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Periodic domain patterns in tetragonal ferroelectrics are explored using a phase field model calibrated for barium titanate. In this context, we discuss the standard periodic boundary condition and introduce the concept of reverse periodic boundary conditions. Both concepts allow the assembly of cubic cells in accordance with mechanical and electrical conditions. However, application of the reverse periodic boundary condition is due to an increased size of the RVE and enforces more complex structures compared to the standard condition. This may be of particular interest for other multiphysics simulations. Additionally, we formulate mechanical side conditions with minimal spherical (hydrostatic) stress, or conditions with controlled average strain. It is found that in sufficiently small periodic cells, only a uniform single domain, or the simplest stripe domains constitute equilibrium states. However, once the periodic cells are of order 20 domain wall widths in size, more complex, 3-dimensional patterns emerge. Some of these patterns are known from prior studies, but we also identify other domain patterns with long, ribbon-like domains threaded through them and some vortex-like structures.

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.

A sharp-interface model based on the linear constrained theory of laminates identifies eight distinct rank-2 periodic patterns in tetragonal ferroelectrics. While some of the periodic solutions, such as the herringbone and stripe patterns are commonly observed, others such as the checkerboard pattern consisting of repeating polarization vortices are rarely seen in experiments. The linear constrained theory predicts compatible domain arrangements, but neglects gradient effects at domain walls and misfit stresses due to junctions of domains. Here, we employ a phase-field model to test the stability of the periodic domain patterns with in-plane polarizations, under periodic boundary conditions which impose zero average stress and electric field. The results indicate that domain patterns containing strong disclinations are of high energy and typically unstable in the absence of external stresses or electric fields. The study also provides insight into the internal stresses developed in the various domain patterns.

Comprehensive optimization of truss and frame structures is intended to optimize all aspects of a structure: the topology, shape and size of members. However, simultaneous optimization of all mentioned aspects increases the complexity of the problem and leads to high demands for the used optimization algorithm. We propose to decompose the optimization task and to use specialized algorithms for each stage of the process. First, a phase field model evolves the topology as basic design. Then, we use an interface to set up a simplified beam model, where connection topologies and nodal positions as well as cross sections are simultaneous optimization parameters of the Evolutionary Algorithm.