Isogeometric Collocation: Cost Comparison with Galerkin Methods and Extension to Adaptive Hierarchical NURBS Discretizations

By Dominik Schillinger, John A. Evans, Alessandro Reali, Michael A. Scott, Thomas J. R. Hughes. 2013

327 citations recorded by the source corpus.

Subfields: Computational Mechanics, Mechanical Engineering

View Paper: https://www.semanticscholar.org/paper/73acfd410a42a83d7f326f195c1ddb23957f8356

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In Computational Mechanics

In Mechanical Engineering

Citations in the corpus

The variational collocation method (2016, methodology): “However, collocation methods are notably less accurate on a per degree of freedom basis [19].”
Isogeometric analysis: An overview and computer implementation aspects (2012, background): “(iv) In an attempt to compete with low order FEs with one-point quadrature that are extensively used in industrial applications, isogeometric collocation methods were developed [3, 4, 113].”
Single-variable formulations and isogeometric discretizations for shear deformable beams (2015, methodology): “Isogeometric collocation has been recently proposed as a high-order, lowcost alternative to Galerkin-based isogeometric analysis [12,18–25].”
A NURBS-based immersed methodology for fluid–structure interaction (2015, methodology): “This is of course a viable approach, but we find more appealing (and, very likely, more efficient) the possibility of approximating directly the strong form of (20) using an isogeometric collocation-like approach [69, 70, 71, 72].”
Generalized locally Toeplitz sequences : Theory and applications (2017, methodology): “To avoid this issue, isogeometric collocation methods have been recently introduced in [3, 83].”
Isogeometric collocation methods for Cosserat rods and rod structures (2017, unknown): “It has already been shown that collocation methods are computationally more efficient than Galerkin methods [29].”
Isogeometric collocation methods for the Reissner–Mindlin plate problem (2015, unknown): “Within the isogeometric collocation context, several promising significant studies have been recently proposed, including phase-field modeling [19], contact [20], and hierarchical local refinement [18].”
Fast formation and assembly of finite element matrices with application to isogeometric linear elasticity (2019, methodology): “Firstly, collocating the strong form of the equations [4, 5, 21] is a 35 viable alternative to the Galerkin method that decreases the formation and assembly time by orders of magnitude at the expense of decreased accuracy per degree of freedom and sub-optimal convergence rates [42].”
Image-based goal-oriented adaptive isogeometric analysis with application to the micro-mechanical modeling of trabecular bone (2015, background): “[3,4]) analysis for a wide variety of applications, encompassing the domains of fluid dynamics, (non-linear) solid mechanics, and multi-physics modeling.”
Optimal and reduced quadrature rules for tensor product and hierarchically refined splines in isogeometric analysis (2017, methodology): “Figure 18 plots the accuracy in terms of the relative error in the H1 semi-norm versus the number of degrees of freedom for uniform mesh refinement and local refinement through hierarchical B-splines.”
Isogeometric collocation using analysis-suitable T-splines of arbitrary degree (2016, methodology): “We follow [38] and manufacture the source term f such that the exact solution to the problem is u(x, y) = θ(x, y)(2)[ρ(x, y)(2) − 1][ρ(x, y)(2) − 16]sin(x), (33)”
Integration by interpolation and look-up for Galerkin-based isogeometric analysis (2015, methodology): “Consequently, our method restores the appeal of the Galerkin approach by making its performance similar to the recently established isogeometric collocation method [1,19,29].”