Bone Model - Revision history

Malhotra: infobox string

2023-11-30T09:25:40Z

infobox string

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 [[Category:time invariant]]
 [[Category:second differential order]]
 ==Description: Trabecular Bone Micro-Finite Element Models==

Mlinaric: Edit categories

2023-08-29T16:43:20Z

Edit categories

← Older revision		Revision as of 16:43, 29 August 2023
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	[[Category:benchmark]]		[[Category:benchmark]]
	[[Category:Oberwolfach]]		[[Category:Oberwolfach]]
			[[Category:linear]]
			[[Category:time invariant]]
			[[Category:second differential order]]

	==Description: Trabecular Bone Micro-Finite Element Models==		==Description: Trabecular Bone Micro-Finite Element Models==

Mlinaric: Typos, N->n

2023-08-29T13:27:50Z

Typos, N->n

← Older revision		Revision as of 13:27, 29 August 2023
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	<figure id="fig2">[[File:Bone2.jpg\|490px\|thumb\|right\|Figure 2]]</figure>		<figure id="fig2">[[File:Bone2.jpg\|490px\|thumb\|right\|Figure 2]]</figure>

	Three-dimensional serial reconstruction techniques allow us to develop very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture.		Three-dimensional serial reconstruction techniques allow us to develop a very detailed micro-finite element (micro-FE) model of bones that can very accurately represent the porous bone micro-architecture.
	Fig. 1 sketches the micro finite element analysis <ref name="rietbergen1995"/>.		Fig. 1 sketches the micro finite element analysis <ref name="rietbergen1995"/>.
	Micro [[wikipedia:CT_scan\|computed tomography]] (CT) is employed to make 3D high-resolution images (~50 microns) of a bone.		Micro [[wikipedia:CT_scan\|computed tomography]] (CT) is employed to make 3D high-resolution images (~50 microns) of a bone.
	Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone [[wikipedia:Voxel\|voxels]] to equally sized 8-node brick elements.		Then the 3D reconstruction is directly transformed into an equally shaped micro finite element model by simply converting all bone [[wikipedia:Voxel\|voxels]] to equally sized 8-node brick elements.
	This results in [[wikipedia:Finite_Element_Method\|finite element]] (FE) models with a very large number of elements.		This results in [[wikipedia:Finite_Element_Method\|finite element]] (FE) models with a very large number of elements.
	Such models can be used, for example to study differences in bone tissue loading between healthy and [[wikipedia:Osteoporosis\|osteoporotic]] human bones during quasi static loading <ref name="rietbergen2003"/>.		Such models can be used, for example, to study differences in bone tissue loading between healthy and [[wikipedia:Osteoporosis\|osteoporotic]] human bones during quasi-static loading <ref name="rietbergen2003"/>.

	There is increasing evidence, however, that bone responds in particular to dynamic loads <ref name="lanyon1984"/>.		There is increasing evidence, however, that bone responds in particular to dynamic loads <ref name="lanyon1984"/>.
	It has been shown that the application of high-frequency, very low magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic.		It has been shown that the application of high-frequency, very low-magnitude strains to a bone can prevent bone loss due to osteoporosis and can even result in increased bone strength in bones that are already osteoporotic.
	In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading.		In order to better understand this phenomenon, it is necessary to determine the strain as sensed by the bone cells due to this loading.
	This would be possible with the micro-FE analysis, but ~~then~~ such an analysis ~~need~~ to be a dynamic one.		This would be possible with the micro-FE analysis, but such an analysis needs to be a dynamic one.

	The present benchmark presents six bone models varying in dimension from about two-hundred-thousand to twelve ~~millions~~ equations with the goal to research on scalability of model reduction software.		The present benchmark presents six bone models varying in dimension from about two hundred thousand to twelve million equations with the goal to research on scalability of model reduction software.
	Each model represents a second order system of the form		Each model represents a second-order system of the form

	:<math>		:<math>
	\begin{~~array}{rcl~~}		\begin{align}
	M \ddot{x}(t) + K x(t) &=& B \\		M \ddot{x}(t) + K x(t) &= B u(t), \\
	y(t) &=~~& Cx~~(t)		y(t) &= C x(t),
	\end{~~array~~}		\end{align}
	</math>		</math>

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	The input matrix contains a single column with <math>B(1) = 1</math>.		The input matrix contains a single column with <math>B(1) = 1</math>.
	The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math> and <math>UZ</math> for the first node.		The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math>, and <math>UZ</math> for the first node.

