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#### Keywords

- Eulerian-Lagrangian formulation (1)
- FEM (1)
- Lagrange formalism (1)
- Mesh-less methods (1)
- Order of printed copy (1)
- film casting process (1)
- flexible bodies (1)
- free surface Stokes flow (1)
- nonlinear model reduction (1)
- optimal control (1)

#### Faculty / Organisational entity

- Fraunhofer (ITWM) (3) (remove)

An easy numerical handling of time-dependent problems with complicated geometries, free moving boundaries and interfaces, or oscillating solutions is of great importance for many applications, e.g., in fluid dynamics (free surface and multiphase flows, fluid-structure interactions [22, 18, 24]), failure mechanics (crack growth and propagation [4]), magnetohydrodynamics (accretion disks, jets and cloud simulation [6]), biophysics and -chemistry. Appropriate discretizations, so-called mesh-less methods, have been developed during the last decades to meet these challenging demands and to relieve the burden of remeshing and successive mesh generation being faced by the conventional mesh-based methods, [16, 10, 3]. The prearranged mesh is an artificial constraint to ensure compatibility of the mesh-based interpolant schemes, that often conflicts with the real physical conditions of the continuum model. Then, remeshing becomes inevitable, which is not only extremely time- and storage consuming but also the source for numerical errors and hence the gradual loss of computational accuracy. Apart from this advantage, mesh-less methods also lead to fundamentally better approximations regarding aspects, such as smoothness, nonlocal interpolation character, flexible connectivity, refinement and enrichment procedures, [16]. The common idea of mesh-less methods is the discretization of the domain of interest by a finite set of independent, randomly distributed particles moving with a characteristic velocity of the problem. Location and distribution of the particles then account for the time-dependent description of the geometry, data and solution. Thereby, the global solution is linearly superposed from the local information carried by the particles. In classical particle methods [20, 21], the respective weight functions are Dirac distributions which yield solutions in a distributional sense.

This contribution presents a model reduction method for nonlinear problems in structural mechanics. Emanating from a Finite Element model of the structure, a subspace and a lookup table are generated which do not require a linearisation of the equations. The method is applied to a model created with commercial FEM software. In this case, the terms describing geometrical and material nonlinearities are not explicitly known.

This work deals with the optimal control of a free surface Stokes flow which responds to an applied outer pressure. Typical applications are fiber spinning or thin film manufacturing. We present and discuss two adjoint-based optimization approaches that differ in the treatment of the free boundary as either state or control variable. In both cases the free boundary is modeled as the graph of a function. The PDE-constrained optimization problems are numerically solved by the BFGS method, where the gradient of the reduced cost function is expressed in terms of adjoint variables. Numerical results for both strategies are finally compared with respect to accuracy and efficiency.