POD surrogates for real-time multi-parametric sheet metal forming problems

Our approach aims at coupling the ever increasing off-line computing power of mainframe computers with the interactive on-line possibilities of ubiquitous low computing power devices at the early design stages in order to provide insight into the design problems and to search for candidate optimal design points. In the off-line phase, the method under investigation relies on combining an optimized space-filling sampling plan on the design parameter space with extensive finite elements (FE) simulations yielding a learning set of displacement fields. The objective of this paper is the on-line phase. We provide a rigorous mathematical presentation of a family of non-intrusive, bi-level surrogates. We focus on displacement field approximation by Proper Orthogonal Decomposition (POD) combined with kriging interpolation of coefficients. The method is illustrated with two simple, easily reproduced numerical examples of quality assessment of deep-drawing process of a cylindrical cup by on-the-fly plotting forming limit diagrams (FLDs) and related quantities enabling thus to spot improved design points.     

Parameter sensitivity analysis in frictional contact problems of sheet metal forming

Influence of frictional contact effects on parameter variations of some nonlinear behaviour is studied. The flow approach to deep drawing simulation is taken as the underlying nonlinear mechanics problem. Theoretical considerations are followed by the discussion of computational aspects. In particular, difficulties resulting from parameter nondifferentiability of the response at some points along the deformation path are indicated and discussed in the computational context. An advanced numerical illustration is given.The research on which this paper is based was partly supported by the Polish Committee for Scientific Research (KBN) under grants No 3P40401804 and 7 T08B 007 08 to the Institute of Fundamental Technological Research, Polish Academy of Sciences, and by the International Center for Numerical Methods in Engineering (CIMNE), University Polytechnics of Catalunya, Barcelona, Spain.     

Investigation of anisotropy problems in sheet metal forming using finite element method

This paper discusses the results of the numerical study of rectangular cup drawing of steel sheets using finite element methods. To be able to verify the results of the numerical solutions, an experimental study was done where the material behavior under deformation was analyzed. A 3D parametric finite element (FE) model was built using the commercial FE-package ABAQUS/Standard. ABAQUS allows analyzing physical models of real processes putting special emphasis on geometrical non-linearities caused by large deformations, material non-linearities and complex friction conditions. Friction properties of the deep drawing quality steel sheet were determined by using the pin-on-disc tribometer. The results show that the friction coefficient depends on the measured angle from the rolling direction and corresponds to the surface topography. A quadratic Hill anisotropic yield criterion was compared with von Mises yield criterion having isotropic hardening. The sensitivity of constitutive laws to the initial data characterizing material behavior is also presented. It is found out that plastic anisotropy of the matrix in ductile sheet metal has influence on deformation behavior of the material. When the material and friction anisotropy are taken into account in the finite element analysis, this approach gives better approximate numerical results for real processes.     

Development of a new failure prediction criterion in sheet metal forming

In industrial try-out processes in sheet metal forming usually the forming limit curve is used as failure criterion in order to describe the onset of localized necking. Forming limits, however, are strain-path dependent. Up to today many different approaches how the strain-path dependent behavior of the forming limit curve can be avoided have been published. Best known are the approaches based on forming limit stress curves published by Arrieux, and the approach of Müschenborn published in 1975. An overview over existing failure criteria is given in this contribution. The failure criterion forming limit stress curve as well as several failure criteria based on Müschenborn’s approach will be evaluated with newly recorded experimental data on forming limit curves for non-proportional loading. A new approach, that is in contrast to the two mentioned approaches not based on assumptions but on experimental observations, is presented herein. The suggested approach is presented as failure surface where strains above the surface indicate the onset of localized necking. The failure surface is given as function of the loading mode and the level of effective pre-stretching. Different suggestions how to use and simplify the new approach are given in this paper. The prediction accuracy of the addressed approaches as well as of the newly suggested approach is evaluated by transforming the data of the approaches back into the conventional forming limit curve for several non-linear strain paths. The comparison of the described approaches shows that forming limits for non-proportional loading can be well-predicted with the suggested approach.     

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.     

