A space-time continuous finite element method for 2D viscoelastic wave equation

Li, Hong; Zhao, Zhonglong; Luo, Zhendong

doi:10.1186/s13661-016-0563-1

Cited by 19 publications

(23 citation statements)

References 13 publications

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“…For Problem , we have the following conclusion on the existence, uniqueness, and stability of the weak solution.Theorem If

f \in L^{2} (0, T, L_{ω}^{2} (Ω))

G \in L_{ω}^{2} (Ω)

, and

H \in H_{ω}^{1} (Ω)

, then there exists a unique generalized solution for the variational formulation satisfying the following stability:

{(‖‖, \nabla u)}_{0, ω} ⩽ \overset{true˜}{C} ({(‖‖, G)}_{0, ω} + {(‖‖, \nabla H)}_{0, ω} + {(‖‖, f)}_{L^{2} (L_{ω}^{2})}),

where

\overset{true˜}{C} = \sqrt{\max \{1, γ^{- 1}, 1 / (2 italicϵ γ C_{p}^{2})\}}

. Proof Because the system of equations has an other form generalized solution u ∈ H 2 (0, T ; H 2 (Ω)) just as obtained in satisfying that

\{\begin{matrix} left \\ \int_{Ω} [u_{italictt} vdxdy + ϵ \nabla u_{t} \cdot \nabla v + γ \nabla u \cdot \nabla v] dxdy = \int_{Ω} fvdxdy, \forall v \in H_{0}^{1} (Ω), \\ u (x, y, 0) = H (x, y), ... \end{matrix}

…”

Section: Some Useful Sobolev Spaces and The Weak Solution For The 2d mentioning

confidence: 99%

“…Although FDS, FEM, and FVEM have been used to solve the viscoelastic wave equations [6][7][8][9][15][16][17][18][19], as far as we know, the spectral method, especially the CS method, has yet not been used to study the 2D viscoelastic wave equations. Therefore, in this paper, we propose a Crank-Nicolson CS (CNCS) model to solve the 2D viscoelastic wave equations.…”

Section: Introductionmentioning

confidence: 99%

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A Crank–Nicolson collocation spectral method for the two‐dimensional viscoelastic wave equation

Jin

Luo

2018

Numerical Methods Partial

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In this paper, we first establish the Crank–Nicolson collocation spectral (CNCS) method for two‐dimensional (2D) viscoelastic wave equation by means of the Chebyshev polynomials. And then, we analyze the existence, uniqueness, stability, and convergence of the CNCS solutions. Finally, we use some numerical experiments to verify the correctness of theoretical analysis. This implies that the CNCS model is very effective for solving the 2D viscoelastic wave equations.

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“…For Problem , we have the following conclusion on the existence, uniqueness, and stability of the weak solution.Theorem If

f \in L^{2} (0, T, L_{ω}^{2} (Ω))

G \in L_{ω}^{2} (Ω)

, and

H \in H_{ω}^{1} (Ω)

, then there exists a unique generalized solution for the variational formulation satisfying the following stability:

{(‖‖, \nabla u)}_{0, ω} ⩽ \overset{true˜}{C} ({(‖‖, G)}_{0, ω} + {(‖‖, \nabla H)}_{0, ω} + {(‖‖, f)}_{L^{2} (L_{ω}^{2})}),

where

\overset{true˜}{C} = \sqrt{\max \{1, γ^{- 1}, 1 / (2 italicϵ γ C_{p}^{2})\}}

. Proof Because the system of equations has an other form generalized solution u ∈ H 2 (0, T ; H 2 (Ω)) just as obtained in satisfying that

\{\begin{matrix} left \\ \int_{Ω} [u_{italictt} vdxdy + ϵ \nabla u_{t} \cdot \nabla v + γ \nabla u \cdot \nabla v] dxdy = \int_{Ω} fvdxdy, \forall v \in H_{0}^{1} (Ω), \\ u (x, y, 0) = H (x, y), ... \end{matrix}

…”

Section: Some Useful Sobolev Spaces and The Weak Solution For The 2d mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Crank–Nicolson collocation spectral method for the two‐dimensional viscoelastic wave equation

