A new strategy using the Proper Generalized Decomposition to model time evolving spatial domains

In this paper, we propose a new approach to adapt the Proper Generalized Decomposition (PGD) to problems containing space domains that are evolving over time. PGD shines with high-order parametrized and nonlinear problems, drastically reducing their computational time. It was proven highly effective in a wide range of problems, but the space domain has always remained fixed within the parametric manifold of interest. In this work, we adapt the PGD to non-constant domains that change over time at given discrete time instances. More specifically, we focus on time evolving space domains and separate the solution along space and time. The space modes are calculated in an expanded space that comprises all the degrees of freedom throughout the simulation. To visualize the solution, the modes are then projected onto the current physical representation. The time modes are solved in a piecewise manner, dividing the time domain into intervals and initializing the time modes to zero at the beginning of each interval. This work is illustrated with an additive manufacturing-inspired example in which the hot boundary elements are sequentially activated to simulate the addition of material. This aligns perfectly with the previously described strategy as it involves an expanding boundary. The impact of the mesh division and the initialization of the new points is discussed.