In software engineering, eliciting non-functional requirements (NFR) remains a critical yet often overlooked task in requirements engineering practice. Traditional NFR elicitation methods predominantly rely on the experience and manual analysis of requirements engineers, leading to inefficiency, omissions, and inconsistencies. Recent breakthroughs in large language models (LLM) in natural language processing have provided new technological means for the automated NFR elicitation. However, directly employing LLM for NFR generation often faces challenges such as hallucination and insufficient domain expertise. To address these issues, this study proposes an automated NFR elicitation method based on LLM to achieve high-quality NFR generation. A structured and correlated dataset comprising 3856 functional requirements and 5723 NFR is constructed, establishing 22647 FR-NFR association pairs. The proposed method integrates retrieval-augmented generation (RAG) technology through three core modules: a semantic case retrieval module based on the maximum marginal relevance algorithm, a prompt engineering module designed for NFR generation, and an optimized LLM generation module. Through professional evaluation by software engineering experts and automatic metrics including BLEU and ROUGE, experimental results demonstrate that the proposed method outperforms existing approaches in terms of completeness, accuracy, and testability of requirements.