SeismicSource

This class applies a seismic source at a given point.

Description

The orientation of an earthquake fault is described using three directions - strike ( $var element = document.getElementById("moose-equation-c1b3b4a3-4d26-4429-bf38-c86b63de733a");katex.render("\\phi_s", element, {displayMode:false,throwOnError:false});$ ), dip ( $var element = document.getElementById("moose-equation-8f713404-285a-4388-94d2-0d29ac7773f8");katex.render("\\delta", element, {displayMode:false,throwOnError:false});$ ) and slip direction ( $var element = document.getElementById("moose-equation-22967c33-531d-4173-88d5-0bb0d141da9a");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ ) as shown in Figure 1.

Figure 1: Definition of the fault-orientation parameters - strike $var element = document.getElementById("moose-equation-0a317adb-3512-4991-94b2-3ab671b078fa");katex.render("\\phi_s", element, {displayMode:false,throwOnError:false});$ , dip $var element = document.getElementById("moose-equation-4e63a174-a1d3-4c0c-8ac6-11e6af5c535c");katex.render("\\delta", element, {displayMode:false,throwOnError:false});$ and slip direction $var element = document.getElementById("moose-equation-15d0a0c6-a065-4e67-b264-d1523437ad46");katex.render("\\lambda", element, {displayMode:false,throwOnError:false});$ . The slip direction is measured clockwise around from north, with the fault dipping down to the right of the strike direction. Strike direction is also measured from the north. $var element = document.getElementById("moose-equation-256e24ac-6f43-4a51-8f99-545b26e062d6");katex.render("\\delta", element, {displayMode:false,throwOnError:false});$ is measured down from the horizontal (image courtesy Aki and Richards (2012)).

The seismic moment ( $var element = document.getElementById("moose-equation-fd9e9f44-15ed-4364-ad3d-44824c2c72ff");katex.render("M_o", element, {displayMode:false,throwOnError:false});$ ) of the earthquake point source in the fault specific coordinate system is:

$var element = document.getElementById("moose-equation-98317fb8-263c-4a5f-bdae-d9b78c620b3a");katex.render("M_o(t) = \\mu A \\bar{u}(t),", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-2ff7481d-640e-486a-9e87-f460638c8849");katex.render("\\mu", element, {displayMode:false,throwOnError:false});$ is the shear modulus of the soil, $var element = document.getElementById("moose-equation-30b58f25-292d-44d7-8e68-af20461ef038");katex.render("A", element, {displayMode:false,throwOnError:false});$ is the area of fault rupture and $var element = document.getElementById("moose-equation-c0c0db65-798a-4192-9bde-3a3c55272059");katex.render("\\bar{u}(t)", element, {displayMode:false,throwOnError:false});$ is the spatially averaged slip rate of the fault.

When this seismic moment is converted into the global coordinate system (x, y and z) with the x direction oriented along the geographic north and z direction along the soil depth, the resulting moment can be written in a symmetric $var element = document.getElementById("moose-equation-695237bc-5de8-427c-bcf0-b2294b7adb7d");katex.render("3 \\times 3", element, {displayMode:false,throwOnError:false});$ matrix form whose components are as follows:

$var element = document.getElementById("moose-equation-b55fafe4-9f15-417c-a898-e546189a9b15");katex.render("M_{xx}(t) = -M_o(t)(\\sin \\delta \\cos \\lambda \\sin2 \\phi_s + \\sin 2\\delta \\sin\\lambda \\sin^2 \\phi_s\\\\ M_{xy}(t) = M_o(t)(\\sin\\delta \\cos \\lambda \\cos 2 \\phi_s + \\frac{1}{2} \\sin 2\\delta \\sin \\lambda \\sin 2 \\phi_s) = M_{yx}(t) \\\\ M_{xz}(t) = -M_o(t)(\\cos \\delta \\cos \\lambda \\cos \\phi_s + \\cos 2\\delta \\sin \\lambda \\sin 2\\phi_s = M_{zx}(t)\\\\ M_{yy}(t) = M_o(t)(\\sin \\delta \\cos \\lambda \\sin 2 \\phi_s - \\sin 2 \\delta \\sin \\lambda \\cos^2 \\phi_s \\\\ M_{yz}(t) = -M_o(t)(\\cos \\delta \\cos \\lambda \\sin \\phi_s - \\cos 2\\delta \\sin\\lambda \\cos\\phi_s) = M_{zy}(t) \\\\ M_{zz}(t) = M_o(t) \\sin 2\\delta \\sin \\lambda", element, {displayMode:true,throwOnError:false});$

Each component of the above matrix is a force couple with the first index representing the force direction and the second index representing the direction in which the forces are separated (see Figure 2).

