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In this research, three dimensional grain burn-back of solid rocket motors is simulated based on level set method and its accuracy is increased according to marching cube algorithm (MCA). To that end and according to requirements of level set method, grain burn-back is simulated during three steps included grid generation, distance function determination, and calculation of burn-back parameters. In this article, with focus on last step, we will compare strengths and weaknesses of grain burn-back analysis for common methods such as captured cell, three dimensional cut cell, sectional, and Heaviside Delta Dirac and in following, we introduce and run MCA. In order to validation, firstly three simple grains such as cylindrical, quad and hexahedron are considered and the performance of capture cell, 3D cut cell and sectional methods are compared with MCA in terms of accuracy and CPU time. Then, to evaluate the new method facing complex and practical grains, burn-back results of conocyle and NAWC N.13 grains is compared with MCA and heaviside/dirac delta methods. The obtained results show that MCA has a better performance at CPU time and accuracy.

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One of obstacles in simulation of two phase flow is parasite currents. These currents cause unphysical distortion at interface which impairs interface capturing and numerical results. In present study, two methods (using Filter and s-CLSVOF) are implemented in OpenFOAM two phase flow solver called interFoam to reduce parasite current. 3 filters are added to color function volume of fluid (CF-VOF) method. These filters reduce parasite current in different ways, one smoothes color function, one smoothes curvature and the other one compresses the interface. The original and the modified solvers are tested with a quiescent bubble bench mark to investigate the effect of each filter on parasite currents. Then optimum arrangement of filters is compared with s-CLSVOF method and interFoam. Present study shows parasite current magnitude can be reduced at least up to 50% in the modified solvers. Also, the comparison of pressure jump from numerical results and analytical result with Young-Laplace equation shows modified solvers can predict pressure jump better than original solver. The pressure jump error is reduced up to 400% in the modified solvers. Also present study shows filters have better performance than s-CLVOF method and it can be considered as a suitable substitution of coupled methods.

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Extended finite element method (X-FEM) has been recently emerged as an approach to implicitly create a discontinuity based on discontinuous partition of unity enrichment (PUM) of the standard finite element approximation spaces. Despite numerous progresses in mesh generating updating of finite element mesh during crack propagation remain extremely heavy and difficult. This problem becomes more complicate, when there are many discontinuities in the finite element domain. However, the extended finite element method (X-FEM) in the combination with level set method (LSM) could overcome this cumbersome issue. In this contribution, predefined cracks and internal boundaries are created using level set functions and also the effects of soft/hard inclusions (interfaces) and voids are considered on crack propagation schemes. In fact, the interaction of crack and heterogeneities are considered. The level set functions are utilized to represent the locations and the evolutions of internal interfaces. In addition, the stress intensity factors for mixed mode crack problems are numerically calculated by using the interaction integral method. Different crack growth paths are simulated automatically for different oriented edge and center cracks and the interactions of internal boundaries on crack propagations are shown. All numerical examples are demonstrated the flexibility and capabilities of X-FEM in the applied fracture mechanics.

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The encounter between bubble pairs can be happened in the bubble flows and may result in coalescence, which is one of the most important elementary physical processes occurring in liquid columns. Sufficient knowledge of the coalescence process of two bubbles can lead to a better description of the bubbly flow’s behavior. Effects of uniform magnetic fields on the interactions and coalescence of dielectric bubbles were not studied up to now; therefore in this research, interactions and coalescence of two bubbles in a viscous stagnant liquid has been simulated numerically. Considered bubbles are spherical and fluids are stagnant, initially. Both liquid and gas phases considered being incompressible and dielectric where applied magnetic field is uniform. In the numerical simulation of the problem, the Finite Volume method was applied using the SIMPLE algorithm to discretizing the governing equations while the finite difference method was used for discretizing of the magnetic field equation. For simulating the interface of two phases, the level set method has been incorporated. The results outlined in the present study well agree with the existing experimental and numerical results. Obtained results show that applied uniform magnetic field affects shape, dynamics and also interactions and coalescence of bubble pairs. Applied magnetic field enhances coalescence between in-line rising bubbles. Therefore, the external uniform magnetic field could be used for contactless control of the coalescence process between bubbles.

