Since concrete is cheaper and more resistant to corrosion than steel, the wide-shallow concrete bucket foundation is being used extensively in ocean engineering. By adding the inner bucket and cruciform skirts, both the bearing capacity and rigidity of the wide-shallow concrete bucket foundation increase significantly. When compared to the hollow steel bucket foundation, the inclusion of thicker skirts, as well as the addition of inner bucket and cruciform skirts, would cause changes to the soil flow mechanism, resulting in soil heave within each compartment and changes in soil strength evolution and penetration resistance during installation in clay. In order to study the influence of the addition of the inner bucket and cruciform skirts on the soil heaving inside each compartment, soil softening and penetration resistance, three-dimensional large deformation finite element (LDFE) models for the bucket foundation with and without inner bucket, and cruciform skirts considering soil remolding were established using the Coupled Eulerian–Lagrangian (CEL) approach. It was found that the inner bucket significantly changes the soil flow and softening of the soil during penetration of the bucket foundation. According to the theoretical analysis and numerical results, the diameter of the optimal inner bucket is equal to 5/8 of the outer diameter. The adhesion coefficient observed in this study falls within the range of 0.5 to 0.8, which is higher than the theoretical value of 0.25 that assumes the soil is fully remolded. The reason for this discrepancy is that the soil is only partially remolded during the actual installation of the bucket foundation. The neglect of the softening of the soil or considering the soil as completely softened will result in significant variation in the predicted penetration resistance; hence, partial softening of the soil should be taken into account.