1. The construction of boundary type cubature is a problem that has been noticed for a very long time in numerical integration.

2. The boundary type cubature is quite the type of formulae that only use numerical values on the boundary of the domain.

3. When we calculate reserves with traditional cubature, parameters of reservoir bed we use are average values. This ignores anisotropic factors of reservoirs, so that affects accuracy of calculate.

4. This paper presents a free vibration study of moderately thick super-elliptical plates with various boundary constraints using the differential cubature method.

5. The main research content is as follows:(1) Reduced collocation response surface model (RCRSM) is constructed using polynomial chaos expansion (PCE) with points of monomial cubature rule (MCR).

6. By using the lowering dimensionality expansions, we construct boundary type cubature formulas with the algebraic precision for integrals.