Kinematic Mapping

To transition between kinetic states, we derive a mapping procedure similar to that used by Noid et al. in their development of a generalised CG procedure for atomic systems [102]. Given two equilibrium structures X and Y with P and Q nodes respec- tively, we need to be able to construct the structural topology of Y using the details of structure X. In other words, we need to generate a Q node target FFEA structure (Y ) from a P node base structure (X). We may write the positions of the target nodes, ~y, as a function of the positions of the base nodes, ~x,

Chapter 4. Kinetic Scheme 104

Figure 4.9: The process by which we generate a coordinate mapping between two kinetic states. a) The two different kinetic states we want to simulate. Both have different equilibrium structures as shown, but also differing material parameters. b) The set up for the simulation required to generate the map. We firstly overlay the structures, and then attach linear restraints between user defined ‘equivalent posi- tions’. All of the corners and the center points of each edge are attached here. c) The simulation with linear restraints included. We can see that the restraints are strong enough to overcome the elasticity of the continuum, forcing them to overlap to the

global energy minimum subject to these configurational constraints.

where Mαβ is a Q × P linear mapping operator which we must construct. Notice

that the matrix operates on the nodes indices, indexed by α and β, and not to each individual directional component, indexed by k i.e. the same mapping is performed on each of the Cartesian directions. Figure 4.9a provides an example of two possible structural states, the cube and the parallelohedron.

For a stable FFEA simulation, the only constraint on M is that following its operation on X, structure Y must contain no inverted elements. Any form of M which exhibits this property will allow a kinetic state change to occur within an FFEA simulation with no disruption. However, the total set of possible mappings include center of mass translations, rotations and volumetric and shear structural deformations. The FFEA kinetic framework was designed to include the effect of how the dynamics change due to their energetic landscape, with kinetic transitions acting as instantaneous changes in the energy environment. The application of these types of structural maps constitute instantaneous changes in both energy and structure, something the kinetic framework was not specifically designed for. In the dumbbell model (Section 4.4.2), a kinetic transition involved the material parameters changing instantaneously whilst the spring extension was kept constant.

As kinetic transitions occur instantaneously within the FFEA framework as well, we must consider some form of structural alignment between the base and target structure

Chapter 4. Kinetic Scheme 105

before constructing the mapping matrix.

4.5.3.1 Deformable Structure Alignment

Making the mapping free from translations and rotations is relatively straightforward. A number of algorithms are available for point cloud alignment; alignment of rigid sets of nodes with no topological information. This type of alignment is shown in Figure 4.9b. One choice of mapping would be to define a structural map directly between these two equilibrium structures. This would allow us to transition directly between the initial and final equilibrium states of the process, but misses out the mechanical process of the transition itself. We call this type of structural transition ‘equilibrium mapping’. However, as is clear for the hexahedral structures in Figure4.9, FFEA objects are inherently deformable, and so we can further align the two structures by volumetric and shear deformations.

For two mechanically deformable structures, with different equilibrium structures but representing the same object, we would expect there to be a continuous mechanical transformation that would change one structure into the other composed of transla- tions, rotations and shear and volumetric deformation. We can use the dynamic part of FFEA to take advantage of this and align the two structures using an energy min- imisation procedure. The ideal case would be to define an energy penalty for the two structures being out of alignment, and run a dynamic FFEA simulation in the absence of noise and simply allow the two structures to move into alignment. However, how two structures ‘should’ align is a non-trivial problem. Volume overlap is an insufficient metric to define overlap as can be seen with our hexahedral test objects in Figure4.9b, where we notice that there are a number of rotational symmetries which would have the exact same volume overlap. For more complex structures, a volume overlap algo- rithm may instead get caught in a local overlap minimum and converge to a completely incorrect alignment.

