Setting Material Point Initial Velocity - Revision history

Nairnj: /* Initial Velocity Gradient */

2024-04-27T01:11:34Z

Initial Velocity Gradient

← Older revision		Revision as of 01:11, 27 April 2024
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	where <math>\Delta X</math>, <math>\Delta Y</math>, and <math>\Delta Z</math> are the material point dimensions.		where <math>\Delta X</math>, <math>\Delta Y</math>, and <math>\Delta Z</math> are the material point dimensions.


	~~where velocity are give by~~
	and not angular velocity. The setting to get a particular angular velocity (ω radians per [[ConsistentUnits Command#Legacy and Consistent Units\|time units]]) depends are particle locations and dimensions. For particles at typical starting positions, the angular momentum for given ω would be:

	* 2D, 4 particles per cell: <math>L_{p,z} = {3\over 16}\left(\Delta x^2+\Delta y^2\right)\omega</math>
	* 2D, 1 particle per cell: <math>L_{p,z} = {1\over 4}\left(\Delta x^2+\Delta y^2\right)\omega</math>

	where Δx and Δy are cell dimensions for the grid cell containing the particle. Simulations with more particles per cell would need to set different angular momenta for different particle locations, which is currently not possible. One alternative is to use an average value noting that fraction in the above angular momenta will range from 1/8 to 1/4 depending on particle location.

	For 3D simulations, the equations are the same except may involve all three axes. The equation for each axis (e.g., <math>L_{p,x}</math>) depends on cell dimensions in the other two directions (e.g., Δy and Δz).

Nairnj: /* Initial Velocity Gradient */

2024-04-27T01:11:07Z

Initial Velocity Gradient

← Older revision		Revision as of 01:11, 27 April 2024
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	<math>\nabla\vec V_p = \left(\begin{array}{cc} 0 & -\omega_z \\ \omega_z & 0 \end{array}\right)		<math>\nabla\vec V_p = \left(\begin{array}{cc} 0 & -\omega_z \\ \omega_z & 0 \end{array}\right)
	\qquad {rm and} \qquad		\qquad {\rm and} \qquad
	\nabla\vec V_p = \left(\begin{array}{ccc} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right)</math>		\nabla\vec V_p = \left(\begin{array}{ccc} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right)</math>

	The initial particle angular momentum of a single particle due to ~~intial~~ angular velocities is:		The initial particle angular momentum of a single particle due to initial angular velocities is:

	~~     <math>\left(\begin{array}{ccc} 0 & L_{p,z} & -L_{p,y} \\~~
	~~-L_{p,z} & 0 & L_{p,x} \\~~
	~~L_{p,y} & -L_{p,x} & 0 \end{array}\right) = \frac{M_p}{6}~~
	~~\left(\begin{array}{ccc} 0 & -\Delta X^2\omega_z & \Delta X^2\omega_y \\~~
	~~\Delta Y^2\omega_z & 0 & -\Delta Y^2\omega_x \\~~
	~~-\Delta Z^2\omega_y & \Delta Z^2\omega_x & 0 \end{array}\right)~~
	~~</math>~~

	<math>\left(L_{p,x},L_{p,y},L_{p,z}\right) =		<math>\left(L_{p,x},L_{p,y},L_{p,z}\right) =
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	\left(\Delta X^2+\Delta Z^2\right)\omega_y,\left(\Delta X^2+\Delta Y^2\right)\omega_z\right]		\left(\Delta X^2+\Delta Z^2\right)\omega_y,\left(\Delta X^2+\Delta Y^2\right)\omega_z\right]
	</math>		</math>

			where <math>\Delta X</math>, <math>\Delta Y</math>, and <math>\Delta Z</math> are the material point dimensions.

Nairnj: /* Initial Velocity Gradient */

2024-04-27T01:09:33Z

Initial Velocity Gradient

← Older revision		Revision as of 01:09, 27 April 2024
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	<math>\left(L_{p,x},L_{p,y},L_{p,z}\right) =		<math>\left(L_{p,x},L_{p,y},L_{p,z}\right) =
	\frac{M_p}{12}		\frac{M_p}{12}\left[\left(\Delta Y^2+\Delta Z^2\right)\omega_x,
			\left(\Delta X^2+\Delta Z^2\right)\omega_y,\left(\Delta X^2+\Delta Y^2\right)\omega_z\right]
	</math>		</math>

