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This paper is concerned with quadrature/cubature rules able to deal withmultiple subspaces of functions, in such a way that the integration points arecommon for all the subspaces, yet the weights are tailored to each specificsubspace. These subspace-adaptive weights cubature rules can be used toaccelerate computational mechanics applications requiring efficientlyevaluating spatial integrals whose integrand function dynamically switchesbetween multiple pre-computed subspaces. One of such applications is localhyperreduced-order modeling (HROM), in which the solution manifold isapproximately represented as a collection of basis matrices, each basis matrixcorresponding to a different region in parameter space. We pose theoptimization problem and propose two algorithms for its resolution. The firstone is a greedy strategy based on an enhanced version of the Empirical CubatureMethod (ECM) developed by the authors elsewhere (we call it theSubspace-Adaptive Weights ECM, SAW-ECM for short), while the second method isbased on a convexification of the cubature problem so that it can be addressedby linear programming algorithms. We show in a toy problem involvingintegration of polynomial functions that the SAW-ECM clearly outperforms theother method both in terms of computational cost and optimality. We illustratethe performance of the SAW-ECM in the construction of a local HROMs in a highlynonlinear equilibrium problem (large strains regime). We demonstrate that,provided that the subspace-transition errors are negligible, the errorassociated to hyperreduction using adaptive weights can be controlled by thetruncation tolerances of the SVDs used for determining the basis matrices. ThePython source codes of the proposed SAW-ECM are openly accessible in the publicrepository https://github.com/Rbravo555/localECM.

A subspace-adaptive weights cubature method with application to the local hyperreduction of parameterized finite element models