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.     

Shape design sensitivities improvement using convected unstructured meshes

This paper focuses on various forms of direct differentiation methods for design sensitivity computation in the shape optimisation of continuum structures and the role of convected meshes on the accuracy of the sensitivities. A Pseudo-Analytical Sensitivity Analysis (P-ASA) method is presented and tested. In this method the response analysis component uses unstructured finite element meshes and the sensitivity algorithm entails shape-perturbation for each design variable. A material point is convected during a change of shape and the design sensitivities are therefore intrinsically associated with the mesh-sensitivities of the finite element discretization. Such mesh sensitivities are obtained using a very efficient boundary element point-tracking analysis of an affine notional underlying elastic domain. All of the differentiation, with respect to shape variables, is done exactly except for the case of mesh-sensitivities: hence the method is almost analytical. In contrast to many other competing methods, the P-ASA method is, by definition, independent of perturbation step-size, making it particularly robust. Furthermore, the sensitivity accuracy improves with mesh refinement. The boundary element point-tracking method is also combined with two popular methods of sensitivity computation, namely the global finite difference method and the semi-analytical method. Increases in accuracy and perturbation range are observed for both methods.     

Adaptiver 3D-Vernetzungsalgorithmus zur FEM-Analyse von Massivumformvorgängen

In the numerical simulation of bulk metal forming process by the finite element method (FEM) extremely degenerated meshes result due to high local deformation. The finite elements distort so much that they lose their regular shape. Remeshing and rezoning techniques are necessary to avoid the resulting numerical problems. For remeshing it is important to create a new mesh with regular elements, and to refine the generated grid in dependency of geometric features. An automatic remeshing-module independent of the FE-code is developed at the Institute of Metal forming and Metal forming Machine tools (IFUM). This program creates a mesh that meets all requirements of the object by combination of adaptive meshing and optimisation of the correct grid distortion. In this paper an algorithm to mesh complex 3D geometries with tetrahedron elements is introduced, by taking into account the specific of metal forming processes and their implication in FE-Analysis.     

金属体积成形的无网格Galerkin模拟

为避免金属体积成形有限元法模拟中网格畸变造成网格重划和模拟精度降低,采用无网格法模拟金属体积成形.利用无网格法近似位移场,建立金属体积成形的无网格法连续性控制方程,采用罚函数法施加本征边界条件和体积不变条件,基于Markov变分原理推导了金属体积成形的无网格Galerkin求解列式.用数值计算法求解该列式,实现金属体积成形的无网格模拟.数值结果表明,无网格法能有效处理金属体积成形中出现的大变形,避免了网格畸变和重划,具有较高的模拟精度.     

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.     

A modified finite difference sensitivity analysis method allowing remeshing in large strain path‐dependent problems

Finite difference sensitivity analysis is simple and general yet usually inefficient and inaccurate compared to the analytical sensitivity approach. Although its high computational cost is not an issue in iteratively solved problems, its inaccuracies are critical in path‐dependent problems when remeshing is required. In this case, the errors caused by parametric inversion and interpolation in variables transfer to the new mesh can be as large as the gradient components. This paper presents an efficient modified finite difference approach that allows remeshing either in path‐independent or path‐dependent problems, not being affected by the aforementioned errors. The strategy to cope with remeshing is extensive to the semi‐analytical method which, for non‐linear analyses, is shown to be a particular case of the proposed finite difference sensitivity approach. With this implementation, the finite difference, the semi‐analytical and the analytical sensitivity methods all have comparable computational costs. The perturbation of unstructured meshes is performed with an inverse power Laplacian smoothing. The low cost and the accuracy of the sensitivity fields obtained after remeshing are shown in two examples, considering shape and constitutive design variables. Copyright © 2004 John Wiley & Sons, Ltd.     

