Galerkin Approach For 1d Heat Conduction

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Kylee Mccandrew

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Aug 3, 2024, 12:03:12 PM8/3/24

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The accurate and efficient analysis of anisotropic heat conduction problems in complex composites is crucial for structural design and performance evaluation. Traditional numerical methods, such as the finite element method (FEM), often face a trade-off between calculation accuracy and efficiency. In this paper, we propose a quasi-smooth manifold element (QSME) method to address this challenge, and provide the accurate and efficient analysis of two-dimensional (2D) anisotropic heat conduction problems in composites with complex geometry. The QSME approach achieves high calculation precision by a high-order local approximation that ensures the first-order derivative continuity. The results demonstrate that the QSME method is robust and stable, offering both high accuracy and efficiency in the heat conduction analysis. With the same degrees of freedom (DOFs), the QSME method can achieve at least an order of magnitude higher calculation accuracy than the traditional FEM. Additionally, under the same level of calculation error, the QSME method requires 10 times fewer DOFs than the traditional FEM. The versatility of the proposed QSME method extends beyond anisotropic heat conduction problems in complex composites. The proposed QSME method can also be applied to other problems, including fluid flows, mechanical analyses, and other multi-field coupled problems, providing accurate and efficient numerical simulations.

In this study, the interpolating element-free Galerkin (IEFG) method for solving three-dimensional (3D) transient heat conduction problem is presented. By using the improved interpolating moving least-squares (IIMLS) method to form the shape function, and using the weak form of 3D transient heat conduction problems to obtain the discretized equations, the formulae of the IEFG method are obtained. The shape function of the IIMLS method satisfies the property of Kronecker delta function, and then the IEFG method can apply essential boundary conditions directly, which can result in higher computational speed and accuracy. Some examples are given to discuss the convergence and advantages of the IEFG method. By analyzing the numerical results obtained by IEFG method and improved EFG method, we conclude that IEFG method has clear advantages in computational speed and accuracy.

N2 - A novel Overset Improved Element-Free Galerkin-Finite Element Method (Ov-IEFG-FEM) for solving transient heat conduction problems with concentrated moving heat sources is introduced in this communication. The method is a mesh- less/mesh-based chimera-type approach that utilises a coarse finite element mesh to discretise the problem geometry, while a separate set of overlapping nodes (patch nodes) moves with the heat source to capture the marked thermal gradients with higher accuracy using the Improved Element-Free Galerkin (IEFG) technique. Outside of the heat source area, where accuracy requirements are significantly lower, the thermal problem is solved using the Finite Element Method (FEM). The approach involves solving the problem over these two overlapping computational domains and transferring numerical information between the approximations performed on both. Such transfer of information occurs through immersed boundaries that are properly defined, enabling straightforward achievement of accurate results. The proposed Ov-IEFG-FEM is conceived to provide an enriched solution by appropriately coupling the temperature fields computed on the patch nodes and the coarse background mesh using IEFG and FEM, respectively. A comprehensive explanation concerning the appropriate coupling between the temperature fields of both the coarse background finite element mesh and the fine arrangement of moving patch nodes for the IEFG computations, is also provided in this communication. Numerical experiments demonstrate the method effectiveness in accurately and efficiently solving transient heat conduction problems with concentrated moving heat sources.