Nairnj: /* Initial Velocity Gradient */

2024-04-27T01:06:48Z

Initial Velocity Gradient

← Older revision		Revision as of 01:06, 27 April 2024
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	\nabla\vec V_p = \left(\begin{array}{ccc} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right)</math>		\nabla\vec V_p = \left(\begin{array}{ccc} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right)</math>

	The initial particle angular momentum of a single particle due to intial angular velocities~~, expressed as a tensor,~~ is		The initial particle angular momentum of a single particle due to intial angular velocities is:

	<math>\left(\begin{array}{ccc} 0 & L_{p,z} & -L_{p,y} \\		<math>\left(\begin{array}{ccc} 0 & L_{p,z} & -L_{p,y} \\
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	-\Delta Z^2\omega_y & \Delta Z^2\omega_x & 0 \end{array}\right)		-\Delta Z^2\omega_y & \Delta Z^2\omega_x & 0 \end{array}\right)
	</math>		</math>

			<math>\left(L_{p,x},L_{p,y},L_{p,z}\right) =
			\frac{M_p}{12}
			</math>


	where velocity are give by		where velocity are give by

Nairnj: /* Initial Velocity Gradient */

2024-04-27T01:03:12Z

Initial Velocity Gradient

← Older revision		Revision as of 01:03, 27 April 2024
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	<math>\left(\begin{array}{ccc} 0 & L_{p,z} & -L_{p,y} \\		<math>\left(\begin{array}{ccc} 0 & L_{p,z} & -L_{p,y} \\
	-L_{p,z} & 0 & L_{p,x} \\		-L_{p,z} & 0 & L_{p,x} \\
	L_{p,y} & -L_{p,x} & 0 \end{array}\right) = ~~M_p~~\frac{1}{6} \left(\begin{array}{ccc}		L_{p,y} & -L_{p,x} & 0 \end{array}\right) = \frac{M_p}{6}
	\Delta X^2 &~~0 &0~~ \\		\left(\begin{array}{ccc} 0 & -\Delta X^2\omega_z & \Delta X^2\omega_y \\
	~~0 &~~ \Delta Y^2 ~~& 0~~ \\		\Delta Y^2\omega_z & 0 & -\Delta Y^2\omega_x \\
	0 & 0 & \Delta Y^2		-\Delta Z^2\omega_y & \Delta Z^2\omega_x & 0 \end{array}\right)
	\~~end{array}~~\~~right)~~
	~~\left(\begin{array}{ccc} 0 &~~ -\~~omega_z &~~ \omega_y ~~\\ \omega_z~~ & ~~0 & -~~\~~omega_x \\ -\omega_y &~~ \omega_x & 0 \end{array}\right)
	</math>		</math>

Nairnj: /* Initial Velocity Gradient */

2024-04-27T01:00:35Z

Initial Velocity Gradient

← Older revision		Revision as of 01:00, 27 April 2024
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	The initial particle angular momentum of a single particle due to intial angular velocities, expressed as a tensor, is		The initial particle angular momentum of a single particle due to intial angular velocities, expressed as a tensor, is

	<math> ~~L_p~~ = M_p\frac{1}{6} \left(\begin{array}{ccc}		<math>\left(\begin{array}{ccc} 0 & L_{p,z} & -L_{p,y} \\
			-L_{p,z} & 0 & L_{p,x} \\
			L_{p,y} & -L_{p,x} & 0 \end{array}\right) = M_p\frac{1}{6} \left(\begin{array}{ccc}
	\Delta X^2 &0 &0 \\		\Delta X^2 &0 &0 \\
	0 & \Delta Y^2 & 0 \\		0 & \Delta Y^2 & 0 \\

Nairnj: /* Initial Velocity Gradient */

2024-04-27T00:59:37Z

Initial Velocity Gradient

← Older revision		Revision as of 00:59, 27 April 2024
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	0 & \Delta Y^2 & 0 \\		0 & \Delta Y^2 & 0 \\
	0 & 0 & \Delta Y^2		0 & 0 & \Delta Y^2
	\end{array}\right)\left(\begin{array}{cc} 0 & -\omega_z \\ \omega_z & 0 \end{array}\right)		\end{array}\right)
			\left(\begin{array}{ccc} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right)
	</math>		</math>

