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This project shows a shape optimization of a solid structure. The investigated guidance rail was optimized in a cooperation project with the Adam Opel AG. The optimization goal was to minimize stress concentrations at the surface of the structure.

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Model dataOpel XXX Führungsschiene

Model format: NASTRAN

Mechanically equivalently supported substructure

Number of FE-nodes: 181.000


  • 108000 TETRA10
  • 29000 TRIA6

Loading by central RBE

The geometry was discretized by solid elements with shell elements on the surface. The shell elements are utilized to compute surface stresses.


Structural analysis

The guidance rail model was investigated by a linear analysis in order to determine the properties of the initial design. The boundary conditions for the analysis are fixed support at the outer edges and nodal load at the central RBE node. The applied elements use quadratic shape functions in order to ensure accurate analysis results. The following pictures show the scaled displacements and the surface stresses in the notch region.

Opel XXX Führungsschiene, Deformation Opel XXX Führungsschiene, Spannungen


Optimization modelOpel XXX Führungsschiene, Designraum

The design space for this optimization problem spans over the emphasized regions at the upper and the lower parts of the notch region. The surface normals of all FE-nodes in the design space are defined as optimization variables. This results in a total amount of 2006 optimization variables. The optimization goal is the minimization of the surfaces stresses (vMises equivalence stress). All stresses of the surface elements are summarized in a Kreisselmeyer-Steinhauser function. During the optimization the surface mesh as well as the volume mesh have to be adapted in order to minimize mesh deformations. Otherwise the solution would be disturbed by numerical stiffening effects.



Optimization result

Comparing initial geometry (left figure) and optimized geometry (right figure) shows a significant strengthening of the cross section in the design space. Additionally, the mesh quality of the optimized geometry becomes visible. The mesh quality ensures minimal element deformations as well as reliable analysis results.

Opel XXX Führungsschiene, Ausgangskontur

Opel XXX Führungsschiene, optimierte Kontur

The consequences of the strengthened cross section on the stresses of the upper (left figure) and lower surface (right figure) are shown in the following figures. The contourplot uses the same range as the stress plot of the initial design. The optimized cross section results in a tremendous reduction of the surface stresses. This example shows a stress reduction of 33% compared to the initial design. Furthermore the optimal geometry exhibits no local stress concentrations.

Opel XXX Führungsschiene, optimierte Kontur, Spannung oben Opel XXX Führungsschiene, optimierte Kontur, Spannung unten



This application example shows the efficient minimization of surface stresses by the application of numerical shape optimization. The optimization results in a moderate but effective strengthening of the cross section which finally yields to a 33% reduction of the maximum surface stress as well as to a continuous stress level. The lower surface stresses reduce the material load and improves the lifetime of the component.