Jin

Luo

2018

Numerical Methods Partial

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“…However, the viscoelastic wave equations in the actual engineering generally involve the intricate given data, such as the given boundary and initial values and the given source function, so that we cannot find their analytic solutions. We have to rely on finding their solutions by means of numerical methods (see, e.g., ).…”

Section: Introductionmentioning

confidence: 99%

“…The finite volume element (FVE) method, the finite element (FE) method, the finite difference (FD) scheme, and the spectral method are deemed as four most common means to solve the viscoelastic wave equations(see, e.g., ), but comparison with the FD scheme, the FE method, and the FVE method, the spectral method has higher accuracy since it employs the whole smooth functions (e.g., Jacobi's, Legendre's, and Chebyshev's polynomials, and trigonometric functions) to approximate unknown functions, while the FE and FVE methods generally employ normal polynomials to approximate unknown functions and the FD scheme employs difference quotient to approximate differential quotient. In particular, the classical Crank–Nicolson collocation spectral (CNCS) method of the 2D viscoelastic wave equations can attain the super‐convergence about spatial variables .…”

Section: Introductionmentioning

confidence: 99%

A reduced‐order extrapolated Crank–Nicolson collocation spectral method based on proper orthogonal decomposition for the two‐dimensional viscoelastic wave equations

Luo

Jin

2019

Numerical Methods Partial

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In order to reduce the order of the coefficient vectors of the solutions for the classical Crank–Nicolson collocation spectral (CNCS) method of two‐dimensional (2D) viscoelastic wave equations via proper orthogonal decomposition, we first establish a reduced‐order extrapolated CNCS (ROECNCS) method of the 2D viscoelastic wave equations so that the ROECNCS method has the same basis functions as the classical CNCS method and maintains all advantages of the classical CNCS method. Then, by means of matrix analysis, we discuss the existence, stability, and convergence for the ROECNCS solutions so that the theory analysis becomes very concise. Finally, we utilize some numerical experiments to validate that the consequences of numerical computations are accordant with the theoretical analysis such that the effectiveness and viability of the ROECNCS method are further verified. Therefore, both theory and method of this paper are new and totally different from the existing reduced‐order methods.

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“…By solving this system, the response at all locations and all times can be easily calculated. There are many articles about the application of STCD based on the finite element method in various engineering problems: Varoglu and Finn in the Burgers' equation, Nguyen and Reynen in the advection‐diffusion equation, Lewis et al in parabolic problems, French and Peterson in the wave equation, Li et al in the viscoelastic wave equation, Karyofylli et al in flow simulation, etc. On the other hand, the space‐time discontinuous discretization approach is a powerful method, which leads to A ‐stable, higher‐order accurate solutions, especially, in the case of problems with discontinuities (or sharp gradient) in the solution .…”

Section: Introductionmentioning

confidence: 99%

Weight‐adaptive isogeometric analysis for solving elastodynamic problems based on space‐time discretization approach

Izadpanah

Shojaee

Hamzehei‐Javaran

2019

Numerical Meth Engineering

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Summary In this paper, a novel adaptive isogeometric analysis (IGA) is introduced and its application in the numerical solution of two‐dimensional elastodynamic problems based on the space‐time discretization (STD) approach is studied. In the STD approach, the time is considered as an additional dimension and is discretized the same as the spatial domain. The weights of control points play the main role in the proposed method. In the conventional IGA, the same set of weights is used in the modeling of geometric and solution spaces. The idea is to define two groups of weights: geometric and solution weights. Geometric weights are known and can be determined based on the position of control points, but the solution weights are considered to be unknown and can be determined using a proper strategy so that the accuracy of the solution is optimized. This strategy is based on the minimization of an error function. The results obtained from the proposed method are compared with those obtained from the conventional IGA.

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A space-time continuous finite element method for 2D viscoelastic wave equation

Cited by 19 publications

References 13 publications

A Crank–Nicolson collocation spectral method for the two‐dimensional viscoelastic wave equation

A Crank–Nicolson collocation spectral method for the two‐dimensional viscoelastic wave equation

A reduced‐order extrapolated Crank–Nicolson collocation spectral method based on proper orthogonal decomposition for the two‐dimensional viscoelastic wave equations

Weight‐adaptive isogeometric analysis for solving elastodynamic problems based on space‐time discretization approach

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