AB - A novel Overset Improved Element-Free Galerkin-Finite Element Method (Ov-IEFG-FEM) for solving transient heat conduction problems with concentrated moving heat sources is introduced in this communication. The method is a mesh- less/mesh-based chimera-type approach that utilises a coarse finite element mesh to discretise the problem geometry, while a separate set of overlapping nodes (patch nodes) moves with the heat source to capture the marked thermal gradients with higher accuracy using the Improved Element-Free Galerkin (IEFG) technique. Outside of the heat source area, where accuracy requirements are significantly lower, the thermal problem is solved using the Finite Element Method (FEM). The approach involves solving the problem over these two overlapping computational domains and transferring numerical information between the approximations performed on both. Such transfer of information occurs through immersed boundaries that are properly defined, enabling straightforward achievement of accurate results. The proposed Ov-IEFG-FEM is conceived to provide an enriched solution by appropriately coupling the temperature fields computed on the patch nodes and the coarse background mesh using IEFG and FEM, respectively. A comprehensive explanation concerning the appropriate coupling between the temperature fields of both the coarse background finite element mesh and the fine arrangement of moving patch nodes for the IEFG computations, is also provided in this communication. Numerical experiments demonstrate the method effectiveness in accurately and efficiently solving transient heat conduction problems with concentrated moving heat sources.

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This research is conducted to investigate heat and mass transport past over a stretched surface having pores in a pseudo-plastic model. To study porosity effect, Darcy Forchheimer relation is used. Thermal and mass transport expressions are derived by engaging the double diffusion theories as extensively used by researchers proposed by Cattaneo and Christov. Furthermore, the thermal performance is studied by mixing the tri-hybrid nanoparticles in a pseudo-plastic material. The phenomenon of boundary layer is used to derive the complex model. The correlation for tri-hybrid nanoparticles is used to convert the model partial differential equations into ordinary differential equations (ODE) along with appropriate similarity transformation. The transfigured ODEs are coupled nonlinear in nature, and the exact solution is not possible. To approximate the solution numerically, finite element scheme (FES) is used and code is developed in MAPLE 18.0 for the graphical results, grid independent survey, and tabular results. The obtained results are compared with the published findings that confirm the accuracy and authenticity of the solution and engaged scheme. From the performed analysis, it is concluded that FES can be applied to complex engineering problems. Furthermore, it is monitored that nanoparticles are essential to boost the thermal performance and higher estimation of Schmidt number control the mass diffusion.

Section 4 reports detailed discussion and description of obtained solution against numerous involved parameters, and Section 5 lists important findings. Developing approach of tri-hybrid nanoparticles is shown in Figure 1.

Step I: Equations (6) and (7) within BCs are called the strong form. It is noticed that collecting all terms of equations (6) and (7) on one side and integrating it over each elements of domain are residuals. Such procedure is known as weighted (residual method) for the development of weak forms. The residuals are as follows:

Step III: The assembly approach is utilized for the development of stiffness element, whereas assembly approach is performed via assembly procedure of FEA. Stiffness elements are as follows:

The comparison among present analysis and published results [33,34,35] are derived in Table 3. It is noticed that the present problem is reduced into published problems [33,34,35] by considering φ a = φ b = φ c = 0 , F r = ε = 0 . A good agreement between the results of the present problem and published works is noticed.

Temperature gradient, Sherwood number, and drag force coefficient are simulated by incorporating influences of Forchheimer number, heat generation number, chemical reaction number, and Sc including the study of ternary hybrid nanoparticles. Impacts related to these parameters on temperature gradient, Sherwood number, and drag force coefficient are simulated as shown in Table 4. Role of F r boosts surface force but the concentration and temperature gradients are declined when F r is increased. H h diminishes the rate of thermal energy, mass diffusion, and surface force. It is estimated that the dual trends (heat absorption and heat generation) are addressed. In both cases for H h are observed significantly for establish maximum production in rate of concentration and temperature gradients. A declination is predicted in view of surface force versus an impact of chemical reaction (generative and destructive) but in terms of Sherwood and Nusselt, numbers are inclined. For the case of Sc , rates of mass diffusion and thermal energy are significantly increased.

This thesis summarizes certain boundary element methods applied to some initial and boundary value problems. Our model problem is the two-dimensional homogeneous heat conduction problem with vanishing initial data. We use the heat potential representation of the solution. The given boundary conditions, as well as the choice of the representation formula, yield various boundary integral equations. For the sake of simplicity, we use the direct boundary integral approach, where the unknown boundary density appearing in the boundary integral equation is a quantity of physical meaning.