We propose an efficient formulation using the framework of Generalized Finite Element Method (GFEM) for the solutions of transient heat diffusion problems having multiple heat sources in the solution domain. The purpose here is to use minimal computational resources and rely on coarse mesh grids to capture the sharp variations of temperature field. We use Gaussian functions of global nature to enrich the GFEM approximation space which ensure efficient solution in the whole solution domain. To capture steep thermal gradients at multiple locations, a multiplicity of enrichment functions is used and the peaks of enrichment functions are centred at the cores of heat sources. The advantage of this approach is that no further degrees of freedom (DOFs) are added to capture the solution at multiple locations. Besides, the enrichment functions are time-independent; the temporal variation of temperature is embedded in the definition of the enrichment functions. This formulation requires the assembly of system matrix once and only the right-hand side of the system of equations is updated at subsequent time steps which results in a significant reduction in the overall computational time. We consider problems in two-dimensional (2D) and three-dimensional (3D) domains to show the effectiveness of the proposed approach. A reduction of more than 95% in DOFs and more than 80% in computational time is achieved as compared to the h-version of finite element method.

我们提出使用广义有限元方法（GFEM）框架的有效解决方案，用于解决在解决方案域中具有多个热源的瞬态热扩散问题。这里的目的是使用最少的计算资源，并依靠粗糙的网格来捕获温度场的急剧变化。我们使用具有全局性质的高斯函数来丰富GFEM逼近空间，从而确保在整个求解域内都能高效求解。为了在多个位置捕获陡峭的热梯度，使用了多种富集函数，并且富集函数的峰值位于热源的中心。这种方法的优势在于，无需添加更多自由度（DOF）即可在多个位置捕获解决方案。除了，富集功能与时间无关；温度的时间变化被嵌入到富集函数的定义中。这种表述需要一次组装系统矩阵，并且仅在随后的时间步长更新方程组系统的右侧，这导致整体计算时间的显着减少。我们考虑二维（2D）和三维（3D）域中的问题，以证明所提出方法的有效性。与之相比，DOF减少了95％以上，计算时间减少了80％以上。这种表述需要一次组装系统矩阵，并且仅在随后的时间步长更新方程组系统的右侧，这导致整体计算时间的显着减少。我们考虑二维（2D）和三维（3D）域中的问题，以证明所提出方法的有效性。与之相比，DOF减少了95％以上，计算时间减少了80％以上。这种表述需要一次组装系统矩阵，并且仅在随后的时间步长更新方程组系统的右侧，这导致整体计算时间的显着减少。我们考虑在二维（2D）和三维（3D）域中的问题，以证明所提出方法的有效性。与之相比，DOF减少了95％以上，计算时间减少了80％以上。h-有限元方法的版本。