On background Cells during the Analysis of Bulk Forming Processes by the Reproducing Kernel Particle Method

The reproducing kernel particle method based on the irreducible flow formulation is utilised to perform the numerical simulation of bulk metal forming processes. Emphasis is given on analysing the influence of employing triangular or quadrilateral background cells on the predictions of material flow, forming load and distribution of strain. A new proposal to smooth the distribution of average stresses during stress computations in the background cells is also included. The effectiveness of the proposed method is discussed by comparing its numerical predictions with a benchmark test case, finite element calculations and experimental data. The benchmark test case is included with the objective of illustrating the influence of several theoretical and numerical subjects such as; order of the basis correction functions, dimension of the compact support and computation of the volume associated to each nodal point. Experimental data was acquired from metal forming controlled laboratory-based tests that were designed so that the proposed method could be tested on its ability to efficiently handle large plastic deformations. It is shown that adaptive arbitrary triangular background cells are capable of efficiently handling large plastic deformations without remeshing.     

Discrete ductile fracture modelling for the metal blanking process

 Design and realisation of a metal blanking process in current industrial practice are mainly based on empirical knowledge. For more sophisticated applications, involving high accuracy geometry specifications, or non-standard materials and product shapes, this empirical approach often fails. This paper presents a set of interrelated numerical techniques resulting in a finite element model of the metal blanking process, focusing on the prediction of the shape of the cut edge of a blanked product. The large, localised deformations are handled by an Operator Split Arbitrary Lagrange Euler (OS-ALE) method supplemented by full remeshing. Transport of the state variables between subsequent meshes for the OS-ALE and remeshing methods is accomplished by the Discontinous Galerkin (DG) method and an interpolation procedure, respectively. Ductile fracture is incorporated using a discrete cracking approach, which is shown to generate mesh independent results. Received 4 February 2000     

Fast resolution of incremental forming processes by the Multi-Mesh method. Application to cogging

A Multi-Mesh Multi-Physics (MMMP) method is developed to reduce the very long computational time required for simulating incremental forming processes such as cogging or ring rolling. It consists in using several finite element meshes on the same domain to solve the different physics of the problem. A reference mesh is used to accurately store the results and history variables, while the different computational meshes are optimized to solve each physic of the problem. The MMMP algorithm consists in two main key-steps: the generation of the different unstructured meshes and the data transfer between the meshes. The accuracy of the method is supported by using meshes that are embedded by nodes. The method is applied to the simulation of the cogging metal forming process for which it shows as accurate and more than ten times faster than the standard method with a single mesh.     

Mesh objective continuum damage models for ductile fracture

During machining processes, the work piece material is subjected to high deformation rates, increased temperature, large plastic deformations, damage evolution and fracture. In this context the Johnson‐Cook failure model is often used even though it exhibits pathological mesh size dependence. To remove the mesh size sensitivity, a set of mesh objective damage models is proposed based on a local continuum damage formulation combined with the concept of a scalar damage phase field. The first model represents a mesh objective augmentation of the well‐established element removal model, whereas the second one degrades the continuum stress in a smooth fashion. Plane strain plate and hat specimens are used in the finite element simulations, with the restriction to the temperature and rate independent cases. To investigate the influence of mesh distortion, a structured and an unstructured meshes were used for the respective specimen. For structured meshes, the results clearly show that the pathological mesh size sensitivity is removed for both models. When considering unstructured meshes, the mesh size sensitivity is more complex as revealed by the considered hat‐specimen shear test. Nevertheless, the present work indicates that the proposed models can predict realistic ductile failure behaviors in a mesh objective fashion. Copyright © 2015 John Wiley & Sons, Ltd.     

Analysis of superplastic metal forming by a finite element method

A finite element velocity method for analysing the superplastic sheet metal forming process is presented. This method is developed from the principle of virtual work and is based on the use of isoparametric continuum elements. The large inelastic deformation of the superplastic material is modelled as the behaviour of an incompressible non-linear viscous flow material. The contact and friction problem is solved by using the compatibility load step method, which is an extension of an earlier work. The finite element method is applied to selected problems to illustrate the applicability of the solution procedure.     

h‐Adaptive FE methods applied to single‐ and multiphase problems

A major problem in using the finite element method for solving numerous engineering problems in the framework of single‐ and multiphase materials is the assessment of discretization errors and the design of suitable meshes. To overcome this problem, adaptive finite element methods have been developed. Based on the error indicator by Zienkiewicz and Zhu, it is the goal of the present paper to present a new error indicator which is especially designed for multiphase problems. Furthermore, efficient h‐adaptive strategies concerning both the generation of new meshes in the framework of independent and hierarchical remeshing strategies and the data transfer between old and new meshes are pointed out. Finally, numerical examples are given to exhibit the efficiency and the quality of the presented h‐adaptive methods and to compare the different strategies to each other. Copyright © 2002 John Wiley & Sons, Ltd.     

