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Mass lumping with dual test functions in IGA dynamics

  • The focus of this study is on providing an efficient and highly accurate mass lumping scheme for explicit dynamic calculations in isogeometric analysis (IGA). Computer-Aided design models make use of higher-order polynomials, usually Non-Uniform Rational Splines (NURBS), to build up the geometry model. In IGA they are chosen as shape functions to take advantage of their high continuity and to keep the description of geometry exact. Thus, compared to standard Finite Element Method (FEM), a smaller number of elements is required to gain results of the same quality level. In general, explicit time integration methods require a huge number of time steps to obtain numerical stable results of reasonable quality. Hence, diagonal mass matrices are preferred to reduce the computational costs within each time step. Row-sum lumping and diagonal scaling method are two well-known techniques, which have been developed as mass lumping schemes for standard FEM. Unfortunately, these standard FEM mass lumping schemes deteriorate convergence rates inThe focus of this study is on providing an efficient and highly accurate mass lumping scheme for explicit dynamic calculations in isogeometric analysis (IGA). Computer-Aided design models make use of higher-order polynomials, usually Non-Uniform Rational Splines (NURBS), to build up the geometry model. In IGA they are chosen as shape functions to take advantage of their high continuity and to keep the description of geometry exact. Thus, compared to standard Finite Element Method (FEM), a smaller number of elements is required to gain results of the same quality level. In general, explicit time integration methods require a huge number of time steps to obtain numerical stable results of reasonable quality. Hence, diagonal mass matrices are preferred to reduce the computational costs within each time step. Row-sum lumping and diagonal scaling method are two well-known techniques, which have been developed as mass lumping schemes for standard FEM. Unfortunately, these standard FEM mass lumping schemes deteriorate convergence rates in IGA dynamics significantly. With rising order of the basis functions, accuracy of the results decreases. The reason for this is that shape functions of higher order lead to more dense mass matrices and thus the error caused by lumping the mass matrices increases. Therefore, new mass lumping schemes have to be developed, which are more suitable for the use of higher-order shape functions. Several types of dual basis functions (duals), which have already been used in the isogeometric mortar method, are considered for a new mass lumping scheme. Keeping the initial NURBS as shape functions, the duals are applied as test functions for the variational form. Depending on the chosen type of duals, diagonal or banded mass matrices are obtained without any additional lumping. Compared to lumping the original mass matrices based on NURBS as test functions, applying additional row-sum lumping to the banded matrices causes significantly smaller errors. The presented new mass lumping scheme is a very promising approach, in which duals are used as test functions in IGA dynamics with explicit time integration, e.g. the Central Differences Method. The implementation into existing codes can be easily done. The assembled system matrices, if based on NURBS as shape and test functions, only have to be multiplied with a transformation matrix. Numerical examples show that if using this dual lumping scheme within IGA methods, explicit dynamic analysis can be performed efficiently with higher-order shape functions.show moreshow less

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Metadaten
Author: Susanne HeldORCiD, Wolfgang DornischORCiD
URL:https://drive.google.com/file/d/1CYA-jTeVREnLcY2XIY1HuYiYutElGxXE/view
Title of the source (English):Isogeometric Analysis 2022 (IGA 2022) - Book of Abstracts
Editor: Artem Korobenko, John Evans, Ming-Chen Hsu
Document Type:Conference publication not peer-reviewed
Language:English
Year of publication:2022
Contributing Corporation:USACM
First Page:44
Faculty/Chair:Fakultät 6 Architektur, Bauingenieurwesen und Stadtplanung / FG Statik und Dynamik
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