Figure 2: The nine different force couples required to model an earthquake source (image courtesy Aki and Richards (2012)).

The total force in global coordinate direction $var element = document.getElementById("moose-equation-761942be-707d-487b-910e-aedc0e156694");katex.render("i", element, {displayMode:false,throwOnError:false});$ resulting from an earthquake source applied at point $var element = document.getElementById("moose-equation-9b78f4ac-a95b-43f2-b66b-0850744cc567");katex.render("\\vec{\\zeta}", element, {displayMode:false,throwOnError:false});$ in space is then: $var element = document.getElementById("moose-equation-6ef3aa0e-bd15-4f8f-8ffd-771769ef5126");katex.render("f_i(\\vec{x}, t) = - \\sum_{j=1}^{3} \\frac{\\partial M_{ij}(\\vec{x}, t)}{\\partial x_j} = \\sum_{j=1}^{3} M_{ij}(t) \\frac{\\partial \\delta (\\vec{x} - \\vec{\\zeta})}{\\partial x_j},", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-bdaa9ae8-4adc-4bd9-ad71-b287ab2bafc1");katex.render("\\delta(.)", element, {displayMode:false,throwOnError:false});$ is the delta function in space.

If an asynchronous fault rupture is being simulated, the rupture time of each point source is calculated based on the distance of the point source from the epicenter and the rupture speed. The slip time history is then shifted by the rupture time to simulate asynchronous fault rupture.

Input Parameters

areaThe area over which slip is distributed.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The area over which slip is distributed.
dipThe dip angle of the fault in degrees.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The dip angle of the fault in degrees.
displacementsThe vector of displacement variables.
C++ Type:std::vector<NonlinearVariableName>
Unit:(no unit assumed)
Controllable:No
Description:The vector of displacement variables.
rakeThe rake angle of the fault in degrees.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The rake angle of the fault in degrees.
shear_modulusThe shear modulus of the soil around the seismic source.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The shear modulus of the soil around the seismic source.
slipThe function describing slip as a function of time.
C++ Type:FunctionName
Unit:(no unit assumed)
Controllable:No
Description:The function describing slip as a function of time.
strikeThe strike angle of the fault in degrees.
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The strike angle of the fault in degrees.

Required Parameters

active__all__ If specified only the blocks named will be visited and made active
Default:__all__
C++ Type:std::vector<std::string>
Controllable:No
Description:If specified only the blocks named will be visited and made active
alpha0The Hilber Hughes Taylor (HHT) time integration parameter.
Default:0
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The Hilber Hughes Taylor (HHT) time integration parameter.
epicenterThe x, y and z coordinates of the epicenter.
C++ Type:std::vector<double>
Unit:(no unit assumed)
Controllable:No
Description:The x, y and z coordinates of the epicenter.
inactiveIf specified blocks matching these identifiers will be skipped.
C++ Type:std::vector<std::string>
Controllable:No
Description:If specified blocks matching these identifiers will be skipped.
numberThe number of point sources. This number should be same as number of entries in the function files describing the position of the point source.
C++ Type:unsigned int
Controllable:No
Description:The number of point sources. This number should be same as number of entries in the function files describing the position of the point source.
pointThe x, y and z coordinate of a single point source.
C++ Type:std::vector<double>
Unit:(no unit assumed)
Controllable:No
Description:The x, y and z coordinate of a single point source.
position_functionThe vector of function names that describes the x, y and z coordinates of the source. Only the x positionfunction is required for 1 dimensional problems, x and ypositions are required for 2 dimensional problems and x, y and z positions are required for 3D problems. The firstcolumn in these functions gives source number (startingwith 1) and the second column contains the coordinate.
C++ Type:std::vector<FunctionName>
Unit:(no unit assumed)
Controllable:No
Description:The vector of function names that describes the x, y and z coordinates of the source. Only the x positionfunction is required for 1 dimensional problems, x and ypositions are required for 2 dimensional problems and x, y and z positions are required for 3D problems. The firstcolumn in these functions gives source number (startingwith 1) and the second column contains the coordinate.
rupture_speedThe speed at which the earthquake rupture propogates through the fault (usually around 0.8 * shear wave speed).
C++ Type:double
Unit:(no unit assumed)
Controllable:No
Description:The speed at which the earthquake rupture propogates through the fault (usually around 0.8 * shear wave speed).

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.

Advanced Parameters

References

K. Aki and P. G. Richards. Quantitative Seismology. University Science Books; Second edition, 2012.

Description
Input Parameters
References

Usage

Development

Demonstration