In this two-part paper, a stable and efficient nodally-integrated reproducing kernel particle method (RKPM) is introduced for solving the governing equations of generalized thermomechanical theories. Part I investigates quadrature in the weak form using coupled and uncoupled classical thermoelasticity as model problems. It is first shown that nodal integration of these equations results in spurious oscillations in the solution many orders of magnitude greater than pure elasticity. A naturally stabilized nodal integration is then proposed for the coupled equations. The variational consistency conditions for nth order exactness and convergence in the two-field problem are then derived, and a uniform correction on the test function approximations is proposed to achieve these conditions. Several benchmark problems are solved to demonstrate the effectiveness of the proposed method. In the sequel, these methods are developed for generalized thermoelasticity and generalized finite-strain thermoplasticity theories of the hyperbolic type that are amenable to efficient explicit time integration.

在这篇由两部分组成的论文中，介绍了一种稳定有效的节点积分再生核粒子方法 (RKPM)，用于求解广义热力学理论的控制方程。第一部分使用耦合和非耦合经典热弹性作为模型问题研究弱形式的正交。首先表明，这些方程的节点积分导致解中的虚假振荡比纯弹性大许多数量级。然后为耦合方程提出自然稳定的节点积分。n的变分一致性条件然后推导出二域问题中的 th 阶精确性和收敛性，并建议对测试函数近似值进行统一校正以实现这些条件。解决了几个基准问题以证明所提出方法的有效性。在后续中，这些方法被开发用于适用于有效显式时间积分的双曲线类型的广义热弹性和广义有限应变热塑性理论。