An explicit finite volume element method for solving characteristic level set equation on triangular grids

Level set methods are widely used for predicting evolutions of complex free surface topologies,such as the crystal and crack growth,bubbles and droplets deformation,spilling and breaking waves,and two-phase flow phenomena.This paper presents a characteristic level set equation which is derived from the two-dimensional level set equation by using the characteristic-based scheme.An explicit finite volume element method is developed to discretize the equation on triangular grids.Several examples are presented to demonstrate the performance of the proposed method for calculating interface evolutions in time.The proposed level set method is also coupled with the Navier-Stokes equations for two-phase immiscible incompressible flow analysis with surface tension.The Rayleigh-Taylor instability problem is used to test and evaluate the effectiveness of the proposed scheme.

PREDICTION OF ACOUSTIC PERFORMANCE IN EXPANSION CHAMBER MUFFLERS WITH MEAN FLOW BY FINITE ELEMENT METHOD

The equation of wave propagation in a circular chamber with mean flow is obtained. Computational solution based on finite element method is employed to determine the transmission loss of expansive chamber. The effect of the mean flow and geometry (length of expansion chamber and expansion ratio)on acoustic attenuation performance is discussed, the predicted values of transmission loss of expansion chamber without and with mean flow are compared with those reported in the literature and they agree well. The accuracy of the prediction of transmission loss implies that finite element approximations are applicable to a lot of practical applications.

A SPLIT-CHARACTERISTIC FINITE ELEMENT MODEL FOR 1-D UNSTEADY FLOWS

An efficient and accurate solution algorithm was proposed for 1-D unsteady flow problems widely existing in hydraulic engineering. Based on the split-characteristic finite element method, the numerical model with the Saint-Venant equations of 1-D unsteady flows was established. The assembled f＇mite element equations were solved with the tri-diagonal matrix algorithm. In the semi-implicit and explicit scheme, the critical time step of the method was dependent on the space step and flow velocity, not on the wave celerity. The method was used to eliminate the restriction due to the wave celerity for the computational analysis of unsteady open-channel flows. The model was verified by the experimental data and theoretical solution and also applied to the simulation of the flow in practical river networks. It shows that the numerical method has high efficiency and accuracy and can be used to simulate 1-D steady flows, and unsteady flows with shock waves or flood waves. Compared with other numerical methods, the algorithm of this method is simpler with higher accuracy, less dissipation, higher computation efficiency and less computer storage.

A CHARACTERISTIC FINITE ELEMENT METHOD FOR CONSTRAINED CONVECTION-DIFFUSION-REACTION OPTIMAL CONTROL PROBLEMS

In this paper, we develop a priori error estimates for the solution of constrained convection-diffusion-reaction optimal control problems using a characteristic finite element method. The cost functional of the optimal control problems consists of three parts： The first part is about integration of the state over the whole time interval, the second part refers to final-time state, and the third part is a regularization term about the control. We discretize the state and co-state by piecewise linear continuous functions, while the control is approximated by piecewise constant functions. Pointwise inequality function constraints on the control are considered, and optimal a L2-norm priori error estimates are obtained. Finally, we give two numerical examples to validate the theoretical analysis.

A CHARACTERISTIC MIXED FINITE ELEMENT TWO-GRID METHOD FOR COMPRESSIBLE MISCIBLE DISPLACEMENT PROBLEM

A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium.The concentration equation is treated by a mixed finite element method with characteristics(CMFEM)and the pressure equation is treated by a parabolic mixed finite element method(PMFEM).Two-grid algorithm is considered to linearize nonlinear coupled system of two parabolic partial differential equations.Moreover,the L q error estimates are conducted for the pressure,Darcy velocity and concentration variables in the two-grid solutions.Both theoretical analysis and numerical experiments are presented to show that the two-grid algorithm is very effective.

The MFE-CFE-GFE Method for the Fully Coupled Quasi-Static Thermo-Poroelastic Problem

In this work,we consider a combined finite element method for fully coupled nonlinear thermo-poroelastic model problems.The mixed finite element(MFE)method is used for the pressure,the characteristics finite element(CFE)method is used for the temperature,and the Galerkin finite element(GFE)method is used for the elastic displacement.The semi-discrete and fully discrete finite element schemes are established and the stability of this method is presented.We derive error estimates for the pressure,temperature and displacement.Several numerical examples are presented to confirm the accuracy of the method.