(‘Tian-Yau problem’) Let $\bar{X}$ be a smooth $n$-dimensional Fano manifold, and $D=\sum D_{i}$ be an anticanonical divisor, which we assume to be simple normal crossing, with only multiplicity one components. Let $X=\bar{X}\setminus D$, then $X$ has a nowhere vanishing holomorphic volume form $\Omega$. When does $X$ admit a complete Calabi-Yau metric?