Finite deformation formulation of a shell element for problems of sheet metal forming

An elastic-plastic thin shell finite element suitable for problems of finite deformation in sheet metal forming is formulated. Hill's yield criterion for sheet materials of normal anisotropy is applied. A nonlinear shell theory in a form of an incremental variational principle and a quasi-conforming element technique are employed in the Lagrangian formulation. The shell element fulfills the inter-element C ¹ continuity condition in a variational sense and has a sufficient rank to allow finite stretching, rotation and bending of the shell element. The accuracy and efficiency of the finite element formulation are illustrated by numerical examples.     

Semi-analytical approach for plane strain sheet metal forming using a bending-under-tension numerical model

In this paper, we revisit the plane strain deep-drawing process. We show that a detailed analysis of the physical process may result in a dramatic reduction of computing time when the problem is split into several regions undergoing well-defined loading paths. The proposed approach allows us to assess the springback of the formed sheet in a quasi-instant time and is thus suitable in the initial design phase and provides a fast and economical way to determine the influence of the numerous parameters involved in sheet metal forming. We present a semi-analytical model that has been developed for sheet metal forming mainly subjected to plane strain bending-under-tension and involving large strains. The sheet is considered to be an assembly of regions where the loading is considered homogeneous in the length direction. A handful of finite elements or even a single element is sufficient to compute the loading path followed by each region. The contact is circumvented by constraining the kinematics with appropriate boundary conditions and the approach is valid for any material behavior law. The semi-analytical model is applied to standard test cases and then compared with full-scale simulations.     

薄板类件多点成形过程的数值模拟

以希尔关于弹塑性材料唯一性的充分性条件为理论基础,采用虚功率增率型原理,建立了多点成形的有限元模型,探讨了板材厚度、材质及弹性介质等因素对成形结果的影响,通过数值模拟找出参数合适的弹性介质,并对薄板类件的多点成形过程进行了实验验证.研究表明,通过使用参数合适的弹性介质,可以有效抑制压痕现象的产生,并能保证工件的成形精度;实验验证表明对马鞍形件多点成形过程中压痕现象的模拟是合理的.     

微薄板塑性成形本构关系研究

尺寸效应的影响使得传统的成形理论和变形机制不再适用于微塑性成形.在考虑尺寸效应对微薄板成形性能影响的基础上,对已有的CuZn36黄铜薄板微拉伸实验结果进行处理,提出了一种研究微塑性成形本构关系的方法.根据弹性和塑性变形过程,分阶段分析了t/d(板厚/晶粒大小)对屈服强度和切线模量的影响,修正了双线性弹塑性本构关系,获得了考虑尺寸效应的微塑性成形本构关系.借鉴宏观增量本构关系,结合微拉伸实验,采用Mises屈服准则和随动强化模型,得出适合微塑性成形的弹塑性增量本构方程,为微塑性成形的理论研究和实际应用奠定了基础.     

Thinning as a failure criterion during sheet metal forming

Thinning during forming is often considered a failure criterion in the metal forming industry. It is believed that a critical amount of thinning takes place in a sheet metal before failure. In this study, varying widths of low-carbon steel sheets were punch stretched under laboratory conditions. Thinning during punch stretching was measured at various locations along the steel sheets. These measurements demonstrated that thinning during forming is not constant, but that it is a function of the strain path followed by the sheet. Hence, thinning should not be used as a failure criterion during forming of sheet metals.     

The effects of forming history on sheet metal assembly

In this study a simulation-based sensitivity study is performed in order to investigate the influence of the forming history on the properties of an assembly. In the study the assembly properties are predicted by sequentially simulating the manufacturing process chain of a sheet metal assembly. Several simulations of the assembly stage are performed in which different combinations of forming histories are transferred from the forming stage. It is found that the forming history affects the properties of the assembly and that the residual stress state is the most influential history variable. This demonstrates the importance of utilising the complete final mechanical state of each manufacturing step as the initial condition for the subsequent step in the manufacturing process chain in order to achieve accurate predictions of the assembly properties. Furthermore, if more reliable predictions can be made concerning the manufacturability of a product and its in-service behaviour, more design alternatives can be evaluated during product development while a considerably smaller number of physical prototypes are needed.     

板材数字化成形加工路径规划研究

在板材数字化成形过程中零件的直壁部分常因为金属变形量过大而极易发生破裂,故在加工前需要进行适当的路径规划以控制成形中直壁部分金属过度变形,使板材整体壁厚变化缓慢而均匀,优化其成形过程,最终完成薄板材的直壁部分成形.研究了具体的路径规划方法及其加工轨迹自动生成,主要从工艺角度对直壁部分成形作了有益的探索.     

