Some studies on mechanics of continuous mediums viewed as differential manifolds

On the instability of offshore foundations: theory and mechanism

“…The safety of offshore piles in sand could be significantly compromised by waves/currents-induced scour due to the relatively large pile diameter and small slenderness ratio [26]. Understanding the transient and localized turbulence around the structures, especially in the shear flows [27], would be of benefit to the reveal of corresponding fluid-structure interaction mechanisms [28]. Scour reduces the pile embedment depth and increases the load eccentricity, inducing much larger deformations of the pile and changing the profiles of the lateral soil resistance.…”

Vorticity vector-potential method for 3D viscous incompressible flows in time-dependent curvilinear coordinates

“…It can be realized through tensor field analysis in general curvilinear coordinates [19,20]. The details of finite deformation theory with the curvilinear coordinates of physical configurations including time explicitly can be referred in [17]. An essential conclusion follows from this theory that the general material derivative of any tensor field (x, t) takes the following forṁ…”

“…Relevant works dealing with moving coordinate systems include the study of temporal derivative of vectors [14][15][16]. The finite deformation theory with curvilinear coordinates of the current physical configuration including time explicitly was systematically studied in [17]. The essential feature of this kind of curvilinear coordinates (transformations) is that the geometrically irregular and instantaneously varying domain in the physical space can be mapped onto a regular and fixed parametric domain in the diffeomorphism manner.…”

“…The essential feature of this kind of curvilinear coordinates (transformations) is that the geometrically irregular and instantaneously varying domain in the physical space can be mapped onto a regular and fixed parametric domain in the diffeomorphism manner. Subsequently, the governing equations of the general conservation laws can be set up in the parametric domain according to the theory in [17]. Since the deformable boundary in the physical domain is mapped strictly to a plane in the parametric domain, interpolations of variables near the boundary used in many other methods are avoided, which ensures the precision in computation.…”

“…Furthermore, to deal with the varying physical domain, time-dependent curvilinear coordinates are adopted. Properties of the finite deformation theory with time-dependent curvilinear coordinates [17] are used to obtain the component form of the governing equations. The method is therefore applicable to flow in the presence of an arbitrarily moving and continuously deforming surface.…”