Patrick W. Dondl, Patrina S. P. Poh, Martin Rumpf and Stefan Simon Simultaneous elastic shape optimization for a domain splitting in bone tissue engineering Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 475(2227): 20180718 2019 10.1098/rspa.2018.0718
2016
Patrick W. Dondl, Behrend Heeren and Martin Rumpf Optimization of the branching pattern in coherent phase transitions C. R. Math. Acad. Sci. Paris, 354(6): 639--644 2016 https://arxiv.org/abs/1512.06620
Abstract: Branching can be observed at the austenite-martensite interface of martensitic phase transformations. For a model problem, Kohn and Müller studied a branching pattern with optimal scaling of the energy with respect to its parameters. Here, we present finite element simulations that suggest a topologically different class of branching patterns and derive a novel, low dimensional family of patterns. After a geometric optimization within this family, the resulting pattern bears a striking resemblance to our simulation. The novel microstructure admits the same scaling exponents but results in a significantly lowered upper energy bound.