86,767 research outputs found
Piece-wise constant approximations in variational problems via W^1,p estimates
No full textPiece-wise constant approximation of first order variational problems via W −1, p estimates
C. Davini, R. Paroni
Dipartimento di Georisorse e Territorio, Via Cotonificio 114, 33100 Udine, Italy
DAP, Università di Sassari, Palazzo del Pou Salit, 07041 Alghero, Italy
A classical problem in the calculus of variations is: find the minimizers of the functional
F(v) =
W (x, v, ∇v) dx – (f , v)
Ω
among all functions v ∈ W 1, p(Ω) with trace equal to w ∈ W 1, p(Ω) over a subset (of positive length) ∂uΩ
of the boundary of Ω, where 1 < p < +∞, Ω ⊂ R2 is an open bounded set with Lipschitz boundary,
f ∈ Lq(Ω) and W : Ω × R × R2 → R is a Carathéodory function convex in the last variable and satisfying
a standard p-growth from below and above.
Different schemes have been developed in order to find an approximation of the minimizer(s) of the problem above. Probably, the most popular is the technique based on the use of continuous piece-wise affine finite elements. Higher order approximants have also been used. These on one hand give a better rate of convergence but on the other hand make the numerical scheme more complex.
Our point of view here is to consider the space which makes the numerical scheme as simple as possible, which is the space of piece-wise constant functions over triangulations of the base domain.
By using piece-wise constant functions the first problem at our hand is to define what we mean by gradient. At each nodal point xi of the triangulation of the base domain we call generalized gradient of a piece-wise constant function a suitable mean of the distributional gradient on a dual element around the point xi. This notion is then extended to the full triangulation by taking the generalized gradient constant on each dual element. Despite the given name, the generalized gradient is not a gradient, even though we show that it has some of the properties which are peculiar to a gradient. In particular we prove that if a sequence of generalized gradients weakly converges in Lp then the weak limit is a gradient. The generalized gradient instead does not have the "imbedding property" of a gradient, which is: a sequence of piece-wise constant functions which weakly converges together with the sequence of the generalized gradients does not necessarily strongly converges. This property is recovered by requiring that a certain weighted Lp norm of the jumps of the piece-wise constant function across the edges of the mesh should
tend to zero as the size of the triangulation goes to zero. This is proved by working in the W −1, p space
and using an inequality due to Necˇas. The lacking of this imbedding property strongly influences the definition of the discrete functionals which approximate the original one.
References
[1] Davini, C.; Paroni, R. External approximation of first order variational problems via W −1, p estimates. ESAIM Control Optim. Calc. Var. 14 (2008), no. 4, 802–824
Anisotropic inhomogeneous rectangular thin-walled beams
No full textThis paper is devoted to the asymptotic analysis of the problem of linear elasticity for an anisotropic and inhomogeneous body occupying, in its reference configuration, a cylindrical
domain with a rectangular cross section with sides proportional to ε and ε^2 and clamped on one of its
bases. The sequence of solutions uε of the equilibrium problem is shown to converge in an appropriate
topology, as ε goes to zero, to the solution of a problem for a beam in which the extensional, flexural,
and torsional effects are all coupled together
Saint-Venant's theory for beams with multi-connected cross-section: justification and error estimate
No full textThe aim of the paper is twofold. First, starting from the three-dimensional theory of linear elasticity, we give a
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Saint-Venant's theory for beams with multi-connected cross-section: justification and error estimate
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No full textAnisotropic inhomogeneous rectangular thin-walled beams
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On Hessian matrices in the space BH
No full textAn extension of Alberti's result to second order derivatives is obtained. Precisely, if Ω is an open subset of RN and if f ∈ L1(Ω; RN × N)) is symmetric-valued, then there exist u ∈ W1,1 (Ω) with ∇u ∈ BV(Ω RN) and a constant C > 0 depending only on N such that D 2u = f LN [Ω + [∇u] ⊗ ν∇u HN-1 ∫ S(∇u), and ∫Ω |u|+ |∇u|dx + ∫s(∇u) |[∇u]|dHN-1 ≤ C ∫Ω |f| dx
On lower semicontinuity in BHp and 2-quasiconvexification
No full textIt is proved that if u ∈ BHp(Ω;Rd), with p > 1, if {un} is bounded in BHp(Ω; Rd), |Ds2un|(Ω) → 0, and if un → u in W1, 1(Ω; Rd), then ∫ Ω f(x, u(x), ∇u(x), ∇2u(x)), dx ≤ lim infn→+∞ ∫Ω f(x, un (x), ∇un(x), ∇2un(x)) dx provided f(x, u, ξ, ·) is 2-quasiconvex and satisfies some appropriate growth and continuity condition. Characterizations of the 2-quasiconvex envelope when admissible test functions belong to BHP are provided
Nonlinear thin-walled beams with a rectangular cross-section - Part I
No full textOur aim is to rigorously derive a hierarchy of one-dimensional models for thin-walled
beams with rectangular cross-section, starting from three-dimensional nonlinear elasticity.
The different limit models are distinguished by the different scaling of the elastic
energy and of the ratio between the sides of the cross-section. In this paper we report
the first part of our results
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