A gradient continuous smoothed GFEM (SGFEM) is proposed to solve the heat transfer and thermoelasticity problem. The SGFEM is based on the idea of the PU method, whose composite shape function is composed of an element shape function and a local nodal shape function. The higher-order finite-element shape function is employed to ensure the continuity of the gradient between the elements. The local nodal shape function is obtained by introducing the gradient smoothed meshfree shape function in the Taylor expansion of the nodal function. The composite shape function satisfies many valuable properties such as the Kronecker-delta property, gradient continuity; unit decomposition and linear independence. More importantly, the SGFEM retains the Kronecker-delta property and is independent of the meshfree approximation. All these properties guarantee the excellent performance of the proposed method in practical examples. Four typical numerical examples including steady, transient heat transfer and thermoelasticity are calculated by SGFEM together with three other common numerical methods including the finite-element method with triangular element (FEM-T3), the finite-element method with quadrilateral element (FEM-Q4), and the node-based finite-element method with triangular element (NSFEM-T3). All examples demonstrate significant advantages of the SGFEM in accuracy, error convergence rate, stability and efficiency.

提出了梯度连续平滑GFEM（SGFEM）来解决传热和热弹性问题。SGFEM 基于 PU 方法的思想，其复合形状函数由单元形状函数和局部节点形状函数组成。采用高阶有限元形状函数来保证单元间梯度的连续性。通过在节点函数的泰勒展开中引入梯度平滑的无网格形状函数，得到局部节点形状函数。复合形状函数满足许多有价值的性质，如 Kronecker-delta 性质、梯度连续性；单元分解和线性独立。更重要的是，SGFEM 保留了 Kronecker-delta 属性并且独立于无网格近似。所有这些特性保证了所提出的方法在实际例子中的优异性能。SGFEM 计算稳态、瞬态传热和热弹性四个典型数值实例，并结合三角单元有限元法 (FEM-T3)、四边形单元有限元法 (FEM-Q4) 等三种常用数值方法进行计算)，以及基于节点的三角形单元有限元方法 (NSFEM-T3)。所有例子都展示了 SGFEM 在准确性、误差收敛速度、稳定性和效率方面的显着优势。瞬态传热和热弹性由 SGFEM 与其他三种常用数值方法一起计算，包括三角形单元有限元法 (FEM-T3)、四边形单元有限元法 (FEM-Q4) 和基于节点的三角单元有限元法 (NSFEM-T3)。所有例子都展示了 SGFEM 在准确性、误差收敛速度、稳定性和效率方面的显着优势。瞬态传热和热弹性由 SGFEM 与其他三种常用数值方法一起计算，包括三角形单元有限元法 (FEM-T3)、四边形单元有限元法 (FEM-Q4) 和基于节点的三角单元有限元法 (NSFEM-T3)。所有例子都展示了 SGFEM 在准确性、误差收敛速度、稳定性和效率方面的显着优势。