In [13], we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ2u=f in a cracked domain whose crack faces have clamped boundary conditions. In this paper, we extend our investigation to the derivation of the SIF extraction formulas of the biharmonic equation in non-convex polygonal domains containing cracks or reentrant corners when various BC such as clamped (CC), simply supported (SS), free (FF), or mixed conditions (CS, FC, SF) are imposed on the boundary. We prove that the SIF is expressed as the integral of fΨsk⁎−uΔ2Ψsk⁎ for a cut-off function Ψ and a dual singular function sk⁎ on a small neighborhood of the singularity. The dual singular function is determined in this paper. For a numerical approximation of u, we proposed an iteration method as well as a direct finite element method. For a finite element solution of Δ2u=f, we have to use C1-continuous basis functions. For continuous differentiable basis functions, we use either C1-continuous B-spline basis functions or the conventional Hermite basis function. Because of several advantages of B-spline functions in imposing complex boundary conditions, we choose B-spline basis functions for the finite element approximation of u. Moreover, in the direct numerical method, we propose to use implicitly enriched basis functions that resemble the singularities.

We consider an optimal control problem for the Poisson equation on a non-convex polygonal domain with the corner singularity. Previously, we proposed a novel algorithm for the accurate numerical solution for the Poisson equation on a polygonal domain with the domain singularity. Then, we investigated the error estimate and its efficient procedure for the numerical algorithm. In this article, we propose an efficient algorithm and perform an error estimate for a distributed optimal control problem of the Poisson equation. The solutions of the optimality system with such singularity have singular decompositions: regular part plus singular part for each state variable and adjoint variable. The coefficient of the singular function is usually called stress intensity factor and can be computed by the extraction formula. We introduced a modified optimality system which has "zero" stress intensity factors using this stress intensity factor, from whose solutions we can compute very accurate solution of the original optimality system simply by adding a singular part. We give a precise error analysis and provide numerical results which justify the results therein.

Studies on detoxification of asbestos and recovery of valuable metals from detoxified asbestos were conducted. First, asbestos was detoxified by a microwave heat treatment that used silicon carbide balls, which are inorganic heating elements that absorb microwaves and release heat at room temperature. For efficient heat treatment, the asbestos containing waste (ACW) was powdered by crushing and grinding processes and then a detoxification heat treatment was performed. Microwave heat treatment temperature and time variables were adjusted to investigate the detoxification properties according to heat treatment conditions. After heat treatment, treated ACW was analyzed for detoxification properties through crystal structure and microstructure analysis using X‐ray diffraction (XRD) and scanning electron microscopy. Complete asbestos detoxification was confirmed when the microwave heat treatment was performed at 1200°C for over 60 min and at 1300°C for over 10 min. Second, recovery of valuable metals from detoxified asbestos was carried out. The main components, Si, Ca, and Mg, of detoxified asbestos‐containing waste (DACW) were separated and recovered in the form of SiO 2 , CaSO 4 , and Mg(OH) 2 . SiO 2 was separated first after treatment of DACW with HCl. Ca in the form of sulfate, CaSO 4 , was recovered when subsequent treatment of remaining aqueous acid solution with H 2 SO 4. The remaining Mg was recovered by precipitation as Mg(OH) 2 form under strong basic conditions. Each separated component was confirmed by XRD and Inductively Coupled Plasma (ICP) analysis.

In [8, 9] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with corner singularities, compute the finite element solutions using standard Finite Element Methods and use the extraction formula to compute the stress intensity factor(s), then they posed new PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor(s), which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. The error analysis was given in [5]. In their approaches, the singular functions and the extraction formula which give the stress intensity factor are the basic elements. In this paper we consider the biharmonic problems with the cramped and/or simply supported boundary conditions and get the singular functions and its duals and find properties of them, which are the cornerstones of the approaches of [8, 9, 10].

We derive formulas extracting stress intensity factors of the biharmonic equations on cracked domains with clamped (or simply supported or free) boundary conditions along the crack faces. Each of these formulas can be written in terms of the integral of the given source function multiplied by the cut-off dual singularity and the integral of the unknown true solution multiplied by the cut-off dual singularity over the unit disk. The unknown true solution in the extraction formulas is calculated by either the Implicitly Enriched Galerkin Method or the Iteration Methods. The former was developed by the authors (Kim, Oh, Palta, Kim) and the latter proposed in this paper is the sum of the solution of a regular biharmonic equation and singular functions with iteratively estimated stress intensity factors as coefficients. We show the Iteration Methods quickly converge and the proposed Enrichment Method yields highly accurate stress intensity factors. We also demonstrate that for a known true solution, the extraction formulas yield exact stress intensity factors.