An integrated continuous-discontinuous approach towards damage engineering in sheet metal forming processes

This paper addresses the simulation of ductile damage and fracture in metal forming processes. A combined continuous-discontinuous approach has been used, which accounts for the interaction between macroscopic cracks and the surrounding softening material. Softening originates from the degradation processes taking place at a microscopic level, and is modelled using continuum damage mechanics concepts. To avoid pathological localisation and mesh dependence and to incorporate length scale effects due to microstructure evolution, the damage growth is driven by a non-local variable via a second order partial differential equation. The two governing equations, i.e. equilibrium and non-local averaging, are solved in an operator-split manner. This allows one to use a commercial finite element software to solve the equilibrium problem, including contact between the tools and work piece. The non-local averaging equation is solved on a fixed configuration, through a special purpose code which interacts with the commercial code. A remeshing strategy has been devised that allows: (i) to capture the localisation zone, (ii) prevent large element distortions and (iii) accommodate the crack propagation. To illustrate the capabilities of the modelling tool obtained by combining these continuum mechanics concepts and computational techniques, process simulations of blanking, fine-blanking and score forming are presented.     

Sensitivity analysis of viscoplastic deformation process with application to metal preform design optimization

The sensitivity analysis of rigid viscoplastic deformation processes with application to metal preform design optimization is investigated. For viscoplastic constitutive models, the deformation process is path-dependent in nature and thus the sensitivity analysis of the deformation history is formulated in an incremental procedure. To this end, an algorithm is derived on the basis of the time integration scheme used in the primary finite element analysis, where the contact conditions are treated with the penalty method. The discretized equilibrium equations, as well as the time integration equations, are directly differentiated with respect to the design variables. The discrete form of the sensitivity equations is then solved with procedures similar to those used in the direct analysis, where the secant matrix decomposed in the direct analysis can also be utilized at each time instant. Thus the sensitivity of the deformation history is evaluated in a step-wise procedure. The present algorithm can be employed for the optimization of metal forming processes. The accuracy of the proposed sensitivity analysis as well as its applicability are demonstrated by numerical examples with reference to preform design optimization problems, where the aggregate function method is employed for converting the non-smooth Min–max type objective function into a numerically tractable one.     

An updated Lagrangian method with error estimation and adaptive remeshing for very large deformation elasticity problems

Accurate simulations of large deformation hyperelastic materials by the FEM is still a challenging problem. In a total Lagrangian formulation, even when using a very fine initial mesh, the simulation can break down due to severe mesh distortion. Error estimation and adaptive remeshing on the initial geometry are helpful and can provide more accurate solutions but are not sufficient to attain very large deformations. The updated Lagrangian formulation where the geometry is periodically updated is then preferred. However, it requires data transfer from the old mesh to the new one and this is a very delicate issue. In this paper, we present an updated Lagrangian formulation where the error is estimated and adaptive remeshing is performed in order to reach high level of deformations while controlling both the accuracy of the solution and mesh distortion. Special attention is given to data transfer methods and a very accurate cubic Lagrange projection method is introduced. A continuation method is used to automatically pilot the complete algorithm including load increase, error estimation, adaptive remeshing, and data transfer. A number of examples will be presented and analyzed. Copyright © 2014 John Wiley & Sons, Ltd.     

Design sensitivity analysis and optimization of non‐linear transient dynamics. Part I—sizing design

A continuum‐based sizing design sensitivity analysis (DSA) method is presented for the transient dynamic response of non‐linear structural systems with elastic–plastic material and large deformation. The methodology is aimed for applications in non‐linear dynamic problems, such as crashworthiness design. The first‐order variations of the energy forms, load form, and kinematic and structural responses with respect to sizing design variables are derived. To obtain design sensitivities, the direct differentiation method and updated Lagrangian formulation are used since they are more appropriate for the path‐dependent problems than the adjoint variable method and the total Lagrangian formulation, respectively. The central difference method and the finite element method are used to discretize the temporal and spatial domains, respectively. The Hughes–Liu truss/beam element, Jaumann rate of Cauchy stress, rate of deformation tensor, and Jaumann rate‐based incrementally objective stress integration scheme are used to handle the finite strain and rotation. An elastic–plastic material model with combined isotropic/kinematic hardening rule is employed. A key development is to use the radial return algorithm along with the secant iteration method to enforce the consistency condition that prevents the discontinuity of stress sensitivities at the yield point. Numerical results of sizing DSA using DYNA3D yield very good agreement with the finite difference results. Design optimization is carried out using the design sensitivity information. Copyright © 2000 John Wiley & Sons, Ltd.     

