Nonconforming P1 Finite Element for the Biharmonic Problem and Its Application to Stokes Problem

The aim of this paper is to simulate the two-dimensional stationary Stokes problem. In vorticity-Stream function formulation, the Stokes problem is reduced to a biharmonic one; this approach leads to a formulation only based on the stream functions and therefore can only be applied to two-dimensional problems. The idea developed in this paper is to use the discretization of the Laplace operator by the nonconforming [Formula: see text] finite element. For the solutions which admit a regularity greater than [Formula: see text], in the general case, the convergence of the method is shown with the techniques of compactness. For solutions in [Formula: see text] an error estimate is proved, and numerical experiments are performed for the steady-driven cavity problem.

Research on the Coupled Water–Soil SPH Algorithm Improvement and Application Based on Two-phase Mixture Theory

An improved water–soil coupling algorithm was proposed based on the two-phase mixture theory within the framework of smoothed particle hydrodynamics (SPH). In this algorithm, the buoyant density was considered in saturated soil and the stress of two phases was completely exfoliated with the Terzaghi’s effective stress principle. Then the interaction between water and soil was only constituted by viscous drag force. The proposed algorithm was validated by several numerical tests to effectively solve a series of numerical problems caused by the truncation of the kernel approximation on the interface between submerged soil and water, and it can also be a feasible measure to simulate underwater soil excavation problems without drainage and underwater landside problems. Meanwhile, combined with frictional sliding contact algorithm, the interaction between water/soil and structure which was considered as rigid can be effectively modeled, and the calculated contact forces acting on the structure are more accurate. Furthermore, this improved algorithm can be applied to deal with large deformation problems involving complex water–soil–structure interaction in hydraulic and geotechnical engineering such as underwater excavation, shield dig, caisson sinking and other practical engineering problems. It is also significant to engineering design and the improvement of construction level.