What affects the performance of POD for the simulation of heat transfer through building component?

The capability of the proper orthogonal decomposition for the simulation of heat transfer in building components is investigated via three applications: linear heat transfer (not coupled to mass transfer), mildly non-linear heat transfer (coupled to air and moisture transfer (hygroscopic)) and highly non-linear heat transfer (coupled to moisture transfer (capillary)). It is shown that increasing non-linearity leads to an increasing number of required construction modes. To further investigate the reason for this degrading performance of POD, the singular values’ decay progress from the different training snapshots is addressed in this paper. The results confirm that a fast decay of the singular values implies a high interrelation of the snapshots and a better performance of the POD method.

Error Constants for the Semi-Discrete Galerkin Approximation of the Linear Heat Equation

In this paper, we propose $$L^2(J;H^1_0(\Omega ))$$ L 2 ( J ; H 0 1 ( Ω ) ) and $$L^2(J;L^2(\Omega ))$$ L 2 ( J ; L 2 ( Ω ) ) norm error estimates that provide the explicit values of the error constants for the semi-discrete Galerkin approximation of the linear heat equation. The derivation of these error estimates shows the convergence of the approximation to the weak solution of the linear heat equation. Furthermore, explicit values of the error constants for these estimates play an important role in the computer-assisted existential proofs of solutions to semi-linear parabolic partial differential equations. In particular, the constants provided in this paper are better than the existing constants and, in a sense, the best possible.

Estimation of Welding-Induced Plastic Zone Size and Residual Stress Levels: Linear Heat Input Approximation

Welding is a highly nonlinear temperature distribution process, where the presence of high-temperature gradients leads to the development of significantly high residual stress levels, up to and/or beyond the material yield strength magnitude and localized plastic deformation. To achieve the desired dimensional accuracy, determination of plastic zone size, shape, and location is critically important in reducing or controlling final distortions, decreasing the residual stress according to length scale, and defining the optimum sequence of the welding process. The plastic zone caused by welding has been found to be directly proportional to linear heat input, defined in (J/mm). The use of actual linear heat input in the estimation of welding-induced residual stress in finite element models often results in an overestimation of heat transferred to the fusion zone of the metal. This manuscript highlights the importance of estimating plastic zone, developed during thermal processes like welding, and its role in mitigating final distortion by using a 3-bar model for the determination of final residual stresses. In the second part, previously developed analytical linear heat input solution for 2D residual stress models is discussed and further demonstrated using examples from open literature. Lastly, a sequentially uncoupled thermal and thermo-mechanical finite element analysis (FEA) is performed, using a generalized plane strain element, and concluded by validation of the numerically developed plastic zone size with analytically developed solutions.

Understanding the thermal problem of variable gradient functionally graded plate based on hybrid numerical method under linear heat source

The thermal problem of functionally graded materials (FGM) under linear heat source is studied by a hybrid numerical method. The accuracy of the analytical method and the efficiency of the finite element method are taken into account. The volume fraction of FGM in the thickness direction can be changed by changing the gradient parameters. Based on the weighted residual method, the heat conduction equation under the third boundary condition is established. The temperature distribution of FGM under the action of linear heat source is obtained by Fourier transform. The results show that the closer to the heat source it is, the greater the influence of the heat source is and the influence of the heat source is local. The temperature change trend of the observation points is consistent with the heat source, showing a linear change. The results also show that the higher the value of gradient parameter is, the higher the temperature of location point is. The temperature distribution of observation points is positively correlated with gradient parameter. When the gradient parameter value exceeds a certain value, it has a little effect on the temperature change in the model and the heat conduction in the model tends to be pure metal heat conduction, the optimal gradient parameters combined the thermal insulation property of ceramics and the high strength toughness of metals are obtained.