Problems of Fluid-Structure Interaction (FSI) can either be solved in a partitioned/segregated manner or a monolithic/fully-coupled manner, recently the latter has been regarded in literature as a robust way to cope with a variety of FSI problems. The classical monolithic approaches have the drawback of dealing with a large non-linear equation system combining fluid and solid, which is difficult to solve. However, this negative feature can be elegantly improved by the one-field idea: expressing the solid equation in terms of velocity, thus reducing the FSI system to a fluid-like problem. We implement this idea based upon the Fictitious Domain method (FDM) \cite{yongxing1,yongxing2}, the distinguish feature of this methodology is that it only solves for one velocity field for the whole FSI domain; the interactions remain decoupled until solving the final linear algebraic equations. To achieve this the finite element procedures are carried out separately on two different meshes for the fluid and solid respectively, and the assembly of the final linear system brings the fluid and solid parts together via an isoparametric interpolation matrix between the two meshes. In our recent new research, we apply this one-field idea to an Arbitrary Lagrangian-Eulerain (ALE) formulation \cite{yongxing3}. The distinguish feature of this one-field ALE is that it only solves for one velocity field in the whole FSI domain, and it solves in a monolithic manner so that the fluid-solid interface conditions are satisfied automatically. We prove that the proposed scheme is unconditionally stable through energy analysis. We validate the two numerical schemes, one-field FDM and one-field ALE, through a selection of two and three dimensional numerical examples. These include validation of energy stability, comparison to benchmarks and experimental results, and comparing with existing numerical methods, typically with Immersed Finite Element Methods (IFEM).