BGM is a shape optimization method whose driving force is the magnitude of surface stresses on structural components. This method is based on the observation that biological structures, such as tree trunks and animal bones, actually add layers of biological material to the surface areas affected by activation stress. In the 1990s, Mattheck proposed extending this concept to CAE and structural part design by adding material to surfaces with high stresses and/or removing it from surfaces where stresses were low.
A volumetric growth ε based on the actual von Mises stress σ_{von Mises} and a reference activation stress σ_ref was proposed:

The optimization behavior (add/remove material) is controlled by setting the reference stress value σref. The method proposed in this study is based on a more versatile and evolved version of the Mattheck model and prescribes displacing the node in the direction normal to the surface:

where σ^node is the stress evaluated at each node, σ^th is a threshold value for the stress defined by the analyst, and σ^max and σmin respectively are the maximum and minimum stress values in the current set. d is the maximum offset between the nodes on which the maximum and the minimum stresses are evaluated; this parameter is defined by the user to control the displacement of the nodes whilst limiting the possible distortion of the mesh. Nodes on the surface to be optimized can either be moved inward, if the stress on node is less than the threshold value, or outward, if the evaluated stress is greater. The proposed BGM implementation can be guided by a variety of stress/strain definitions (such as maximum principal stress, maximum plastic strain, accumulated equivalent plastic strain, etc.), so that the method can be tailored to the specific design requirements. Propagation to the full computational mesh of the BGM field calculated on the surfaces can be achieved using radial basis functions (RBF) mesh morphing. RBFs were introduced in the early 1960s to interpolate scattered multidimensional data and to allow interpolation of a scalar field anywhere in the defined space from known values at discrete points called source points. Since the data to be interpolated at x comprises scattered scalar values at N source points Xsi, the interpolating function s(x) can be obtained as

by summing the interactions at the probe location x with all the source points that have been computed by weighting all the radial interactions (calculated as Euclidean distances between each source point) with γi coefficients, and the probe location transformed by the radial function φ(r). The N unknown γi are calculated by imposing the passage of the s(x) function through all the (known) source points.
In mesh morphing, the source points are the surface mesh nodes over which the displacement is controlled, whilst the entire set of nodes within the mesh is updated by receiving the interpolated deformation field. Individual RBFs interpolate the three components of the displacement. The above mesh morphing technique and BGM can be successfully coupled in an optimization approach consisting of the following steps:

1. The baseline geometry is discretized into finite elements; load and constraints are applied, and the FEM solution is evaluated;
2. From the FEM solution, the nodal stress on the surfaces to be optimized are retrieved, σth and d are set by the user and Snode displacement along the surface normal for each node is evaluated for each selected node;
3. The evaluated displacements are used to configure the RBF problem by imposing them as values to be interpolated (values on source points), the user can optionally set additional source point values to complete the morphing configuration (i.e., points to maintain fixed);
4. The FEM model mesh is morphed, and the FEM solution is evaluated again;
5. The stress values on the surface to be optimized are analyzed: if the new stress levels can be further optimized, the procedure is iterated from step 2; otherwise, the geometry can be considered optimized.

In the methodology described above, the stress analyst must set two BGM parameters: the threshold stress σth and the maximum displacement d. The first parameter represents the stress level value on which the optimization procedure will try to converge; the second parameter represents the maximum displacement allowed within a single iteration of optimization: the lower its value, the greater the number of iterations that will be required to reach the optimum, and the lower the risk of mesh distortions that could invalidate the FEM model will be.

The proposed approach has been implemented using the RBF Morph line of software. The RBF Morph ACT extension for Ansys Mechanical provides Ansys FEA users with a comprehensive advanced mesh morphing toolkit, and the BGM-based parameter-less shape optimization workflow is among these. This specific feature has undergone intense development driven by successful applications obtained from academic and industrial users. As shown in Fig. 2, the user is asked to specify the areas to be reshaped according to the BGM method and the maximum desired action within each iteration. Overall optimization is then achieved by simply defining the required number of intermediate iterations as individual design points in Workbench. The complete table will then be populated automatically with time depending on the cost of a single FEA run. For complex industrial applications, such as the nonlinear thermomechanical fatigue (TMF) case in the example in the image, full optimization may take several hours. Typically, a dozen FEA solutions are required to obtain a good refined optimal shape.

Fig. 3 - Evolution of the shape of a cantilever beam: the final shape has a parabolic profile and the stress levels on the variety of purposes that demonstrate how BGM external surface are uniform.

Fig. 4 - Optimized cross-sectional shape of a coil spring. The stress levels on the inner surface are uniform and the optimized shape leads to higher spring efficiency and lower maximum stress.

The proposed approach has been used for a variety of purposes that demonstrate how BGM can be useful – not only for minimizing von Mises stress, but also for obtaining lighter parts and mitigating more complex results, as in the case of accumulated non-linear equivalent plastic strain (NLEPEQ), so that component life optimization can be effectively achieved. Firstly, the algorithm’s reliability is evaluated for simple benchmarks that are thoroughly explored in solid mechanics textbooks: a cantilever beam (Fig. 3) and a coil spring (Fig. 4). It is worth noting that a bioinspired automatic method can adequately address even quite simple engineering problems with straightforward but not obvious solutions.

Fig. 5 - Tree trunk evolution study: based on Mattheck’s work in the 1990s, it was possible to reproduce the same shape evolution in a 3D model, highlighting the reduction of the hot-spot stress level

Fig. 5 shows the study of the evolution of a tree trunk explored by Mattheck in 1990. After 30 years, the simple 2D model originally proposed was replaced by a detailed 3D model that takes into account the wood’s orthotropic behavior. The predicted shape corresponds very well with previous FEA calculations and with the experimentally observed evolution of the real tree.

Fig. 6 - Original baseline configuration and optimized configuration after 10 iterations: maximum stress level reduced by 20.8%, increasing whole volume by 2.5%.

Fig. 6 shows an industrial automotive application to optimize an aftermarket motorcycle part (the Motocorse San Marino’s wheel hub for the Ducati Panigale V4); in this case, manufacturing constraints were imposed to maintain the full shape of the bracket and of the central cylindrical part of the hub.
Fig. 7 shows a TMF case that demonstrates the workflow in use at SVS FEM for a variety of automotive applications, including thermal fatigue of high-performance turbocharger housings, on a valve model.

Fig. 7 - Final configuration of the valve subject to thermomechanical fatigue, which resulted in a 49% reduction in NLEPEQ.