This paper focuses on topology optimization for additive manufacturing, where the minimum length scale of designs must be greater than the nozzle's length scale. We present a general framework for controlling the minimum length scale in topology optimization using the phase field method, which is applicable to mechanical, heat transfer, and thermal-mechanical coupled topology optimization. The proposed method introduces an energy functional as the objective function of topology optimization, which includes the surface free energy, strain energy, volume constraints, minimum length scale constraints, and necessary boundary terms to satisfy the equilibrium condition of elasticity. A newly defined convolution operation $ {\mathcal B}_r $ (where $ r $ is the radius related to the length scale) is combined with established skeleton extraction techniques to achieve control over the minimum length scale. At the same time, the skeletons of new designs are preserved in a way that is consistent with those obtained from designs without size control. To find the minimum of the constructed energy functional, the augmented Lagrangian method is first used to derive the extremum functional. Then the stationary point of the extremum functional is transformed into the solution of a time-dependent Allen-Cahn type equation, which effectively converts the topology optimization problem into solving partial differential equations. An operator-splitting-based hybrid numerical method is proposed to solve the Allen-Cahn type equation. The numerical results of mechanical, heat transfer, and thermal-mechanical coupled topology optimizations are presented to demonstrate the efficiency and scalability of the proposed minimum length scale control framework.
Citation: Qian Yu, Yuyan Liu, Mingyuan Yang. Minimum length scale control in thermal-mechanical coupled topology optimization based on phase field method[J]. AIMS Mathematics, 2025, 10(7): 16720-16743. doi: 10.3934/math.2025750
This paper focuses on topology optimization for additive manufacturing, where the minimum length scale of designs must be greater than the nozzle's length scale. We present a general framework for controlling the minimum length scale in topology optimization using the phase field method, which is applicable to mechanical, heat transfer, and thermal-mechanical coupled topology optimization. The proposed method introduces an energy functional as the objective function of topology optimization, which includes the surface free energy, strain energy, volume constraints, minimum length scale constraints, and necessary boundary terms to satisfy the equilibrium condition of elasticity. A newly defined convolution operation $ {\mathcal B}_r $ (where $ r $ is the radius related to the length scale) is combined with established skeleton extraction techniques to achieve control over the minimum length scale. At the same time, the skeletons of new designs are preserved in a way that is consistent with those obtained from designs without size control. To find the minimum of the constructed energy functional, the augmented Lagrangian method is first used to derive the extremum functional. Then the stationary point of the extremum functional is transformed into the solution of a time-dependent Allen-Cahn type equation, which effectively converts the topology optimization problem into solving partial differential equations. An operator-splitting-based hybrid numerical method is proposed to solve the Allen-Cahn type equation. The numerical results of mechanical, heat transfer, and thermal-mechanical coupled topology optimizations are presented to demonstrate the efficiency and scalability of the proposed minimum length scale control framework.
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Qian Yu, Yuyan Liu, Mingyuan Yang. Minimum length scale control in thermal-mechanical coupled topology optimization based on phase field method[J]. AIMS Mathematics, 2025, 10(7): 16720-16743. doi: 10.3934/math.2025750
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