The scaled boundary finite element method based on the hybrid quadtree mesh for solving transient heat conduction problems

Inverse heat conduction analysis provides an efficient approach for estimating the thermophysical properties of materials, the boundary conditions, or the initial conditions. In this paper, two-dimensional transient nonlinear inverse heat conduction problems are investigated, for estimating time- and space-dependent boundary heat flux, as well as the temperature-dependent thermal conductivities. Modifications are carried out to extend the previous one-dimensional inversion algorithm to solve the two-dimensional transient nonlinear heat conduction problem, to overcome the frequently occurring divergence issues, and to improve the stability of the inversion algorithm. Boundary-only measurements are used as additional information, and a dimensionless objective function is adopted. In the direct problem, formulations for solutions to the two-dimensional transient nonlinear heat conduction problem are derived and validated. Numerical examples show that the inversion algorithm is effective, efficient, accurate, and robust, for recovering multiple parameters, with and without a functional form.

On the basis of reproducing kernel particle method(RKPM), the complex variable reproducing kernel particle method (CVRKPM) is discussed. The advantage of the CVRKPM is that the correction function of a 2-D problem is formed with 1-D basis function when the shape function is obtained. Then, we apply the complex variable method to two-dimensional transient heat conduction problems. In combination with the Galerkin weak form of transient heat conduction problems, the penalty method is employed to enforce the essential boundary conditions, the CVRKPM for transient heat conduction problems is investigated and the corresponding formulae are obtained. Compared with the conventional RKPM, the CVRKPM introduced in this paper has a higher precision and a lower computation cost. Some examples given in this paper verify the effectivity of the proposed method.

The numerical manifold method for 2D transient heat conduction problems in functionally graded materials

This paper presents a technique to evaluate the fracture parameters by combining the edge based smoothed finite element method (ESFEM) and the scaled boundary finite element method (SBFEM). A semi-analytical solution is sought in the region close to the vicinity of the crack tip using the SBFEM, whilst, the ESFEM is used for the rest of the domain. As both methods satisfy the partition of unity and the compatibility condition, the stiffness matrices obtained from both methods can be assembled as in the conventional finite element method. The stress intensity factors (SIFs) are computed directly from their definition. Numerical examples of linear elastic bodies with cracks are solved without requiring additional post-processing techniques. The SIFs computed using the proposed technique are in a good agreement with the reference solutions. A crack propagation study is also carried out with minimal local remeshing to show the robustness of the proposed technique. The maximum circumferential stress criterion is used to predict the direction of propagation.

A novel approach to solve inverse heat conduction problems: Coupling scaled boundary finite element method to a hybrid optimization algorithm

The dynamic response of the host structure to a high-frequency actuation is usually used for the detection of tiny damage in structures in the form of breathing crack. The simulation of the microcrack’s effect on the response is essential for several damage identification targets. The conventional finite element method suffers from very small mesh size requirements to address the high-frequency problems, resulting in very large mass and stiffness matrices. In this study, the scaled boundary finite element method was applied to model different schemes of structural health monitoring of a structure with breathing cracks based on high-frequency vibration. The scaled boundary finite element method discretizes only the boundary of the model and thus substantially reduces the size of structural matrices. The node-to-node contact strategy was introduced to the scaled boundary finite element method to capture the contact problem that occurs during the vibration of the breathing crack. As breathing crack vibration results in some nonlinear effects, the simulation of three phenomena was of interest: higher harmonic generation, frequency shift, and vibro-acoustic modulation. A shooting method was used for efficient time integration and description of the frequency response function in the nonlinear regime. According to the results, the scaled boundary finite element method is of great power, efficiency, and accuracy to treat the contact problems, especially in high-frequency regimes. Moreover, the nonlinear methods provide certain advantages over the linear techniques in the early detection of incipient damage.

Rapid urbanization and climate change have intensified urban flood risks, necessitating resilient upstream infrastructure to ensure metropolitan water security and effective flood mitigation. Gravity dams, as critical components of urban flood protection systems, regulate discharge to downstream urban areas. Gravity dams are critical for regulating flood discharge, yet their seismic vulnerability poses significant challenges, particularly under compound effects involving concurrent seismic loading and climate-induced elevated reservoir levels. This study introduces a novel seismic analysis framework for gravity dams using the scaled boundary finite element method (SBFEM), which efficiently models dam–water and dam–foundation interactions in infinite domains. A two-dimensional numerical model of a concrete gravity dam, subjected to realistic seismic loading, was developed and validated against analytical solutions and conventional finite element method (FEM) results, achieving discrepancies as low as 0.95% for static displacements and 0.21% for natural frequencies. The SBFEM approach accurately captures hydrodynamic pressures and radiation damping, revealing peak pressures at the dam heel during resonance and demonstrating computational efficiency with significantly reduced nodal requirements compared to FEM. These findings enhance understanding of dam behavior under extreme loading. The proposed framework supports climate-adaptive design standards and integrated hydrological–structural modeling. By addressing the seismic safety of flood-control dams, this research contributes to the development of resilient urban water management systems capable of protecting metropolitan areas from compound climatic and seismic extremes.