	The archive [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++\|C++]] to assemble the dynamic system.		The archive [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++\|C++]] to assemble the dynamic system.
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	:<math>		:<math>
	\begin{~~array}{rcl~~}		\begin{align}
	M \ddot{x}(t) + K x(t) &=~~& b~~ \\		M \ddot{x}(t) + K x(t) &= B u(t) \\
	y(t) &=~~& Cx~~(t)		y(t) &= C x(t)
	\end{~~array~~}		\end{align}
	</math>		</math>

	System dimensions:		System dimensions:

	<math>M ~~\in \mathbb{R}^{N \times N}</math>~~,		<math>M, K \in \mathbb{R}^{n \times n}</math>,
	~~<math>~~K \in \mathbb{R}^{N \times N}</math>,		<math>B \in \mathbb{R}^{n \times 1}</math>,
	<math>b \in \mathbb{R}^{N \times 1}</math>,		<math>C \in \mathbb{R}^{3 \times n}</math>.
	<math>C \in \mathbb{R}^{3 \times N}</math>.

	System variants:		System variants:

	<tt>BS01</tt>: <math>N = 127224</math>,		<tt>BS01</tt>: <math>n = 127224</math>,
	<tt>BS10</tt>: <math>N = 914898</math>,		<tt>BS10</tt>: <math>n = 914898</math>,
	<tt>B010</tt>: <math>N = 986703</math>,		<tt>B010</tt>: <math>n = 986703</math>,
	<tt>B025</tt>: <math>N = 2159661</math>,		<tt>B025</tt>: <math>n = 2159661</math>,
	<tt>B050</tt>: <math>N = 4934244</math>,		<tt>B050</tt>: <math>n = 4934244</math>,
	<tt>B120</tt>: <math>N = 11969688</math>.		<tt>B120</tt>: <math>n = 11969688</math>.

	==Citation==		==Citation==

Lund: /* Data */ Correct typos

2023-04-25T15:46:33Z

Data: Correct typos

← Older revision		Revision as of 15:46, 25 April 2023
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	Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.		Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.

	Files (in .mat) ~~format~~ can be found for the three smaller systems ([https://sparse.tamu.edu/Oberwolfach/boneS01 BS01], [https://sparse.tamu.edu/Oberwolfach/boneS10 BS10], and [https://sparse.tamu.edu/Oberwolfach/bone010 B010]) on SuiteSparse		Files (in .mat format) can be found for the three smaller systems ([https://sparse.tamu.edu/Oberwolfach/boneS01 BS01], [https://sparse.tamu.edu/Oberwolfach/boneS10 BS10], and [https://sparse.tamu.edu/Oberwolfach/bone010 B010]) on SuiteSparse.

	==Dimensions==		==Dimensions==

Lund: Added links to SuiteSparse

2023-04-25T15:46:10Z

Added links to SuiteSparse

← Older revision		Revision as of 15:46, 25 April 2023
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	Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.		Model reduction for models <tt>BS010</tt> and <tt>BS10</tt> was performed in <ref name="rudnyi2005"/>. The benchmarking of the parallel [http://graal.ens-lyon.fr/MUMPS/ MUMPS] direct solver <ref name="amestoy2006"/> for the stiffness matrices is described in <ref name="rudnyi2006"/>.

			Files (in .mat) format can be found for the three smaller systems ([https://sparse.tamu.edu/Oberwolfach/boneS01 BS01], [https://sparse.tamu.edu/Oberwolfach/boneS10 BS10], and [https://sparse.tamu.edu/Oberwolfach/bone010 B010]) on SuiteSparse

	==Dimensions==		==Dimensions==

Lund: add s to BoneModel-assemble download link

2023-04-24T15:03:29Z

add s to BoneModel-assemble download link

← Older revision		Revision as of 15:03, 24 April 2023
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	The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math> and <math>UZ</math> for the first node.		The output matrix takes the first three components of the state vector, that is, three displacements <math>UX</math>, <math>UY</math> and <math>UZ</math> for the first node.