Nairnj: /* Initial Velocity Gradient */

2024-04-27T00:59:12Z

Initial Velocity Gradient

← Older revision		Revision as of 00:59, 27 April 2024
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	0 & \Delta Y^2 & 0 \\		0 & \Delta Y^2 & 0 \\
	0 & 0 & \Delta Y^2		0 & 0 & \Delta Y^2
	\end{array}\right)		\end{array}\right)\left(\begin{array}{cc} 0 & -\omega_z \\ \omega_z & 0 \end{array}\right)
	</math>		</math>

	where velocity are give by		where velocity are give by
	and not angular velocity. The setting to get a particular angular velocity (ω radians per [[ConsistentUnits Command#Legacy and Consistent Units\|time units]]) depends are particle locations and dimensions. For particles at typical starting positions, the angular momentum for given ω would be:		and not angular velocity. The setting to get a particular angular velocity (ω radians per [[ConsistentUnits Command#Legacy and Consistent Units\|time units]]) depends are particle locations and dimensions. For particles at typical starting positions, the angular momentum for given ω would be:

Nairnj: /* Initial Velocity Gradient */

2024-04-27T00:58:31Z

Initial Velocity Gradient

← Older revision		Revision as of 00:58, 27 April 2024
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	=== Initial Velocity Gradient ===		=== Initial Velocity Gradient ===

	Note that the above commands are setting angular velocity that correspond to ~~any~~ antisymmetric velocity gradient. The initial velocity gradients in 2D and 3D ~~are~~:		Note that the above commands are setting angular velocity x, y, and z components that correspond to an antisymmetric velocity gradient. The initial velocity gradients in 2D and 3D become:

	<math>\nabla\vec V_p = \left(\begin{array}{cc} 0 & -\omega_z \\ \omega_z & 0 \end{array}\right)		<math>\nabla\vec V_p = \left(\begin{array}{cc} 0 & -\omega_z \\ \omega_z & 0 \end{array}\right)
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	\nabla\vec V_p = \left(\begin{array}{ccc} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right)</math>		\nabla\vec V_p = \left(\begin{array}{ccc} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right)</math>

			The initial particle angular momentum of a single particle due to intial angular velocities, expressed as a tensor, is

			<math> L_p = M_p\frac{1}{6} \left(\begin{array}{ccc}
			\Delta X^2 &0 &0 \\
			0 & \Delta Y^2 & 0 \\
			0 & 0 & \Delta Y^2
			\end{array}\right)
			</math>
			where velocity are give by
	and not angular velocity. The setting to get a particular angular velocity (ω radians per [[ConsistentUnits Command#Legacy and Consistent Units\|time units]]) depends are particle locations and dimensions. For particles at typical starting positions, the angular momentum for given ω would be:		and not angular velocity. The setting to get a particular angular velocity (ω radians per [[ConsistentUnits Command#Legacy and Consistent Units\|time units]]) depends are particle locations and dimensions. For particles at typical starting positions, the angular momentum for given ω would be:

Nairnj: /* Initial Velocity Gradient */

2024-04-27T00:51:49Z

Initial Velocity Gradient

← Older revision		Revision as of 00:51, 27 April 2024
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	Note that the above commands are setting angular velocity that correspond to any antisymmetric velocity gradient. The initial velocity gradients in 2D and 3D are:		Note that the above commands are setting angular velocity that correspond to any antisymmetric velocity gradient. The initial velocity gradients in 2D and 3D are:

	<math>\nabla \vec V_p = \left(\begin{array}{cc} 0 & -\omega_z \\ \omega_z & 0 \end{array}\right)</math>		<math>\nabla\vec V_p = \left(\begin{array}{cc} 0 & -\omega_z \\ \omega_z & 0 \end{array}\right)
			\qquad {rm and} \qquad
			\nabla\vec V_p = \left(\begin{array}{ccc} 0 & -\omega_z & \omega_y \\ \omega_z & 0 & -\omega_x \\ -\omega_y & \omega_x & 0 \end{array}\right)</math>

	and not angular velocity. The setting to get a particular angular velocity (ω radians per [[ConsistentUnits Command#Legacy and Consistent Units\|time units]]) depends are particle locations and dimensions. For particles at typical starting positions, the angular momentum for given ω would be:		and not angular velocity. The setting to get a particular angular velocity (ω radians per [[ConsistentUnits Command#Legacy and Consistent Units\|time units]]) depends are particle locations and dimensions. For particles at typical starting positions, the angular momentum for given ω would be: