COMPUTER AIDED ENGINEERING A non-invasive form finding method with application to metal forming Philipp Landkammer 1 • Thomas Schneider 2 • Robert Schulte 2 • Paul Steinmann 1 • Marion Merklein 2 Received: 15 January 2016 / Accepted: 20 January 2016 / Published online: 28 January 2016 Ó German Academic Society for Production Engineering (WGP) 2016 Abstract Inverse form finding aims in determining the optimal material configuration of a workpiece for a specific forming process. A gradient- and parameter-free (nodal- based) form finding approach has recently been developed, which can be coupled non-invasively as a black box to arbitrary finite element software. Additionally the algo- rithm is independent from the constitutive behavior. Con- sequently, the user has not to struggle with the underlying optimization theory behind. Benchmark tests showed already that the approach works robustly and efficiently. This contribution demonstrates that the optimization algorithm is also applicable to more sophisticated forming processes including orthotropic large strain plasticity, combined hardening and frictional contact. A cup deep drawing process with solid-shell elements and a combined deep drawing and upsetting process to form a functional component with external teeth are investigated. Keywords Inverse form finding  Shape optimization  Cup deep drawing  Sheet-bulk metal forming 1 Introduction Within metal forming processes a trend in the direction of increasingly complex and closely-tolerated functional components is undisputed. This becomes clearly evident in the new process class of sheet-bulk metal forming (SBMF), which enables manufacturing of near net-shaped variants in direction of the sheet thickness, see Merklein et al. [1]. Thereby, finite element (FE) simulations are nowadays a fundamental tool for analyzing forming processes. This leads to an improved process understanding, reduces trial- and error-procedures and strongly accelerates the devel- opment process of functional components. With regard to simulative investigations, a crucial issue is to determine the optimal preform of a workpiece, which results exactly in the desired shape after the forming process. To address this problem, a couple of different form finding methods have been developed for sheet and bulk metal forming in the last decades. A short overview is given in the following. In the context of sheet metal forming, pioneering work was done by Guo et al. [2]. Thereby, the triangle-meshed material configuration of the sheet is iteratively updated within a non-linear analysis after testing the nodal dis- tances between a computed and a target spatial configura- tion. Also with application to sheet metal forming, Kim et al. [3] proposed a roll-back-method to iteratively update the automatically triangle-meshed cut-out of the flat sheet. Padmanabhan et al. [4] presented an iterative trimming scheme to determine the optimal design of a flat sheet for deep drawing processes and even achieved a coupling of their strategy to external FE software [5]. Since an automated meshing for three-dimensional geometries is still a major issue, different strategies for inverse form finding have to be considered for bulk metal forming. Thereby, a distinction is commonly made between parameter-based and parameter-free approaches. In the former, the design variables are the control points of a computer aided design (CAD) model, as proposed by Braibant and Fleury [6]. Also Fourment and Chenot [7] pursued the parameter-based strategy with control points of & Philipp Landkammer philipp.landkammer@fau.de 1 Institute of Applied Mechanics (LTM, FAU Erlangen- Nu¨rnberg), Egerlandstrasse 5, 91058 Erlangen, Germany 2 Institute of Manufacturing Technology (LFT, FAU Erlangen- Nu¨rnberg), Egerlandstrasse 13, 91058 Erlangen, Germany 123 Prod. Eng. Res. Devel. (2016) 10:93–102 DOI 10.1007/s11740-016-0659-6