An existence and uniqueness theorem for the dynamics of flexural shells

Piersanti, Paolo

doi:10.1177/1081286519876322

Cited by 11 publications

(18 citation statements)

References 30 publications

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“…In particular, we recall, again, that in the proof of the latter (cf. [24]), the following convergence was obtained (cf. (2)):…”

Section: Numerical Experiments: Cylindrical Shellsmentioning

confidence: 97%

“…for almost all (a.a. in what follows) t ∈ (0, T ). This operator is well-defined, linear, and continuous (cf., [24]).…”

Section: Prepared Using Sagejclsmentioning

confidence: 99%

“…Letting κ → 0, we obtain that the family (ζ ε κ ) κ>0 of solutions to Problem P ε F,κ (ω) converges to the unique strong solution ζ ε of Problem P ε F (ω) with the following modes of convergences (cf., Theorem 6.1 of [24]):…”

Section: Penalisation Of the Considered Problemmentioning

confidence: 99%

“…We conduct our third, and last, set of numerical tests in the case where the linearly elastic shell under consideration is an elliptic membrane shell (cf., Section 7 of [24]). Following the terminology proposed in Section 4.1 of [21], such a shell is said to be an elliptic membrane shell if the following two additional assumptions are satisfied: first, γ 0 = γ, i.e., the homogeneous boundary condition of place is imposed over the entire lateral face γ × [−ε, ε] of the shell, and second, its middle surface θ(ω) is elliptic, according to the definition given in Section 2.…”

Section: Final Remarks: Numerical Experiments For the Elliptic Membramentioning

confidence: 99%

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Numerical simulations for the dynamics of flexural shells

Shen

Piersanti

2020

Mathematics and Mechanics of Solids

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In this paper, we study a model describing the displacement of a linearly elastic flexural shell subjected to given dynamic loads from the computational point of view. The model under consideration takes the form of a set of hyperbolic variational equations posed over the space of admissible linearised inextensional displacements, and a set of initial conditions. Since the original model is not suitable for the implementation of a finite element method, we conduct the experiments on the corresponding penalised model. It was recently shown that the solution to such a penalised model is a good approximation of the solution to the original model. Numerical tests are therefore conducted on the the penalised model; the approximation of the solution to the penalised model is obtained via Newmark’s scheme, which is then implemented and tested for shells having the following middle surfaces: a portion of a cylinder and a portion of a cone. For the sake of completeness, we also present the results of the numerical tests related to a model describing the displacement of a linearly elastic elliptic membrane shell under the action of given dynamic loads.

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“…In particular, we recall, again, that in the proof of the latter (cf. [24]), the following convergence was obtained (cf. (2)):…”

Section: Numerical Experiments: Cylindrical Shellsmentioning

confidence: 97%

“…for almost all (a.a. in what follows) t ∈ (0, T ). This operator is well-defined, linear, and continuous (cf., [24]).…”

Section: Prepared Using Sagejclsmentioning

confidence: 99%

Section: Penalisation Of the Considered Problemmentioning

confidence: 99%

Section: Final Remarks: Numerical Experiments For the Elliptic Membramentioning

confidence: 99%

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Numerical simulations for the dynamics of flexural shells

Shen

Piersanti

2020

Mathematics and Mechanics of Solids

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“…It is easy to see that such an operator is well-defined, linear and continuous. It can be easily verified (cf., e.g., [27]) that the continuity constant is independent of t ∈ (0, T ).…”

Section: Introductionmentioning

confidence: 99%

A time-dependent obstacle problem in linearised elasticity

Piersanti

2020

Nonlinear Analysis

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In this paper we establish the existence of solutions to a time-dependent problem for a linearly elastic body subjected to a confinement condition, expressing that all the points of the deformed reference configuration remain confined in a prescribed half space. This problem takes the form of a set of hyperbolic variational inequalities. The fact that any solution of the studied problem takes the form of a vector field instead of a real-valued function, the generality of the confinement condition under consideration, the fact that the integration domain is a subset of R 3 , and the choice of the function space where solutions are sought make the analysis substantially more complicated, thus requiring the adoption of new resolution strategies. external force. In this paper, we study the existence of solutions to an obstacle problem modelling the displacement of a three-dimensional linearly elastic body 5 confined in a half space.Obstacle problems arise in many applicative fields: For instance, the motion of three valves of the Aorta, that can be regarded as linearly elastic shells (cf., e.g., [1]), is governed by a mathematical model built up in a way such that each valve remains confined in a certain portion of space without colliding with the 10 remaining two valves. A substantial contribution to the theory of hyperbolic obstacle problems can be found in the seminal papers [2] and [3]. Other important contributions in this field can be found in the references [4], [5], [6], and [7], where the problems are set out as follows: the integration domain ω is a subset of R 2 , and the unknown 15 function, at almost all time instants, is a real-valued function that belongs to H 2 0 (ω). It is also worth mentioning the papers [8], [9], [10], [11], [12], [13], [14] and [15]. The nonlinear analysis tools used in this paper can be found in the recent monograph [16].The first main novelty of this paper is that the unknown, represented by 20 displacement of the linearly elastic body under consideration, is a vector field that, at almost all time instants, belongs to a nonempty, closed , and convex subset of the Sobolev space H 1 (Ω)×H 1 (Ω)×H 1 (Ω), where Ω ⊂ R 3 is a domain.This will require the implementation of a more general argument to recover the energy estimates in the Galerkin method. 25The second main novelty is given by the generality of the confinement condition under consideration, which comprises at once all of the three components of the displacement vector field. Such a confinement conditions were first considered in the papers [17], [18], [19], and [20]. Other types of confinement conditions which are more amenable in the context of numerical simulations are discussed 30 in the paper [21]. Finally, the method we propose here for recovering the initial condition for the first derivative in time of the displacement slightly differs from the one used in [4], [5], [6], and [7]. 2 The paper is organised as follows. First, some notations and background are 35 provided. Secondly, the main existence theorem for a dyna...

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Asymptotic Analysis of Linearly Elastic Flexural Shells Subjected to an Obstacle in Absence of Friction

Piersanti

2023

J Nonlinear Sci

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An existence and uniqueness theorem for the dynamics of flexural shells

Cited by 11 publications

References 30 publications

Numerical simulations for the dynamics of flexural shells

Numerical simulations for the dynamics of flexural shells

A time-dependent obstacle problem in linearised elasticity

Asymptotic Analysis of Linearly Elastic Flexural Shells Subjected to an Obstacle in Absence of Friction

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