Isogeometric analysis for explicit elastodynamics using a dual-basis diagonal mass formulation

authored by: Cosmin Anitescu, Thanh Chuong Nguyen, Timon Rabczuk, Xiaoying Zhuang
Abstract: We propose a method to obtain diagonal mass matrices for NURBS-based approximation spaces by a “dual lumping” method. The use of lumped mass matrices is of great importance in elastodynamics problems, as they can be employed in explicit time integration schemes which do not require the solution of a linear system. In finite elements, several well-established methods, such as row-sum, diagonal scaling, or nodal quadrature methods have been used to obtain lumped mass matrices for different applications. However, for higher-order and higher continuity approximation spaces such as those derived from NURBS, these approaches have only limited (second-order) accuracy. In this work, we derive a dual basis which has optimal approximation and dispersion properties, while maintaining local support. The dual space has discontinuities at the element boundaries (knots) and it is used to provide the test functions in the context of a Petrov–Galerkin method. This results in a general framework for the study of lumped mass matrices which can be employed in explicit time integration schemes with high-order accuracy. Numerical experiments are presented to demonstrate the applicability of the method to problems with smooth solutions as well as to wave propagation problems with reduced regularity.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Bauhaus-Universität Weimar
Ton Duc Thang University
Type: Article
Journal: Computer Methods in Applied Mechanics and Engineering
Volume: 346
Pages: 574-591
No. of pages: 18
ISSN: 0045-7825
Publication date: 13.12.2018
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Mechanics of Materials, Mechanical Engineering, General Physics and Astronomy, Computer Science Applications
Electronic version(s): https://doi.org/10.1016/j.cma.2018.12.002 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{4b509cb6a3d64d6cb9439a2d61e53b1e,
title = "Isogeometric analysis for explicit elastodynamics using a dual-basis diagonal mass formulation",
abstract = "We propose a method to obtain diagonal mass matrices for NURBS-based approximation spaces by a “dual lumping” method. The use of lumped mass matrices is of great importance in elastodynamics problems, as they can be employed in explicit time integration schemes which do not require the solution of a linear system. In finite elements, several well-established methods, such as row-sum, diagonal scaling, or nodal quadrature methods have been used to obtain lumped mass matrices for different applications. However, for higher-order and higher continuity approximation spaces such as those derived from NURBS, these approaches have only limited (second-order) accuracy. In this work, we derive a dual basis which has optimal approximation and dispersion properties, while maintaining local support. The dual space has discontinuities at the element boundaries (knots) and it is used to provide the test functions in the context of a Petrov–Galerkin method. This results in a general framework for the study of lumped mass matrices which can be employed in explicit time integration schemes with high-order accuracy. Numerical experiments are presented to demonstrate the applicability of the method to problems with smooth solutions as well as to wave propagation problems with reduced regularity.",
keywords = "B{\'e}zier extraction, Dual-basis, Elastodynamics, Isogeometric analysis, Lumped mass",
author = "Cosmin Anitescu and Nguyen, {Thanh Chuong} and Timon Rabczuk and Xiaoying Zhuang",
year = "2018",
month = dec,
day = "13",
doi = "10.1016/j.cma.2018.12.002",
language = "English",
volume = "346",
pages = "574--591",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier",
}