Conference Agenda

One phenomenon to consider in metal forming nowadays is the accumulation of ductile damage during the forming process. Ductile damage, i.e. the nucleation, growth and subsequent accumulation of micro-defects such as voids, is inherently present in any formed part. Therefore, it is advisable to reduce these damage effects and in turn produce parts with reduced damage accumulation and thus higher safety factors. Herein, optimisation is a very useful tool to enhance already established processes with damage minimisation in mind. By defining process dependent parameters as the design variables of the optimisation, improved tool sets can be generated to reduce the ductile damage of the formed part.

An important aspect to consider when dealing with simulation of forming processes is the underlying necessity of contact algorithms. Approaches to handle these discontinuous problems are available in literature, by e.g. utilising sub-gradients, however, their application mainly see academic use. The problems for the proposed optimisation are however very complex in nature and therefore require robust and efficient implementation. Consequently, the commercial finite element software Abaqus is used to simulate the processes and solve the necessary contact problems.

In this submission, a framework defined in Matlab is presented, which utilises Abaqus as the finite element program to solve the stated optimisation problems. The framework is applied to different forming processes, such as rod extrusion and deep drawing-like processes, in order to optimise them with regard to their accumulated damage. Different sets of geometric parameters are defined, which in turn result in optimal design for the work tools of the analysed problems. Due to the nature of the framework, the optimisation is not limited to process optimisation and further examples regarding curve fitting for experimental setups are also presented.

When molding processes, such as injection molding, are used to produce plastics parts, it can be difficult to achieve the correct product shape. As part of the process, the material must cool down and solidify. Since this can happen in an inhomogeneous way, residual stresses can remain in the material. These lead to warpage, after the part is ejected from the machine.

There are several aspects of the process that could be adjusted to improve the resulting product shape. The focus of this work is on the shape of the mold cavity. If suitable adjustments are made to this cavity, the product shape can be improved although shrinkage and warpage still occur. In order to estimate the effects of certain cavity shape changes, a numerical simulation method for the process is required.

This cavity design problem can then be treated either as a shape optimization problem or as an inverse problem. In the former case, a suitable shape parameterization and objective function need to be found. Both options profit from the use of splines, since this allows the shape to be transferred back to a CAD format. The method of Isogeometric Analysis (IGA) offers a convenient way of using splines as a geometry representation in the Finite Element Method. We will discuss the different design approaches and explain the benefits and challenges involved with the spline representations.

In the typical product development process of an automotive part, multiple disciplinary teams collaborate to converge on a final design. Structural mechanics, design, crashworthiness and manufacturability are relevant disciplines that mutually influence one another. Sheet metal forming operations are the cornerstone of automotive part production, as a significant portion of the individual components of the Body-in-White (BiW) are fabricated through stamping and deep-drawing processes. Manufacturability assurance for sheet metal forming is commonly addressed by engineering experience and heuristic rules based on geometrical constraints. This work explores the idea of formulating analytical manufacturing constraints for stamped and deep-drawn parts and its inclusion into existing multidisciplinary shape optimization workflows to address formability and performance objectives simultaneously.

As discussed by [1], gradient methods based on adjoint sensitivity analysis, together with a filtering technique as Vertex-Morphing are powerful tools for the typical large and very large optimization use cases in the industry. In this contribution, we present the current progress in the formulation of a constraint for shape optimization that accounts for the manufacturing process, discuss the definition of a meaningful objective function and present details regarding the calculation of adjoint-based sensitivities and its combination with Vertex-Morphing. The formulations of the primal and adjoint problems are also presented, based on the simplified Finite Element Analysis for sheet metal forming proposed by [2].

[1] Kai-Uwe Bletzinger. A consistent frame for sensitivity filtering and the vertex assigned morphing of optimal shape. Structural and Multidisciplinary Optimization, 49, 01 2014.

[2] Y. Q. Guo, J. L. Batoz, J. M. Detraux, and P. Duroux. Finite element procedures for strain estimations of sheet metal forming parts. International Journal for Numerical Methods in Engineering, 30(8):1385–1401, 1990.

This work aims to gain a deeper understanding of sensitivity information through the use of principal component analysis (PCA). By decomposing sensitivity matrices, it is possible to explore and analyze the underlying relationships between variables and the impact of their changes on the overall structure. The approach for this analysis is discussed in [1]. PCA allows us to analyze the eigenvectors of the covariance matrix, which are known as the principal components. The first principal component is considered the most significant mode of variation as it indicates the direction with the highest variance in the data. Similarly, the second principal component represents the direction with maximum variance, but this time it must be orthogonal to the first principal component. This process continues for the remaining principal components. The work at hand makes use of gradient-based sensitivity analysis [2] for dynamic structures and compares two different methods for shape design: Isogeometric Analysis (IGA) [3] and Finite Element Method (FEM). The main focus is on using direct differentiation, but if analytical gradients are not available, numerical differentiation methods such as complex-step method (CSM) can be used as alternatives. We utilize different types of basis functions, such as Bernstein polynomials, B-Splines, and Non-Uniform Rational B-Splines (NURBS), to describe the shape of the structure. IGA has several advantages over traditional FEM-based approaches. These advantages include the ability to accurately describe geometry using fewer control points, high-order continuity, and increased flexibility due to control point weights. These characteristics have a significant impact on shape sensitivity analysis. IGA is used during the structural optimization process to avoid costly remeshing and design velocity field calculations. It is more efficient and effective than traditional FEM approaches for these tasks. In contrast to static analysis, the response of a structure to time-dependent loads is significantly affected by inertia and damping effects. The necessary computational characteristics for this type of problem are discussed and the full solution algorithm is presented.

References

[1] N. Gerzen and F.-J. Barthold, “Design space exploration based on variational sensitivity analysis,” PAMM, vol. 14, no. 1, pp. 783–784, Dec. 2014. DOI: 10.1002/pamm.201410374.

[2] F.-J. Barthold, N. Gerzen, W. Kijanski, and D. Materna, “Efficient variational design sensitivity analysis,” in Mathematical Modeling and Optimization of Complex Structures (Computational Methods in Applied Sciences), P. Neittaanmäki, S. Repin, and T. Tuovinen, Eds., Computational Methods in Applied Sciences. DOI: 10.1007/978-3-319-23564-6_14.

[3] T. Hughes, J. Cottrell, and Y. Bazilevs, “Isogeometric analysis: Cad, finite elements, nurbs, exact geometry and mesh refinement,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 39-41, pp. 4135–4195, 2005. DOI: 10.1016/j.cma.2004.10.008.

Topology optimization is an effective numerical tool to design high-performance, efficient and economical lightweight structures. In this talk the solution procedure for a two material topology optimization problem constrained by a scalar second order PDE is presented. The approach relies on a numerical topological-shape derivative as a main ingredient for the gradient-based solution algorithm.

We state the optimization problem in the continuous setting and subsequently discretize it. On the continuous level we review the classical shape derivative where the perturbation is realized by the action of a vector field and the classical topological derivative where the perturbation is done by means of sets. In contrast to this, in the presented approach the geometry is represented by the zero level-set of a scalar function. Based on this representation we suggest a topological-shape derivative unifying the concepts of shape derivative and topological derivative. In a next step we consider the discretization of the PDE as well as the level-set function by linear triangular Lagrange finite elements. In this numerical setting we can now consider the perturbation of the level-set function by the perturbation of its nodal values. Based on this we give explicit formulas for the computation of the numerical topological-shape derivative. This derivative information is used in an algorithm to update the level-set function where no distinction between shape changes and topological changes is made. The algorithm is tested in a numerical example, where the shape of two circles with different radii is recovered.