dynamic_validation.md

Validation of MetaVoxels Dynamics

Here's a running notebook of my work to perform validation of the dynamic characteristics of the MetaVoxels beam solver. The key questions are:

1. Are we calculating the mass matrix correctly?
   • The issue here probably crops up from the implementation of voxcad routines where nodes are the "1st class" object and beams are the abstraction, vs. a true slender beam solver where the nodes are an abstraction and the beams are physical
   • Lumped vs distributed approximations?
     • Probably have to excavate from Frame3DD :-(
   • Are we ignoring important couplings?
2. How are we calculating our various forms of damping?
   • There is both local damping of bonds and global damping of motion to calculate and validate
3. Accuracy of the solution over time
   • Current integrator is a symplectic euler scheme, should preserve total energy (in absence of damping)
   • Can we reframe the code to leverage DifferentialEquations.jl library?

The proposed model system

As a first pass for validation let's use a cantilevered beam made of a line of voxels. Our one hard constraint on beam parameters is that the length/height or width ratio be > 10, so lets use 100. Properties of the beam are as follows:

• L = 1 m
• w = h = 0.01 m
• A = 0.001 m^2
• Iyy, Izz = 0.01^4/12 m^4
• Izz = 0.01^4/6 m^4
• E = 6.9e10 N/m^2
• nu = 0.33
• rho = 2700 kg/m^3
• m = 2.7 kg/m

We'll discretize the beam into n segments from 5 to 10 to 20. To test the beam dynamics we'll apply a series of initial unit impulses to the beam tip using the externalforces! method. The impulses will be a transverse impulse to excite beam bending, an axial impulse to excite the axial modes, and a torsional impulse about the x-axis.

How will we do the comparison?

There are analytical solutions to these beam loadings which we can compare to, but we must decide where we want to spend our effort. One option is to calculate the exact time solution to these unit impulsive loadings and compare pointwise through time. The other option, which may be quicker, is to extract the natural frequencies and mode shapes from the metavoxels response and comparing to analytical frequencies and mode shapes. Because metavoxels is matrix-free we can't solve the eigenvalue problem directly.

Predicted natural frequencies

Bending : wn = an^2 * sqrt(EI/(mL^4)) : an^2 = 3.516, 22.035, 61.697 ... Axial : wn = n*pi/2 * sqrt(EA/(mL^2)) Torsional : wn = n*pi/2 * sqrt(GJ/(IzzL^2))

Measurements

1st Bending

n	Exact	Solidworks Beam	MetaVoxels
5	8.17 Hz	8.02 Hz	10.6 Hz
10	-	8.13 Hz	8.52 Hz
20	-	8.16 Hz	6.37 Hz

2nd Bending

n	Exact	Solidworks Beam	MetaVoxels
5	51.2 Hz	48.1 Hz	68.9 Hz
10	-	50.4 Hz	52.4 Hz
20	-	51.0 Hz	38.6 Hz

3rd Bending

n	Exact	Solidworks Beam	MetaVoxels
5	143 Hz	130 Hz	194 Hz
10	-	140 Hz	147 Hz
20	-	142 Hz	109 Hz

1st Axial

n	Exact	Solidworks Beam	MetaVoxels
5	1264 Hz	1259 Hz
10	-	1263 Hz
20	-	1264 Hz

What's Wrong?

The issue is how the lumped mass is being calculated. Solidworks is a good reference for us because it uses a lumped model (indicated by frequencies converging from below). Metavoxels is failing because the lumped masses are based solely on nodal cross-section properties and not the edges which comprise the actual structure. This introduces 2 errors:

1. Lumped mass doesn't change with number of elements (length of edges) in a beam

The code for calculating the lumped mass in run.jl is :

nomSize = round(sqrt(node["material"]["area"]);digits=10)
nomSize = nomSize * 2.0
mass = round(nomSize * nomSize * nomSize * rho;digits=10)

This explains why adding nodes decreases the natural frequencies so much, the total mass of the beam is increasing as we add nodes.

2. The nodes at beam ends are not treated appropriately

The first and last element of the beam should have half the lumped mass as the interior elements because they have only one edge attached instead of two. MetaVoxels doesn't do this currently. We can confirm that with the code and by comparing our MetaVoxels natural frequencies with the expected values. For n=5 elements the metavoxels lumped mass was 0.0216 kg/node while the correct lumped mass was 0.054 kg/interior node. The natural frequency should be proportional to mass^(1/2) and we would expect the metavoxels frequencies to be ~1.6x higher than theory but due to the end mass they were only ~1.3x higher.