This paper describes a three-step finite element method and its applications to unsteady incompressible fluid flows. Stability analysis of the one-dimensional pure convection equation shows that this method has third-order accuracy and an extended numerical stability domain in comparison with the Lax-Wendroff finite element method. The method is cost-effective for incompressible flows because it permits less frequent updates of the pressure field with good accuracy. In contrast with the Taylor-Galerkin method, the present method does not contain any new higher-order derivatives, which makes it suitable for solving non-linear multidimensional problems and flows with complicated boundary conditions. The three-step finite element method has been used to simulate unsteady incompressible flows. The numerical results obtained are in good agreement with thosc in the literature. K~Y WORDS Three-step method Convection-dominated flows Unsteady incompressible flows Density flows 794 C. B. JlANG A N D M. KAWAHARAthan 1/J3. The cost of using small time steps is especially burdensome for the Navier-Stokes equations, where a pressure update is required at each time step. To ease this problem, a subcycling technique was proposed by Gresho et ~1 . '~ which permits less frequent updates of the pressure field with little loss of accuracy. The third-order Taylor-Galerkin scheme has an extended stability domain which is cost-efficient for incompressible flows. However, its applications are mainly to hyperbolic problems and some convection4iffusion equations,?-because too many terms are introduced in the third-order time derivative term, especially for non-linear multidimensional equations, and treatments of the boundary integrations arising from high-order time derivative terms are too complicated.A three-step finite element method based on a Taylor series expansion in time is proposed in the present study. It is not necessary to calculate any new higher-order spatial derivatives here. This makes it convenient to simulate non-linear multidimensional flows. The ideas are almost the same as those of the two-step finite element method,'"-'' but the present scheme retains the good accuracy and uniform CFL property of the Taylor-GalerkinThe method is cost-effective for incompressible flows since it allows less frequent updates of the pressure field with good accuracy.Stability analysis of the one-dimensional pure convection equation is performed. The results show that the present method has third-order accuracy and an extended stability domain compared with the Lax-Wendroff finite element method. The present three-step finite element method is applied to solve incompressible laminar flows. The same order of interpolation is used for the velocity and pressure.Before introducing the three-step finite element method, it is necessary to have a brief statement of the two-step Lax-Wendroff finite element method. TWO-STEP LAX-WENDROFF FINITE ELEMENT METHODLet us consider the convection-diffusion equation where f is the concentration, ui is the v...

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Application of othotropic composite materials makes the engineer to get ultra-light structure with higher load-bearing capability easily. This paper gives the optimization algorithm considering both the topology and the optimal configuration of the fiber in composite material. In the procedure of optimal design of mechanical part, to get structure with high ratio of performance-cost, the layout of structure (include topology, material and shape etc.) and the properties of material in different othotropic direction should be designed simultaneously, thus the full capability of material can be used efficiently. An ideal optimal structure should possess good macroscopic material layout and microscopic distributed properties. This paper gives the mathematical formulation of optimization problem, in which the topology and the layout of fiber are considered simultaneously. Implementting procedure and solving stratigies are given. For non-convex of the optimization formulation, mathematical programming strategies are given in the paper. The effects of the different composite material property on optimal topology of structure are discussed. Topology optimization of orthotropic material structure gives good result. All numerical examples show the feasibility and validity of topology optimization formulation and solving approach proposed in the paper.

A finite element method is applied to solve the two-dimensional turbulent channel flows. Based on the fractional step techniques, the momentum and the k -€ equations arc split into convection and diffusion equations. The convection equations are solved by the second-order Taylor-Galerkin finite element method, which can overcome the spurious oscillations with minimal artificial diffusion, and the diffusion equations are solved by the fully explicit Galerkin method. Since the same order interpolation is used for the velocity, pressure and turbulent quantities, the present method is computationally efficient. The sudden expansion flow and the obstructed turbulent channel flow arc studied. The results arc in good agreement with experimental observations.

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