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In this study, we explored the effect of an active afterheater on the growth of gallium oxide single crystals using the EFG method. We analyzed the temperature distribution of the crystal under the growing process through multiphysics simulations of the models with and without an active afterheater and investigated the morphology of crystals by applying each model to real experimental growths. The afterheater is a component in the growing furnace that activates radiant heat transfer, and its performance depends on its location, size, material, and shape. The simulation results showed that the afterheater applied in this study was found to be effective in obtaining good temperature distribution in the reactor. Through experimental crystal growth corresponding to the simulation approaches, it was confirmed that an appropriate afterheater reduces thermal stress at the growth front and provides a thermal annealing effect on the post-grown crystals during the growing process to improve crystal quality.

The projector-augmented wave (PAW) method is used to calculate electric field gradients (EFG) for various PAW potentials. A variety of crystals containing reactive nonmetal, simple metal, and transition elements, are evaluated in order to determine the predictive ability of the PAW method for the determination of nuclear quadrupole resonance frequencies in previously unstudied materials and their polymorphs. All results were compared to experimental results and, where possible, to previous density functional theory (DFT) calculations. The EFG at the 14N site of NaNO2 is calculated by DFT for the first time. The reactive nonmetal elements were not very sensitive to the variation in PAW potentials, and calculations were quite close to experimental values. For the other elements, the various PAW potentials led to a clear spread in EFG values, with no one universal potential emerging. Within the spread, there was agreement with other ab initio models.

For solving a partial different equation by a numerical method, a possible alternative may be either to use a mesh method or a meshless method. A flexible computational procedure for solving 1D linear elastic beam problems is presented that currently uses two forms of approximation function (moving least squares and kernel approximation functions) and two types of formulations, namely the weak form and collocation technique, respectively. The approximations functions constructed in continuous or in discrete way are used as approximations of the strong forms of partial differential equations (PDEs), and those serving as approximations of the weak forms of PDEs to set up a linear system of equations to reproduce Element Free Galerkin (EFG) and Smooth Particle Hydrodynamics (SPH) meshless methods. To approximate the strong form of a PDE, the partial differential equation is usually discretized by specific collocation technique. The SPH is a representative method for the strong form collocation approach. To approximate the weak form of a PDE, Galerkin weak formulation is used.

An equilibrium Element-Free Galerkin (EFG) based formulation for limit analysis of rigid-perfectly plastic plane strain problems is presented. In the formulation pure stress fields are approximated using a moving least squares technique, and a stabilized conforming nodal integration scheme is used in combination with the collocation method, ensuring that the equilibrium equations only need to be fulfilled at the nodes and instability problems can be eliminated. The von Mises yield criterion is enforced by introducing second-order cone constraints, ensuring that the resulting optimization problem can be solved using efficient interior-point solvers. Finally, the efficacy of the procedure is demonstrated by applying it to a benchmark Prandtl problem.

Electric field gradients (EFG’s) at the In sites and spin magnetic moments at the Ce sites were calculated for the case of solid CeIn3. The calculations were performed by increasing pressure gradually from -5 to +22 GPa within the density functional theory (DFT) using the augmented plane waves plus local orbital (APW+lo) method employing the well-known PBE-GGA+U and WC-GGA+U schemes. The results almost show a linear reduction of spin magnetic moments of Ce by gradually increasing the pressure from -5 GPa to 22 GPa. However, from our results one can see that the calculated electric field gradients at the In site are growing up by increasing the pressure. We have compared the EFG’s at zero pressure with experimental and theoretical results of the others. The comparison shows that at ambient pressure our EFG’s are more consistent with experiment than the results of the other groups. Our result shows that the calculated spin magnetic moments are suppressed in the vicinity of some quantum critical point.

The main idea of meshless methods is to approximate the unknown field by a linear combination of shape functions built without having recourse to a mesh of the domain. The computational domain is discretized using a set of scattered nodes. The shape functions associated with a given node is then built considering the weight functions whose support overlaps the one of the weight function of this node; thus, there is actually no need to establish connectivities between the different nodes as in the finite element method. Monte-Carlo integration techniques are promising schemes in the context of meshless techniques. The purpose of the present paper is to implement in EFG a new body integration technique for the evaluation of the stiffness matrix that does not rely on a partition of the domain into cells, but rather points. Numerical examples based on three-dimensional elasticity problems are presented to examine the accuracy and convergence of the proposed method. In this context, Quasi-Monte Carlo integration techniques are used. The results are compared to traditional EFG. Conclusions are drawn concerning the proposed techniques and its capabilities.