Evolution of mesh refinement rules for impact dynamics

Genetic programming (GP) was used in an experiment to investigate the possibility of learning rules that trigger adaptive mesh refinement. GP detected mesh cells that required refinement by evolving a formula involving cell quantities such as material densities. Various cell variable combinations were investigated in order to identify the optimal ones for indicating mesh refinement. The problem studied was the high speed impact of a spherical ball on a metal plate. 1 The problem of optimal mesh refinement It is sometimes impossible or too costly to carry out physical experiments to investigate an engineering design. Instead, fast computers can undertake a computational experiment, or simulation of the Physics of a real experiment, which implements a numerical method to approximate the solution of the partial differential equations (PDEs) that describe the physical situation. Usually PDEs are numerically solved with the Weighted Residuals Method (WRM) [3]. Examples of WRM are the Finite Element Method (FEM) [1] and the Finite Difference Method (FD) [2]. These divide the geometry of the experiment into a number of points or cells that are collectively known as a ‘grid’ or a ‘mesh’. Over these points the PDE derivatives are defined and integrated using approximating functions. The WRM approximation should converge to the analytical solution of the PDE as the spacing between grid points decreases and also as the order of the functions increases (h-p method). By definition the WRM requires a choice for its ‘test’ or ‘weighting’ function and a choice for its ‘trial’ or ‘shape’ functions. However, the most significant choice is whether the test function itself is symmetric or skewed and this is completely determined by the type of PDE that needs to be solved. The PDEs that describe structural engineering problems are elliptic and contain even order derivative terms. With these, WRMs with a symmetric test function possess a mathematical property that ensures that they converge in an ‘optimal’ fashion to the analytical solution. Examples of these WRMs are the FD with central differencing and the Galerkin FEM. Under some assumptions the Galerkin FEM can be shown to be equivalent to the Ritz method [5]. The approximation is called ‘optimal’ as it equates to the minimization of a quadratic functional the Hessian matrix in the system of FEM equations is positive definite, making the FEM approximation a global minimiser of the functional . The structural engineering PDEs tend to have smooth solutions. And so consequently, the WRM that discretises the domain into a number of equally spaced points, the ‘uniform grid’ or ‘uniform mesh’, converges to an accurate result. Uniform grids, however, are not appropriate for PDEs that describe Fluid Dynamics, Heat Transfer, and Combustion because these PDEs contain first order derivatives. The equivalence of the Galerkin FEM and the Ritz method breaks down for these problems as the Hessian is no longer positive definite. Two strategies exist to alleviate this problem: (a) refining the grid by reducing the distance between the grid points in places where the solution gradient is large to reduce the contribution of the discrete advection terms to , and/or (b) skewing the test function along characteristic directions, streak lines or streamlines, to recover a more elliptic system. Failure to adopt either or sometimes both strategies results in unstable and inaccurate solutions. Strategy (b) presents mathematical difficulties. Though we know how to construct the optimal test function for a linear ODE case (1D), i.e. the Hemker function for the steady-state convection-diffusion equation, an optimal test function for the linear PDE case, i.e. 2D, 3D, is not a tensor product of the 1D function. Nor in CFD, save for trivial examples, is it is a function skewed along a streamline. As showed by Morton [4], the optimal test function is extremely complex. FD practicing engineers have never used it confusing the issue with discussion of truncation error. FEM schemes, e.g. SUPG [6], are equally sub-optimal in general. Solving the PDEs with a sub-optimal test function means solving the wrong PDE. Usually it gives a solution which is deceptively smooth [8] losing the power of prediction, i.e. a numerical result will not match an experimental result. Strategy (a) refines the grid selectively and is usually implemented iteratively. For example, the transport terms in the Navier-Stokes equations are usually linearized via a Picard iteration or Newton method, and a grid refinement rule is evaluated at each iteration based on current gradients. In timetransient PDEs a pseudo time stepping explicit method, or a time stepping implicit method the mesh refinement rule can be embedded in the time stepping. Strategy (a) without strategy (b) requires excessive mesh refinement to control sharp boundary layers. And for problems that involve shocks, strategy (b) must be in force if we wish to get any result at all, and is popularly referred to as an artificial viscosity scheme. On some PDEs with certain boundary conditions the grid may need to be refined gradually the cell size should not change abruptly or may need to avoid distorting the shape of the computational cells as this could promote wave reflections and introduce artificial numerical effects. It is also usually required to design the grid for computational efficiency, i.e. the implementation minimizes indirect memory addressing, and/or inter process communication for solution of the PDEs on parallel supercomputers. 2 Discovery of mesh refinement rule with Genetic Programming A grid refinement rule [9] compares the magnitude of the gradients in the current solution to the local grid spacing and decides how and where to refine the mesh: either in the next iterative step, or perhaps refine the grid and re-compute the current step. It may also de-refine a grid, i.e. remove grid points or move them farther apart. The grid is called ‘structured’ when grid cells are of the same geometry (e.g. quadrilateral, triangular) and each cell is always surrounded by the same number of neighbouring cells. And a grid is ‘unstructured’ when cells have various numbers of neighbouring cells. However, a structured grid of square cells can be locally refined in a nested fashion by sub-dividing square cells into internal square cells, and the resulting mesh is ‘unstructured’ as described in reference [9]. Popular grid refinement rules are not sophisticated. They compare the current localised solution gradient in a cell to a threshold to decide whether to refine or de-refine. Grid refinement rules are ideal candidates for optimisation with an evolutionary method. Rules need to be more sophisticated to address the following list which is not exhaustive: (a) computational efficiency, (b) non-linear continuation and accuracy. In the rest of the paper Genetic Programming (GP) is used to discover a rule that satisfies a modest example objective from this list.