Korn’s Inequalities and Their Applications in Continuum Mechanics

Horgan, Cornelius O.

doi:10.1137/1037123

Cited by 198 publications

(149 citation statements)

References 36 publications

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“…At this point we remark that, differently from (3.13), the constant k D in (3.22) is not known in general. However, as described in Section 5 of the review article [23], there is a large class of two-dimensional domains for which k D can be estimated explicitly in terms of geometric constants. The corresponding results include star-shaped and simply connected regions (see [24,25]).…”

Section: Mixed Boundary Conditionsmentioning

confidence: 99%

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Gatica¹

2006

ESAIM: M2AN

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Abstract. We present a new stabilized mixed finite element method for the linear elasticity problem in R 2 . The approach is based on the introduction of Galerkin least-squares terms arising from the constitutive and equilibrium equations, and from the relation defining the rotation in terms of the displacement. We show that the resulting augmented variational formulation and the associated Galerkin scheme are well posed, and that the latter becomes locking-free and asymptotically locking-free for Dirichlet and mixed boundary conditions, respectively. In particular, the discrete scheme allows the utilization of Raviart-Thomas spaces of lowest order for the stress tensor, piecewise linear elements for the displacement, and piecewise constants for the rotation. In the case of mixed boundary conditions, the essential one (Neumann) is imposed weakly, which yields the introduction of the trace of the displacement as a suitable Lagrange multiplier. This trace is then approximated by piecewise linear elements on an independent partition of the Neumann boundary whose mesh size needs to satisfy a compatibility condition with the mesh size associated to the triangulation of the domain. Several numerical results illustrating the good performance of the augmented mixed finite element scheme in the case of Dirichlet boundary conditions are also reported.Mathematics Subject Classification. 65N12, 65N15, 65N30, 74B05.

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Section: Mixed Boundary Conditionsmentioning

confidence: 99%

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Gatica¹

2006

ESAIM: M2AN

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“…The k ∼ model provides an evolution inequality for both |k| 2 g and | | 2 g from equations (13) and (14).…”

Section: The "K ∼ " Modelmentioning

confidence: 99%

On properties of turbulence models

Moulden

2010

WIT Transactions on Engineering Sciences

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Many applications of fluid mechanics are to turbulent flows. The practical computation of such flows call upon turbulence models to provide the information needed to close the mean motion equations. The present comments concern those properties of turbulence models that may be obtained by direct study of the field equations defining these models. The second moment model and the k ∼ model are the focus of attention and norm inequalities are written for these models. The stability properties of the associated dynamical system are determined. Some difficulties associated with obtaining norm estimates are also discussed.

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“…[30]- [32]). Для отслеживания зависимостей кон-стант Корна от разнообразных геометрических параметров весьма полезен сле-дующий результат, заимствованный из работы [7] и относящийся к областям Ξ, звездным относительно шара B R = {x : |x| < R}.…”

Section: структура статьиunclassified

Неравенства Корна Для Упругих Сочленений Массивных Тел, Тонких Пластин И Стержней

Назаров¹

2008

Uspekhi Mat. Nauk

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Неравенства Корна для упругих сочленений массивных тел, тонких пластин и стержней С. А. НазаровДля сочленений массивных упругих тел и тонких пластин и стержней -во всевозможных их комбинациях -получены неравенства Корна, асимп-тотическая точность которых достигается путем введения разнообразных весовых множителей в L2-нормы смещений и их производных. Поскольку тонкие тела по-разному реагируют на растяжение и изгиб, такие нера-венства Корна по необходимости становятся анизотропными. Допуска-ются сочленения упругих тел с контрастными жесткостями, но постоян-ные в установленных неравенствах не зависят от обоих параметров: от-носительной толщины h ∈ (0, 1] и относительной жесткости µ ∈ (0, +∞). Нормы, отвечающие жестко защемленным элементам конструкции, суще-ственно отличаются от норм, отвечающих малоподвижным или подвиж-ным элементам, которые не закреплены непосредственно, но только при помощи соседних элементов, -поэтому адекватная структура весовых ани-зотропных норм определяется геометрией всего сочленения. Каждый ва-риант неравенства Корна сопровождается примером, подтверждающим оптимальность подбора весовых множителей.Библиография: 77 названий.

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Korn’s Inequalities and Their Applications in Continuum Mechanics

Cited by 198 publications

References 36 publications

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

Analysis of a new augmented mixed finite element method for linear elasticity allowing $\mathbb{RT}_0$-$\mathbb{P}_1$-$\mathbb{P}_0$ approximations

On properties of turbulence models

Неравенства Корна Для Упругих Сочленений Массивных Тел, Тонких Пластин И Стержней

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