Conference Agenda

The present study concerns the laminar mixed convection flow over a heated or cooled horizontal plate of finite length at a zero angle of attack and a small Richardson number. The plate is located either in a channel or in a semi-infinite space behind a flow straightener. In the limit of a small Prandtl number, these conditions correspond to the boundary-layer solutions of Müllner and Schneider (2010), and Schneider (2000), respectively.

The hydrostatic pressure difference between the plate's lower and upper sides and the Kutta condition at the trailing edge induce a circulation with a global effect on the flow around the plate. In contrast to the classical aerodynamics problem of an isothermal flow around an inclined plate, the thermal wake also contributes to the circulation in the outer flow. This circulation can lead to flow separation at the bottom side of a heated plate (or an upper side of a cooled plate) when the Richardson number exceeds a certain threshold, depending on the Reynolds and Prandtl numbers.

The steady two-dimensional solution of the governing equations under the Boussinesq approximation is computed with the Finite Element solver FEniCS. Goal-oriented adaptive mesh refinement is employed in order to resolve both the viscous and the thermal boundary layers.

In the talk, the numerical solution will be compared to the boundary-layer solutions. The effect of the governing parameters on the flow will be investigated, also beyond the range of validity of the boundary-layer solutions. For a plate inside a channel, the flow separates close to the leading edge even for relatively low values of the Richardson number. The threshold Richardson number for separation decreases with increasing Reynolds number. For certain parameters, multiple steady two-dimensional solutions come into existence, differing by the size of the separation bubble. We show that the separation can be suppressed by bending a short leading section of the plate. Finally, we consider the effect of a heat source at the leading edge of a cooled plate.

References

M. Müllner and W. Schneider, Heat Mass Transf. 46, 1097-1110 (2010)

W. Schneider, Proc. 3rd Eur. Therm. Sci. Conf., 195-198 (2000)

The virtual element method (VEM) has been demonstrated to be effective in a variety of engineering problems. In recent years, it has gained high interest in both mathematics and engineering communities. In this work, a low order virtual element method for the treatment of three-dimensional contact problems with non-conforming interface meshes is presented. The contact conditions can be employed on different enforcing strategies. For non-conforming meshes, a node-to-surface enforcement leads to wrong force distributions at the contact interface. Here, we utilise a mesh adaptivity strategy, which leads to conforming meshes at the contact interface, without introducing new elements or changing the ansatz. In fact, we take advantage of the useful feature of the virtual element method, which allows to introduce new topological nodes during the simulation. It allows to employ a very simple node-to-node contact formulation for the treatment of contact. Thus, this work presents a simple geometrical approach to cut element faces and introduce new nodes into the existing mesh. Beside a node-to-node contact formulation, this also allows to treat the contact pairs as polygonal pairs and thus to use a surface-to-surface contact formulation. To validate the presented methodology, numerical examples in 3D are performed, including the contact patch test and Hertzian contact problem.

To precisely model crack growth, accurate calculations of crack front loading and crack deflection angles are essential. These calculations require solutions of the underlying boundary value problems (BVPs), which are typically obtained by applying numerical methods, e.g., the finite element method (FEM). However, since accuracy and computational cost of the analyses are in general competing aspects, compromises often must be made to generate satisfactory results in acceptable times. In contrast, the use of more efficient methods, both for the solution of the BVP as well as for the subsequent crack tip loading analyses, can substantially lower the computational effort while maintaining desired accuracies. The virtual element method (VEM) is a fairly new discretization scheme for the numerical solution of BVPs, and can be interpreted as a generalization of the FEM. Since the VEM can handle arbitrary polytopal meshes in a straightforward manner, it provides a higher degree of flexibility in the discretization process than the FEM, which turns out to be profitable in terms of both computing times and accuracy.

In the context of numerical applications of fracture mechanics, the probably most attractive feature of the VEM results from the possibility to employ elements of complex shapes, which may be convex as well as concave. Consequently, crack growth simulations benefit from the fact that incremental changes in the geometry of a crack do not require any remeshing of the structure, but rather crack paths can run through already existing elements. Although the method has already proved to provide an efficient tool for crack growth simulations in plane problems, there is still further research required regarding the efficient and precise evaluation of crack front loading quantities and the extension towards spatial crack problems.

This work aims to discuss aspects of the virtual element method for crack tip loading analyses and crack growth simulations. Classical as well as advanced concepts of numerical fracture mechanics are adopted and implemented in connection with the VEM, carefully investigating and exploiting the advantages and opportunities the discretization method offers in this regard. Crack growth simulations based on the VEM are performed and results are compared to reference solutions as well as solutions obtained by the FEM.

The finite element method can become susceptible to mesh distortion and numerical instabilities at huge deformations. As an alternative numerical method, the Material Point Method (MPM) can be used for this purpose, combining the advantages of the Lagrangian description of the bodies while solving the equations of interest on the Eulerian grid, see [1]. In the MPM, bodies are discretized as material points while their mechanical properties are mapped to the background grid on which the equations are solved. In this contribution, numerical examples are presented that are subject to large deformations in the context of dynamic processes. These examples exhibit a kind of mesh-dependence in different quantities. Therefore, the focus of this contribution is on the improvement and increase in stability of the numerical results, which is achieved by translations of the grid. Within this method, the origin of the background grid is shifted randomly at the beginning of each time step in a small manner in each direction. This shifting procedure can be interpreted as smearing the grid over time, eliminating the mesh-dependence shown in the resulting quantities.

References:

[1] D. Sulsky, Z. Chen und H. Schreyer. “A particle method for history-dependent materials”. In: Comput. Method Appl. M. 118.1-2 (1994), S. 179-196. doi: 10.1016/0045-7825(94)90112-0.