How Red Bull simulate front-wing aeroelasticity

When it first became apparent in 2010 that Red Bull were using aeroelasticity to generate extra front-wing downforce, a number of eagle-eyed observers pointed out that the Milton Keynes based team had established a relationship with MSC Software in 2009, precisely to develop simulation software capable of representing this type of Fluid-Structure Interaction:

Using MSC's latest MD (multi-discipline) software versions of Nastran and Adams, we already combine mechanism and deformable finite element simulations. We also increasingly use aerodynamic output directly from CFD analysis to generate more accurate loads for the structural simulations. There are rule restrictions to limit this, but multi-physics coupling of these effects allows us to legally enhance the performance of deformable components, for example to optimise down-force and drag characteristics for flexible wing components.

It's clear that Red Bull's front wings are still flexing under load this year, so let's dig a little bit deeper to try and understand what has been achieved here, and what the other teams need to do to respond. Careful scrutiny of MSC's website reveals a detailed explanation of exactly how their software works in this respect:

The objective of modeling fluids in a structural analysis is to account for the influence of fluid pressures on the structure and for improved accuracy in structural response prediction. Structures are generally modeled using Lagrangian scheme where material is tied to a finite element mesh. On the other hand, fluids are solved with Eulerian scheme with material being independent of the mesh, but instead flowing through the mesh. Dual schemes are required because of the way structures and fluids behave.

When fluids and structures need to be modeled in a single analysis, the challenge is running these different schemes in a single run. This is accomplished through an automatic coupling algorithm, where two meshes – one for structure and another for fluid, exist. A coupling surface is created between these two domains which acts as a boundary to the flow of material in Eulerian mesh, while enabling transfer of the stresses to the Lagrangian structural mesh causing it to deform.

So we can hypothesise that Red Bull are representing the solid interior of their front-wing with a Lagrangian simulation, and coupling it to a Eulerian hydrocode to represent the airflow around it. Some further explanation of these terms is perhaps in order.

Both Lagrangian and Eulerian computer simulations divide a continuous domain, occupied by a solid or fluid, into a discrete mesh of cells. The corners of the cells are called the nodes of the mesh. In a Lagrangian simulation, the mesh moves with the motion of the solid or fluid, whereas in a Eulerian simulation, the mesh is fixed in space, and the solid or fluid moves through the mesh. A computer simulation also divides the continuous flow of time into a sequence of discrete time-steps, using the final data from one time-step as the initial data for the next.

If Red Bull are using a Lagrangian simulation to represent the solid front-wing, then they are using a mesh which moves with the wing as it deforms. At the beginning of each time-step in such a simulation, one has the coordinates of each node, the velocity of each node, and the stress and strain associated with each cell. The strain represents the relative displacement, or stretch, which the points inside the solid body are subjected to under the influence of external loads. A solid body responds to strain by generating internal restoring forces, called stresses. The stress and the strain each have isotropic components, and when the isotropic components are subtracted, what remains are essentially the shear stresses and shear strains. These are dubbed the deviatoric stress and deviatoric strain.

The task of a Lagrangian simulation is to go from the nodal positions and velocities, and the stresses and strains inside each cell, at the beginning of each time-step, to the positions, velocities, stresses and strains at the end of the time-step. In the case of Red Bull's front-wing, the aerodynamic pressures at the beginning of each time-step will simply be part of the boundary data.

The basic method for solving this problem is as follows: Use the stresses to calculate the forces on the nodes, thence the acceleration of the nodes, and from this update the velocities of the nodes; use the velocities of the nodes to update the positions of the nodes; calculate the rate-of-strain from the velocities, infer the rate-of-stress from the rate-of-strain, and update the stress and the strain. (In reality there's a lot of shuttling back-and-forth with half time-steps and the like to minimise numerical errors, but these are the basic ideas).

The method of updating the stress is quite involved. Given the nodal velocities, one can simply take the gradient of the velocity field, (and symmetrise it), to obtain the rate-of-strain. (In the case of a racing-car front-wing, all the strain will be elastic, so there is no need to worry about plastic strain). The rate-of-strain can be divided into the deviatoric rate-of-strain and the isotropic rate-of-strain. The rate of deviatoric stress can then be calculated using the shear modulus of the material, and the rate of isotropic stress can be calculated using the bulk modulus. (In the case of Red Bull's front-wing, one can expect the bulk modulus and shear modulus to vary with position across the wing). The deviatoric and isotropic stresses can then be updated.

With the Lagrangian time-step for the solid front-wing deformation completed, the boundary of the deformed configuration can be fed as initial data to the next time-step in the Eulerian hydrocode representing the airflow over the front-wing. (Although, once again, one presumes there is a more sophisticated shuttling back-and-forth between the Lagrangian and Eulerian codes to minimise errors). The Eulerian code will calculate new pressure forces on the boundary of the front-wing, and the cycle will begin all over again.

Of course, realising the desired front-wing performance not only requires the development of this type of simulation technology, but also an understanding of how to implement the requisite elasticity via the orientation of the carbon-fibre plies. Nevertheless, it remains a surprise that so many F1 teams rely on off-the-shelf simulation software rather than developing their own.