A phase field model for stress‐based evolution of load‐bearing structures

Abstract

Summary We present a continuum‐type optimality algorithm for the evolution of load‐bearing solid structures with linear elastic material. The objective of our model is to generate structures with help of a sensitivity function accounting for equivalent stress. Similar to Evolutionary Structural Optimization, a threshold of equivalent stress is evaluated. However, we do not consider a material rejection ratio. The evolution process is governed by an Allen‐Cahn equation in the context of phase field modeling. The steady state of phase transition is the final geometry of the structure. The model accounts for the desired filling level, geometry of the design space, static loadings, and boundary conditions. Our variational approach drops conservation of mass and couples density as well as stiffness of the continuum by a sophisticated function to the phase field variable. Continuous regions of voids and material evolve from the initial state of homogeneously distributed material. The complexity of evolving structures depends on two numerical parameters, which we discuss in several examples.

Phase field model for the martensitic transformation: comparison of the Voigt/Taylor and Khachaturyan approach

“…While the linear and cubic functions satisfy equation ( 33) perfectly, however, the tangent hyperbolicus function does not. If θ = 14 a residual error of 1 − h tanh (φ = 1) = h tanh (φ = 0) ≈ 10 −6 remains [38].…”

Topology Optimization with Isogeometric Analysis and Phase Field Modeling

“…To compare results between standard FEM and IGA, we restrict ourself to linear strain ε = SymGrad[u] as proposed in [5,6]. However, we refer to [7,8], if Green or Cosserat strain is of interest.…”

Martensitic transformation at a crack under mode I and II loading

“…However, we find out, that in some cases where singularities occur, the order parameter φ can exceed its valid range. We suggest therefore another interpolation function such as in [2] with the properties s(φ ≤ 0) ≈ 0 and s(φ ≥ 1) ≈ 1. Compared to a linear interpolation exceeding has no effect on the energy landscape, which in turn means the order parameter will stay in its valid range.…”