Campus Units

Mechanical Engineering

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Accepted Manuscript

Publication Date


Journal or Book Title

International Journal for Numerical Methods in Engineering





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We demonstrate the potential of collocation methods for efficient higher-order analysis on standard nodal finite element meshes. We focus on a collocation method that is variationally consistent and geometrically flexible, converges optimally, embraces concepts of reduced quadrature, and leads to symmetric stiffness and diagonal consistent mass matrices. At the same time, it minimizes the evaluation cost per quadrature point, thus reducing formation and assembly effort significantly with respect to standard Galerkin finite element methods. We provide a detailed review of all components of the technology in the context of elastodynamics, that is, weighted residual formulation, nodal basis functions on Gauss–Lobatto quadrature points, and symmetrization by averaging with the ultra-weak formulation. We quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings. Our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort, such as higher-order elastostatic analysis. Furthermore, we illustrate the potential of collocation for efficient higher-order explicit dynamics. Throughout this work, we advocate a straightforward implementation based on simple modifications of standard finite element codes. We also point out the close connection to spectral element methods, where many of the key ideas are already established.


This is the peer reviewed version of the following article: Schillinger, Dominik, John A. Evans, Felix Frischmann, René R. Hiemstra, Ming‐Chen Hsu, and Thomas JR Hughes. "A collocated C0 finite element method: Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics." International Journal for Numerical Methods in Engineering 102, no. 3-4 (2015): 576-631, which has been published in final form at doi: 10.1002/nme.4783. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving. Posted with permission.

Copyright Owner

John Wiley & Sons, Ltd.



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Published Version