Triply periodic minimal surface (TPMS) structures have a very good lightweight potential, due to their surface-to-volume ratio, and thus are contents of various applications and research areas, such as tissue engineering, crash structures, or heat exchangers. While TPMS structures with a uniform porosity or a linear gradient have been considered in the literature, this paper focuses on the investigation of the mechanical properties of gyroid structures with non-linear porosity gradients. For the realisation of the different porosity gradients, an algorithm is introduced that allows the porosity to be adjusted by definable functions. A parametric study is performed on the resulting gyroid structures by performing mechanical simulations in the linear deformation regime. The transformation into dimensionless parameters enables material-independent statements, which is possible due to linearity. Thus, the effective elastic behaviour depends only on the structure geometry. As a result, by introducing non-linear gradient functions and varying the density of the structure over the entire volume, specific strengths can be generated in certain areas of interest. A computational design of porosity enables an accelerated application-specific structure development in the field of engineering.

A distributed Lagrange multiplier/fictitious domain method in a phase‐field formulation for the simulation of rigid bodies in incompressible fluid flow is presented. The phase‐field method yields an implicit representation of geometries and thus rigid body particulate flows within arbitrary geometries can be simulated based on a fixed Cartesian grid. Therefore, a phase‐field based collision model is introduced in order to address contact of particles with arbitrary solid structures as boundaries. In addition, grain growth within the boundary geometry can be considered leading to changes in its shape during the simulation. The method is validated on benchmark problems and a convergence study is performed. Multiple numerical experiments are carried out in order to show the methods' capability to simulate problems with differently shaped rigid bodies and particulate flows involving complex boundary geometries like foam structures.

The interaction of immersed rigid bodies with two‐phase flow is of high interest in many applications. A model for the coupling of a Hohenberg–Halperin type model for two‐phase flow and a fictitious domain method for consideration of rigid bodies is introduced leading to a full multiphase‐field method to address the overall problem. A normalized phase variable is used alongside a method for application of wetting boundary conditions over a diffuse fluid‐solid interface. This enables the representation of capillary effects and different wetting behavior based on Young's law. A number of simulations is conducted in order to validate the model and highlight its ability to handle a variety of setups for two‐phase particulate flow. This includes dynamic wetting situations, the motion of multiple particles within the two‐phase flow and the interaction with arbitrarily shaped solid structures inside the domain.