Generalization of the Pythagorean Eigenvalue Error Theorem and Its Application to Isogeometric Analysis

© 2018, Springer Nature Switzerland AG. This chapter studies the effect of the quadrature on the isogeometric analysis of the wave propagation and structural vibration problems. The dispersion error of the isogeometric elements is minimized by optimally blending two standard Gauss-type quadrature rules. These blending rules approximate the inner products and increase the convergence rate by two extra orders when compared to those with fully-integrated inner products. To quantify the approximation errors, we generalize the Pythagorean eigenvalue error theorem of Strang and Fix. To reduce the computational cost, we further propose a two-point rule for C1 quadratic isogeometric elements which produces equivalent inner products on uniform meshes and yet requires fewer quadrature points than the optimally-blended rules

Low-cost origami fabrication of 3D self-aligned hybrid microfluidic structures

[EN] 3D microfluidic device fabrication methods are normally quite expensive and tedious. In this paper, we present an easy and cheap alternative wherein thin cyclic olefin polymer (COP) sheets and pressure sensitive adhesive(PSA) were used to fabricate hybrid 3D microfluidic structures, by the Origami technique, which enables the fabrication of microfluidic devices without the need of any alignment tool. The COP and PSA layers were both cut simultaneously using a portable, low-cost plotter allowing for rapid prototyping of a large variety of designs in a single production step. The devices were then manually assembled using the Origami technique by simply combining COP and PSA layers and mild pressure. This fast fabrication method was applied, as proof of concept, to the generation of a micromixer with a 3D-stepped serpentine design made of ten layers in less than 8 min. Moreover, the micromixer was characterized as a function of its pressure failure, achieving pressures of up to 1000 mbar. This fabrication method is readily accessible across a large range of potential end users, such as educational agencies (schools,universities), low-income/developing world research and industry or any laboratory without access to clean room facilities, enabling the fabrication of robust, reproducible microfluidic devices.Fernando Benito-Lopez acknowledges the Ramón y Cajal Programme (Ministerio de Economía y Competitividad), Spain. This project has received funding from the European Union´s Seventh Framework Programme (FP7) for Research, Technological Development and Demonstration under Grant agreement no. 604241. LBD personally acknowledges to Elkartek (KK-2015/00088) Grant form the Gobierno Vasco. JS and FBL personally acknowledge Marian Martínez de Pancorbo for let them use her laboratory facilities at UPV/EHU. Authors also acknowledge Adhesive Research for the donation of the PSA samples and to Iñaki Veci for the drawing of the 3D scheme