Generative algorithm-based conceptual design has been innovatively applied as an emerging digital design paradigm for early-stage design ideation. With powerful large language models (LLMs), designers can enter an initial prompt as a design requirement to generate using machine reasoning capability descriptive natural language content. The machine-generated output can be used as stimuli to inspire designers during design ideation. However, the lack of transparency and insufficient controllability of LLMs can limit their effectiveness when assisting humans on a generative conceptual design task. This generation process lacks theoretical guidance and a comprehensive understanding of design requirements, which may potentially lead to generated concepts that are mismatched or lack creativity. Inspired by the Function-Behavior-Structure (FBS) model, this paper proposes a task-decomposed AI-aided approach for generative conceptual design. We decompose a conceptual design task into three sub-tasks including functional reasoning, behavioral reasoning, and structural reasoning. Prompt templates and specification signifiers are specified for different steps to guide the LLMs to generate reasonable results, controllably. The output of each step becomes the input of the next, aiding in aggregating gains per step and embedding the selection preferences of human designers at each stage. A conceptual design experiment is conducted, and the results show that the conceptual design ideation with our method are more reasonable and creative in comparison to a baseline.