The finite element method is used to model elastic bodies. In this method, the body under consideration is divided into a finite number of tetrahedral mesh elements, on which the unknown functions are approximated locally. For this approximation we use Bernstein basis polynomials, defined in terms of the barycentric coordinates of a tetrahedron. We present their fundamental properties, derivatives, and integrals, and use them to define Bézier simplex mappings. The mechanical behaviour of the body is described in terms of displacements, strains, stresses, and pressure. We consider a linear elastic model and a nearly incompressible model, and derive the corresponding variational formulations and finite element discretizations. The derived models are implemented on tetrahedral meshes. Numerical examples are used to analyse displacements, approximation accuracy, volume preservation, and the computational complexity of the method.
|
