The **leak** **before** **break**(LBB) concept is difficult to apply to a structure with a thin tube that is immersed in a water environment. A heat exchanger in a nuclear power plant is such a structure. The present paper addresses an **application** of the LBB concept to a heat exchanger in a nuclear power plant. The minimum leaked coolant amount containing the radioactive material which can activate the radiation detector device installed in near the heat exchanger is assumed. The postulated initial flaw size that can not grow to the critical flaw size within the time period to activate the radiation detector is justified. In this case, the radiation detector can activate the warning signal caused by coolant leakage from initially postulated flaws of the heat exchanger. The nuclear plant can safely shutdown when this occurs. Since the postulated initial flaw size can not grow to the critical flaw size, the structural integrity of the heat exchanger is not impeded. Particularly the informational scenario presented in this paper discusses an actual nuclear plant.

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Ignalina NPP contains RBMK-1500 type reactor. RBMK reactor is graphite-moderated with a water-cooled reactor core. The fuel cell assembly is located in the centre of the moderator column and consists of a pressure tube into which the fuel **element** assembly is inserted and through which the coolant flows. As a constructional material for manufacturing of pressure tubes zirconium alloys Zr + 2.5 % Nb are used. The hydrogen absorbed by zirconium alloy during corrosion process is one of the factors determining lifetime of pressure tube. When hydrogen concentration in pressure tube exceeds solubility limit initiation and development of delayed hydride cracking (DHC) can take place. The formation of hydrides under certain conditions can reduce resistance to brittle fracture. In this work the evaluation of the influence of hydrides on the fracture of pressure tube and **application** of **leak** **before** **break** (LBB) for these tubes with DHC was performed. The deterministic analysis of the pressure tube employing LBB concept was carried out using experimental data. Performed deterministic LBB analysis confirms that the pressure tube comply with LBB requirements.

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A **finite** **element** model which utilizes the extended **finite** **element** **method** (XFEM) to model leaks through cracks is presented in this paper. Preliminary calculations using Ecrevisse in Code Aster indicates there is a case to include thermo-mechanical effects in **Leak**-**before**-**Break** analysis. This is because the **leak** rate can reduce due to crack closure effects. The preliminary work motivated the development of a specific **finite** **element** which incorporated a **leak** rate model and was coupled to the structure through a convection law and pressure boundary condition. Convergence studies were performed on the XFEM thermal model to validate its suitability and optimum convergence rates were seen. The mechanical model was validated by comparing the **finite** **element** approximated crack opening displacement with analytical solutions and the errors were less than 1% for a relative **element** size of more than 0.01. Thermo-mechanical simulations were then carried out using a simple 2D plate with a central crack. The **leak** rate was shown to decrease by about 30% when the fluid was 10 o C hotter than the

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Abstract. Smoothed **Finite** **Element** **Method** (SFEM) was introduced by **application** of the stabilized conforming nodal integration in the conventional nite **element** **method**. In this **method**, integration is performed on \smoothing domains" rather than elements. Smoothing domains are created based on cells, nodes, or edges for two-dimensional problems. Based on the smoothing domain creation **method**, dierent types of SFEM are developed that have dierent properties. It has been shown that these methods are insensitive to mesh distortion and are generally more computationally ecient than mesh- free and nite **element** methods for the same accuracy level. Because of their interesting features, they have been used to solve dierent problems. This paper investigates the performance of these methods in coupled hydro-mechanical (consolidation) analysis by solution to some problems using a developed SFEM/FEM code. Biot's consolidation theory is reviewed, and after introduction of the idea and formulation of SFEMs, discretized form of equations is given. Requirements for creation of stable coupled hydro-mechanical models are discussed and based on them, two methods for creation of stable SFEM models are introduced. To investigate the eectiveness of the methods, a number of examples are solved and results are compared with the nite **element** and analytical ones.

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Structure of this paper can be summarized as follows. Section overviews some related methods brieﬂy. In Section we express the model problem as an inequality and Laplacian constrained variational optimization problem. In the same section, we present a theorem which shows the connection between the optimization problem and the second kind vari- ational inequality problems. We discretize the VI problems using a ﬁnite **element** **method**. We complete the paper with a conclusion section where we discuss the **method** presented in this paper and point some possible future extensions.

