Unlocking the secrets of locking: Finite element analysis in planar linear elasticity
Finite element methods have been the subject of active research for the last six decades. However, gaps remain in the understanding of even the simplest applications, including linear elasticity — the area where finite elements were first developed as a serious technique for the approximation of par...
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| Published in | Computer methods in applied mechanics and engineering Vol. 395; p. 115034 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier B.V
15.05.2022
Elsevier BV |
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| Abstract | Finite element methods have been the subject of active research for the last six decades. However, gaps remain in the understanding of even the simplest applications, including linear elasticity — the area where finite elements were first developed as a serious technique for the approximation of partial differential equations by engineers interested in the stress analysis of structures. For nearly incompressible materials, such as rubber, the standard, conforming finite element method sometimes exhibits suboptimal convergence rates for the energy and/or stresses. This type of behavior, termed “locking” or “non-robustness”, is still not completely understood despite receiving a great deal of investigation. This paper reviews the concept of locking in conforming finite element approximations to planar linear elasticity and seeks to quantify the effects of locking for the h- and p-version finite element method with a particular emphasis on quantifying the effect of mesh topology and geometry. As a by-product, we show that the inf–sup constant, which is responsible for locking, is independent of the mesh size h and polynomial degree p when p≥4. Numerical examples are provided throughout to illustrate the theoretical results. |
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| AbstractList | Finite element methods have been the subject of active research for the last six decades. However, gaps remain in the understanding of even the simplest applications, including linear elasticity - the area where finite elements were first developed as a serious technique for the approximation of partial differential equations by engineers interested in the stress analysis of structures. For nearly incompressible materials, such as rubber, the standard, conforming finite element method sometimes exhibits suboptimal convergence rates for the energy and/or stresses. This type of behavior, termed "locking" or "non-robustness", is still not completely understood despite receiving a great deal of investigation. This paper reviews the concept of locking in conforming finite element approximations to planar linear elasticity and seeks to quantify the effects of locking for the h- and p- version finite element method with a particular emphasis on quantifying the effect of mesh topology and geometry. As a by-product, we show that the inf–sup constant, which is responsible for locking, is independent of the mesh size h and polynomial degree p when p ≥ 4. Numerical examples are provided throughout to illustrate the theoretical results. Finite element methods have been the subject of active research for the last six decades. However, gaps remain in the understanding of even the simplest applications, including linear elasticity — the area where finite elements were first developed as a serious technique for the approximation of partial differential equations by engineers interested in the stress analysis of structures. For nearly incompressible materials, such as rubber, the standard, conforming finite element method sometimes exhibits suboptimal convergence rates for the energy and/or stresses. This type of behavior, termed “locking” or “non-robustness”, is still not completely understood despite receiving a great deal of investigation. This paper reviews the concept of locking in conforming finite element approximations to planar linear elasticity and seeks to quantify the effects of locking for the h- and p-version finite element method with a particular emphasis on quantifying the effect of mesh topology and geometry. As a by-product, we show that the inf–sup constant, which is responsible for locking, is independent of the mesh size h and polynomial degree p when p≥4. Numerical examples are provided throughout to illustrate the theoretical results. |
| ArticleNumber | 115034 |
| Author | Parker, Charles Ainsworth, Mark |
| Author_xml | – sequence: 1 givenname: Mark surname: Ainsworth fullname: Ainsworth, Mark email: mark_ainsworth@brown.edu – sequence: 2 givenname: Charles orcidid: 0000-0003-0767-5732 surname: Parker fullname: Parker, Charles email: charles_parker@brown.edu |
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| Cites_doi | 10.1051/m2an/1987210201991 10.1103/PhysRevB.80.132104 10.1016/0045-7825(84)90160-9 10.1051/m2an:2008002 10.1090/S0025-5718-04-01711-9 10.1007/BF01396303 10.1137/20M1366587 10.1007/s11565-016-0247-x 10.1023/A:1007639413619 10.1137/20M1359110 10.1090/S0002-9904-1943-07818-4 10.1137/20M1359109 10.1017/S0962492906210018 10.1137/0726045 10.1007/s00033-003-3211-4 10.1007/BF01396238 10.1016/0377-0427(84)90038-4 10.1137/0710058 10.1090/mcom/3346 10.1016/j.cam.2019.06.007 10.1137/0728034 10.1051/m2an/1985190101111 10.1137/1036141 10.1090/S0025-5718-1990-1011446-7 10.1051/m2an/1990240202651 10.1002/fld.1407 10.1007/s002110100308 10.1007/BF01400318 10.1007/BF01396304 10.1016/0045-7825(92)90165-G 10.1137/18M1182450 10.1137/0729075 10.1002/nme.1620180302 10.1090/S0025-5718-1992-1140646-2 10.1016/0045-7825(95)00947-7 10.5802/smai-jcm.38 |
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| Snippet | Finite element methods have been the subject of active research for the last six decades. However, gaps remain in the understanding of even the simplest... |
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| SubjectTerms | Elasticity Finite element analysis Finite element method Linear elasticity Locking Mathematical analysis Partial differential equations Polynomials Robustness Robustness (mathematics) Stress analysis Topology |
| Title | Unlocking the secrets of locking: Finite element analysis in planar linear elasticity |
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