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• W3148465934 abstract "A challenge in the design of functional parts is the determination of the initial, undeformed shape such that under a given load a part will obtain the desired deformed shape. This is an inverse form finding problem and it is posed as follows: the deformed shape, the mechanical loading, and the boundary conditions are given, whereas the inverse deformation map that determines the material configuration, i.e., the undeformed shape, is sought. Inverse form finding methods are useful tools in conceiving designs in less time and at lower cost than with experiments or direct computational design. In the present work two inverse form finding methods are presented for the optimal determination of the initial shape of formed functional components, considering anisotropic hyperelastic and elastoplastic behaviours. The material is modelled by a macroscopic phenomenological approach in the logarithmic strain space for large strains based on the small strains theory. This model uses the laws of thermodynamics to describe the macroscopic behaviour of the material. The anisotropy in the material is formulated through the eight crystal systems according to the spectral decomposition of the fourth-order elasticity tensor using the Kelvin modes. A Cauchy formulation of the boundary value problem, called inverse mechanical problem, allows to determine the undeformed configuration of a functional component. All quantities are parametrised in the spatial coordinates. This formulation is suitable when dealing with hyperelastic materials. For elastoplastic behaviour, the provided deformed configuration, load, and boundary conditions are no longer sufficient to compute the wanted undeformed configuration. The set of internal variables corresponding to the deformed configuration is equally required in this case. Usually the set of internal variables at the deformed state is unknown before the computation of the undeformed configuration in elastoplasticity. Therefore a gradient-based shape optimisation is used in this work according to an inverse problem via successive iterations of a direct mechanical problem. The objective function of the inverse form finding problem is defined by a least squares minimisation of the difference between the target and the current deformed configuration of the workpiece. The design variables are defined by the discretised nodes of the functional component with the finite element method (node-based shape optimisation). This choice leads, however, to mesh distortions in the undeformed shape, which are avoided by using a recursive algorithm. Between two iterative steps of the algorithm the current optimised undeformed configuration is used in the computation of the next value of the objective function. The total applied force is then split over all entities. In the computation of both inverse form finding methods, deformed workpieces with different geometries, material parameters and crystal systems were used. The inverse mechanical problem and the shape optimisation formulation in hyperelasticity gave identical results with respect to the geometry of the obtained undeformed shape. Nevertheless the computational costs of the inverse mechanical formulation were about 2000 times lower. For elastoplastic behaviours the shape optimisation formulation has to be computed with the recursive algorithm in order to avoid mesh distortions. All the results were validated by the comparison between the given deformed configuration of the workpiece and the directly computed deformed configuration of the workpiece. A difference of about 10-6 to 10-24 mm was achieved with both inverse form finding methods." @default.
• W3148465934 created "2021-04-13" @default.
• W3148465934 creator A5072506833 @default.
• W3148465934 date "2013-01-01" @default.
• W3148465934 modified "2023-09-23" @default.
• W3148465934 title "On Inverse Form Finding for Anisotropic Materials in the Logarithmic Strain Space" @default.
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