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The CANDU industry has claimed that **leak** detection systems installed at the CANDU plants are more sensitive compared to those at other types of nuclear plants as a result of the costs associated with upgrading of heavy water and the presence of tritium in the primary coolant. Economic considerations lead to the initiation of shutdowns due to unidentified leaks from the primary heat transport system circuit at rates of less than 360 kg/hr and station procedures are typically more conservative using rates of about 100 kg/hr or less Three different methods for **leak** detection are employed: (i) heat transport system inventory monitored via storage tank level, (ii) vapour recovery systems utilizing drier collection and powerhouse exhaust, and (iii) liquid detection systems using beetles (liquid collection trays) and the D 2 O recovery

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The **finite** **element** **method** is a numerical solution technique. It’s used to solve complex problems. This **method** becomes popular over the last decade. The scientists and engineers have used the **finite** **element** **method** (FEM) for the modeling of the complex problem. This **method** having applicability in many areas of engineering, physical problem (stiffness, density and more) and physics such heat transfer (conduction, convection and radiation), fluid flow, electrical potential problem and many more. **Finite** **element** is a mathematical technique for obtaining approximate numerical solutions to the abstract equations of calculus that predicts the response of physical systems subjected to external influence.

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The limitation of the two flux model is that only the propagation of diffuse intensities, but not of a collimated source can be described, because the loss of intensity of the collimated beam due to scattering will not completely be converted into backward flux, but be dispersed over all directions due to the scattering characteristics of the medium. In the four flux theory the flow of intensity is broken up into a forward and backward collimated flux and F^., and a forward and backward diffuse flux F+ and F.. All four fluxes decrease due to absorption, but scattering causes a permanent conversion from collimated to diffuse fluxes. We will not go deeper into the theory of multiple flux theory here. The main advantage of these models is their simplicity which allows to find analytical solutions in good agreement with experimental data, but they are limited to simple plane-parallel geometries. In the following we concentrate on the diffusion equation on which our **finite** **element** model for light transport is based. Although analytical solutions of the diffusion equation are also restricted to simple cases, numerical approaches allow the **application** to arbitrarily complex and inhomogeneous media.

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The LBB concept was used for the Main Vessel (MV) and pipings of the SPX ([12] to [14]) and PHENIX plants. As far as the MV is concerned, the LBB approach has been applied to verify the absence of risk as regards the core supporting function and to help in the definition of the In-Service Inspection (ISI) program. Generally speaking, the low level of membrane primary stresses, which is favourable for the integrity of the vessel, makes the **application** of LBB more difficult due to small crack opening areas.

The test specimen is filled with water introducing a cover gas space in the upper portion of the straight vertical pipe attached with the Tee. By connecting the cover gas space to the constant nitrogen pressure tanks, the required internal pressure is sustained in the specimen, even under leaked condition in the course of test (Fig.4). The required bending moment is imposed on the specimen by applying cyclic vertical force at the end of the straight horizontal pipe welded to the Tee. Servo controlled hydraulic actuators with computerized control system is used for the **application** of cyclic loads.

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If the piping system fails to satisfy either the Level 1 acceptance criteria or all of the elements of the Level 1 specific screening criteria, the applicant’s next logical step would be to try to demonstrate LBB using the Level 2 approach. As an illustrative example, when a Level 1 analysis was applied to an actual surge line (using data gleaned from an actual LBB submittal), it was found that the Level 1 margin on crack size was less than 2.0, i.e., the critical crack size was less than twice the postulated leakage crack sizes, [16]. As such, this piping system failed to meet one of the acceptance criteria for a Level 1 **application** 1 . However, when this same piping system was analyzed using the Level 2 criteria, it was found that the resultant margin on crack size was approximately 3, which easily satisfies this **element** of the acceptance criteria.

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The **leak**-**before**-**break** (LBB) analysis of pressurized piping and vessels, which is developed during the past three decades, is an important rule for assessing pressure vessels,especially nuclear equipment. LBB is an advanced technology for the safety and dependability of structures in nuclear reactors, as well as in the petrochemical industry. This paper introduced the concept of **leak**-**before**-**break** with its analytical process, and enumerating the some LBB analytical methods,then introduces the theories of Fuzzy reliability to evaluation the reliability of structures. LBB technology has a good prospect in development and **application**, which is of great importance to the modern industry.

