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The progress of LiDAR-based 3D object detection has significantly enhanced developments in autonomous driving and robotics. However, due to the limitations of LiDAR sensors, object shapes suffer from deterioration in occluded and distant areas, which creates a fundamental challenge to 3D perception. Existing methods estimate specific 3D shapes and achieve remarkable performance. However, these methods rely on extensive computation and memory, causing imbalances between accuracy and real-time performance. To tackle this challenge, we propose a novel LiDAR-based 3D object detection model named BSH-Det3D, which applies an effective way to enhance spatial features by estimating complete shapes from a bird's eye view (BEV). Specifically, we design the Pillar-based Shape Completion (PSC) module to predict the probability of occupancy whether a pillar contains object shapes. The PSC module generates a BEV shape heatmap for each scene. After integrating with heatmaps, BSH-Det3D can provide additional information in shape deterioration areas and generate high-quality 3D proposals. We also design an attention-based densification fusion module (ADF) to adaptively associate the sparse features with heatmaps and raw points. The ADF module integrates the advantages of points and shapes knowledge with negligible overheads. Extensive experiments on the KITTI benchmark achieve state-of-the-art (SOTA) performance in terms of accuracy and speed, demonstrating the efficiency and flexibility of BSH-Det3D. The source code is available on https://github.com/mystorm16/BSH-Det3D.