Final answer:
The T4 basis functions in finite element analysis are linear and of the first polynomial degree. Properties include being Lagrangian and continuous, and a T4 finite element has 12 degrees of freedom.
Step-by-step explanation:
Derivation of T4 Basis Functions
The question appears to be inquiring about theoretical aspects related to finite element analysis in engineering, specifically regarding T4 basis functions.
The T4 element is a four-node tetrahedral element used in three-dimensional meshing. Unfortunately, without additional context, deriving all T4 basis functions is beyond the scope of this platform, as it requires knowledge of the specific formulation and element shape functions used within the finite element software or analysis.
Polynomial Degree of T4 Basis Functions
T4 basis functions are linear, which means they are of the first polynomial degree. This is because each of the four vertex nodes has a basis function that varies linearly with the spatial coordinates within the tetrahedron.
Properties of T4 Basis Functions
The properties of T4 basis functions typically include being Lagrangian, where the function is 1 at its associated node and 0 at all other nodes within the element. They are also continuous across element boundaries, ensuring the compatibility of displacements between adjacent elements.
Degrees of Freedom of T4 finite elements
A T4 finite element generally has 12 degrees of freedom, as each of its four nodes can move in three independent spatial directions (x, y, z).