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In this paper, an efficient numerical model for solution of the two-dimensional unsteady dam- **break** problem is described. The model solves the shallow water equations through Characteristic-Based Split (CBS) **finite** **element** **method**. The formulation of the model is based upon the fractional time step technique primarily used in the **finite** difference **method** for the incompressible Navier-Stokes equations. In addition to well-known advantages of the **finite** **element** discretization in introducing complex geometries and making accurate results near the boundaries, the CBS utilizes interesting advantages. These include the ability of the **method** to simulate both compressible and incompressible flows using the same formulation. Improved stability of the CBS algorithm along with its capability to simulate both sub- and super-critical flows are other main advantages of the **method**. These useful advantages of the algorithm introduce the CBS as a unique procedure to solve fluid dynamics problems under various conditions. Since dam-**break** problem has principally a high non-linear nature, the model is verified firstly by modeling one-dimensional problems of dam-**break** and bore formation problems. Furthermore, **application** of the model to a two-dimensional hypothetical dam-**break** problem shows the robustness and efficiency of the procedure. Despite the high non-linearity nature of the solved problems, the computational results, compared with the analytical solutions and reported results of other numerical models, indicate the favorable performance of the used procedure in modeling the dam-**break** problems.

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Abstract: **Finite** **element** **method** is a popular computer aided numerical **method** based on the discretisation of the domain, structure or continuum into number of elements and obtaining the solution. It converts a infinite number of degree problem to a **finite** number of degree problem using discretization. On the other hand, **finite** strip **method** is of semi-numerical and semi-analytical nature. The **finite** strip **method** is applicable to problems which may have complex geometry in their cross-section, but are simple along the length. the **application** of this **method** has been extended to thin and thick rectangular and annual sector plates, cylindrical shells, straight and curved box girders and folded plates etc. In this paper, the behaviour of plate subjected to different loading condition with various boundary conditions is studied.

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The use of **Leak** **Before** **Break** (LbB) arguments is well established in the nuclear industry. A 'detectable leakage' LbB procedure, similar to that of NUREG 1061, is commonly used. Such a procedure is included in the UK R6 defect assessment procedures and involves achieving an adequate margin between the evaluated critical crack length and the detectable leakage crack length. However, experience in applying LbB methods with the effects of creep included in the analysis is still limited. This paper contains results of calculations aimed at assessing how these influences impact upon the safety margins achieved in a low temperature **application**.

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The authors proposed an elasto-plastic COD assessment **method** applicable to thin wall pipes with large diameter made of modified 9Cr-1Mo steel [4] based on the conventional GE/EPRI **method** [5]. Parametric **finite** **element** analyses (FEA) were conducted to determine the coefficients using the **finite** **element** nonlinear structural analysis system “FINAS” [6]. For the material which has a small work hardening coefficient, such as modified 9Cr-1Mo steel, COD produced **before** fully plastic condition is relatively large. Therefore, the local plastic COD should be taken into account explicitly in the assessment. The local plastic COD δ LP was calculated using the following

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Further improvements are foreseen in order to calculate thoroughly the COA, at internal and external sides, and by considering inelastic deformations. Recent development of eXtended **Finite** **Element** techniques shows quite interesting results as it allows accounting for complex geometries of cracks and structures.

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It should be pointed out that the works mentioned above all gave one order accuracy in time increment Δt. That is to say, the first order characteristic **method** in time was analyzed. As for higher order characteristic **method** in time, Rui and Tabata 12 used the second order Runge-Kutta **method** to approximate the material derivative term for convection- diﬀusion problems. The scheme presented was of second order accuracy in time increment Δt, symmetric, and unconditionally stable. Optimal error estimates were proved in the framework of L 2 -theory. Numerical analysis of convection-diﬀusion-reaction problems with

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(leading to mass conservation and flux continuity) and in particular the approach from [39] for a posteriori error estimates in mixed methods without flux continuity. We first build a flux reconstruction that is globally H(div, Ω)-conforming and locally conservative in each mesh **element**. In a first stage, a simple coarse balancing problem, with one unknown per interface and two unknowns (in two space dimensions) per each subdomain boundary lying in 𝜕Ω, is solved. Then we adopt the construction of [39, Section 3.5.2] and solve a local Neumann problem in a band around the interfaces in each subdomain by the mixed **finite** **element** **method**. Finally, two 𝐻 0 1 (Ω)-conforming potential reconstructions are built. One

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