  In high-order and high-dimensional finite elements, ill-conditioned nodal
distributions are often computationally cost-prohibitive. As a result, uniform
distributions quickly fall apart. For tensor-product like elements,
Gauss-Legendre-Lobatto (GLL) nodal distributions are often used as a
substitute. Besides these, other efficient nodal distributions are difficult to
create due to a desired symmetry within elements and conformity with
neighboring elements. In this paper, we provide a general framework to
construct symmetric, well-conditioned, cross-element compatible nodal
distributions which can be used for high-order and high-dimensional finite
elements. Starting from the inherent symmetries in any potential element, the
framework is used to build up nodal groups in a structured and efficient manner
utilizing the natural coordinates of each element, while ensuring nodes stay
within the elements. By constructing constrained symmetry groups, the vertices,
edges, and faces, of all elements are required to conform to their respective
lower-dimensional distributions. Optimizing over these groups yields the
desired optimized nodal distributions. We demonstrate the strength of this
framework by creating and comparing optimized nodal distributions with GLL
distributions (in elements such as the line, quadrilateral, and hexahedron),
and its robustness by generating optimized nodal distributions for otherwise
difficult elements (such as the triangle, tetrahedron, and triangular prism).
