POD for Real-Time Simulation of Hyperelastic Soft Biological Tissue Using the Point Collocation Method of Finite Spheres

Figure 1

Schematic of the PCMFS for dynamic surgery simulation. Discretization of a domain 𝐵⊂ℜ³ by the PCMFS using a set of nodal points. 𝐵_𝐼 is the open sphere at node 𝐼. Natural boundary conditions are defined on 𝜕𝐵_𝑓, and the essential boundary conditions are defined on 𝜕𝐵_𝑢:𝜕𝐵=𝜕𝐵_𝑢∪𝜕𝐵_𝑓, and 𝜕𝐵_𝑢∩𝜕𝐵_𝑓=0. Note that 𝜕𝐵_𝑢=𝜕𝐵_𝑢1∪𝜕𝐵_𝑢2, where 𝐮(𝐗)=𝟎 for 𝐗∈𝜕𝐵_𝑢1 and 𝐮=𝐮_tooltip on 𝜕𝐵_𝑢2×(0,𝑇).