Hierarchical Rank-One Sequence Convexification for the Relaxation of Variational Problems with Microstructures

TL;DR

The work tackles nonconvex variational problems in nonlinear solid mechanics by approximating the rank-one convex envelope $W^{ ext{rc}}$ via hierarchical rank-one sequences. The proposed Hierarchical Rank-One Sequence Convexification (HROC) algorithm builds a binary lamination tree using locally optimal rank-one directions and provides first- and second-derivative information for constitutive updates, enabling concurrent relaxation in finite-element simulations. Benchmark studies show good pointwise accuracy and linear-time scaling in the one-dimensional convexifications, while a counterexample highlights the method’s local nature and potential limitations in global optimality. Applications to continuum damage mechanics in 2D and 3D demonstrate mesh-independence and feasible three-dimensional simulations, including microstructure reconstruction, with isotropy managed through rotational averaging. The approach offers a practical and efficient route to regularised, dissipative large-strain models, while indicating directions for improving globality, anisotropy handling, and advanced relaxation criteria.

Abstract

This paper presents an efficient algorithm for the approximation of the rank-one convex hull in the context of nonlinear solid mechanics. It is based on hierarchical rank-one sequences and simultaneously provides first and second derivative information essential for the calculation of mechanical stresses and the computational minimization of discretized energies. For materials, whose microstructure can be well approximated in terms of laminates and where each laminate stage achieves energetic optimality with respect to the current stage, the approximate envelope coincides with the rank-one convex envelope. Although the proposed method provides only an upper bound for the rank-one convex hull, a careful examination of the resulting constraints shows a decent applicability in mechanical problems. Various aspects of the algorithm are discussed, including the restoration of rotational invariance, microstructure reconstruction, comparisons with other semi-convexification algorithms, and mesh independency. Overall, this paper demonstrates the efficiency of the algorithm for both, well-established mathematical benchmark problems as well as nonconvex isotropic finite-strain continuum damage models in two and three dimensions. Thereby, for the first time, a feasible concurrent numerical relaxation is established for an incremental, dissipative large-strain model with relevant applications in engineering problems.

Figures (13)

• Figure 2.1: (\ref{['fig:Hsetconstruction']}) construction of $\mathcal{H}$ set, (\ref{['fig:Hset']}) $\mathcal{H}$ set, and (\ref{['fig:tree']}) Rank-one tree. The lines in (\ref{['fig:Hsetconstruction']}) and (\ref{['fig:Hset']}) encode rank-one directions. (\ref{['fig:Hsetconstruction']}) shows the first recursive check for the $\mathcal{H}_7$ condition.
• Figure 3.1: Construction of the $\mathcal{H}$-sequence by convexifying first in the point $\boldsymbol{F}$ (\ref{['fig:h_search_begin']}) and afterwards in $\boldsymbol{F}^{+}$ and $\boldsymbol{F}^{-}$ (\ref{['fig:h_search_end']}). In (\ref{['fig:h_search_begin']}), the given macroscopic deformation gradient $\boldsymbol{F}$ is convexified along all the discretised rank-one lines contained in $\mathcal{R}$ (dashed lines). $\boldsymbol{F}^{+}$ and $\boldsymbol{F}^{-}$ correspond to the supporting points of the line delivering the lowest convexified value of the energy at $\boldsymbol{F}$ (drawn as solid line). The corresponding first-order laminate tree is visualised on the right-hand side in Figure \ref{['fig:h_search_begin']}. Next, in (\ref{['fig:h_search_end']}), all rank-one lines are checked again in both the points $\boldsymbol{F}^{+}$ and $\boldsymbol{F}^{-}$ if a lower convexified function value is possible in these two points, giving a second level laminate. The leaves might then be checked for even further split ups possibly leading to level-three laminates.
• Figure 4.1: (\ref{['fig:KSD_W']}) Kohn-Strang-Dolzmann energy density and (\ref{['fig:KSD_hroc']}) approximation of the rank-one convex envelope by HROC algorithm in the $F_{11}$--$F_{22}$ plane for $F_{12} = F_{21} = 0$. The relative error is $\max_{\boldsymbol{F} \in \mathcal{N}_{\delta, r}} \| \frac{W_{\text{KSD}}^{\text{HROC}}({\boldsymbol{F}}) - W_{\text{KSD}}^{\text{rc}}({\boldsymbol{F}})}{W_{\text{KSD}}^{\text{rc}}({\boldsymbol{F}})}\| = 0.0386$ and the absolute error $0.0472$ with convexification parameter $N = 5000$ on the grid in the $F_{11}$--$F_{22}$ plane represented by the parameters $\delta \approx 1 / 20, r = 1$.
• Figure 4.2: On the left-hand side, the pointwise approximation error of the HROC algorithm in $\hat{\boldsymbol{F}}$ versus the one-dimensional convexification discretisation parameter $N$ is shown for the Kohn--Strang--Dolzmann example. A plateau for an error of $10^{-3}$ is reached for $10^3$ or more points along the discretised rank-one lines. The right-hand side shows how the one-dimensional discretisation points scale in terms of computational time. A discretisation with $N \approx 10^3$ points leads to acceptable computational times of milliseconds which is suited for the use in constitutive models while at the same time preserving possible accuracy of the algorithm, which, however, is already at the limit when it comes to feasibility for realistic boundary value problems.
• Figure 4.3: $F_{11}$--$F_{22}$ plane of the multiwell energy density benchmark problem in Figure \ref{['fig:multi']}. Figure \ref{['fig:multi_hroc']} shows the approximated rank-one convex envelope using the HROC algorithm. Comparing the two, a negligible difference is obtained. Here, the HROC algorithm was used for the pointwise evaluation in $6561$ grid points in the $F_{11}$--$F_{22}$ plane.