the finishing stage, the shear plastic deformation is found mainly in the flash portion. As the shape factor of workpiece increases, the filling of the die cavity is more complete, but the required forming energy increases and the variation of microstructure within the final forged product is intensified. The effect of die velocity also improves die cavity filling.     

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.     

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.     

A continuum approach for second-order shape design sensitivity of elastic solids with loaded boundaries

A second-order shape design sensitivity analysis (DSA) method applicable to the shape change on the loaded boundaries is derived for three-dimensional linear elastic solids using a continuum method with the material derivative. The continuum method is also used to derive mixed second-order variations of stress and displacement performance measures with respect to shape design variables and distribution of non-conservative traction loads, and also with respect to shape design variables and material properties. A shape design acceleration field is defined for the second-order shape design sensitivity. Both direct differentiation and hybrid methods are presented in this paper. A numerical method, which can be implemented using established finite element analysis (FEA) codes, is developed. The feasibility and accuracy of the proposed second-order shape DSA method has been demonstrated by solving a structural example-doubly curved arch dam.     

Topology optimization of finite strain viscoplastic systems under transient loads

A transient finite strain viscoplastic model is implemented in a gradient‐based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark‐beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capability of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. The numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.     

Multiscale analysis method for thermo‐mechanical performance of periodic porous materials with interior surface radiation

This study develops a novel multiscale analysis method to predict thermo‐mechanical performance of periodic porous materials with interior surface radiation. In these materials, thermal radiation effect at microscale has an important impact on the macroscopic temperature and stress field, which is our particular interest in this paper. Firstly, the multiscale asymptotic expansions for computing the dynamic thermo‐mechanical coupling problem, which considers the mutual interaction between temperature and displacement field, are given successively. Then, the corresponding numerical algorithm based on the finite element‐difference method is brought forward in details. Finally, some numerical results are presented to verify the validity and relevancy of the proposed method by comparing it with a direct finite element analysis with detailed numerical models. The comparison shows that the new method is effective and valid for predicting the thermo‐mechanical performance and can capture the microstructure behavior of periodic porous materials exactly.s Copyright © 2015 John Wiley & Sons, Ltd.     

Variational inequality‐based framework of discontinuous deformation analysis

For modeling discrete particle‐block systems, a new framework of discontinuous deformation analysis is established on the basis of finite‐dimensional variational inequality. The presented method takes into account the contacts, the rolling resistance, and the tensile resistance of cemented interface among particles and blocks using the corresponding variational or quasivariational inequalities. The new formulation avoids using the artificial springs that are usually indispensable in many conventional methods dealing with similar discrete problems and conveniently integrates the rigid circle particles, the nonrigid ring particles, and the arbitrary shape blocks into a uniform framework. The proposed discontinuous deformation analysis approach is further coupled with the finite element method using a node‐based composite contact matrix and several simple transformation matrices to solve practical problems. A particle/block‐based composite contact matrix is constructed to further broaden the application of the proposed method. The accuracy, robustness, and capability of the presented method are demonstrated with examples.     

Process optimal design in non‐isothermal,steady‐state metal forming by the finite element method

A new approach to process optimal design in non‐isothermal, steady‐state metal forming is presented. In this approach, the optimal design problem is formulated on the basis of the integrated thermo‐mechanical finite element process model so as to cover a wide class of the objective functions and to accept diverse process parameters as design variables, and a derivative‐based approach is adopted as a solution technique. The process model, the formulation for process optimal design, and the schemes for the evaluation of the design sensitivity, and an iterative procedure for design optimization are described in detail. The validity of the schemes for the evaluation of the design sensitivity is examined by performing a series of numerical tests. The capability of the proposed approach is demonstrated through applications to some selected process design problems. Copyright © 1999 John Wiley & Sons, Ltd.     

Sensitivity analysis of heat conduction for functionally graded materials

A sensitivity analysis is presented for the steady-state and transient heat conduction of functionally graded materials (FGMs). Based on the finite element method, the sensitivity equations of heat conduction are presented by using the direct method and the adjoint method. In the solution of transient problem, the precise time integration (PTI) is employed. The spatial volume fractions of materials of FGM (size problem) and the shape design parameters are considered. Detailed formulations especial for the FGMs are provided. The numerical examples are presented to demonstrate the precision and applicability of the proposed method.     

Finite element formulation for anisotropic coupled piezoelectro‐hygro‐thermo‐viscoelasto‐dynamic problems

A finite element algorithm has been developed for the efficient analysis of smart composite structures with piezoelectric polymer sensors or/and actuators based on piezoelectro‐hygro‐thermo‐viscoelasticity. Variational principles for anisotropic coupled piezoelectro‐hygro‐thermo‐viscoelasto‐dynamic problems have also been proposed in this study. As illustrative studies, dynamic responses in laminated composite beams and plates with PVDF sensors and actuators are obtained as functions of time using the present finite element procedures. The voltage feedback control scheme is utilized. The proposed numerical method can be used for analysing problems in the design of smart structures as well as smart sensors and actuators. Copyright © 1999 John Wiley & Sons, Ltd.     

An efficient 3D implicit approach for the thermomechanical simulation of elastic–viscoplastic materials submitted to high strain rate and damage

This paper aims at presenting a general consistent numerical formulation able to take into account, in a coupled way, strain rate, thermal and damage effects on the behavior of materials submitted to quasistatic or dynamic loading conditions in a large deformation context. The main features of this algorithmic treatment are as follows:

• A unified treatment for the analysis and implicit time integration of thermo‐elasto‐viscoplastic constitutive equations including damage that depends on the strain rate for dynamic loading conditions. This formalism enables us to use dynamic thermomechanically coupled damage laws in an implicit framework.
• An implicit framework developed for time integration of the equations of motion. An efficient staggered solution procedure has been elaborated and implemented so that the inertia and heat conduction effects can be properly treated.
• An operator split‐based implementation, accompanied by a unified method to analytically evaluate the consistent tangent operator for the (implicit) coupled damage–thermo‐elasto‐viscoplastic problem.
• The possibility to couple any hardening law, including rate‐dependent models, with any damage model that fits into the present framework.

All the developments have been considered in the framework of an implicit finite element code adapted to large strain problems. The numerical model will be illustrated by several applications issued from the impact and metal‐forming domains. All these physical phenomena have been included into an oriented object finite element code (implemented at LTAS‐MN ²L, University of Liège, Belgium) named Metafor.Copyright © 2013 John Wiley & Sons, Ltd.     

A multiscale finite element method for modeling fully coupled thermomechanical problems in solids

This article proposes a two‐scale formulation of fully coupled continuum thermomechanics using the finite element method at both scales. A monolithic approach is adopted in the solution of the momentum and energy equations. An efficient implementation of the resulting algorithm is derived that is suitable for multicore processing. The proposed method is applied with success to a strongly coupled problem involving shape‐memory alloys.Copyright © 2012 John Wiley & Sons, Ltd.     

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 .