Mixed-Topology TPMS

Mixed-topology surfaces [JM24] combine multiple implicit functions by taking weighted sums of the functions. By controlling the weights, the resulting surface can be varied continuously between the set of surfaces. Depending on the set of surfaces, this can constitute a low-parameter, topology-varying design space that can be used for design optimization (as in [JM24]).

Triply periodic minimal surfaces (TPMS) are particularly interesting subjects for a mixed-topology approach because the resultant surface will remain triply periodic (though not minimal) as long as the periodicities are compatible.

Such surfaces can be easily generated in MyMesh with the implicit module. Several TPMSs are available as built-in functions in implicit including gyroid(), primitive() (Schwarz P), and diamond() (Schwarz D) [1].

from mymesh import implicit
import numpy as np

functions = [implicit.primitive, implicit.gyroid]
bounds = [0,1,0,1,0,1]
h = 0.04 # element size

weights1 = [0.25, 1]
mixed_topology1 = lambda x,y,z : np.sum([w*f(x,y,z) for w,f in zip(weights1, functions)], axis=0)
M1 = implicit.TetMesh(implicit.thickenf(mixed_topology1,1), bounds, h)
M1.plot(bgcolor='white')

weights2 = [0.5, 0.8]
mixed_topology2 = lambda x,y,z : np.sum([w*f(x,y,z) for w,f in zip(weights2, functions)], axis=0)
M2 = implicit.TetMesh(implicit.thickenf(mixed_topology2,1), bounds, h)
M2.plot(bgcolor='white')

weights3 = [0.8, 0.5]
mixed_topology3 = lambda x,y,z : np.sum([w*f(x,y,z) for w,f in zip(weights3, functions)], axis=0)
M3 = implicit.TetMesh(implicit.thickenf(mixed_topology3,1), bounds, h)
M3.plot(bgcolor='white')

weights4 = [1, 0.25]
mixed_topology4 = lambda x,y,z : np.sum([w*f(x,y,z) for w,f in zip(weights4, functions)], axis=0)
M4 = implicit.TetMesh(implicit.thickenf(mixed_topology4,1), bounds, h)
M4.plot(bgcolor='white')

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RFBOutputContext()
RFBOutputContext()
RFBOutputContext()
RFBOutputContext()

[1]

To be specific, the implicit function TPMS representations are Fourier series approximations (also known as periodic nodal surfaces [vSN91]) and are not truly minimal surfaces (though they are truly triply periodic). This can be seen by measuring the surface curvature (curvature) and observing that the mean curvature is not zero at all points. Nevertheless, these are reasonably accurate and commonly used approximations.