We construct a cubature formula of algebraic degree of exactness n with n 2 /2 + O(n) nodes, on the bidimensional domains generated by linear blending of two arcs of ellipses corresponding to the same angular interval. The construction is based on recent results on "subperiodic" trigonometric quadrature. Our formula generalizes several recent cubature formulas on standard circular sections. Among its numerous possible applications, we quote for example integration of functions with singularities, and integration on nonstandard circular sections arising in optical design or in meshfree methods with compactly supported radial bases. AMS subject classification: 65D32.Keywords: algebraic cubature, subperiodic trigonometric quadrature, product Gaussian quadrature, linear blending of elliptical arcs, singular integrand, nonstandard circular/elliptical sections, obscured and vignetted pupils, meshfree methods.

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A new approach to defining the effective fracture toughness for heterogeneous materials is proposed. This temporal averaging approach is process dependent, incorporating the crack velocity and material toughness. The effectiveness of the new technique is investigated in the context of hydraulic fracture through heterogeneous rock with a periodic material toughness. The plane strain model is considered without fluid leak-off, to more easily investigate different regimes (toughness/viscosity). Numerical simulations are used to examine the effectiveness of the new homogenisation strategy, with comparison against the recently proposed maximum toughness strategy. Simulations are conducted using an extremely effective (in house-built) time–space adaptive solver. The regimes in which each strategy is effective are determined.

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