# 12.1. Variational Nonlocal Method

In the vein of nonlocality, a variational nonlocal method was derived such that one can identify the state variable that controls softening \(Z\) and pose a variational principle such that the stored energy is dependent on a nonlocal state variable \(\bar{Z}\). At a point, a Lagrange multiplier enforces \(\bar{Z} = Z\). When we minimize and discretize, however, we derive an \(L_{2}\) projection for the “coarser” \(\bar{Z}\) and the balance of linear momentum for the “fine” scale. If we assume that the basis functions for the coarser discretization \(D\) are constant and discontinuous, we obtain the nonlocal \(\bar{Z}\) as a simple volume average of \(Z\).

\[\bar{Z} = \frac{1}{\int_{D} dV} \int_{D} Z dV\]

In this particular case, less is more. We do not want to recover the mesh-dependent solution inherent in \(Z\) with a \(\bar{Z}\). Instead, we seek to specify an additional discretization (length scale) independent of the discretization for \(Z\). Because \(\bar{Z}\) is just an average, we can consider a coarse domain to be a patch of fine scale elements having volume \(V\) that is consistent with a prescribed length scale \(l\) where \(V = l^{3}\). For example, one might correlate the mesh dependence in the solution with scalar damage \(\phi\). The variational nonlocal method would construct a \(\bar{\phi}\) for each nonlocal domain \(D\). The stress would then evolve from \(\bar{\phi}\) and not \(\phi\).

Domain decomposition algorithms are invoked to construct coarse scale domains of common volume. For parallel execution, each processor (having nonlocal element blocks) is partitioned during initialization. Nonlocal averages are calculated on the processor and no communication is necessary between processors.

Warning

Because nonlocal domains are initially decomposed on each processor, nonlocal geometries will not (a) be consistent with different parallel decompositions and (b) admit rebalancing. No infrastructure exists to maintain the character of the nonlocal domains during rebalancing.