A 1kW-class horizontal axis wind turbine (HAWT) rotor blade is taken into account to investigate elastic characteristics in 2-D. The elastic blade field is composed of symmetric cross-ply laminated composite material. Blade element momentum theory is applied to obtain the boundary conditions pressuring the blade, and the plane stress elasticity problem is formulated in terms of two displacement parameters with mixed boundary conditions. For the elastic characteristics a fair of differential equations are derived based on the elastic theory. The domain is divided by triangular and rectangular elements due to the complexity of the blade configuration, and a finite element method is developed for the governing equations to search approximate solutions. The results describe that the elastic behavior is deeply influenced by the layered angle of the middle laminate and the stability of the blade can be improved by controlling the layered angle of laminates, which can be evaluated by the mathematical approach.

An algorithm on computing accurate finite element approximation to the Poisson equation on a polygonal domain with corner singularities was studied in Kim and Lee (2016, 2017) numerically. The algorithm requires several iterations depending on singularities of the solution. Each iteration requires a solution of the standard finite element approximation to the Poisson equation with possible different Dirichlet data and the corresponding stress intensity factors. This paper provides an error estimate of the finite element approximation given by the algorithm, and, hence, determine the number of iterations needed to achieve full rates of convergence in both the energy and the L2 norms.

. In [7, 8] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous boundary conditions, compute the finite element solutions using standard FEM and use the extraction formula to compute the stress intensity factor(s), then they posed new PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor(s), which converges with optimal speed. From the solution they could get an accu- rate solution just by adding the singular part. Their algorithm involves an iteration and the iteration number depends on the acuracy of stress intensity factors, which is usually obtained by extraction formula which use the finite element solutions computed by standard Finite Element Method. In this paper we investigate the dependence of the iteration number on the convergence of stress intensity factors and give a way to reduce the iteration number, together with some numerical experiments.

In [8] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities. They consider the Poisson equations with homogeneous Dirichlet boundary condition with one corner singularity at the origin, and compute the finite element solution using standard FEM and use the extraction formula to compute the stress intensity factor, then pose a PDE with a regular solution by imposing the non-homogeneous boundary condition using the computed stress intensity factor, which converges with optimal speed. From the solution they could get an accurate solution just by adding the singular part. This approach uses the polar coordinate and the cut-off function to control the singularity and the boundary condition. In this paper we consider Poisson equations with multiple singular points, which involves different cut-off functions which might overlaps together and shows the way of cording in FreeFEM++ to control the singular functions and cut-off functions with numerical experiments.

In this article, we consider the Poisson equation on a polygonal domain with the domain singularity raised from the changed boundary conditions with the inner angle omega > pi/2. The solution of the Poisson equation with such singularity has a singular decomposition: regular part plus singular part. The singular part is a linear combination of one or two singular functions. The coecients of the singular functions are usually called stress intensity factors and can be computed by the extraction formula. In [11] we introduced a new partial di erential equation which has 'zero' stress intensity factor using this stress intensity factor, from whose solution we can obtain a very accurate solution of the original problem simply by adding singular part. Although the method in [11] works well for the Poisson problem with Dirichlet boundary condition, it does not give optimal results for the case with stronger singularity, for example, mixed boundary condition with bigger inner angle. In this paper we give a revised algorithm which gives optimal convergences for both cases.

In [15] they introduced a new finite element method for accurate numerical solutions of Poisson equations with corner singularities, which is useful for the problem with known stress intensity factor.They consider the Poisson equations with homogeneous Dirichlet boundary condition, compute the finite element solution using standard FEM and use the extraction formula to compute the stress intensity factor, then they pose a PDE with a regular solution by imposing the nonhomogeneous boundary condition using the computed stress intensity factor, which converges with optimal speed. From the solution we could get accurate solution just by adding the singular part. This approach works for the case when we have the accurate stress intensity factor.In this paper we consider Poisson equations with mixed boundary conditions and show the method depends the accrucy of the stress intensity factor by considering two algorithms.

In this article, we consider the Poisson equation with homogeneous Dirichlet boundary conditions, on a polygonal domain with one reentrant corner. The solution of the Poisson equation with a concave corner yields a singular decomposition, u=w+ληs, where w is regular, s is a singular function, and the coefficient λ is the so called stress intensity factor. This stress intensity factor can be computed using the extraction formula. We introduce a new non-homogeneous boundary value problem, which has ‘zero’ stress intensity factor. Using the solution of this new partial differential equation, we can compute an accurate solution of the original problem, simply by adding singular part. We obtain an optimal convergence rate with smaller errors when compared with others.

