We propose in this paper a Proper Generalized Decomposition (PGD) approach for the solution of problems in linear elastodynamics. The novelty of the work lies in the development of weak formulations of the PGD problems based on the Lagrangian and Hamiltonian Mechanics, the main objective being to devise numerical methods that are numerically stable and energy conservative. We show that the methodology allows one to consider the Galerkin-based version of the PGD and numerically demonstrate that the PGD solver based on the Hamiltonian formulation offers better stability and energy conservation properties than the Lagrangian formulation. The performance of the two formulations is illustrated and compared on several numerical examples describing the dynamical behavior of a one-dimensional bar.
翻译:在本文中,我们提出了一种解决线性椭圆体动力学问题的适当通用分解(PGD)方法,这项工作的新颖之处在于根据拉格朗江和汉密尔顿机械学开发出微弱的PGD问题配方,主要目标是设计数字稳定、能源保守的数值方法。我们表明,这种方法允许人们考虑以Galerkin为基础的PGD版本,并以数字方式表明,以汉密尔顿配方为基础的PGD解答器比拉格朗江配方具有更好的稳定性和节能特性。两种配方的性能通过几个数字例子加以说明和比较,说明一维条形条的动态行为。