Instability of Finite Difference Schemes Due to Boundary Conditions in Elastic Media*

Ilan, A.; Loewenthal, Dan

doi:10.1111/j.1365-2478.1976.tb00947.x

Cited by 79 publications

(32 citation statements)

References 10 publications

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“…The dependence on Poisson's ratio was previously observed in the literature [3][4][5][6][7][8][9][10], though not stated explicitly. The transition values of Poisson's ratio previously reported in the literature [4][5][6] coincide with our results for the specific cases under study. The eigenmodes of deformation are obtained directly from the eigenvalue problem at the element level.…”

Section: Introductionmentioning

confidence: 79%

“…(9) and (10). The resulting element mass and stiffness matrices are used to find the eigenvalues from the generalized eigenvalue problem given by (6). The resulting characteristic equation is an eighth-degree polynomial in k e and is complicated to solve for the roots analytically.…”

Section: Consistent Mass Matrix Formulationmentioning

confidence: 99%

“…In the present work, with the help of Mathematica, we obtain the eigenvalues and the corresponding deformation modes (eigenmodes) by directly solving the eigenvalue problem at the element level given by Eq. (6). For the case of lumped mass matrix, the resulting eigenvalues are:…”

Section: Lumped Mass Matrix Formulation: Reduced Integrationmentioning

confidence: 99%

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Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

Ling¹,

Cherukuri²

2002

Computational Mechanics

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Expressions for critical timesteps are provided for an explicit finite element method for plane elastodynamic problems in isotropic, linear elastic solids. Both 4-node and 8-node quadrilateral elements are considered. The method involves solving for the eigenvalues directly from the eigenvalue problem at the element level. The characteristic polynomial is of order 8 for 4-node elements and 16 for 8-node elements. Due to the complexity of these equations, direct solution of these polynomials had not been attempted previously. The commonly used critical time-step estimates in the literature were obtained by reducing the characteristic equation for 4-node elements to a second-order equation involving only the normal strain modes of deformation. Furthermore, the results appear to be valid only for lumped-mass 4-node elements. In this paper, the characteristic equations are solved directly for the eigenvalues using Mathematica and critical time-step estimates are provided for both lumped and consistent mass matrix formulations. For lumped-mass method, both full and reduced integration are considered. In each case, the natural modes of deformation are obtained and it is shown that when Poisson's ratio is below a certain transition value, either shear-mode or hourglass mode of deformation dominates depending on the formulation. And when Poisson's ratio is above the transition value, in all the cases, the uniform normal strain mode dominates. Consequently, depending on Poisson's ratio the critical time-step also assumes two different expressions. The approach used in this work also has a definite pedagogical merit as the same approach is used in obtaining time-step estimates for simpler problems such as rod and beam elements. IntroductionIt is well known known that the critical time-step estimates in fully-discretized standard Galerkin formulations of undamped structural dynamics problems obey the relationwhere x max is the largest natural frequency of any element in the finite element mesh and is obtained by solving the eigenvalue problem at the element level. The characteristic equation resulting from the eigenvalue problem is of the same order as the number of degrees of freedom (DOF)s per element. For problems such as axial loading of bars, beam bending and scalar wave propagation, when lower order elements are used, the maximum eigenvalue can be found readily in symbolic terms since the order of the polynomial characteristic equation is manageable [2]. However, for more complicated problems such as elastic wave propagation in two and three-dimensional media, the characteristic equations tend to be very complicated and hence roots cannot easily be found. For example, in the case of plane elastodynamic problems, the characteristic equation is, of order 8 when 4-node quadrilateral elements (Q4) are used and of order 16 when 8-node quadrilaterals (Q8) are used. To the authors' knowledge and as implied in [2], these equations have not been solved previously for the eigenvalues. However, conservative estimate for stable t...

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Section: Introductionmentioning

confidence: 79%

Section: Consistent Mass Matrix Formulationmentioning

confidence: 99%

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Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

Ling¹,

Cherukuri²

2002

Computational Mechanics

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“…As mentioned above, the fault edges are located 4 km below the free surface. Nevertheless, the free surface boundary condition proposed by Ilan and Loewenthal (1976) is used. The absorption boundary method is used for 3-D finite differences (Higdon 1991).…”

Section: Boundary Conditionsmentioning

confidence: 99%

A Study of Interactions Between Thrust and Strike-slip Faults

Wang¹,

Shieh²

2013

Terr. Atmos. Ocean. Sci.

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A 3-D finite difference method is applied in this study to investigate a spontaneous rupture within a fault system which includes a primary thrust fault and two strike-slip sub-faults. With the occurrence of a rupture on a fault, the rupture condition follows Coulomb's friction law wherein the stress-slip obeys the slip-weakening fracture criteria. To overcome the geometrical complexity of such a system, a finite difference method is encoded in two different coordinate systems; then, the calculated displacements are connected between the two systems using a 2-D interpolation technique. The rupture is initiated at the center of the main fault under the compression of regional tectonic stresses and then propagates to the boundaries whereby the main fault rupture triggers two strike-slip sub-faults. Simulation results suggest that the triggering of two sub-faults is attributed to two primary factors, regional tectonic stresses and the relative distances between the two sub-faults and the main fault.

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“…(1980). 1l ia ,, ,, , , ii, , , ,i , , , , Experimental modeling using specific geometries can be found in Knopoff and Gangi (1960), Martelet at. (1977), and Nathman (1980.…”

Section: Introductionmentioning

confidence: 99%

Development of Ultrasonic Modeling Techniques for the Study of Crustal Inhomogeneities.

Toksöz¹

1983

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Instability of Finite Difference Schemes Due to Boundary Conditions in Elastic Media*

Cited by 79 publications

References 10 publications

Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

A Study of Interactions Between Thrust and Strike-slip Faults

Development of Ultrasonic Modeling Techniques for the Study of Crustal Inhomogeneities.

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