Bucket foundations possess the characteristics of excellent bearing capacity, relatively low economic cost and being suitable for soft clay foundations. These advantages have led to their gradual adoption in various coastal and offshore engineering structures. Recently, bucket foundations with internal compartments have emerged as a preferred choice for the foundations of structures that need to bear large moment loads, such as offshore wind turbines. The inclusion of inner compartments in bucket foundations significantly influences soil flow, soil softening and penetration resistance. In addition, the soil heave and softening generated during their penetration can adversely affect its subsequent bearing capacity, often resulting in a lower bearing capacity than that of the fully embedded foundations. Therefore, it is crucial to investigate the installation process and the bearing capacity of bucket foundations with inner compartments. Based on a series of numerical analyses using the Coupled Eulerian-Lagrangian (CEL) method, this study examined the penetration and installation process and the post-installation bearing capacity of bucket foundations with cruciform inner compartments. Parameters including soil strength heterogeneity, soil sensitivity, relative ductility, soil stiffness coefficient, and the foundation wall thickness ratio have been evaluated. Detailed analyses were conducted on soil flow, soil heave, soil softening and penetration resistance. Furthermore, a formula was proposed to calculate the penetration resistance of bucket foundations with cruciform inner compartments, incorporating the soil strain softening and strain rate effects.