Interaction of Waves with Frictional Interfaces Using Summation-by-Parts Difference Operators: Weak Enforcement of Nonlinear Boundary Conditions
We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions...
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Published in | Journal of scientific computing Vol. 50; no. 2; pp. 341 - 367 |
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Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
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Springer US
01.02.2012
Springer Nature B.V |
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Abstract | We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.
The use of SBP operators also provides a rigorously defined energy balance for the discretized problem that, as the mesh is refined, approaches the exact energy balance in the continuous problem. This enables one to investigate earthquake energetics, for example the efficiency with which elastic strain energy released during rupture is converted to radiated energy carried by seismic waves, rather than dissipated by frictional sliding of the fault. These theoretical results are confirmed by several numerical tests in both one and two dimensions demonstrating the computational efficiency, the high-order convergence rate of the method, the benefits of using strictly stable numerical methods for long time integration, and the accuracy of the energy balance. |
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AbstractList | We present a high-order difference method for problems in elastodynamics involving
the interaction of waves with highly nonlinear frictional interfaces. We restrict our
attention to two-dimensional antiplane problems involving deformation in only one direction.
Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity
across it are of a form closely related to those used in earthquake rupture models and
other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators
and weak enforcement of boundary and interface conditions, a strictly stable method
is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated
in terms of characteristic variables, as opposed to the physical variables in terms of
which they are more naturally stated, the semi-discretized system of equations can become
extremely stiff, preventing efficient solution using explicit time integrators.
The use of SBP operators also provides a rigorously defined energy balance for the discretized
problem that, as the mesh is refined, approaches the exact energy balance in the
continuous problem. This enables one to investigate earthquake energetics, for example the
efficiency with which elastic strain energy released during rupture is converted to radiated
energy carried by seismic waves, rather than dissipated by frictional sliding of the fault.
These theoretical results are confirmed by several numerical tests in both one and two dimensions
demonstrating the computational efficiency, the high-order convergence rate of
the method, the benefits of using strictly stable numerical methods for long time integration,
and the accuracy of the energy balance. We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators. The use of SBP operators also provides a rigorously defined energy balance for the discretized problem that, as the mesh is refined, approaches the exact energy balance in the continuous problem. This enables one to investigate earthquake energetics, for example the efficiency with which elastic strain energy released during rupture is converted to radiated energy carried by seismic waves, rather than dissipated by frictional sliding of the fault. These theoretical results are confirmed by several numerical tests in both one and two dimensions demonstrating the computational efficiency, the high-order convergence rate of the method, the benefits of using strictly stable numerical methods for long time integration, and the accuracy of the energy balance. We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.The use of SBP operators also provides a rigorously defined energy balance for the discretized problem that, as the mesh is refined, approaches the exact energy balance in the continuous problem. This enables one to investigate earthquake energetics, for example the efficiency with which elastic strain energy released during rupture is converted to radiated energy carried by seismic waves, rather than dissipated by frictional sliding of the fault. These theoretical results are confirmed by several numerical tests in both one and two dimensions demonstrating the computational efficiency, the high-order convergence rate of the method, the benefits of using strictly stable numerical methods for long time integration, and the accuracy of the energy balance. |
Author | Kozdon, Jeremy E. Dunham, Eric M. Nordström, Jan |
Author_xml | – sequence: 1 givenname: Jeremy E. surname: Kozdon fullname: Kozdon, Jeremy E. email: jkozdon@stanford.edu organization: Department of Geophysics, Stanford University – sequence: 2 givenname: Eric M. surname: Dunham fullname: Dunham, Eric M. organization: Department of Geophysics, Stanford University – sequence: 3 givenname: Jan surname: Nordström fullname: Nordström, Jan organization: Department of Mathematics, Linköping University |
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Cites_doi | 10.1006/jcph.1994.1057 10.1023/A:1020342429644 10.1016/0021-9991(90)90152-Q 10.1090/S0025-5718-1995-1308459-9 10.1006/jcph.1994.1005 10.1016/j.jcp.2004.03.001 10.1006/jcph.1998.6114 10.1137/060654943 10.1016/S0168-9274(02)00239-8 10.1137/0135035 10.1007/s10915-005-9013-4 10.1016/j.jcp.2007.01.023 10.1016/j.jcp.2007.11.040 10.1016/j.jcp.2006.02.014 10.1137/050639107 10.1016/S0022-5096(01)00042-4 10.1006/jcph.2001.6864 10.1002/cpa.3160230304 10.1016/0021-9991(87)90041-6 10.1007/s10915-009-9305-1 10.1023/A:1021149523112 10.1090/S0025-5718-1975-0386296-7 10.1016/B978-0-12-208350-1.50012-1 |
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Keywords | Nonlinear boundary conditions Wave propagation Friction Summation-by-parts Simultaneous approximation term method Elastodynamics High order finite difference |
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Snippet | We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We... We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We... |
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SubjectTerms | Algorithms Applied mathematics Boundary conditions Computational Mathematics and Numerical Analysis Discretization Earthquakes Elastodynamics Finite differences Finite element method High order finite difference · Nonlinear boundary conditions · Simultaneous Interfaces MATEMATIK Mathematical and Computational Engineering Mathematical and Computational Physics MATHEMATICS Mathematics and Statistics Numerical methods Operators (mathematics) Seismic waves Sliding Strain energy Theoretical Tillämpad matematik Time integration |
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Title | Interaction of Waves with Frictional Interfaces Using Summation-by-Parts Difference Operators: Weak Enforcement of Nonlinear Boundary Conditions |
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