Proportional Boundary Finite-element Method Applied to Two-dimensional Elastostatic Problem
The control equation of the plane elastostatic problem under the scaled boundary coordinate system is derived by method of virtual work principle, and the solution of the governing equation is expounded. In a small example with a theoretical solution, we apply the proportional boundary finite element method to calculate and find that the accuracy of the result is very high. Introduction Proportional boundary finite-element method was first applied to continuum analysis as a novel semi-analytical approach by Wolf and Song[1-2]and particularly suitable for fracture mechanics and unbounded domains[3-5].The governing equation from the virtual work principle does not require fundamental solution in the proportional boundary finite element method. We establish a coordinate system consisting of circumferential and radial directions. The circumferential boundary is discretized by finite elements and the governing equation is reduced to a radial ordinary partial differential equation[6].It is also suitable for elastic static problems[7].An example of applying proportional boundary finite element method to two-dimensional bounded plane elastic static problems is given in the paper. Two-dimensional Elastic Mechanics Governing Equations in Scaled Coordinate System Governing Equations Equilibrium equation for two-dimensional elastic problem as following: 0 ] [    f L  (1) A proportional boundary coordinate system introduces by a scaling centre ( 0 0 , y x ) selected within the domain(figure 1) in proportional boundary finite-element method. Figure 1. The scaling centre seleted within the domain.  runs from the scaling centre towards the boundary is the normalised radial coordinate , so the value at the center of the scale is 0, and the boundary is 1.Circumferential coordinate s specifies any value distance from the original boundary. So Cartesian coordinate ( y x, ) is expressed by coordinates of the scaling center as follow: 173 ) ( 0 ) ( 0 , s s y y y x x x       (2) ] [L can be expressed as following in proportional boundary coordinate system