An ellipticity domain for the distortional Hencky-logarithmic strain energy

We describe ellipticity domains for the isochoric elastic energy $F\mapsto \|{\rm dev}_n\log U\|^2=\bigg\|\log \frac{\sqrt{F^TF}}{(\det F)^{1/n}}\bigg\|^2 =\frac{1}{4}\,\bigg\|\log \frac{C}{({\rm det} C)^{1/n}}\bigg\|^2$ for $n=2,3$, where $C=F^TF$ for $F\in {\rm GL}^+(n)$. Here, ${\rm dev}_n\log {U} =\log {U}-\frac{1}{n}\, {\rm tr}(\log {U})\cdot 1\!\!1$ is the deviatoric part of the logarithmic strain tensor $\log U$. For $n=2$ we identify the maximal ellipticity domain, while for $n=3$ we show that the energy is Legendre-Hadamard elliptic in the set $\mathcal{E}_3\bigg(W_{_{\rm H}}^{\rm iso}, {\rm LH}, U, \frac{2}{3}\bigg)\,:=\,\bigg\{U\in{\rm PSym}(3) \;\Big|\, \|{\rm dev}_3\log U\|^2\leq \frac{2}{3}\bigg\}$, which is similar to the von-Mises-Huber-Hencky maximum distortion strain energy criterion. Our results complement the characterization of ellipticity domains for the quadratic Hencky energy $W_{_{\rm H}}(F)=\mu \,\|{\rm dev}_3\log U\|^2+ \frac{\kappa}{2}\,[{\rm tr} (\log U)]^2$, $U=\sqrt{F^TF}$ with $\mu>0$ and $\kappa>\frac{2}{3}\, \mu$, previously obtained by Bruhns et al

Adaptive Mesh Refinement algorithm based on dual trees for cells and faces for multiphase compressible flows

A novel adaptive mesh refinement method is proposed. The novelty of the method lies in using a dual data structure with two trees: A classical one for the computational cells and an extra one dedicated to computational cell faces. This new dual structure simplifies the algorithm, making the method easy to implement. It results in an efficient adaptive mesh refinement method that preserves an acceptable memory cost. This adaptive mesh refinement method is then applied to compressible multiphase flows in the framework of diffuse-interface methods. Efficiency of the method is demonstrated thanks to computational results for different applications: Transport, shock tube, surface-tension flow, cavitation and water-droplet atomization, in one and multi-dimensions. The test cases are performed with the open-source code ECOGEN and with quantitative comparisons regarding non-adaptive mesh refinement methods to analyze benefits. A discussion specific to parallel computing is also presented