On the numerical integration of trimmed isogeometric elements

• A numerical algorithm is proposed for the design of optimal quadrature rules on trimmed isogeometric elements.
• The quadrature design scheme imposes no restrictions on the shape or topology of the trimmed element domain.
• The quadrature points are in the interior of the trimmed domain and positivity of the weights is preserved.
• Integration rules designed by the proposed algorithm are found to outperform competing techniques in terms of accuracy and efficiency.

A numerical algorithm is proposed to construct efficient quadrature rules for trimmed isogeometric elements of arbitrary shape and topology as part of the standard finite element preprocessing step. The constructed integration rule is unique to an element and it is considered to be optimal in the sense that the final quadrature points and weights satisfy the moment fitting equations within the trimmed domain up to a predefined tolerance. The resulting quadrature points are in the interior of the trimmed domain and the positivity of the weights is preserved. The accuracy and efficiency of the quadrature scheme are assessed and compared to traditionally employed integration techniques. Results indicate that the proposed integration rules are more precise and inhibit the proliferation of quadrature points observed in competing approaches. Selected problems of elastostatics, elastodynamics, and elasto-plastic dynamics are used to further demonstrate the validity and efficacy of the method.