	The archive [~~http~~://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++\|C++]] to assemble the dynamic system.		The archive [https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-assemble.tar.gz BoneModel-assemble.tar.gz] contains the element mass and stiffness matrices as well as the sample code in [[wikipedia:C++\|C++]] to assemble the dynamic system.
	The code can write the dynamic system in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format or can be used as a hook to transform the global matrices to an appropriate format.		The code can write the dynamic system in the [http://math.nist.gov/MatrixMarket/ Matrix Market] format or can be used as a hook to transform the global matrices to an appropriate format.
	The gzipped compressed data files for element assembly as described above can be downloaded from Table 1.		The gzipped compressed data files for element assembly as described above can be downloaded from Table 1.

Lund: correct link to BS10

2023-04-24T15:02:23Z

correct link to BS10

← Older revision		Revision as of 15:02, 24 April 2023
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	\|File		\|File
	\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e6-BS01.dat.gz BoneModel-dim1e6-BS01.dat.gz] (305.1 kB)		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e6-BS01.dat.gz BoneModel-dim1e6-BS01.dat.gz] (305.1 kB)
	\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-~~B010~~.dat.gz BoneModel-dim1e7-BS10.dat.gz] (2.8 MB)		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-BS10.dat.gz BoneModel-dim1e7-BS10.dat.gz] (2.8 MB)
	\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B010.dat.gz BoneModel-dim1e7-B010.dat.gz] (3.9 MB)		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B010.dat.gz BoneModel-dim1e7-B010.dat.gz] (3.9 MB)
	\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B025.dat.gz BoneModel-dim1e7-B025.dat.gz] (8.6 MB)		\|[https://csc.mpi-magdeburg.mpg.de/mpcsc/MORWIKI/Oberwolfach/BoneModel-dim1e7-B025.dat.gz BoneModel-dim1e7-B025.dat.gz] (8.6 MB)

Lund: change b to B

2022-08-31T13:24:02Z

change b to B

← Older revision		Revision as of 13:24, 31 August 2022
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	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	M \ddot{x}(t) + K x(t) &=& b \\		M \ddot{x}(t) + K x(t) &=& B \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}

Lund: Corrected dimension of B120

2022-06-28T13:51:56Z

Corrected dimension of B120

← Older revision		Revision as of 13:51, 28 June 2022
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	<tt>B025</tt>: <math>N = 2159661</math>,		<tt>B025</tt>: <math>N = 2159661</math>,
	<tt>B050</tt>: <math>N = 4934244</math>,		<tt>B050</tt>: <math>N = 4934244</math>,
	<tt>B120</tt>: <math>N = ~~441785526~~</math>.		<tt>B120</tt>: <math>N = 11969688</math>.

	==Citation==		==Citation==

Himpe: load vector fixes

2022-04-23T16:02:49Z

load vector fixes

← Older revision		Revision as of 16:02, 23 April 2022
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	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	M \ddot{x}(t) + K x(t) &=& B \\		M \ddot{x}(t) + K x(t) &=& b \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}
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	:<math>		:<math>
	\begin{array}{rcl}		\begin{array}{rcl}
	M \ddot{x}(t) + K x(t) &=& B \\		M \ddot{x}(t) + K x(t) &=& b \\
	y(t) &=& Cx(t)		y(t) &=& Cx(t)
	\end{array}		\end{array}
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	<math>M \in \mathbb{R}^{N \times N}</math>,		<math>M \in \mathbb{R}^{N \times N}</math>,
	<math>K \in \mathbb{R}^{N \times N}</math>,		<math>K \in \mathbb{R}^{N \times N}</math>,
	<math>B \in \mathbb{R}^{N \times 1}</math>,		<math>b \in \mathbb{R}^{N \times 1}</math>,
	<math>C \in \mathbb{R}^{3 \times N}</math>.		<math>C \in \mathbb{R}^{3 \times N}</math>.