真經一句話，假傳萬卷書。

英語中沒有理解的概念，就會被千奇百怪的亂解讀，獨立結構就是一例！

如題主的例子所示，所謂的獨立結構就是為了簡化語言而已，省略了後句中的時間。

The boy rushed into the room, his face covered with tears.

上句就是下句的簡化：

The boy rushed into the room. His face ( was ) covered with tears.

語言的發展就是越來越簡化、越來越高效，能省則省，能一個字表達，就不要兩個字，能省略時間，就省略時間，正如上句一樣。

更多示例：

The meeting being over, all of us went home. 開完會後我們都回家了。

還原時間：

The meeting (was) over, all of us went home. 開完會後我們都回家了。

2. The work done, we went home. 工作完成後，我們就回家了。

還原時間：

After the work had been done, we went home.

您問的是“非謂語動詞的獨立主格結構“嗎?

以下舉兩例:

Everyone having sat down, the teacher began his class.

Thousands of eyes fixed on him, Nick felt on edge.

備註:近幾年高考淡化對非謂語動詞做獨立主格的考查了，但是理解這一結構，有助於閱讀理解，也有助於寫作中使用高階句型。