SUMMARYFor a wide class of finite element matrices integrated numerically rather than exactly, a definable number of sampling points is found to be sufficient for keeping their theoretical properties unchanged. A systematic criterion limiting the number of possible point configurations for numerical quadrature formulas on triangles is established. Some new high order formulas are presented. Tables containing optimal formulas with respect to minimum number of sampling points and required degrees of accuracy are given. They are arranged so as to assist with selection of suitable quadrature formulas for finite element computer programming.

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SUMMARYSome new moderate degree cubature formulas for tetrahedra are derived. The suitability of various types of rules for the finite-element technique is discussed exhaustively. Consequently, the newly developed formulas supplemented by lower degree known ones are tabulated for the purpose of direct application in finite-element programming. All the formulas are presented in a virtually machine-independent form.

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