📝 Add contact algorithm theory #58

thiagordonho · 2021-03-01T16:03:03Z

Describe the PR
This contribution incorporates the theory behind the Contact Algorithm to the MPM documentation.

https://deploy-preview-58--mpm.netlify.app/#/theory/contact

…ct_theory

jgiven100 · 2021-03-06T21:37:47Z

@thiagordonho I merged my linear elastic updates into this branch so everything is up-to-date!

kks32

Add a figure like the one in MPM25 years paper to show the algorithm

kks32 · 2021-03-23T21:26:03Z

theory/contact.md

+
+# Contact Algorithm {docsify-ignore}
+
+> at each time step $\Delta t$ from $t$ to $t + \Delta t$, the nodal kinematics are initially computed similarly to the conventional MPM algorithm but considering the distinct materials:


Suggested change

> at each time step $\Delta t$ from $t$ to $t + \Delta t$, the nodal kinematics are initially computed similarly to the conventional MPM algorithm but considering the distinct materials:

> at each time step $\Delta t$ from $t$ to $t + \Delta t$, the nodal kinematics are computed similar to the conventional MPM algorithm while considering distinct materials:

kks32 · 2021-03-23T21:26:16Z

theory/contact.md

@@ -0,0 +1,208 @@
+# Contact Mechanics Between Distinct Bodies
+
+The Material Point Method is naturally capable of modelling distinct bodies because each material point holds the information of its own material. However, the conventional MPM alone does not handle contact mechanics as the surface of distinct bodies meet. Additionally, one may need to identify contact interfaces without the need of prescribing their geometry at the start of a simulation. Therefore, the MPM requires a contact algorithm capable of identifying the contact of distinct bodies and applying their contact relationship. A first approach to deal with frictional contact was introduced by Bardenhagen et. al. (2000). This approach is the one presented within this document. Bardenhagen et. al. (2000) also describes the normal to the interface surface for each material as the normalized gradients of the volume. However, the authors method can lead to normal vectors of both materials that are not always aligned -- i.e., opposite to each other for two materials in contact -- which can lead to further errors of the contact relationship. Therefore, a slight change, as proposed by Nairn (2013), was introduced to the algorithm to handle such cases.


Revise as per comments posted on Google Docs

kks32 · 2021-03-23T21:27:06Z

theory/contact.md

+
+* The state parameters at the material points are initialised at the beginning of every time step in the same manner as it is in the conventional MPM.
+
+* The shape functions $N_i(x_p^t)$ and the gradient of the shape functions $B_i (x_p^t)$ are also computed at each material point as the conventional MPM dictates, with no changes due to the contact algorihtm.


Suggested change

* The shape functions $N_i(x_p^t)$ and the gradient of the shape functions $B_i (x_p^t)$ are also computed at each material point as the conventional MPM dictates, with no changes due to the contact algorihtm.

* The shape functions $N_i(x_p^t)$ and the gradient of the shape functions $B_i (x_p^t)$ are also computed at each material point like in conventional MPM.

kks32 · 2021-03-23T21:30:08Z

theory/contact.md

+
+* The shape functions $N_i(x_p^t)$ and the gradient of the shape functions $B_i (x_p^t)$ are also computed at each material point as the conventional MPM dictates, with no changes due to the contact algorihtm.
+
+* A nodal set of all the material ids (with no repetition) is created by identifying the material ids of all the material points in the cell. Each new material id is appended to this set. The size of this set will indicate whether the node is located at an interface of two or more materials or not.


Suggested change

* A nodal set of all the material ids (with no repetition) is created by identifying the material ids of all the material points in the cell. Each new material id is appended to this set. The size of this set will indicate whether the node is located at an interface of two or more materials or not.

* To identify a contact node, the material ids of all the material points in a cell are mapped to the associated nodes, where a list of unique material ids is maintained. Any node which has more than two material ids is identified as a contact node.

kks32 · 2021-03-23T21:32:20Z

theory/contact.md

+
+* A nodal set of all the material ids (with no repetition) is created by identifying the material ids of all the material points in the cell. Each new material id is appended to this set. The size of this set will indicate whether the node is located at an interface of two or more materials or not.
+
+* The nodal mass and momentum are calculated separately for each body $k$. They are based on the mass and velocity of all the material points in the cell that belong to their respective body and are mapped to the nodes using the shape functions.


Suggested change

* The nodal mass and momentum are calculated separately for each body $k$. They are based on the mass and velocity of all the material points in the cell that belong to their respective body and are mapped to the nodes using the shape functions.