ANALYSIS OF AXISYMMETRIC FORGING OF POROUS MATERIALS BY THE FINITE ELEMENT METHOD

Axisymmetric forging of powdered-metal preforms was analyzed by the finite element method neglecting the effect of temperature on deformation. A remeshing program based on the area-weighted-average technique was developed and tested by simulating forging of the flange-hub shape. Then the analysis was performed for forging of a pulley blank. The detailed deformation characteristics for the different preform shapes under the two frictional conditions were obtained. A comparison between the prediction and the experiment regarding proper preform design to forge defect-free pulley blank demonstrates that the finite element analysis is useful for further developments in preform design in powdered-metal forging.     

An object‐oriented programming approach to the Lagrangian FEM analysis of large inelastic deformations and metal‐forming processes

The general deformation problem with material and geometric non‐linearities is typically divided into a number of subproblems including the kinematic, the constitutive, and the contact/friction subproblems. These problems are introduced for algorithmic purposes; however, each of them represents distinct physical aspects of the deformation process. For each of these subproblems, several well‐established mathematical and numerical models based on the finite element method have been proposed for their solution. Recent developments in software engineering and in the field of object‐oriented C++ programming have made it possible to model physical processes and mechanisms more expressively than ever before. In particular, the various subproblems and computational models in a large inelastic deformation analysis can be implemented using appropriate hierarchies of classes that accurately represent their underlying physical, mathematical and/or geometric structures. This paper addresses such issues and demonstrates that an approach to deformation processing using classes, inheritance and virtual functions allows a very fast and robust implementation and testing of various physical processes and computational algorithms. Here, specific ideas are provided for the development of an object‐oriented C++ programming approach to the FEM analysis of large inelastic deformations. It is shown that the maintainability, generality, expandability, and code re‐usability of such FEM codes are highly improved. Finally, the efficiency and accuracy of an object‐oriented programming approach to the analysis of large inelastic deformations are investigated using a number of benchmark metal‐forming examples. Copyright © 1999 John Wiley & Sons, Ltd.     

Automatic remeshing in 3‐D analysis of forming processes

Finite element modelling, employing updated Lagrangian techniques, is used extensively in the design and analysis of bulk forming processes. However, the full 3‐D capability has not seen widespread use in the automotive, aerospace, and, related industries due to, among other reasons, the need for remeshing, or, representation of the workpiece with a new finite element mesh as the analysis progresses. Automating the remeshing procedure of the deformed workpiece geometry would reduce the time required for a 3‐D analysis by several orders of magnitude. This paper discusses an algorithm for generating a new mesh to represent the deformed workpiece geometry during the analysis. The procedure is used to perform a 3‐D analysis of a valve forging problem. Copyright © 1999 John Wiley & Sons, Ltd.     

Solidification in castings by finite element method

Abstract

A self-contained CAD (computer aided design) system capable of analyzing foundry casting processes in sand and gravity dies is being developed at the University College of Swansea. The work involves preprocessing, postprocessing, and a finite element code with some novel numerical techniques. The solidification of castings is a heat transfer problem involving phase change, which may occur in a narrow range of temperatures. To simulate the phenomena accurately, very fine meshes must be used and the solution of such a system becomes very expensive. In the Swansea system, an adaptive remeshing technique is introduced, which tracks the moving front of the phase change zone. At every time step, a scan is made to determine the points at which phase change is occurring, so that the remeshing may be done to produce a refined mesh at such points. The computing process is then continued. Examples have illustrated that the method is efficient and accurate. In addition, an interfacial heat transfer model is introduced to improve the simulation of the casting process. Advective heat transfer in the liquid is also modelled.

MST/1041