The rapid advancement of Large Language Models (LLMs) is leading many industries to dramatic transformations with artificial intelligence (AI). An LLM possesses strong generative and reasoning capabilities, showcasing immense potential in creative and other mental tasks traditionally performed by humans. However, due to its inherent black-box nature of the end-to-end generation results, the application of an LLM to industrial design remains relatively limited. To address this issue, we propose an LLM-leveraged industrial design automation (LLM-IDA) framework with explainability. The LLM-IDA framework uses multiple LLM modules for pertinent dialogues, starting from the input of requirements, and combines problem analysis and code generation without the need for fine tuning on domain-specific data. The LLM-IDA enhances generation solely through the use of few-shot prompt templates for the entire optimization process. To verify the feasibility of the LLM-IDA framework, we apply it to a mechanical design case study. The experiments show that it successfully achieves automated design without human intervention. To assess more real-world applicability, its success rate is also compared with the Retrieval Augmented Generation (RAG) technique currently used in LLM-based generation through the pass@k metric, which verifies that our LLM-IDA approach delivers better results.