In this paper, a coupled model based on finite element method (FEM), boundary element method (BEM) and scaled boundary FEM (SBFEM) (also referred to as the consistent infinitesimal finite element cell method) for dynamic response of 2D structures resting on layered soil media is presented. The SBFEM proposed by Wolf and Song (Finite-element Modelling of Unbounded Media. Wiley: England, 1996) and BEM are used for modelling the dynamic response of the unbounded media (far-field). The standard FEM is used for modelling the finite region (near-field) and the structure. In SBFEM, which is a semi-analytical technique, the radiation condition at infinity is satisfied exactly without requiring the fundamental solution. This method, also eliminates the need for the discretization of interfaces between different layers. In both SBFEM and BEM, the spatial dimension is decreased by one. The objective of the development of this coupled model is to combine advantages of above-mentioned three numerical models to solve various soil-structure interaction (SSI) problems efficiently and effectively. These three methods are coupled (FE-BE-SBFEM) via substructuring method, and a computer programme is developed for the harmonic analyses of SSI systems. The coupled model is established in such a way that, depending upon the problem and far-field properties, one can choose BEM and/or SBFEM in modelling related far-field region(s). Thus, BEM and/or SBFEM can be used efficiently in modelling the far-field. The proposed model is applied to investigate dynamic response of rigid and elastic structures resting on layered soil media. To assess the proposed SSI model, several problems existing in the literature are chosen and analysed. The results of the proposed model agree with the results presented in the literature for the chosen problems. The advantages of the model are demonstrated through these comparisons.

The scaled boundary finite element method (SBFEM) is a semianalytical computational scheme based on the characteristics of the finite element method (FEM) and boundary element method that combines their respective advantages. In this paper, the SBFEM and polygonal mesh technique are integrated into a new approach to solve steady-state and transient seepage problems. The proposed method is implemented in Abaqus employing a user-defined element (UEL). A detailed implementation of the procedure is presented in which the UEL element is defined, the internal variables RHS and AMATRX are updated, and the stiffness/mass matrix is solved using eigenvalue decomposition. Several benchmark problems are solved to verify the proposed implementation. The results show that the polygonal element of the polygonal SBFEM (PSBFEM) is more accurate than the standard FEM element of the same element size. For transient problems, the results for the PSBFEM and FEM are in excellent agreement. Hence, the proposed method is robust and accurate for solving steady-state and transient seepage problems. The developed UEL source code and the associated input files can be downloaded from GitHub.

In this paper an analysis is carried out to study the efficiency of heat distribution through fins using finite element method (FEM) and differential quadrature method ( DQM). Numerical solutions are obtained using both methods for temperature distribution over the fin surface. Analysis of surface solutions in terms of temperature and error distributions have been presented here. Solutions are also obtained using equally spaced (conventional) and non-equally spaced (optimum) mesh (nodal points) distribution techniques. Two-dimensional heat conduction problem has been solved in both equal and non-equal nodal points distribution cases (called here conventional FEM (CFEM) and conventional DQM (CDQM) and optimum FEM (OFEM) and optimum DQM (ODQM)). The obtained results are compared and investigated with exact results. The best results are found in OFEM solutions. It is also found that, ODQM solutions reach very close to OFEM solutions with negligible error, whereas CFEM and CDQM solutions diverges with increasing number of nodal(mesh) points.

The scaled boundary finite element method (SBFEM) is a novel semi-analytical technique. combining the advantage of the finite element method (FEM) and the boundary element method (BEM) with unique properties of its own. In this paper, the SBFEM is used for computing wave passing a submerged breakwater, and the reflection coefficient and the transmission coefficient are given for the case of wave passing by a rectangular submerged breakwater and a rigid submerged barrier breakwater in a constant water depth. The results are compared with the analytical solution and experimental results. Good agreement is observed. Through compared with the results using the dual boundary element method (DBEM), it is found that the SBFEM could obtain the higher accuracy with fewer elements. Many submerged breakwaters with different demission are computed by the SBFEM, and the changing character of the reflection coefficient and the transmission coefficient are given in the current study.

A crack propagation modelling technique combining the scaled boundary finite element method and quadtree meshes is developed. This technique automatically satisfies the compatibility requirement between adjacent quadtree cells irrespective of the presence of hanging nodes. The quadtree structure facilitates efficient data storage and rapid computations. Only a single cell is required to accurately model the stress field near crack tips. Crack growth is modelled by splitting the cells in the mesh into two. The resulting polygons are directly modelled by the scaled boundary formulation with minimal changes to the mesh. Four numerical examples demonstrate the salient features of the technique.

A combined virtual element method and the scaled boundary finite element method for linear elastic fracture mechanics

This article employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in one- and two-dimensional transient heat conduction problems. The relationship between the CHNN synaptic connection weights and the governing equations of the problems is established and a corresponding network connectivity circuit design scheme proposed. The CHNN algorithm is initially applied to the solution of conventional problems and is then used to solve more complicated problems involving time-varying heat flux profiles. The results confirm that the CHNN scheme provides an accurate means of solving the transient temperature distributions of heat conduction problems on a real-time basis.

The scaled boundary finite element method (SBFEM) is a semi-analytical method, whose versatility, accuracy, and efficiency are not only equal to, but potentially better than the finite element method and the boundary element method for certain problems. This paper investigates the possibility of using an efficient high-order polynomial element in the SBFEM to form the approximation in the circumferential direction. The governing equations are formulated from the classical linear elasticity theory via the SBFEM technique. The scaled boundary finite element equations are formulated within a general framework integrating the influence of the distributed body source, mixed boundary conditions, contributions the side face with either prescribed surface load or prescribed displacement. The position of scaling center is considered for modeling problem. The proposed method is evaluated by solving two-dimensional linear problem. A selected set of results is reported to demonstrate the accuracy and convergence of the proposed method for solving problems in general boundary conditions.