A boundary thin obstacle problem for a wave equation

We summarize recent achievements in applying Nitsche's method to some contact and friction problems. We recall the setting of Nitsche's method in the case of unilateral contact with Tresca friction in linear elasticity. Main results of the numerical analysis are detailed: consistency, well-posedness, fully optimal convergence in H 1 (Ω)-norm, residual-based a posteriori error estimation. Some numerics and some recent extensions to multibody contact, contact in large transformations and contact in elastodynamics are presented as well.

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“…To our knowledge, the well-posedness of Problem (53)- (54) is still an open issue. For partial results, see e.g., [70,66,31,37].…”

Section: Contact In Elastodynamicsmentioning

confidence: 99%

An Overview of Recent Results on Nitsche’s Method for Contact Problems

Chouly

Fabre

Hild

et al. 2017

Lecture Notes in Computational Science and Engineering

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“…With the help of Lemma 1.4 from [17] (essentially Gagliardo-Nirenberg type estimates), the convergence…”

Section: Definition 31 Assume the Initial Datamentioning

confidence: 99%

A contact problem in thermoviscoelastic diffusion theory with second sound

2017

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We consider a contact problem in thermoviscoelastic diffusion theory in one space dimension with second sound. The contact is modeled by the Signorini's condition and the stress-strain constitutive equation is of Kelvin−Voigt type. The thermal and diffusion disturbances are modeled by Cattaneo's law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier's law. The system of equations is a coupling of a hyperbolic equation with four parabolic equations. It poses some new mathematical difficulties due to the nonlinear boundary conditions and the lack of regularity. We prove that the viscoelastic term provides additional regularity leading to the existence of weak solutions. Then, fully discrete approximations to a penalized problem are considered by using the finite element method. A stability property is shown, which leads to a discrete version of the energy decay property. A priori error analysis is then provided, from which the linear convergence of the algorithm is derived. Finally, we give some computational results.

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“…This interval will be denoted by I C when the constitutive law is continuous and I D in the discontinue case. The Signorini problem for the wave equation was studied by Kim [8], there the author proved the existence of at least one solution, by using the Divergent-Rotational Lemma. Similarly, Andrews et al [2] considered the one-dimensional contact problem for the Euler Bernoulli beam model.…”

Section: Introductionmentioning

confidence: 99%

Signorini's problem for the Bresse beam model with localized Kelvin-Voigt dissipation

Munoz Rivera,

Baldez,

Cordeiro

2024

ejde

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We prove the existence of a global solution to Signorini's problem for the localized viscoelastic Bresse beam model (circular arc) with continuous and discontinuous constitutive laws. We show that when the constitutive law is continuous, the solution decays exponentially to zero, and when the constitutive law is discontinuous the solution decays only polynomially to zero. The method we use for proving our result is different the others already used in Signorini's problem and is based on approximations through a hybrid model. Also, we present some numerical results using discrete approximations in time and space, based on the finite element method on the spatial variable and the implicit Newmark method to the discretized the temporal variable. For more information see https://ejde.math.txstate.edu/Volumes/2024/17/abstr.html

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A boundary thin obstacle problem for a wave equation

Abstract: International audienceIn this paper we shall discuss a hyperbolic variational inequality associated with an initial-boundary value problem

Cited by 85 publications

References 10 publications

An Overview of Recent Results on Nitsche’s Method for Contact Problems

An Overview of Recent Results on Nitsche’s Method for Contact Problems

A contact problem in thermoviscoelastic diffusion theory with second sound

Signorini's problem for the Bresse beam model with localized Kelvin-Voigt dissipation

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