Instead, we consider that it is more often the case that a user will have some idea of certain points in the structure in one state corresponding to given points of the structure in the second state (e.g. the location of a binding domain might be obvious in both structures, and so the mapping we use should make these domains overlap). We have designed a semi-automatic method of alignment and mapping for the FFEA toolkit which introduces linear restraints between user-defined ‘equivalent points’. This gives us an energy penalty function and enables an FFEA simulation, in the absence of steric interactions and thermal noise, to relax and overlap to a joint minimum energy configuration based on their shape. In Figure4.9b, we have added a number of linear

Chapter 4. Kinetic Scheme 106

(a) Example of the equilibrium mapping pro-

cedure (b) Example of the full mapping procedure Figure 4.10: The idealised case of the equilibrium and full mapping procedures for a single degree of freedom undergoing a kinetic transition from mesostate 1 to mesostate 2. For the equilibrium mapping protocol, we see that there should be no mechanical energy change affecting the mapping. In the full mapping case however, the size of

the mechanical energy change depends upon the position in phase space.

restraints between the equivalent corners and sides of the cube and parallelohedron, which are visualised as springs. In Figure 4.9c, we then perform consecutive FFEA simulations with further restraints introduced between each run, giving us an adaptive procedure to generate a user-defined overlap. In principle one could continue to add springs to this to increase the degree of overlap. Alternatively, once the two states have been adequately aligned, an additional overlap energy function could be added, though this has not been implemented within the framework.

We can then define a structural map between the two deformed structures following the energy minimisation procedure. This type of map would usually introduce a significant amount of energy to the system as the state transition occurs, following which the new state would relax into its equilibrium state through the dynamic part of the simulation. This is the method we used in Section4.4, which we will refer to as ‘full mapping’. We show the effect of both types of map in Figure4.10.

4.5.3.2 Building the Mapping Matrix

Now we have two suitably overlapping structures (either for equilibrium or full map- ping), we can begin to build the mapping matrix. However, with P possible base nodes to choose from to determine the position of any target node, there is no unique structure for Mαβ. In the case that Q > P , Equation 4.47 represents an overdeter-

mined system of equations with the likely scenario of no solutions for a specific matrix M. For Q < P , Equation 4.47 is instead under-determined with the likely case being

Chapter 4. Kinetic Scheme 107

infinite possible solutions. However, we can exploit the constant topology of the target FFEA structure to build M with a unique mapping between any two structures with the same underlying topologies. We accomplish this by choosing to make each target node position a function of only the nodes making up the base element that contains it,

~

yα= mαγ~xγ, (4.48)

where mαγ is an Q × 4 sub-matrix of Mαβ and ~xγ are the base nodes forming the

containing element. Hence, 0 < γ < 4. If the two structures are such that there are some nodes with no containing element, Equation 4.48 still applies but becomes an extrapolation from the coordinates of the nearest element rather than an interpolation from the containing element. The initial alignment procedure minimises the errors introduced by extrapolation by aligning the structures based upon their shape and local deformability.

We now wish to specify the position yα of node α that is within (or close to) an element

of structure X using the coordinates of the containing element, {~x0, ~x1, ~x2, ~x3}. As we

require the mapping to be translationally invariant, we can disregard one additional degree of freedom and use the local coordinates of the element to determine a unique value of yα with respect to a local origin, the node ~x0,

~ yα= ~x0+ 3 X δ=1 cαδ(~xδ− ~x0) . (4.49)

The vectors ~xδ− ~x0are the three edge vectors joined to the origin node. Three linearly

independent vectors are sufficient to uniquely determine the position of any point in 3D space, and so expanding the vectors into their directional components, k, gives,

yαk− x0k = 3 X δ=1 cαδ(xδk− x0k) , (4.50) = 3 X δ=1 cαδXδk. (4.51)

Xδk is a 3 × 3 matrix, which can be quickly inverted to find the mapping pre-factors,

cαδ, for this particular target node α,

cαδ= (yαk− x0k) X_kδ−1, (4.52)

where the inverse X_kδ−1 must exist due to the linear independence of the element edge vectors. Using the values for cαδ, we can convert back into the global coordinates

Chapter 4. Kinetic Scheme 108 mαγ=        1 − 3 X δ=0 cαδ γ = 0 cα(γ−1) 1 ≤ γ ≤ 3

mαγ contains the non-zero components of a single row of the overall mapping matrix

Mαβ, and so a computational loop over all target nodes will fully populate the global

mapping matrix and give us an interpolative, and therefore stable, mapping matrix. As the global matrix is not square, we cannot simply calculate the inverse to find the equivalent mapping from structure Y to structure X. Therefore, we begin from the aligned state of the two structures and perform the same procedure a second time to build the reverse map. The non-square nature of the mapping matrix presents certain problems in regards to energetics and equipartition which we will discuss in the following section. As a final point, if we take the derivative of Equation 4.47with respect to time, where M is constant, we find that the mapping matrix is suitable for the velocity components as well.