Abstract
BackgroundInverse form finding methods allow conceiving the design of functional components in less time and at lower costs than with direct experiments. The deformed configuration of the functional component, the applied forces and boundary conditions are given and the undeformed configuration of this component is sought.MethodsIn this paper we present a new recursive formulation for solving inverse form finding problems for isotropic elastoplastic materials, based on an inverse mechanical formulation written in the logarithmic strain space. First, the inverse mechanical formulation is applied to the target deformed configuration of the workpiece with the set of internal variables set to zero. Subsequently a direct mechanical formulation is performed on the resulting undeformed configuration, which will capture the path-dependency in elastoplasticity. The so obtained deformed configuration is furthermore compared with the target deformed configuration of the component. If the difference is negligible, the wanted undeformed configuration of the functional component is obtained. Otherwise the computation of the inverse mechanical formulation is started again with the target deformed configuration and the current state of internal variables obtained at the end of the computed direct formulation. This process is continued until convergence is reached.ResultsIn our three numerical examples in isotropic elastoplasticity, the convergence was reached after five, six and nine iterations, respectively, when the set of internal variables is initialised to zero at the beginning of the computation. It was also found that when the initial set of internal variables is initialised to zero at the beginning of the computation the convergence was reached after less iterations and less computational time than with other values. Different starting values for the set of internal variables have no influence on the obtained undeformed configuration, if convergence can be achieved.ConclusionsWith the presented recursive formulation we are able to find an appropriate undeformed configuration for isotropic elastoplastic materials, when only the deformed configuration, the applied forces and boundary conditions are given. An initial homogeneous set of internal variables equal to zero should be considered for such problems.
Highlights
Inverse form finding methods allow conceiving the design of functional components in less time and at lower costs than with direct experiments
In this work we present a recursive method for the determination of the undeformed configuration of a functional component, when only the deformed configuration of a workpiece, the applied forces and the boundary conditions are previously known
This is commonly known as an inverse form finding problem, which is inverse to the standard direct kinematic analysis in which the undeformed sheet of metal, the applied forces and boundary conditions are known while the deformed state is sought
Summary
Inverse form finding methods allow conceiving the design of functional components in less time and at lower costs than with direct experiments. When dealing with metal forming processes, this set of internal variables is not known at the deformed state To overcome this problem in anisotropic elastoplasticity, [15,17] proposed a numerical method based on shape optimisation in order to solve inverse form finding problems. 2011 and 2012 [20,21] compared the inverse mechanical and the shape optimisation formulation in terms of computational costs and accuracy of the obtained undeformed functional component. 2002 [24] proposed an approach to optimal shape design in forging In their recursive formulation the inverse problem is formulated as an optimisation problem, where the objective function sensitivity is calculated by the accumulated sensitivities of the nodal coordinates throughout the entire simulation of the process. [26] proposed an inverse-motion-based form finding for electroelasticity to improve the design and accuracy in electroelastic applications such as grippers, sensors and seals
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