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High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling.

Banks HT, Birch MJ, Brewin MP, Greenwald SE, Hu S, Kenz ZR, Kruse C, Maischak M, Shaw S, Whiteman JR - Int J Numer Methods Eng (2014)

Bottom Line: If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1.Werder et al. found that the space-time coupling matrices are diagonalizable over [Formula: see text] for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled.Copyright © 2014 The Authors.

View Article: PubMed Central - PubMed

Affiliation: Center for Research in Scientific Computation, North Carolina State University Raleigh, NC 27695-8212, USA.

ABSTRACT

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685-6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over [Formula: see text] for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin-Voigt and Maxwell-Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.

No MeSH data available.


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Plots of the L2(Ω) error in ∇ u(T) against N (equal in space and time) for the viscodynamic problem in Example 7 (Galerkin-in-space).
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fig10: Plots of the L2(Ω) error in ∇ u(T) against N (equal in space and time) for the viscodynamic problem in Example 7 (Galerkin-in-space).

Mentions: For this 2D calculation, we choose the same domain as previously mentioned, Ω = {0.005 < x < 0.15 and 0 < y < 0.3} but now with T = 0.5 and the problem given by 47 and 48. We set ϱ = 1010 kg/m3, E = 167 kPa, Poisson's ratio ν = 0.44 and include viscoelastic effects by choosing ϕ0 = 0.2, ϕ1 = 0.8 and τ1 = 0.05 in 41. The load, г, boundary and initial data are chosen so that the exact solution is (u1,u2)T = (sin(2πx) sin(2πy) cos(2πt + π/4),cos(2πx) sin(2πy) sin(2πt − π/4))T and Dirichlet data were prescribed on bottom edge of ∂Ω with tractions prescribed elsewhere The results are given in Figures 9, 10 and 11 for the Galerkin-in-space scheme and, as before, analogous results for the SEM are in 6. These results are for the case where equal polynomial degrees in space and time were used.


High-order space-time finite element schemes for acoustic and viscodynamic wave equations with temporal decoupling.

Banks HT, Birch MJ, Brewin MP, Greenwald SE, Hu S, Kenz ZR, Kruse C, Maischak M, Shaw S, Whiteman JR - Int J Numer Methods Eng (2014)

Plots of the L2(Ω) error in ∇ u(T) against N (equal in space and time) for the viscodynamic problem in Example 7 (Galerkin-in-space).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4376204&req=5

fig10: Plots of the L2(Ω) error in ∇ u(T) against N (equal in space and time) for the viscodynamic problem in Example 7 (Galerkin-in-space).
Mentions: For this 2D calculation, we choose the same domain as previously mentioned, Ω = {0.005 < x < 0.15 and 0 < y < 0.3} but now with T = 0.5 and the problem given by 47 and 48. We set ϱ = 1010 kg/m3, E = 167 kPa, Poisson's ratio ν = 0.44 and include viscoelastic effects by choosing ϕ0 = 0.2, ϕ1 = 0.8 and τ1 = 0.05 in 41. The load, г, boundary and initial data are chosen so that the exact solution is (u1,u2)T = (sin(2πx) sin(2πy) cos(2πt + π/4),cos(2πx) sin(2πy) sin(2πt − π/4))T and Dirichlet data were prescribed on bottom edge of ∂Ω with tractions prescribed elsewhere The results are given in Figures 9, 10 and 11 for the Galerkin-in-space scheme and, as before, analogous results for the SEM are in 6. These results are for the case where equal polynomial degrees in space and time were used.

Bottom Line: If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1.Werder et al. found that the space-time coupling matrices are diagonalizable over [Formula: see text] for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled.Copyright © 2014 The Authors.

View Article: PubMed Central - PubMed

Affiliation: Center for Research in Scientific Computation, North Carolina State University Raleigh, NC 27695-8212, USA.

ABSTRACT

We revisit a method originally introduced by Werder et al. (in Comput. Methods Appl. Mech. Engrg., 190:6685-6708, 2001) for temporally discontinuous Galerkin FEMs applied to a parabolic partial differential equation. In that approach, block systems arise because of the coupling of the spatial systems through inner products of the temporal basis functions. If the spatial finite element space is of dimension D and polynomials of degree r are used in time, the block system has dimension (r + 1)D and is usually regarded as being too large when r > 1. Werder et al. found that the space-time coupling matrices are diagonalizable over [Formula: see text] for r ⩽100, and this means that the time-coupled computations within a time step can actually be decoupled. By using either continuous Galerkin or spectral element methods in space, we apply this DG-in-time methodology, for the first time, to second-order wave equations including elastodynamics with and without Kelvin-Voigt and Maxwell-Zener viscoelasticity. An example set of numerical results is given to demonstrate the favourable effect on error and computational work of the moderately high-order (up to degree 7) temporal and spatio-temporal approximations, and we also touch on an application of this method to an ambitious problem related to the diagnosis of coronary artery disease. Copyright © 2014 The Authors. International Journal for Numerical Methods in Engineering published by John Wiley & Sons Ltd.

No MeSH data available.


Related in: MedlinePlus