Abstract A new and effective computational approach is presented for analyzing transient heat conduction problems. The approach consists of a meshless Fragile Points Method (FPM) being utilized for spatial discretization, and a Local Variational Iteration (LVI) scheme for time discretization. Anisotropy and nonhomogeneity do not give rise to any difficulties in the present implementation. The meshless FPM is based on a Galerkin weak-form formulation and thus leads to symmetric matrices. Local, very simple, polynomial and discontinuous trial and test functions are employed. In the meshless FPM, Interior Penalty Numerical Fluxes are introduced to ensure the consistency of the method. The LVIM in the time domain is a combination of the Variational Iteration Method (VIM) and a collocation method in each finitely large time interval. The present methodology represents a considerable improvement to the current state of science in computational transient heat conduction in anisotropic nonhomogeneous media. In this first part of the two-paper series, we focus on the theoretical formulation and implementation of the proposed methodology. Numerical results and validation are then presented in Part II.

中文翻译：

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一种新的无网格“脆弱点法”和各向异性非均匀介质中一般瞬态热传导的局部变分迭代法。第一部分：理论与实现

摘要 提出了一种新的有效的计算方法来分析瞬态热传导问题。该方法包括用于空间离散化的无网格脆弱点方法 (FPM) 和用于时间离散化的局部变分迭代 (LVI) 方案。各向异性和非均匀性在本实施中不会引起任何困难。无网格 FPM 基于 Galerkin 弱形式公式，因此导致对称矩阵。使用局部的、非常简单的多项式和不连续的试验和测试函数。在无网格 FPM 中，引入了内部惩罚数值通量以确保方法的一致性。时域中的LVIM是Variational Iteration Method（VIM）和每个有限大时间间隔内的搭配方法的结合。本方法代表了对各向异性非均匀介质中计算瞬态热传导的科学现状的显着改进。在两篇论文系列的第一部分中，我们专注于所提出方法的理论制定和实施。数值结果和验证随后在第二部分中介绍。