Multiscale Modeling

1
Multiscale Modeling

• Atomistic mechanics critical for understanding
  fundamental processes, particularly process
  initiation
• Surface roughening
• Columnar growth
• Vacancy and impurity trapping
• Island prediction
• Etc.
• Continuum mechanics needed for solving problems
  when the number of discrete entities becomes too
  large
• Prediction of topology and composition
• Prediction of spatial variation of physical and
  electrical properties
• Complete simulation capability mustinclude fully
  linked atomistic andcontinuum modeling

Scalebridging
2
Domain Representations

• Discrete Models
• Atoms are placed on lattice
• Atoms properties
• Interactions
• Material information
• Orientation
• Numerical methods solve interaction equations
  using stochastic methods

3
Continuum Domain Model Definition

• Non-manifold boundary representation
• Set of topological types and adjacencies
• Shape information associated with topological
  types
• Adjacencies relationships between topological
  types
• Provide abstraction for defining domain
• Allow association of analysis attributes
• Supports needs of domain discretization
  techniques
• Classification relationship between meshand
  geometric domain
• Functional interface to non-manifold modeler
  allows tointerface with multiple commercial
  geometric modelers

4
Create Discrete Domain to Continuum Domain

• Given the discrete model perform scale bridging
  in two steps
• Create continuum model
• Create continuum discretization
• Continuum model must be
• Complete and properly defined
• Reflecting the discrete model structure
  accurately(depending on application, but in
  general)
• Grain structure
• Atomistic properties (e.g. material)
• Provide means to transfer data back to discrete
  model
• Continuum discretization
• Based on continuum model
• Granularity independent of atomic scale

5
Create Discrete Domain from Continuum Domain

• Initial Domain is defined through discrete
  objects (voxel data).
• Options for transfer
• Procedure working directly with discrete data
• For instance
• Coarsen atomistic data
• Construct continuum discretization of coarsened
  atomistic data
• Procedure working with geometric model
• Construct boundary representation
• Construct continuum discretization
• Second option allows easier control of geometric
  constraints
• E.g. control volume/surface of grains duringmesh
  coarsening
• Prevent construction of invalid meshes (e.g.all
  nodes of tet classified on boundary for Level
  Sets)
• Provides abstraction to combine models from
  differentsources

6
Transfer Discrete Domain to Continuum Domain

• Currently developed/under development
• Construction of a 3D substrate grain structure
  from 2D data
• Construction of a 3D substrate grain structure
  from 3D voxel data
• Construction of island depositions on 2D/3D
  grain structures

7
Construction of grain structure from voxel data

• Discrete model
• Defined in terms of 3D voxel data
• Each voxel has a material association(grain
  association)

Voxels representing shape

• Geometric model
• Defined in terms of
• Topology
• Discrete geometry
• Keeps material association ofvoxel data

8
Construction of grain structure from voxel data

• Step 1 Create voxel quads to separate grains
• Loop over pairs of atoms
• If atoms are oriented differently -gt create quad
• Classify quad as interior
• Loop over boundary atoms
• Create boundary quads
• Classify quad as boundary

9
Construction of grain structure from voxel data

• Step 2 Create voxel edges from voxel quads to
  separate grains
• Loop over all edges of voxel quads
• More than two faces connected -gt create mesh edge
• Postprocess to eliminate invalid(dangling) edges

Invalid edge Separating areaof same material
10
Construction of grain structure from voxel data

• Step 3 Create model vertices
• Loop over all voxel edges.
• If more than two edges connected to any of its
  vertices -gt create model vertex
• Create classification of voxel vertices on model
  vertices

voxel vertices
Model vertex
11
Construction of grain structure from voxel data

• Step 4 Create model edges
• For each model vertex walk along voxel edges up
  to next model vertex.
• Create model edge topology along path.
• Create classification of mesh edges and mesh
  vertices on model edges.

Classified on
Model edge topology
12
Construction of grain structure from voxel data

• Step 5 Create model faces
• For each voxel face connected to a model vertex
• Follow edge until start quad is found again
  (circle is closed)
• Process interior voxel faces, edges and vertices
  and classify on created model face
• Processed model edges form model face

13
Construction of grain structure from voxel data

• Step 6 Create model regions
• For each model vertex
• Scan through atoms connected to it with same
  grain association
• Collect all model faces that have been visited
• Processed model faces form shell of model region

14
Construction of island models and meshes

• Substrate is given by proceduredescribed before
• Island structure computed using e.gMonte Carlo
  methods. Islands described as x,y,z coordinates
  of an atom
• z0 is island boundary
• zgt0 is island interior
• Island boundary is coarsened by eliminating
  small features
• Interior points are checked to verify they are
  interior to new boundary
• Boundary coarsening looses volume,I.e.interior
  nodes can become exterior
• Construct boundary representation of islands
• Allows to combine islands and substratebased on
  operators for non-manifold models

Nodes arecollapsed onneighbor nodeif d lt eps
d
d
d
15
Intersect island topology with substrate

• Combine islands models and substrate model
• Need to create valid geometric model
• Compute intersection vertices between island
  boundary edges and substrate model edges
• Split substrate edges and island edges
• Split substrate faces
• Procedures need to perform book keeping
• Identify the split faces that form the bottom of
  the island volume to be built
• Update island boundaries with split edges to
  form the island top face
• Keep track of face normal to be able to
  construct island volumes

16
Construct shape for island model

• Shape of model is described by triangulation of
  model faces
• Grain structure is being triangulated using an
  algorithm to triangulate non convex polygons
• No interior nodes in the model faces are created
• Islands are triangulated using Delaunay
  triangulization
• Each atom of the island is a node of the
  triangulization
• Need to represent height information

17
Construct Finite Element Mesh

• Construct Surface Mesh
• Input to mesh generator
• Surface triangulization
• Mesh controll attributes
• Output Valid Finite Element surface mesh
  consisting of triangles
• Construct Volume Mesh
• Input to volume mesher
• Surface mesh
• Constraints
• No interior faces with all nodes on the boundary
• No tetrahedral elements with all nodes on the
  boundary
• Output Valid Finite Element volume mesh to solve
  Level Set problem

18
Conclusions

• Atomistic and Continuum mechanics supplement each
  other
• Multiscale modeling has the potential to connect
  the scales
• For scale bridging it is important to have a
  clear understanding of domains
• 3D Voxel Data for atomistic simulations
• Non manifold boundary representation for
  continuum simulations
• Tools have been constructed to create non
  manifold boundary models from 3D voxel data.