The thermoelastic behaviors of such as temperature distribution, displacements, and stresses in thermal barrier coatings (TBCs) are seriously influenced by top coat thickness and edge conditions, which were investigated based on the thermal and mechanical properties of plasma-sprayed TBCs. A couple of governing partial differential equations were derived based on the thermoelastic theory. Since the governing equations are too involved to solve analytically, a finite volume method was developed to obtain approximations. The thermoelastic characteristics of TBCs with the various thicknesses and microstructures were estimated through mathematical approaches with different edge conditions. The results demonstrated that the top coat thickness and the edge condition in theoretical analysis are crucial factors to be considered in controlling the thermoelastic characteristics of plasma-sprayed TBCs.

A graded layer was introduced at the interface between the top and bond coats to reduce the risk of failure in a thermal barrier coating (TBC) system, and the thermoelastic behavior was investigated through mathematical approaches. Two types of TBC model with and without the graded layer, subject to a symmetric temperature distribution in the longitudinal direction, were taken into consideration to evaluate thermoelastic behaviors such as temperature distribution, displacement, and thermal stress. Thermoelastic theory was applied to derive two governing partial differential equations, and a finite volume method was developed to obtain approximations because of the complexity. The TBC with the graded layer shows improved durability in thermoelastic characteristics through mathematical approaches, in agreement with the experimental results. The results will be useful in discovering technologies for enhancing the thermomechanical properties of TBCs.

The thermoelastic characteristics of plasma-sprayed thermal barrier coatings (TBCs) have been analyzed using mathematical modeling. Two types of TBC model, cylinder and circular disk which are commercial plasma-sprayed TBCs, subjecting to symmetric temperature distribution to the radial and longitudinal directions, respectively, were taken into consideration. Based on the thermoelastic theories, a second order ordinary differential equation was derived for the cylinder model and a pair of partial differential equations were set up for the circular disk model. The analytic solution was obtained from the ordinary differential equation, while a finite volume method was developed for numerical solutions to the pair of partial differential equations due to the complexity of governing equations. The thermoelastic characteristics of TBC models, such as temperature distributions, displacements, and stresses, were displayed according to the obtained solutions. The rate of heat conduction in the section of the top coat is relatively slow in comparison with the substrate, and no profound difference appears in the temperature distribution between two TBC models. The highest longitudinal tensile stress is expressed at the bond coat of both models, and the substrate is under the compressive stresses to the circumferential direction. While the cylinder expands to the positive longitudinal direction only, the expansion in the circular disk occurs to both the positive and negative longitudinal directions. Relatively large displacement and stresses exhibit in the cylinder as compared with the circular disk. In the circular disk, the stresses to the radial direction undulate at each section, and the displacement profile displays that the width of the circular disk is slightly narrowed. The results demonstrate that the mechanical and thermal properties of the top and bond coats are the crucial factors to be considered in controlling the thermoelastic characteristics of plasma-sprayed TBCs.

A 1kW-class horizontal axis wind turbine (HWAT) rotor blade is taken into account to investigate elastic characteristics in 2-D. The elastic blade field is composed of symmetric cross-ply laminated composite material. Blade element momentum theory is applied to obtain the boundary conditions pressuring the blade, and the plane stress elasticity problem is formulated in terms of two displacement parameters with mixed boundary conditions. For the elastic characteristics a fair of differential equations are derived based on the elastic theory. The domain is divided by triangular and rectangular elements due to the complexity of the blade configuration, and a finite element method is developed for the governing equations to search approximate solutions. The results describe that the elastic behavior is deeply influenced by the layered angle of the middle laminate and the stability of the blade can be improved by controlling the layered angle of laminates, which can be evaluated by the mathematical approach. EQUATIONS, TABLES AND FIGURES Blade element momentum theory is developed under the assumptions: (i) No aerodynamic interactions appear between different blade elements, (ii) The forces on the blade elements are solely determined by the lift and drag coefficients, (iii) the flow is frictionless, which gives the axial force x F rdr a a U dFx   2 )] 1 ( 4 [ 2 1 2    , (1a) the tangential force  F dr r U a a dF 2 ) 1 ( ' 4       , (1b) and the tangential force T dr r U a a dT 3 ) 1 ( ' 4      [1, pp. 41]. (1c) By the considering of a symmetric cross-ply laminated composite field the average stressstrain relations in principal material coordinates in y x  plane (see Figure. 1) for a lamina can be formulated as

