Essay About Finite Element Method And Partial Differential Equations

Thin Film Membrane Simulation
Effect of support structure on permeance of thin film composite membranes-SimulationIntroduction1.1 membrane introductionComposite membrane formed with selective film and support film or selective layer with gutter layer and support film has been increasingly discussed.Thin film composite membranes are semipermeable membranes. In manufactured area, they are significant products used in water purification or water desalination systems. They also have use in chemical applications such as batteries and fuel cells. A Thin film composite membrane can be considered as a molecular sieve constructed in the form of a film from two or more layered materials[1].Thin film composite membranes can be classified as nano-filtration and reverse osmosis (RO) membranes. Both of them are made of thin polymer layer which the thickness thinner than 200nm on a top of porous support. The two or three layer configuration gives the desired properties of high rejection of undesired materials (like salts), high filtration rate, and good mechanical strength[1]. The top layer of membrane are considered the rejection of material.1.2 Calculation introductionThe finite element method is popular in finding approximate solution of boundary condition problems using partial differential equations. Finite elements, which is a method that simpler part can be considered as a subdivision of a whole problem domain. It solving problem by minimizing the associated error function. The idea comes out from connecting a bunch of small straight lines can make an approximated large circle; finite element method is using this conception to connect bunch of small simple element equations for the tiny subdomains to make an approximation of a complex equation for a whole problem domain[2]. It can make the representation of complex geometry more accuracy and easier to represent a total solution.The method work out related to divide the whole problem domain, which can be very large, into the connection of many subdomain. And a series of element equations represent the subdomains. In order to calculate the final numerical answer from the initial values of original boundary problem, the method combining the element equations into the solution techniques, which has known as the global system. Thus, we here can obtain the final numerical answer.The element equations has been talked above are simple equations that approximate the original complex equations. The original equations are often partial differential equations of the original boundary problem. In order to complete the approximation, the method construct the integral to zero of the residual of inner product and weight functions, which are the polynomial of the residual. The function of the integration here is to minimize the error of the approximation from fitting trial of partial differential equations.The global system of equations is generated from the element equations through a transformation of coordinates from the subdomains local nodes to the domains global nodes[2].