Semi-analytical implicit direct time integration scheme on example of 1-D wave propagation problem

Abstract

The most common approach in dynamic analysis of engineering structures and physical phenomenas consists in finite element discretization and mathematical formulation with subsequent application of direct time integration schemes. The space interpolation functions are usually the same as in static analysis. Here on example of 1-D wave propagation problem the original implicit scheme is proposed, which contains the time interval value explicitly in space interpolation function as results of analytical solution of differential equation for considered moment of time. The displacements (solution) at two previous moments of time are approximated as polynomial functions of position and accounted for as particular solutions of the differential equation. The scheme demonstrates the perfect predictable properties as to dispersion and dissipation. The crucial scheme parameter is the time interval – the lesser the interval the more correct results are obtained. Two other parameters of the scheme – space interval and the degree of polynomial approximation have minimal impact on the general behavior of solution and have influence on small zone near the front of the wave.

Semi-analytical implicit direct time integration method for 1-D gas dynamic problem

“…Which eventually allows to obtain give the following solution for two main parameters of the problem within each elementary mass (section) [21]:…”

“…Where ( ) i D x is second order polynomial. The values of coefficients i α , i β and polynomial ( ) i D x can easily be got by comparing (8b) with (9a) [21]. Next step is expansion of Krylov's functions into polynomial series.…”

“…Start from the governing equations of the gas dynamic problem. Contrary to [21] analysis of behavior of the rod, where only two parameters were used, here 4 parameters, which characterize the state of the gas in any point, are used. Other difficulty, as compared with [21], lies in nonlinearity of the governing equations, which need to be linearized in calculation process.…”

“…Here we will use the previous finding [21] as to the method stability; the spurious oscillation might occur when the time step is lower than allowable. Due to employment of three term polynomial expansion of function ( )…”

“…The goal of this work is further elaboration of our semi analytical scheme [21], but now in application to gas dynamic problem. Accounting for the principal novelty of the method it will be verified only on example of the Sod's task, so many equations are considered in simplified form only.…”