In the context of mass personalization, the dynamic and erratic nature of customer requirements (CRs) is increasingly pronounced. Amidst intensifying market competition, enterprises are compelled to swiftly adjust their products to satisfy the ever-changing CRs. However, the complicated design processes often result in prolonged product update and iteration cycles. Thus, it is urgent for enterprises to promote product rapid design, in which, it is essential to map CRs to functional requirements (FRs), then to design parameters (DPs). Traditional mapping methods are time-consuming and labor-intensive, struggle to keep pace with the ever-changing CRs. Moreover, enhancing the efficiency of design knowledge management and reuse is imperative for achieving product rapid design. Considering the strengths of knowledge graphs (KGs) in knowledge management and mining, we establish a multi-domain product design KG (MPDKG) including a CR domain, a FR domain, and a DP domain, and facilitate the mapping process by constructing mapping relations between domains. Firstly, CR entities in the CR domain are extracted through a triple-task cascaded CRs identification model from online reviews and complaints. Secondly, FR entities are identified from product functional lists and instructions, and the Sentence-Bert model is employed to establish the mapping relations between CR entities and FR entities. Subsequently, a corpus comprising product design knowledge is constructed and a cross-domain joint entity relation extraction model is developed to extract DP entities and construct their mapping relations with FR entities. Lastly, the multi-domain mapping of BYD Qin is implemented as a case study to elucidate the practical utility and effectiveness of the proposed methodology.