The demilitarization of TNT, DNT and ammonium picrate in Russia and the West is producing million pounds of surplus energetic materials. These have been disposed by open burning/detonation. The use of open burning/ detonation is becoming unacceptable due to public concern and environmental regulations. Therefore, the chemical conversion study of surplus energetic materials to higher value products would be highly desirable. The West have used TNT as a versatile starting material for the design of different types of valuable products. During our research on conversion of TNT and DNT into high valuable raw materials in fine chemical industry, we reported synthesis of Phloroglucinol from TNT. Herein we describe a synthesis of nitroaryl halides from TNB by substitution of nitro group with halogen atom. The introduction of halogen atoms into aromatic compounds is one of the difficult operations in organic chemistry. The conventional methods proceeded by the Sandmeyer reaction are unsatisfactory from the point of view of industrial application. Another substitution method promoted by potassium fluoride from nitroaryl compound showed the unsatisfactory result (Figure 1). On the other hand, the acid halide-aided substitution reactions of nitroaryl compounds with KF have been received great attention in the field of fluorine chemistry due to relatively easy introduction of fluorine atom. However, only KF have been utilized for this transformation and this conversion method is still required to improve the yield of product. We examined the reactions of TNB prepared from TNT with LiF, NaF, KF and CsF in the presence of phthaloyl chloride and sulfolane. As expected, the use of CsF gave good yield of the 1-fluoro-3,5-dinitrobenzene (1) and 1,3difluoro-5-nitrobenzene (2), while the use of KF, NaF and LiF gave poor yield or no products (Table 1, Figure 2). In case of X− (X: halogens) in the protic solvents, nucleophilicity increases going down the periodic table; for example, nucleophilicity decreases in the order I− > Br− > Cl− > F−. However, in case of X− in the aprotic solvents such as DMSO, DMF and sulfolane, nucleophilicity decreases in the order F− > Cl− > Br− > I−. Because the cations associated with nucleophilic anions are strongly solvated in aprotic solvents, the anions are dissociated from the cations, which further enhance their nucleophilicity. In the presence of protic solvent, presumed intermediate was decomposed by protic solvent (Figure 3). Therefore, we got fluorobenzene compounds easier than other aromatic halogen compounds. To evaluate the scope of this system, the range of metal salts was extended to various metal halides. Among the different metal halides, chlorides and bromides were found to afford a only 1-chloro-3,5-dinitrobenzene (3) and 1bromo-3,5-dinitrobenzene (4) respectively with poor yield (Table 1, Figure 2). It was also observed that reaction of iodine metal salts did not give any product. Therefore, the aryl fluoride is more easily prepared than other aryl halides. In the presence of protic solvent, the

During spray coating, especially in an air plasma spray (APS), pores, cracks, and splat boundaries are developed and those factors exert influence on thermomechanical properties such as elastic modulus, thermal conductivity, and coefficient of thermal expansion. Moreover, the thermo mechanical properties are crucial elements to determine the thermoelastic characteristics, for instance, temperature distribution, displacements, and stresses. Two types of thermal barrier coating (TBC) model, the dense and porous microstructures, are taken into account for the analysis of microstructural characterizations. TriplexPro TM -200 system was applied to prepare TBC samples, and the METECO 204 C-NS powder is adopted for the relatively porous microstructure and METECO 204 NS powder for the dense microstructure in the top coat of TBCs. Governing partial differential equations were derived based on the thermoelastic theory and approximate estimates for the thermoelastic characteristics were obtained using a finite volume method for the governing equations.

Thermoelastic characteristics of thermal barrier coatings (TBCs) with vertical cracks were analyzed through mathematical approaches to investigate the thermoelastic behaviors of TBCs in a service temperature. TriplexPro™-200 system was applied to prepare the relatively dense TBC using METECO 204NS powder. The microstructure of top coat in the TBC was just controlled to create vertical type cracks by reheating without powder feeding in same equipment and rapid cooling process. A couple of governing partial differential equations were derived based on the thermoelastic theory, and a finite volume model was developed to the governing equations to evaluate the thermoelastic characteristics, such as temperature distribution profile, displacement, and stress, inducing a thermal fatigue. For the specimen with two or more vertical type cracks, smaller displacement appears to longitudinal direction and larger displacement to radial direction as the number of crack increases. In the longitudinal stress distribution profiles to z-direction, the tensile stress at the interface between the bond coat and the substrate converts into the compressive stress when the specimen has vertical cracks more than two, while larger magnitude undulation develops for the specimen with smaller number of crack in the radial stress distribution profiles. The results obtained demonstrate that multiple vertical cracks enhance the thermal durability and extend the lifetime of TBCs.