Bernstein-Bézier representations for facial surgery simulation

The finite element method and its application to the simulation of static linear elasticity has a long research history. The same applies for Bernstein–Bézier representations of curves and surfaces in computer aided geometric design. However, the combination of both to build tetrahedral Bernstein–Bézier finite elements presents an inspiring and fruitful challenge. The theory and implementation of these elements and their application in the context of facial surgery simulation is the main focus of this thesis. Both for patients and surgeons, thorough planning is an absolute prerequisite for successful surgical procedures. Therefore, attention is turning to computer–assisted planning systems. The three–dimensional physically–based simulation of facial surgery is envisioned to replace or complement on current surgical planning techniques. After a short motivation and overview of existing deformable models in computer graphics and surgery simulation, we give an introduction to the finite element method. Its application in the context of static elasticity is one of the main building blocks of the envisioned tissue model for surgery simulation. Besides classical linear elasticity, incompressibility and nonlinear stress–strain relations are taken into account. The representation of surfaces and volumes by means of Bernstein–Bézier patches is revisited. Emphasis is put on barycentric representations and on the construction of smooth patch transitions. Further, multivariate hermite interpolants are investigated and evaluated with respect to their suitability for finite element modeling. The construction of a globally trivariate tetrahedral interpolant based on a multi–dimensional generalization of the well–known Clough–Tocher split is presented. As a next step, Bernstein–Bézier techniques are put into the context of finite element analysis of static elastomechanics. A –continuous tetrahedral finite element is derived from the trivariate Clough–Tocher construction. The complex assembly procedure resulting from the construction is given special emphasis. In a thorough test series, –continuous tetrahedral elements are compared with the Clough–Tocher element. Degree elevation and mesh refinement are opposed to the effect of imposing higher level continuity constraints. The construction scheme is shown to invalidate neither the approximation properties nor the locality of the Bernstein finite element basis. At the same time it preserves the integral nature of the basis and therefore allows for analytical integration. Aiming at the evaluation of the physical tissue model and its finite element solution, we describe the implementation of a highly automatic surgery simulation prototype designed to post–simulate actual surgery. We propose methods and solutions needed in the model build–up and we describe an automatic computation of surgery displacement fields corresponding to real surgical procedures. The presentation of results achieved on the example of a test patient concludes the thesis. C1