TY - JOUR
T1 - A collocated C0 finite element method
T2 - Reduced quadrature perspective, cost comparison with standard finite elements, and explicit structural dynamics
AU - Schillinger, Dominik
AU - Evans, John A.
AU - Frischmann, Felix
AU - Hiemstra, René R.
AU - Hsu, Ming Chen
AU - Hughes, Thomas J R
N1 - Publisher Copyright:
© 2014 John Wiley & Sons, Ltd.
PY - 2014/4/1
Y1 - 2014/4/1
N2 - 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.
AB - 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.
KW - Collocation methods
KW - Higher-order explicit Dynamics
KW - Method of weighted residuals
KW - Nodal Gauss–Lobatto basis functions
KW - Reduced Gauss–Lobatto quadrature
KW - Ultra-weak formulation
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U2 - 10.1002/nme.4783
DO - 10.1002/nme.4783
M3 - Article
AN - SCOPUS:84987657140
SN - 0029-5981
VL - 102
SP - 576
EP - 631
JO - International Journal for Numerical Methods in Engineering
JF - International Journal for Numerical Methods in Engineering
IS - 3-4
ER -