For projection-based reduced order models (ROM), the use of a relatively small number of modes might not translate into large enough computational savings. The culprit is the required reconstruction of certain terms in order, for example, to evaluate nonlinearities. Hyper-reduction (HROM) techniques aim to obtain approximations that prevent fulldimensional evaluations. In HROM, only a subset of mesh entities are probed to construct the reduced system of equations. The selected mesh entities and their corresponding weights are given as the solution to an optimisation problem. In this presentation we focus on modifications to such optimisation problem, and propose two approaches: a continuous and a local one. The continuous method we propose considers as optimisation variables not only the points and weights, but also the locations of such points. We use as starting point a discrete solution obtained by standard mesh-sampling algorithms. Then, by recurrently eliminating points with small assigned weights and rearranging the locations of the remaining points, an optimal quadrature is obtained. The method demonstrates significant reduction factors ranging from 2 for 1D functions up to 4 for 3D functions. On the other hand, the local method proposed considers the ROM basis functions not in a global fashion, but clustered in small batches. The algorithm is applied on each cluster to obtain multiple sets of weights while imposing redundancy in the selected indexes. This results in very small sets of selected points, with examples demonstrating reduction factors of several hundreds with respect to standard global methods. Our proposed methods are implemented taking as starting point the Empirical Cubature Method (ECM) [1, 2]. They provide efficient and effective solutions for the assembly of reduced systems in finite element analysis, moreover they have the potential to be applied in a wide range of fields. REFERENCES [1] Hern´andez, J. A. A multiscale method for periodic structures using domain decomposition and ECM-hyperreduction. Computer Methods in Applied Mechanics and Engineering, 368, 113192, 2020. [2] Hernandez, J. A., Caicedo, M. A., & Ferrer, A. Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Computer methods in applied mechanics and engineering, 313, 687-722, 2017.