Phase-field Approaches to Structural Topology Optimization

Abstract

Abstract. The mean compliance minimization in structural topology optimization is solved with the help of a phase field approach. Two steepest descent approaches based on L 2 -and H −1 -gradient flow dynamics are discussed. The resulting flows are given by Allen-Cahn and Cahn-Hilliard type dynamics coupled to a linear elasticity system. We finally compare numerical results obtained from the two different approaches. Mathematics Subject Classification (2000). 74P15, 74P05, 74S03, 35K99.

Sharp Interface Limit for a Phase Field Model in Structural Optimization

“…As was already discussed in section 2.1, the Ginzburg-Landau energy E ε , compare (14), appearing in the objective functional is essential for the existence of a minimizer and (E ε ) ε Γ-converge to a multiple of the perimeter functional as ε tends to zero; cf. [35,36].…”

Allen-Cahn and Cahn-Hilliard Variational Inequalities Solved with Optimization Techniques

“…There the mean compliance penalized with the Ginzburg-Landau energy E (1.1) has to be minimized. The gradient approach can be seen as a pseudo time stepping approach and results in a time discretized Allen-Cahn variational inequality coupled with elasticity and mass constraints, which can be solved with the above method (see [4,5,2]). …”

“…ψ(u) = (1 − u 2 ) 2 or an obstacle potential, e.g. 2) where ψ 0 = 1 2 (1 − u 2 ) or another smooth, non-convex function and I [−1,1] is the indicator function, for the interval [−1, 1]. The interface evolution is then given by the gradient flow equation, i.e.…”

Topology optimization methods for guided flow – comparison of optimality criteria vs. phase‐field method

“…Different optimization methods are shown bei Klimetzek [1], Hinterberger [2] and Pingen [3]. In recent years the phase-field method has been shown to be an applicable method for different kinds of topology optimization [4,5]. We present results of topology optimization methods with optimality criterion and by using a phase-field model in the area of guided fluid flow problems.…”

“…The first method is based on local optimality criterion, preventing the backflow in the flow domain [1,6,7]. The second method is based on a phase field model, which describes a minimization problem and uses a specially constructed driving force to minimize the total energy of the system [4,5]. We investigate the capabilities and limits of both methods and present examples of different resulting geometries.…”