A stream tube based upper bound approach to plane strain metal forming analysis

In an attempt to establish a practicable computer based metal forming analysis capability, the classical upper bound theorem has been revisited. The result is an enhanced methodology which holds promise of providing some of the capabilities associated with more sophisticated finite element formulations without the attendant costs. The procedures developed here culminate with two sample applications.     

Improving geometrical accuracy for flanging by incremental sheet metal forming

Incremental Sheet Forming (ISF) is a manufacturing technology for individualized and small batch production. Among the opportunities this technology provides there is the possibility of a short ramp-up time and to cover the whole production chain of sheet metal parts by using a single reconfigurable machine set-up. Since recent developments proved that manufacturing of industrial parts is feasible, finishing operations such as flanging and trimming gain importance, which are an integral part of manufacturing process chains of many sheet metal parts. This paper analyses the technological capabilities of performing flanging operations by ISF. Due to the localized forming zone and the absence of surrounding clamping devices, ISF exhibits a different material flow than conventional flanging processes. In this paper, the influence of the tool path characteristics, the flange length as well as the flange radius is analysed in order to establish a process window and to compare it to the process limits of conventional flanging operations. Since geometrical deviations occur when flanging operations are performed by ISF, a new adaptive blank holder is developed, which acts in the vicinity of the forming tool and reduces unwanted deformation outside the primary forming zone. The experimental results show the benefits of the adaptive blank holder with respect to geometric accuracy. The established process window and the adaptive blank holder hence contribute to the applicability of incremental flanging operations, such that ISF can be used for all forming and flanging operations along the process chain.     

Radial basis functional model for multi-objective sheet metal forming optimization

Fracture and wrinkling are two major defects in sheet metal forming and can be eliminated via an appropriate drawbead design. This article proposes to adopt a multi-objective particle swarm optimization (MOPSO) approach, which differs from traditional multi-objective optimization with construction of a single cost function. MOPSO shows a certain advantage over other single cost function or population-based algorithms. While radial basis function (RBF) has shown considerable promise in highly non-linear problems, there has been no report in sheet metal forming design. Here RBF is attempted to establish the metamodels for fracture and wrinkling criteria in sheet metal forming design. In this article, a sophisticated automobile inner stamping case is exemplified, which demonstrated that RBF provides a better surrogate accuracy and MOPSO is more effective than the other methods studied. The use of RBF driven MOPSO procedure significantly improved the formability and can be recommended for sheet metal process design.     

板料渐进成形数值模拟与实验研究

为提高渐进成形的成形效率和成形质量,了解板料渐进成形的变形规律及工艺参数对成形的影响,采用有限元方法对板料渐进成形过程进行了数值模拟研究,分析了斜壁盒形件渐进成形过程应力分布和厚度变化趋势,通过对不同进给量和不同成形路径进行数值模拟,分析了工艺参数对成形的影响.结果表明,斜壁盒形件最大应力和最大厚度减薄发生在底面拐角处;成形过程中工具头运动轨迹应尽量采用走螺旋线的方式,可以提高成形件的成形能力和成形质量.渐进成形实验表明,数值模拟结果与实验结果基本吻合.     

Multiobjective robust optimization method for drawbead design in sheet metal forming

It is recognized that fracture and wrinkling in sheet metal forming can be eliminated via an appropriate drawbead design. Although deterministic multiobjective optimization algorithms and finite element analysis (FEA) have been applied in this respect to improve formability and shorten design cycle, the design could become less meaningful or even unacceptable when considering practical variation in design variables and noises of system parameters. To tackle this problem, we present a multiobjective robust optimization methodology to address the effects of parametric uncertainties on drawbead design, where the six sigma principle is adopted to measure the variations, a dual response surface method is used to construct surrogate model and a multiobjective particle swarm optimization is developed to generate robust Pareto solutions. In this paper, the procedure of drawbead design is divided into two stages: firstly, equivalent drawbead restraining forces (DBRF) are obtained by developing a multiobjective robust particle swarm optimization, and secondly the DBRF model is integrated into a single-objective particle swarm optimization (PSO) to optimize geometric parameters of drawbead. The optimal design showed a good agreement with the physical drawbead geometry and remarkably improve the formability and robust. Thus, the presented method provides an effective solution to geometric design of drawbead for improving product quality.