Nutrient Driven Topology Optimization

Abstract: The aim of this thesis is to investigate how a biological structure changes its shape and boundary under different cases of load if flow of nutrients is included, since nutrient flow has not been taken into account in previous studies. In order to simulate such a scenario we construct a model by using topology optimization (the SIMP model) and a balance law which is suitable for biological structures. Moreover, the model is derived by using an analogy with the dissipation inequality and Coleman-Noll’s procedure. The model can be interpreted as bone or some other biological structure, where the growth and remodeling partly occurs due to nutrient flow. The theory is first investigated by selecting an MBB beam with a special boundary condition for the nutrient concentration and inflow of nutrients, and then with a bone-like model. For the analysis with different loads we have observed that the structure becomes thicker were the load is applied. Parameters like beta (β) (reflecting the relation between nutrients and material) and nutrient concentration (c) seem to play an important role in nutrient transport and building of the structure. The result for larger values of β and nutrient concentration (c) gives a thicker structure in the entire domain. We also made an assumption of Fick’s law of diffusion. Fick’s law of diffusion describes the flux from high concentration to low concentration. This phenomenon is observed in analysis with different nutrient concentrations (c): we can see that the structure tends to be built up where the concentration is high and continues to be built in the direction from high to low concentration. In analyses with mu-value (μ), which represents cost of material, the result gives a thinner structure for larger values of μ.