hierarchy.md

Multi-Scale Hierarchical Modeling

Going from thousands of elements (FEA/DEM) to hundred of elements (Beam Theory) to tens of elements (Homogenization and hierarchy).

Example:

3292 Nodes, 11476 edges

830 Nodes, 2111 edges > 75% less nodes and 82% less edges

264 Nodes, 1034 edges > 92% less nodes and 91% less edges

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Hierarchy Implementation

• Automated workflow to extract the elasticity tensor from voxel units

• Detailed explanation here.

• Steps

  • Get Microstructure

  • Apply 6 main harmonic displacements

  • Solve elasticity tensor to get young's modulus and poisson ratio

    S=\frac{1}{E}\begin{vmatrix} 
    1& -\nu & -\nu & 0 & 0 &0\\\ 
    -\nu&  1 & -\nu & 0 & 0 &0\\\ 
    -\nu& -\nu &1 &  0 &0 &0\\\ 
    0&  0 &0 &2(1+\nu)&0 &0\\\ 
    0&  0 &0 &0&2(1+\nu)&  0\\\ 
    0&  0 &0 &0& 0 &2(1+\nu)\end{vmatrix}

  • Use these properties to for coarse simulation

    • Global Properties Using Local Rules
      • Stiffness, poisson ratio and thermal expansion

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Multi-Scale Implementation

1. Best Representation for Multi-scale nodes and edges
   1. add the tag multiscale to setup
   2. nodes
      1. add property scale= 1, 2, 4, 8 ..
   3. edges
      1. add property order (0 to 8) to each source/target (or bake relPos)
         1. 0 is the nodal coordinate: ( 0, 0, 0) i.e. the center
         2. 1 is the nodal coordinate: (-1,-1,-1)
         3. 2 is the nodal coordinate: (-1, 1,-1)
         4. 3 is the nodal coordinate: (-1,-1, 1)
         5. 4 is the nodal coordinate: (-1, 1, 1)
         6. 5 is the nodal coordinate: ( 1,-1,-1)
         7. 6 is the nodal coordinate: ( 1, 1,-1)
         8. 7 is the nodal coordinate: ( 1,-1, 1)
         9. 8 is the nodal coordinate: ( 1, 1, 1)
2. How to turn forces between two nodes into moments (based on this)
   1. transition of center of mass
      1. just add all forces that are applied in updateNodes at the beginning when updating N_intForce and N_intMoment
   2. Rotation
      1. in addition to the moment that is applied (momentPos/momentNeg), add new moment which is the cross product between the relative position and the force
3. How to get edge end position and orientation
   1. orientation is the same as parent node (oVNeg and oVPos)
   2. rotate the nodal coordinates around the center of mass of node
      1. in updateEdges when updating pVNeg and pVPos get position and shifted by the relPos oriented by oVNeg/oVPos (is it as simple as angle normalized * relPos)

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Detailed TODO

• create simple case study
  • one scale 2 node surrounded by 8 scale 1 nodes
  • prescribe/create json in julia/javascript
  • special visualization
    • for bigger scales create 8 smaller cubes that are transparent
  • small force on one of the nodes
  • setup add all stuff written in point 1
• run.jl
  • see what special CUDA arrays have to be added
  • scale to be added to material node
  • multiscale to material node
  • E_relPosSource and E_relPosTarget to be added to edges
  • create special struct for Edges and nodes, or just add properties that do not change like source target or relPos to edge material
• check that when you angle normalized * relPos the position is correct
  • code it (done with quaternions)
• in updateEdges
  • at the beginning update pVNeg and pVPos
  • at the end add moment (cross product) to momentPos or momentNeg
• in updateNodes
  • check N_edgeID to add edges connected
  • add all children forces
  • add all children moments
• Automatic efficient way to switch between both representations
  • generation
  • simulation
  • visualization

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Multi-Scale Hello World

• non axial forces
• new hierarchial representation with nodes having parents and children

10 nodes, 14 edges vs 3 nodes, 2 edges

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Multi-Scale Case Study

	Detailed	Hierarchical	Multi-scale
# Edges	980	127	75
# Nodes	320	62	41
Memory	100%	14%	9%
# Timesteps	50000	1000	1000
Speedup	1x	124x	160x
Accuracy	100%	--	--

dt ∝ particle mass, the smaller the particle the smaller dt, ∴ more timesteps needed

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MADCAT

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Future Steps

• Detailed and hierarchical model comparison
  • Choose interesting case study
    • Madcat (working with Ben)
  • list physical parameters that could be changed to increase accuracy
  • graph computational speedup vs accuracies
• twist a,d anisotropic
• Timescales