Symmetric Discontinuous Galerkin Methods for 1-D Waves

Marica, Aurora; Zuazua, Enrique

doi:10.1007/978-1-4614-5811-1

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…The DGFEM combined with Nitsche's method for inter-elements continuity and applied to the second order wave equation has been analyzed in [16]. The numerical boundary observability of waves for this particular type of finite element discretization has been studied in [27] where it is shown that there is no uniform numerical boundary observability, unless Fourier filtering or a multi-grid strategy is applied. We also note that a formulation close to Nitsche's method combined with a space-time finite element approximation of the wave equation with nonhomogeneous Dirichlet boundary conditions has been analyzed in [2].…”

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“…Alternatively, if we also use Nitsche's method to approximate the controlled wave equation, that is to append the Dirichlet boundary condition y(1, t) = v(t), then we obtain that the associated observability and continuity constants are defined by (27) and (28), and that the controllability problem reduces to minimize the functional…”

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confidence: 99%

“…Values of c N,T (left) and C N,T (right) for Nitsche's method, when the term γN 2 u N (1, t) is dropped in their definitions(27) and(28), with γ = 0.8 and T = 8 7. A numerical example.…”

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Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation

Gagnon¹,

Urquiza²

2021

Evolution Equations &Amp; Control Theory

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We study the boundary observability of the 1-D homogeneous wave equation when using a Legendre-Galerkin semi-discretization method. It is already known that spurious high frequencies are responsible for its lack of uniformity with respect to the discretization parameter [4] which may prevent convergence in the approximation of the associated controllability problem. A classical remedy is to filter out the highest frequency components but this comes with a high computational cost in several space dimensions. We present here three remedies: a spectral filtering method, a mixed formulation (already used in the context of finite element method [14]) and a Nitsche's method. Our numerical results show that the uniform boundary observability inequalities are recovered. On the other hand, surprisingly, none of them seem to provide the trace (or direct) inequality uniformly, a property used to prove the convergence of the numerical controls [11]. However, our numerical tests suggest that convergence of the numerical controls is ensured when the uniform observability inequality holds.

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Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation

Gagnon¹,

Urquiza²

2021

Evolution Equations &Amp; Control Theory

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Approximate and Mean Approximate Controllability Properties for Hilfer Time-Fractional Differential Equations

2020

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Nonlinear modeling and preliminary stabilization results for a class of piezoelectric smart composite beams

Özer¹

2018

Active and Passive Smart Structures and Integrated Systems XII

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Existing smart composite piezoelectric beam models in the literature mostly ignore the electro-magnetic interactions and adopt the linear elasticity theory. However, these interactions substantially change the controllability and stabilizability at the high frequencies, and linear models fail to represent and predict the governing dynamics since mechanical nonlinearities are pronounced in certain applications such as energy harvesting. In this paper, first, a consistent variational approach is used by considering nonlinear elasticity theory to derive equations of motion for a single-layer piezoelectric beam with and without the electromagnetic interactions (fully dynamic and electrostatic). This modeling strategy is extended for the three-layer piezoelectric smart composites by adopting the two widely-accepted sandwich beam theories. For both single-layer and three-layer models, the resulting infinite dimensional equations of motion can be formulated in the state-space form. It is observed that the fully dynamic nonlinear models are unbounded boundary control systems (same in linear theory) ẏ(t) = (A + N)y(t) + Bu(t), the electrostatic nonlinear models are unbounded bilinear control systems ẏ(t) = (A + N)y(t) + (B 1 + B 2 y)u(t) in sharp contrast to the linear theory. Finally, we propose B * −type feedback controllers to stabilize the single piezoelectric beam models. The filtered semi-discrete Finite Difference approximations is adopted to illustrate the findings.

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Symmetric Discontinuous Galerkin Methods for 1-D Waves

Cited by 4 publications

References 14 publications

Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation

Uniform boundary observability with Legendre-Galerkin formulations of the 1-D wave equation

Approximate and Mean Approximate Controllability Properties for Hilfer Time-Fractional Differential Equations

Nonlinear modeling and preliminary stabilization results for a class of piezoelectric smart composite beams

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