Theory NOGRID Simulation Solver Technology

The fundamental basis of almost all CFD problems are the Navier–Stokes equations, which are a statement of the balance of momentum, but sometimes the equations for conservation of mass and the balance of energy are added.

In order to solve the Navier–Stokes equations the fluid domain must be discretized, which means the volume occupied by the fluid is divided into discrete cells (the mesh). The Finite Volume Method (FVM) is a common approach used in CFD codes, as it has an advantage in memory usage and solution speed, especially for large problems, high Reynolds number turbulent flows and source term dominated flows (like combustion). The Finite Element Method (FEM) is used in structural analysis of solids, but is also applicable to fluids. The Finite Difference Method (FDM) has historical importance and is easy to implement. It is currently only used in few specialized codes, which handle complex geometry with high accuracy and efficiency.

Numerical fluid mechanics (Computational Fluid Dynamics, CFD) is a well known and established method within fluid mechanics and is used to solve fluid mechanical problems approximately with numerical methods. The model equations are the Navier-Stokes equations, which are nonlinear partial differential equations of second order, that prescribe a fluid completely.
An important step in handling partial differential equations is to use discrete algebraic replacements for the continuous differential equations. Several methods are currently in use, such as the finite volume (FV), finite element (FE), and finite difference (FD) method. We use a general finite difference (gFD) method to discretize the Navier-Stokes equation. General means that we operate on an unstructured points basis, whereas the classical FD-method operates on structured points only. 

Figure 2: Point cloud versus finite element mesh

Our meshless CFD software NOGRID points, as well as NOGRID pointsBlow,  is based on FPM (Finite Pointset Method), a method, which generates a local defined, non-stationary point cloud distribution for discretization of the Navier-Stokes equations. The FPM method itself is based on the general finite difference method and the key features are: 

• Instead of a mesh NOGRID points discretizes the continuum domain by a point cloud
• The point density is prescribed by a smoothing length defined locally and can vary from one time step to another
• NOGRID points does not use a rigid neighborhood list for a certain point (=node) as it is required in a mesh based method. All neighbors are allowed to move and the neighborhood list is recomputed at each time step

Thus, the simple idea of the NOGRID method is to use a dynamical discretization method.

The point cloud is generated automatically by the software depending on user's settings. That's why there is no need to generate a mesh as required in classical CFD methods. NOGRID finite points are automatically filled, moved, refilled and cleaned depending on user specifications. The point cloud can be defined variously, e.g. constant in the whole flow domain, changing with time or increasing/decreasing locally depending on the flow/geometry situation.