What's the meaning of lai2 in this sentence? 中国观众在看京剧的时候，看得高兴时，就用喝彩，鼓掌的方式来表示自己高兴的心情，他们也用这种方式来赞扬京剧演员


What's the meaning of lai2 in this sentence? Is it a common meaning of this? What are other examples in which I can use it with this meaning? Is "fang1 shi4 lai2" a fixed expression? What does it mean?

It seems to me that it's [description of method] + fang1 shi4 lai2 + [verb] and it means "to do [verb] in a [description] way".
It is a common structure

[用 (object) 来 (verb)] = [use (object) to (Verb)]
[用 (这种方式) 来 (赞扬京剧演员)] = [use (this method) to (praise) Peking Opera actors]
方式 is the object; 赞扬 is the verb; 京剧演员 is the indirect object

Example:
[用(水) 来 (灭)火] = [use (water) to (put out) fire]
水 is the object; 灭 is the verb; 火 is the indirect object
&quot;用这种方式赞扬京剧演员&quot;  and &quot;用水灭火&quot; mean the same as &quot;用这种方式来赞扬京剧演员&quot;  and &quot;用水来灭火&quot;. The difference is adding 来 strengthen the connection between the noun phrase and the verb phrase.
The connection between noun and verb in &quot;use water put out fire&quot; is not as strong as &quot;use water to put out fire&quot;
More example:
&quot;用枪杀敌&quot; (use gun kill enemy)  might be broken into two unrelated phrases if you say it like &quot;用枪， 杀敌&quot; (use gun, kill enemy). While &quot;用枪来杀敌&quot; (use gun to kill enemy) cannot be broken into two unrelated phrase no matter how you say it