Variational Principle for Gun Dynamics with Adjoint Variable Formulation

摘要

Gun dynamics problems involving a moving shell have several delta functions in the forcing terms of the equations of motion. The use of a variational method in conjunction with finite elements smooths the differentiability of the variables in the expression involving the delta functions. This report suggests that solutions of the gun dynamics problems be obtained numerically by a variation principle where the far end conditions in time are not required for purposes of computation. In solving mixed boundary and initial value problems of a high order partial differential equation using spline functions, the computation may be simplified considerably if the variable in time can be truncated into arbitrary sections. Each section may have several node points for the spline functions in the time domain. This is true because we found from previous papers that the initial value problem can be solved in one direction using variational principle and cubic Hermite Polynomials, without worrying about the condition at the far end. The end conditions of the ajoint system can be adjusted according to the end conditions of the original system so that the bilinear concomitant is identically zero. This satisfies the variational principle.