* The nodal mass and momentum are calculated separately for each body $k$ based on the mass and velocity of all the material points corresponding to the body $k$ in the cell. The properties are then mapped to the nodes using the shape functions.

kks32 · 2021-03-23T21:34:39Z

theory/contact.md

+
+* The nodal mass and momentum are calculated separately for each body $k$. They are based on the mass and velocity of all the material points in the cell that belong to their respective body and are mapped to the nodes using the shape functions.
+
+    * Compute nodal mass of each body


Use the convention here to denote $b$ for each body.

📝 Add contact algorithm theory

08cc076

thiagordonho requested review from kks32 and cbgeo March 1, 2021 16:03

kks32 self-assigned this Mar 1, 2021

kks32 added Priority: Medium Priority: Medium Status: Review needed Status: Review needed Type: Documentation Type: Documentation labels Mar 1, 2021

Merge branch 'master' of https://github.com/cb-geo/mpm-doc into conta…

e4e8cb8

…ct_theory

kks32 requested changes Mar 23, 2021

View reviewed changes

thiagordonho added 4 commits March 31, 2021 11:23

📝 Edit contact intro, add figures and revise normal vector description

ebddd81

📝 Add the png files of the contact algorithm

977e964

📝 Change body index in the equations and place it as superscript

54ad031

🔧 Fix normal approaches image

fe7ad31

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📝 Add contact algorithm theory #58

📝 Add contact algorithm theory #58

thiagordonho commented Mar 1, 2021 •

edited by kks32

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jgiven100 commented Mar 6, 2021

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kks32 Mar 23, 2021


		# Contact Algorithm {docsify-ignore}

		> at each time step $\Delta t$ from $t$ to $t + \Delta t$, the nodal kinematics are initially computed similarly to the conventional MPM algorithm but considering the distinct materials:

		@@ -0,0 +1,208 @@
		# Contact Mechanics Between Distinct Bodies

		The Material Point Method is naturally capable of modelling distinct bodies because each material point holds the information of its own material. However, the conventional MPM alone does not handle contact mechanics as the surface of distinct bodies meet. Additionally, one may need to identify contact interfaces without the need of prescribing their geometry at the start of a simulation. Therefore, the MPM requires a contact algorithm capable of identifying the contact of distinct bodies and applying their contact relationship. A first approach to deal with frictional contact was introduced by Bardenhagen et. al. (2000). This approach is the one presented within this document. Bardenhagen et. al. (2000) also describes the normal to the interface surface for each material as the normalized gradients of the volume. However, the authors method can lead to normal vectors of both materials that are not always aligned -- i.e., opposite to each other for two materials in contact -- which can lead to further errors of the contact relationship. Therefore, a slight change, as proposed by Nairn (2013), was introduced to the algorithm to handle such cases.


		* The state parameters at the material points are initialised at the beginning of every time step in the same manner as it is in the conventional MPM.

		* The shape functions $N_i(x_p^t)$ and the gradient of the shape functions $B_i (x_p^t)$ are also computed at each material point as the conventional MPM dictates, with no changes due to the contact algorihtm.


		* The shape functions $N_i(x_p^t)$ and the gradient of the shape functions $B_i (x_p^t)$ are also computed at each material point as the conventional MPM dictates, with no changes due to the contact algorihtm.

		* A nodal set of all the material ids (with no repetition) is created by identifying the material ids of all the material points in the cell. Each new material id is appended to this set. The size of this set will indicate whether the node is located at an interface of two or more materials or not.

	* A nodal set of all the material ids (with no repetition) is created by identifying the material ids of all the material points in the cell. Each new material id is appended to this set. The size of this set will indicate whether the node is located at an interface of two or more materials or not.
	* To identify a contact node, the material ids of all the material points in a cell are mapped to the associated nodes, where a list of unique material ids is maintained. Any node which has more than two material ids is identified as a contact node.


		* A nodal set of all the material ids (with no repetition) is created by identifying the material ids of all the material points in the cell. Each new material id is appended to this set. The size of this set will indicate whether the node is located at an interface of two or more materials or not.

		* The nodal mass and momentum are calculated separately for each body $k$. They are based on the mass and velocity of all the material points in the cell that belong to their respective body and are mapped to the nodes using the shape functions.


		* The nodal mass and momentum are calculated separately for each body $k$. They are based on the mass and velocity of all the material points in the cell that belong to their respective body and are mapped to the nodes using the shape functions.

		* Compute nodal mass of each body

📝 Add contact algorithm theory #58

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📝 Add contact algorithm theory #58

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thiagordonho commented Mar 1, 2021 • edited by kks32 Loading

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thiagordonho commented Mar 1, 2021 •

edited by kks32

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