Table 3 Level-set equations of selected TPMS and related surfaces

From: Machine learning-based mechanics of TPMS architected materials driven by unit-cell geometric features

Design

Level-set Equation φ(x1, x2, x3) = t

GY

\(\varphi ({x}_{1},{x}_{2},{x}_{3}):\sin {x}_{1}\cos {x}_{2}+\sin {x}_{2}\cos {x}_{3}+\sin {x}_{3}\cos {x}_{1}=t\)

IWP

\(\varphi ({x}_{1},{x}_{2},{x}_{3}):2(\cos {x}_{1}\cos {x}_{2}+\cos {x}_{2}\cos {x}_{3}+\cos {x}_{3}\cos {x}_{1})-(\cos 2{x}_{1}+\cos 2{x}_{2}+\cos 2{x}_{3})=t\)

SPC

\(\varphi ({x}_{1},{x}_{2},{x}_{3}):\cos {x}_{1}+\cos {x}_{2}+\cos {x}_{3}=t\)

FKS

\(\varphi ({x}_{1},{x}_{2},{x}_{3}):(\cos {x}_{1}\cos {x}_{2}+\cos {x}_{2}\cos {x}_{3}+\cos {x}_{3}\cos {x}_{1})-(\cos 2{x}_{1}+\cos 2{x}_{2}+\cos 2{x}_{3})=t\)

SCD

\(\varphi ({x}_{1},{x}_{2},{x}_{3}):\cos {x}_{1}\cos {x}_{2}\cos {x}_{3}-\sin {x}_{1}\sin {x}_{2}\sin {x}_{3}=t\)

LD

\(\begin{array}{ll}\varphi ({x}_{1},{x}_{2},{x}_{3}): & \sin 2{x}_{1}\sin {x}_{3}\cos {x}_{2}+\sin 2{x}_{2}\sin {x}_{1}\cos {x}_{3}+\sin 2{x}_{3}\sin {x}_{2}\cos {x}_{1}-\cos 2{x}_{1}\cos 2{x}_{2}\\ & +\cos 2{x}_{2}\cos 2{x}_{3}+\cos 2{x}_{3}\cos 2{x}_{1}+0.3=t\end{array}\)

SLP

\(\begin{array}{ll}\varphi ({x}_{1},{x}_{2},{x}_{3}): & 1.1\sin 2{x}_{1}\sin {x}_{3}\cos {x}_{2}+\sin 2{x}_{2}\sin {x}_{1}\cos {x}_{3}+\sin 2{x}_{3}\sin {x}_{2}\cos {x}_{1}\\ & -0.2\cos 2{x}_{1}\cos 2{x}_{2}+\cos 2{x}_{2}\cos 2{x}_{3}+\cos 2{x}_{3}\cos 2{x}_{1}\\ & -0.4\cos {x}_{1}+\cos {x}_{2}+\cos {x}_{3}=t\end{array}\)

I2Y

\(\begin{array}{ll}\varphi ({x}_{1},{x}_{2},{x}_{3}): & -2\sin 2{x}_{1}\cos {x}_{2}\sin {x}_{3}+\sin {x}_{1}\sin 2{x}_{2}\cos {x}_{3}+\cos {x}_{1}\sin {x}_{2}\sin 2z\\ & +\cos 2{x}_{1}\cos 2{x}_{2}+\cos 2{x}_{2}\cos 2{x}_{3}+\cos 2{x}_{1}\cos 2{x}_{3}=t\end{array}\)

FRD

\(\varphi ({x}_{1},{x}_{2},{x}_{3}):4\cos {x}_{1}\cos {x}_{2}\cos {x}_{3}-(\cos 2{x}_{1}\cos 2{x}_{2}+\cos 2{x}_{2}\cos 2{x}_{3}+\cos 2{x}_{3}\cos 2{x}_{1})=t\)

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