Geometrical design of blade's surface and boundary control of Navier-Stokes equations

In this article a new principle of geometric design for blade's surface of an impeller is provided.This is an optimal control problem for the boundary geometric shape of flow and the control variable is the surface of the blade.We give a minimal functional depending on the geometry of the blade's surface and such that the flow's loss achieves minimum.The existence of the solution of the optimal control problem is proved and the Euler-Lagrange equations for the surface of the blade are derived.In addition,under a new curvilinear coordinate system,the flow domain between the two blades becomes a fixed hexahedron,and the surface as a mapping from a bounded domain in R2 into R3,is explicitly appearing in the objective functional.The Navier-Stokes equations,which include the mapping in their coefficients,can be computed by using operator splitting algorithm.Furthermore,derivatives of the solution of Navier-Stokes equations with respect to the mapping satisfy linearized Navier-Stokes equations which can be solved by using operator splitting algorithms too.Hence,a conjugate gradient method can be used to solve the optimal control problem.

Electrochemical Properties of Electrodes with Different Shapes and Diffusion Kinetic Analysis of Microbial Fuel Cells on Ocean Floor

Microbial fuel cell(MFC) on the ocean floor is a kind of novel energy-harvesting device that can be developed to drive small instruments to work continuously.The shape of electrode has a great effect on the performance of the MFC.In this paper,several shapes of electrode and cell structure were designed,and their performance in MFC were compared in pairs:Mesh(cell-1) vs.flat plate(cell-2),branch(cell-3) vs.cylinder(cell-4),and forest(cell-5) vs.disk(cell-6) FC.Our results showed that the maximum power densities were 16.50,14.20,19.30,15.00,14.64,and 9.95 mWm-2 for cell-1,2,3,4,5 and 6 respectively.And the corre-sponding diffusion-limited currents were 7.16,2.80,18.86,10.50,18.00,and 6.900 mA.The mesh and branch anodes showed higher power densities and much higher diffusion-limited currents than the flat plate and the cylinder anodes respectively due to the low diffusion hindrance with the former anodes.The forest cathode improved by 47% of the power density and by 161% of diffusion-limited current than the disk cathode due to the former's extended solid/liquid/gas three-phase boundary.These results indicated that the shape of electrode is a major parameter that determining the diffusion-limited current of an MFC,and the differences in the elec-trode shape lead to the differences in cell performance.These results would be useful for MFC structure design in practical applica-tions.

SOLUTION OF DIFFERENT HOLES SHAPE BORDERS OF FIBRE REINFORCED COMPOSITE PLATES BY INTEGRAL EQUATIONS

Accurate boundary conditions of composite material plates with different holes are founded to settle boundary condition problems of complex holes by conformal mapping method upon the nonhomogeneous anisotropic elastic and complex function theory. And then the two stress functions required were founded on Cauchy integral by boundary conditions. The final stress distributions of opening structure and the analytical solution on composite material plate with rectangle hole and wing manholes were achieved. The influences on hole-edge stress concentration factors are discussed under different loads and fiber direction cases, and then contrast calculates are carried through FEM.

A Meshless Method for Retrieving Nonlinear Large External Forces on Euler-Bernoulli Beams

We retrieve unknown nonlinear large space-time dependent forces burdened with the vibrating nonlinear Euler-Bernoulli beams under varied boundary data,comprising two-end fixed,cantilevered,clamped-hinged,and simply supported conditions in this study.Even though some researchers used several schemes to overcome these forward problems of Euler-Bernoulli beams;however,an effective numerical algorithm to solve these inverse problems is still not available.We cope with the homogeneous boundary conditions,initial data,and final time datum for each type of nonlinear beam by employing a variety of boundary shape functions.The unknown nonlinear large external force can be recuperated via back-substitution of the solution into the nonlinear Euler-Bernoulli beam equation when we acquire the solution by utilizing the boundary shape function scheme and deal with a smallscale linear system to gratify an additional right-side boundary data.For the robustness and accuracy,we reveal that the current schemes are substantiated by comparing the recuperated numerical results of four instances to the exact forces,even though a large level of noise up to 50%is burdened with the overspecified conditions.The current method can be employed in the online real-time computation of unknown force functions in space-time for varied boundary supports of the vibrating nonlinear beam.

Boundary Shape Control of the Navier-Stokes Equations and Applications

In this paper, the geometrical design for the blade's surface in an impeller or for the profile of an aircraft, is modeled from the mathematical point of view by a boundary shape control problem for the Navier-Stokes equations. The objective function is the sum of a global dissipative function and the power of the fluid. The control variables are the geometry of the boundary and the state equations are the Navier-Stokes equations. The Euler-Lagrange equations of the optimal control problem are derived, which are an elliptic boundary value system of fourth order, coupled with the Navier-Stokes equations. The authors also prove the existence of the solution of the optimal control problem, the existence of the solution of the Navier-Stokes equations with mixed boundary conditions, the weak continuity of the solution of the Navier-Stokes equations with respect to the geometry shape of the blade's surface and the existence of solutions of the equations for the Gateaux derivative of the solution of the Navier-Stokes equations with respect to the geometry of the boundary.

BOUNDARY LAYER ESTIMATION OF A SINGULAR PROBLEM WITH LIMIT EQUATION OF ORDER 2

In this paper, we gave boundary layer estimation of a singular equation of order 4 with limit equation of order 2. The results show that the thickness of boundary layer is intrinsically relative to the reciprocal of the order of equation of and independent of the order of equation of u.

Plate Shape Control Theory and Experiment for 20-high Mill

Roll flattening theory is an important part of plate shape control theories for 20-high mill. In order to improve the accuracy of roll flattening calculation for 20-high mill, a new and more accurate roll flattening model was proposed. In this model, the roll barrel was considered as a finite length semi-infinite body. Based on the boundary integral equation method, the numerical solution of the finite length semi-infinite body under the distributed force was obtained and an accurate roll flattening model was established. Coupled with roll bending model and strip plastic deformation, a new and more accurate plate control model for 20-high mill was established. Moreover, the effects of the first intermediate roll taper angle and taper length were analyzed. The tension distribution calculated by analytical model was consistent with the experimental results.