The Numerical Solution of Ordinary and Partial Differential Equations

Sewell, Granville

doi:10.1002/0471742015

Cited by 99 publications

(63 citation statements)

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“…3. We focus on the development of stable methods, as Crank Nicolson, Galerkin methods (Großmann and Roos 2005;Sewell 2005). An iterative procedure is derived for the coupled problem.…”

Section: Resultsmentioning

confidence: 99%

“…Therefore the dump/restart option from PDE2D (Sewell 2005) has been used. This option in PDE2D is useful for this problem because it makes it easy to stop at some value of the time t, adjust the grid or even the boundary appropriately, and restart.…”

Section: Iterative Solution Strategymentioning

confidence: 99%

“…The initial conditions have been read from the restart file each time and then PDE2D dumps the final solution at the end of each run back into the restart file (Sewell 2005). Four thousand elements of degree 3 have been used for the final triangulation and the grids are most dense near the crack tip.…”

Section: Iterative Solution Strategymentioning

confidence: 99%

“…The time increment t has been chosen as t = 0.00003 s. The time discretization used is the CrankNicolson scheme (Sewell 2005).…”

Section: Iterative Solution Strategymentioning

confidence: 99%

“…There are no body forces, f = 0, and the problem (8) is broken into two first order equations (Sewell 2005) before it can be solved by PDE2D (Sewell). The form of the 2D time-dependent system solved by PDE2D (Sewell) is:…”

Section: Order Reduction For the Time-dependent Problemmentioning

confidence: 99%

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Analytical and numerical treatment of a dynamic crack model

2008

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We discuss the propagation of a running crack in a bounded linear elastic body under shear waves for a simplified 2D-model. This model is described by two coupled equations in the actual configuration: a two-dimensional scalar wave equation in a cracked, bounded domain and an ordinary differential equation derived from an energy balance law. The unknowns are the displacement fields u = u(y, t) and the one-dimensional crack tip trajectory h = h(t). We assume that the crack grows straight. Based on a paper of Nicaise-Sändig, we derive an improved formula for the ordinary differential equation of motion for the crack tip, where the dynamical stress intensity factor occurs. The numerical simulation is an iterative procedure starting from the wave field at time t = t i . The dynamic stress intensity factor will be extracted at t = t i . Its knowledge allows us to compute the crack-tip motion h(t i+1 ) with corresponding nonuniform crack speed assuming (t i+1 − t i ) is small. Now, we start from the cracked configuration at time t = t i+1 and repeat the steps. The wave displacements are A. Lalegname (B) · A.

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“…3. We focus on the development of stable methods, as Crank Nicolson, Galerkin methods (Großmann and Roos 2005;Sewell 2005). An iterative procedure is derived for the coupled problem.…”

Section: Resultsmentioning

confidence: 99%

Section: Iterative Solution Strategymentioning

confidence: 99%

Section: Iterative Solution Strategymentioning

confidence: 99%

“…The time increment t has been chosen as t = 0.00003 s. The time discretization used is the CrankNicolson scheme (Sewell 2005).…”

Section: Iterative Solution Strategymentioning

confidence: 99%

Section: Order Reduction For the Time-dependent Problemmentioning

confidence: 99%

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