Evolution of Load-Bearing Structures with Phase Field Modeling

Abstract

We suggest an algorithm to generate the topology of load-bearing structures with help of a phase field model. The objective function homogenizes equivalent stress within the isotropic elastic material. However, local inhomogeneities in the stress field, e.g., at concentrated loads, do not distract the convergence of the algorithm. Beside a certain threshold in the equivalent stress field, the desired filling level of the design space is the main parameter of our objective function. The phase field parameter describes the density and stiffness of the substance in a closed interval. An Allen-Cahn equation regulates the phase transition, which is not conserving the mass of the system. The model evolves continuous regions of voids or dense material, whereas voids retain an infinitesimal residual stiffness, which is a million times smaller than the stiffness of the dense material. The evolution of structures is discussed by numerical examples.