Speaker
Description
The shape and topology optimization is considered for linear elasticity. There are two methods to compute admissible solutions. The first is the gradient flow dynamical system which leads to the gradient type algorithm. The second is a level set method which is based on numerical solution of the Hamilton-Jacobi equations. The first method is actually superior from mathematical point of view, the convergence of this method can be established to an optimal solution by using the advanced theory of quasilinear partial differential equations. However, the topology is defined in principle by the starting point of numerical procedure. The change of topology could be performed by the application of the topological derivative method. In contrast, the level set method is popular and still poorly analyzed from the mathematical point of view to our best knowledge. This presentation introduces in detail the level set method that combines topological derivatives with shape derivatives and apply them to shape and topology optimization. In our method, the topological derivatives can effectively identify and guide new holes or connection changes that may appear in the structure, while the shape derivatives provide the velocity field for interface evolution, thereby achieving control of complex geometric boundaries. To further improve the efficiency and stability of the algorithm, the gradient flow method is also incorporated into the overall framework and combined with the shape gradient to naturally handle the dynamics of shape evolution in a continuous iteration process. The level set method captures the movement, merging or splitting of geometric shapes by updating the level set function in the embedded domain. The effectiveness of the method is verified by multiple numerical examples.
References:
[1] Plotnikov, P. I., Sokolowski, J. Geometric aspects of shape optimization. J. Geom. Anal. 33 (2023), no. 7, Paper No. 206, 57 pp.
[2] Plotnikov, P. I., Sokolowski, J. Geometric Framework for Gradient Flow in Shape Optimization, Springer Briefs, submitted.
[3] Sokolowski, J., Yixin, Tan., Shape and Topology Optimization of Control Problems in Elasticity, to appear.
