基于分段仿射有限元的$p$ -增长断裂能近似
Approximation of fracture energies with $p$-growth via piecewise affine finite elements
摘要：
几何线弹性断裂问题的建模通常是基于有界变形广义函数空间$GSBD^p(Ω)$、$p\in(1,\infty)$，然而，由于这些函数的正则性很低，而且缺乏适当的密度结果，因而阻碍了对它们的处理。我们在这里构造一个近似的$GSBD^p$功能，$p\in(1,\infty)$，与功能是Lipschitz连续远离跳集，这是一个有限的联合闭子集的$C^1$超曲面。逼近函数的应变在$L^p$中强烈地收敛到目标的应变，并且它们的跳跃集的面积收敛到目标的面积。其关键思想是在一个合适的网格上使用分段仿射函数，该网格是通过立方网格的Freudental划分获得的。
英文摘要：
The modeling of fracture problems within geometrically linear elasticity is often based on the space of generalized functions of bounded deformation $GSBD^p(Ω)$, $p\in(1,\infty)$, their treatment is however hindered by the very low regularity of those functions and by the lack of appropriate density results. We construct here an approximation of $GSBD^p$ functions, for $p\in(1,\infty)$, with functions which are Lipschitz continuous away from a jump set which is a finite union of closed subsets of $C^1$ hypersurfaces. The strains of the approximating functions converge strongly in $L^p$ to the strain of the target, and the area of their jump sets converge to the area of the target. The key idea is to use piecewise affine functions on a suitable grid, which is obtained via the Freudhental partition of a cubic grid.
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