Engineering Analysis with Boundary Elements 119 (2020) 1–12 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound A novel space-time generalized FDM for transient heat conduction problems Jun Lei a,∗ , Qin Wang a , Xia Liu a , Yan Gu b , Chia-Ming Fan c a Department of Engineering Mechanics, Beijing University of Technology, Beijing 100124, China b School of Mathematics and Statistics, Qingdao University, Qingdao 266071, China c Department of Harbor and River Engineering & Computation and Simulation Center, National Taiwan Ocean University, Keelung 20224, Taiwan a r t i c l e i n f o Keywords: Transient heat conduction Heterogeneous Space-time discretization GFDM a b s t r a c t In this paper, a novel space-time generalized finite difference method (GFDM) is proposed for solving transient heat conduction problems by integrating a direct space-time discretization technique into the meshless GFDM. The spatial and temporal dimensions are simultaneously discretized by randomly distributed nodes in the coupled space-time continuum. Transient heat conduction in homogenous and heterogeneous materials are analyzed by this novel meshless space-time GFDM. Examples involving multidimensional spatiotemporal domains and various complex boundary conditions are studied and discussed. The high accuracy and efficiency of this new algorithm are verified by comparing with analytical and other numerical results. 1. Introduction The transport problem involving heat conduction and mass diffu- sion is a typical kind of time-dependent problems occurring widely in the nature and many engineering aspects. The parabolic partial differ- ential equations (PDEs) are commonly employed for describing this de- veloping process in mathematical form. Generally, only linear problems with simple geometries and boundary conditions may lead to analytic results under certain initial conditions. So many works turn to develop effective numerical techniques for solving the time-dependent problems. The standard numerical procedures for time-dependent PDEs are based on semi-discretization, in which the finite difference methodology is employed in time scale and the element discretization in space dimen- sions [1]. As Hulbert and Hughes [2] pointed out, there are two main dis- advantages associated with the semi-discrete approaches. One is the dif- ficulty in designing algorithms to accurately capture discontinuities or sharp gradients in the solution, the other one is hard to achieve an adap- tive local refined mesh for heterogeneous structures. That is because the semi-discrete approach only consists of rectangular subdomains in the space-time manifold. Accordingly, a novel space-time discretization technique was proposed by Nickell and Sackman [3], in which the spa- tial and temporal domains were simultaneously discretized. Inspired by this new idea, various kinds of space-time numerical methods have been developed. The space-time finite element method (FEM) was first proposed by Argyris and Scharpf [4], Fried [5] and Oden [6]. Their finite element ∗ Corresponding author. E-mail address: leijun@bjut.edu.cn (J. Lei). formulations were derived based on Hamilton’s principle for dynamics and elastodynamics. Then, the time-continuous Galerkin FE formula- tions were proposed by directly using the differential equations or vari- ational formulation, in which the unknown quantities were assumed to be continuous with respect to time [7–9]. In contrast to the con- ditional stability of this time-continuous Galerkin technique, the time- discontinuous Galerkin method can lead to stable and higher-order ac- curate ordinary differential equation solvers, which is stemmed from the differential equation viewpoint. It employs finite-element discretiza- tions in space and time simultaneously with bilinear basis functions that are continuous in space and discontinuous in time. Accordingly, the time-discontinuous Galerkin FE technique has been widely used to solve various dynamic problems. For example, it has been successfully applied to the fluid dynamics for linear time-dependent multidimen- sional advective-diffusive systems by Hughes et al [10], the elastody- namic problems involving second-order hyperbolic equations by Hul- bert and Hughes [2,11], the transient structural acoustic problem with time-dependent radiation boundary conditions by Thompson and Pin- sky [12], and the fluid-structure interaction problems by Tezduyar et al [13]. Recently, this time-discontinuous Galerkin technique was further incorporated into the isogeometric analysis for parabolic evolution prob- lems by Langer et al [14]. The direct space-time discretization approach was also associated with the traditional boundary element method (BEM). For example, Ha- Duong et al [15] set up a new system of retarded potential boundary integral equations by using the actual space-time boundary elements. The applications to scattering problems showed its stability in a wide https://doi.org/10.1016/j.enganabound.2020.07.003 Received 5 May 2020; Received in revised form 4 July 2020; Accepted 5 July 2020 0955-7997/© 2020 Elsevier Ltd. All rights reserved.