Computer Aided Optimization

The CAO method (Computer Aided Optimization) is a method for shape optimization from the field of bionics in which the growth behavior of biological power carriers (such as trees , bones ) is simulated. The idea is that the surface of the component to be optimized is allowed to grow virtually in accordance with the biological growth rule, as a bone would, for example, if it had to take over the function of the component. Since the growth rule leads to a shape with homogeneous surface tension, the method can be used to reduce stress peaks and thus to increase strength . As a rule, however, not the entire surface of a component is subjected to the optimization process, but only the critical areas that have been previously identified by the user. The basis for the simulation of the biological growth rule are the stresses in the component to be optimized, which are calculated using a finite element model (FEM). The geometry of the component to be optimized is described solely via the FEM mesh, or - better said - by the position of the nodes on the surface.

The basic sequence of the CAO process is shown in Fig. 1. At the beginning there is an FEM analysis in which the stresses in the component are calculated . Based on the stress values, the biological growth rule is simulated and a resulting growth shift is determined. The new contour is obtained by adding the growth shifts to the corresponding current coordinates. In the next step, a network correction is carried out in order to adapt the network to the new contour and to keep network distortions as low as possible. After that, one growth cycle is complete and the next iteration can begin with stress analysis for the modified structure. The process is terminated when the desired homogeneous surface tension is reached or when geometric restrictions prohibit further growth.

There are two versions for simulating the biological growth rule. In the original version of the CAO method, which was developed at the Karlsruhe Research Center, the growth rule is simulated with the help of a growth layer (also called swelling layer). This is applied in the area to be optimized below the outer contour and should have as constant a thickness as possible. The growth shifts are calculated in an additional FEM analysis in which the growth layer, which is assigned a significantly lower modulus of elasticity than the underlying structure, is subjected to a volumetric threshold law. In this case, a temperature distribution is applied in the growth layer in such a way that it leads to a contraction and thus to a degradation of material in the under-stressed areas, while the layer expands in the highly stressed areas and thus material is deposited. When using the growth layer, the problem arises that in the case of complex CAD models, modeling the growth layer is associated with considerable effort.

Fig. 1: The CAO method (direct method)

With the direct method, the modeling problem no longer exists because there is no growth layer and the growth shifts are calculated directly and applied to the nodes. An existing FEM network can be used without modifications. Furthermore, the FEM analysis for the simulation of the threshold law is saved. At the beginning of the optimization, the user defines the surface nodes in the areas that are to be varied in shape as 'growth nodes'. For each of these growth nodes using the node voltage , a growth shift calculated: ${\ displaystyle \ sigma _ {k} ^ {(i)}}$ ${\ displaystyle {\ vec {d}} _ {k} ^ {(i)}}$

${\ displaystyle {\ vec {d}} _ {k} ^ {(i)} = s (1+ \ alpha) (\ sigma _ {k} ^ {(i)} - \ sigma _ {\ text {ref }}) {\ vec {n}} _ {k} ^ {(i)}}$ . (1)

There is a scaling factor that should be chosen so that in the first step the maximum voltage in the variation range is reduced by approximately half. The direction vector is oriented outwards and usually oriented in the direction of the surface normal . It indicates the direction of the shift in growth. The reference voltage defines which node is considered to be heavily loaded ( ) and which is considered to be low loaded ( ). It is also the tension that is established as a homogeneous surface tension at the end of the optimization. As a rule, several load cases must be taken into account when optimizing . In this case, the maximum stress from all load cases is used for each growth node. It is easy to see that equation (1) precisely describes the desired growth behavior, since in the case of high stress the growth shift is directed outwards and thus material is deposited, while at low stress it points inward and thus material is removed. The correction factor smooths the new contour at strong corners and prevents kinks from forming. ${\ displaystyle s}$ ${\ displaystyle {\ vec {n}} _ {k} ^ {(i)}}$ ${\ displaystyle \ sigma _ {\ text {ref}}}$ ${\ displaystyle \ sigma _ {k} ^ {(i)}> \ sigma _ {\ text {ref}}}$ ${\ displaystyle \ sigma _ {k} ^ {(i)} <\ sigma _ {\ text {ref}}}$ ${\ displaystyle \ alpha}$

Using numerous examples from nature, it was possible to check and demonstrate that the CAO method leads to the observed structures of the biological power carriers and can also simulate growth behavior such as the wandering and turning of bone trabeculae .

Applications

Fig. 2: Shape optimization of a bi-axially loaded perforated plate. When calculating the stresses, the symmetry property was used and only a quarter of the plate was modeled. By mirroring this quarter on the two mirror planes (dashed lines), the entire stress distribution is obtained

As a simple application example, Fig. 2 shows the optimization of the hole shape of a bi-axially loaded perforated plate. The aim was to find a shape with homogeneous tension along the edge of the hole. This was achieved after 10 growth cycles. From a mechanical point of view, the perforated plate shown corresponds approximately to a knothole in a tree, which also grows together elongated and not circular.

As a further application example, the optimization of a rear axle differential is shown in Figs. 3 and 4 . There was a critical point of vibration failure on the flange for fastening the large bevel gear . The radius of the rounding of R = 2 mm used there could not be increased any further in order to reduce the stress. Starting with an undercut with R = 3 mm, an optimization was therefore carried out using the CAO method. The optimized contour was then approximated by two circle segments with radii of R = 2.5 mm and R = 5.5 mm. This can be produced both by turning and by grinding . There was a tension reduction of 28%.

Fig. 3: Cross-section and stress distribution of a differential cage before optimization

Fig. 4: Intermediate steps of shape optimization starting from the undercut (initial model) to the optimized contour and via the smoothing to the optimized final contour in the area critical of vibration failure. For comparison, the shape and stress distribution of the basic contour from Fig. 3 is shown on the left

literature

C. Mattheck: Design and Growth Rule for Biological Structures and their Application in Engineering. Fatigue Fract Eng Mater Struct 13, 5, 1990, 535-550.
C. Mattheck: Design in nature. Rombach GmbH + Co Verlagshaus KG, Freiburg i. B., 1997, ISBN 3793091503
C. Mattheck: Engineering Components Grow Like Trees. Mat.-scientific U. Material Tech. 21, 1990, 143-168
L. Harzheim: Optimization of components with the growth rule of trees and bones. BIONA Report 16, Akad. Wiss. Lit., Mainz, 2003, 83-94
H. Bubenhagen, L. Harzheim: Use of shape optimization to improve the service life of components. Construction 50 1998 H. 11/12, 1998, 40-44
L. Harzheim: Structure optimization, fundamentals and applications. Scientific publishing house Harri Deutsch GmbH, Frankfurt am Main, 2007, ISBN 978-3-8171-1809-0