# MESHFREE – more freedom in CAE

The complexity and difficulties of a flow simulation is highly dimensional. In many cases, we have only little or diffuse knowledge about boundary conditions or material properties. Moreover, the geometry might be extensive and very detailed, it may deform/move/change during the flow process. The flow might contain free surfaces or phase boundaries. It might contain fluid structure interaction (such as airbags, aquaplaning). In many cases, prior to the modelling, we do not know which part of the geometry will be impacted by the flow (rain water management in cars). These and other reasons can make flow simulations very inefficient.

To judge efficiency of a flow simulation, we have to measure not only the CPU-hours taken by a particular task, but also the time and effort it needs for preprocessing, postprocessing, and how well the simulation can be coupled to other tools (fluid-fluid and fluid-structure interaction).

MESHFREE is, as the name suggests, a completely gridless simulation tool for fluid and continuum mechanics, that might considerably improve efficiency and accuracy of flow simulations. By flow, we do not only mean classical CFD, but also flow of non-Newtonian materials such as granular media, polymer melts etc. The tool can overcome some of the difficulties mentioned above and is therefore a very efficient method in many aspects. MESHFREE evolved from the coupling of the classical Finite Pointset Method (FPM), developed in Fraunhofer ITWM, with the Algebraic Multigrid Method (SAMG) of the Fraunhofer SCAI. The flow is represented by a set of numerical points (figure 1). Each point carries problem-relevant physical information, however points do not carry a mass, they are purely numerical. The pointcloud is automatically established and maintained, no need to mesh the flow domain. The user has to simply provide a geometry model of the boundaries (walls, inflows, outflows) by stl- or similar formats.

Figure 1: MESHFREE simulation of Karman vortex street with adaptive point refinement due to the gradient of velocity.

The method is Lagrangian, the numerical points move with flow velocity. In this way, we achieve an almost perfect self-adaptivity of the simulation towards moving geometry or free/phase boundaries. As the points are purely numerical, we can easily add or remove points, which gives rise to a truly adaptive numerical scheme, see again figure 1.

The conservation laws are modelled in their differential form:

This set of partial differential equations (PDE) is solved locally on each numerical point. The d/dt operator is the substantial derivative describing the change of physical quantities of a point moving with fluid velocity. The material properties are settled in the stress tensor S and in the heat conduction k, and we need thermodynamics in order to provide the closure relations.

The solution of the PDE requires the computation of local derivatives, such as and , which is achieved by the so-called moving least squares approach. Here, around each point, local polynomials of user-given order are locally established to fit the discrete function values of the neighbor stencil. The gradients of the polynomials act as approximation of the local derivatives needed for the PDE. The order of approximation coincides with the chosen polynomial order.

MESHFREE convinces by very short preprocessing times. Due to the absence of the computational mesh, the user can concentrate on the definition of boundary conditions and physical/numerical parameters. It is straight forward to categorize simulation tasks, keeping the same simulation set-up for many similar computations, changing only the geometry or significant parameters. For better visualization and comparison, free surfaces and boundaries are triangulated, as for example shown in figure 2.

Figure 2: Simulation and experiment of deep water crossing, cooperation with Volkswagen.

The original implementation of MESHFREE is an explicit upwind formulation, which is used for airbag inflation simulation. For this purpose, ESI and ITWM have developed a generic interface between VPS and FPM/MESHFREE. It has become an inherent feature of VPS and has proven reliability as an industrial tool already since 2004.

The current focus of development is on the implicit formulation, especially questions of computational performance and incorporation of material models. For car development, MESHFREE covers watercrossing applications (figure2), tank filling, sloshing, just to mention the most traditional ones. As a recent development, we put special focus on water/rain management. For example, two separate MESHFREE simulations of airflow around a car and a droplet phase are coupled in order to capture soiling effects.

Another recent development is the modelling of granular flows by the Drucker-Prager model, such as given due to the roll-over interaction of cars with sand or gravel. For this purpose, we employ once more the coupling of MESHFREE with VPS. The sand is modelled by MESHFREE, and the car dynamics is modelled by VPS. The aim is to employ higher quality material models like Barodesy, which is our current focus.