Numerical Analysis of Microstructure

Abstract

Based on a variational model, the numerical analysis of microstructure challenges supercomputers because of enforced oscillations on the length scale of every mesh-size. Local minimizers in a finite element discretization cluster and so complicate the numerical simulation to become a numerical analyst’s nightmare.

The lectures address model examples to study these phenomena, such as the quartic polynomial W(x) = (x ² – 1)² in the real variable x. As a common property of energy densities in models of phase transitions in solids or fluids, the relevant ingredients are that W(x) is non-negative and non-convex and has zeros at a fixed number of points, here x = ±1, the wells or zero-energy phases. Then, the minimization problems are of the form

$$ (M)MinimizeE(u) = \int\limits_{\Omega } {W(Du)dx + (lower\;order\;terms\;in\;u)\;over\;u \in A.} $$

(M)

and typically allow for minimizing (better infimizing) sequences enforcing finer and finer oscillations which we call microstructures.

The numerical simulation of (M) and two related forms (G) and (C) is the subject of the lectures. In the first step, we identify predictable quantities and argue that the macroscopic displacement field, the Young measure for the description of oscillations and the stress field are reasonable computable targets. Typically, infimizing sequences are weakly convergent but not strongly convergent. Hence, relaxation is involved, for instance by measure valued solution concepts in (G) or by convexification in problem (C) (respectively by a quasiconvexification in (Q)).

Once a target quantity is fixed, step two selects one of the discrete counterparts (M _h ), (G _h ), or (C _h ) of problem (M), (G), or (C), respectively. It is not efficient to always rely on (M _h ). Instead, if the convexification is available, i.e. if the convexified or quasiconvexified energy density is known by an analytic formula, then (C _h ) is preferable: standard finite element schemes with a standard solver of the (nonlinear) discrete minimization problem are available and even adaptive mesh-refining algorithms work.

However, there are problems where relaxation is not feasible or inapplicable. This leaves important and difficult applications with enormous numerical difficulties for future research. These lectures may be viewed as an introduction and first step to challenge them.