A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis
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| Format: | Preprint |
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2022
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| _version_ | 1866916159109988352 |
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| author | Voet, Yannis Sande, Espen Buffa, Annalisa |
| author_facet | Voet, Yannis Sande, Espen Buffa, Annalisa |
| contents | Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2212_03614 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis Voet, Yannis Sande, Espen Buffa, Annalisa Numerical Analysis 65M60 65F55 Explicit time integration schemes coupled with Galerkin discretizations of time-dependent partial differential equations require solving a linear system with the mass matrix at each time step. For applications in structural dynamics, the solution of the linear system is frequently approximated through so-called mass lumping, which consists in replacing the mass matrix by some diagonal approximation. Mass lumping has been widely used in engineering practice for decades already and has a sound mathematical theory supporting it for finite element methods using the classical Lagrange basis. However, the theory for more general basis functions is still missing. Our paper partly addresses this shortcoming. Some special and practically relevant properties of lumped mass matrices are proved and we discuss how these properties naturally extend to banded and Kronecker product matrices whose structure allows to solve linear systems very efficiently. Our theoretical results are applied to isogeometric discretizations but are not restricted to them. |
| title | A mathematical theory for mass lumping and its generalization with applications to isogeometric analysis |
| topic | Numerical Analysis 65M60 65F55 |
| url | https://arxiv.org/abs/2212.03614 |