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Numerical simulation of multi material or multi-phase flows are one of the most challenging problems between computational fluid dynamics researches. The main difficulty of these problems is producing some unexpected and non-physical oscillation at material interface which causes entering some error in to computation domain. For eliminating this source of error, many sophisticated algorithm have been proposed recently. By neglecting diffusion processes, Euler equations and HLLC reimann solver are applied. In addition, Level set algorithm is implemented to track interferences between two materials. An accurate, easily developed and low computation cost algorithm, proposed by Abgrall and Karni, is used to prevent generating the oscillations in the interfaces. In the current work, the algorithm is developed to 2 dimensional algorithm. Afterwards, the result of 1 and 2 dimensional code are evaluated to verify the developed algorithm by some standard problems such as sod problem. Finally, shock –bubble (Air – Helium) interaction problem is simulated to investigate the effect of the algorithm in 2 dimensional simulation. The comparison shows that the code and its result have very good accuracy with very low computational cost.

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In this research, interaction and oblique coalescence of bubbles under buoyancy force was simulated, numerically. The governing equations are continuity and momentum equations which have been discretized using the finite volume method and the SIMPLE algorithm. For simulating the interface of two phases, the level set method has been incorporated. Level Set method suffers from poor mass conservation of dispersed phase especially in the case of severe deformation of interface. In order to control of mass conservation of the level set method, re-initialization equations and a geometric mass control loop are used which this loop is implemented in the level set method for the first time in this research. Using proposed geometric mass control loop, mass dissipation drawback of the level set method is handled in simulation of bubbles’ coalescence. The results outlined in the present study well agree with the existing experimental results. Also results of investigation of mass dissipation of the proposed scheme to simulation of oblique coalescence problem show that the maximum amount of this mass dissipation was less than 4%. Therefore, the level set method with proposed geometric mass control loop could be used properly for simulation of oblique interactions and coalescence of bubbles in multiphase flows.

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Shape and topology optimization have become one of the main researches that is widely used in engineering fields. The purpose of topology optimization is to find an appropriate (optimal) distribution of materials in the design domain so that the shape and number of voids is optimized and the objective function is minimized or maximized. In recent decades, noticeable researches and various topology optimization methods were proposed. The level set method is being used successfully in structural shape and topology optimization. This method is an implicit method for moving interior and exterior boundaries, while these boundaries may join together during the process and new voids may be formed. The structural boundary is illustrated by the zero level set and nonzero in the domain. In the above context, the level set function is used as a switch to distinguish between the two domains present in the computing space. This way of illustration has an important feature by which the domain boundaries can be combined together or divided. By using the solution of Hamilton-Jacobi equation resulting from this function, the domain’s boundary starts to move. The control over movement of this boundary is done by velocity vector of Hamilton-Jacobi equation. Now, in order to use this method in topology optimization, it is sufficient to establish a relationship between velocity vector of Hamilton-Jacobi and shape derivation, which is used for optimizing objective function. It is possible to use standard level set for structural topology optimization.
In this paper, the spherical Hankel basis functions are used to optimize the structural topology using the level set method. The proposed functions are a combination of the first and second kind of Bessel functions fields as well as the polynomial ones in complex space and are derived from radial basis functions. Using the spherical Hankel functions, the dependence of the function of the level set method on the space and time is separated, which results in the transformation of the Hamilton-Jacobian partial differential equation into a conventional differential equation. In this way, the difficulties arising from solving partial differential equations are eliminated, and thus there is no need to re-set the function of the level set method in the optimization process. Further, in order to increase the speed and precision of convergence in creating an optimal design, the classic Lagrange shape functions are replaced with the spherical Hankel ones. The proposed shape functions have some properties such as infinite piecewise continuity, the Kronecker delta property, and the partition of unity. Moreover, since they satisfy all three polynomial fields and the first and second kind of Bessel ones in the complex space, they can be effective in improving the accuracy and speed of convergence, while the classic Lagrange shape functions are able to satisfy only the polynomial function fields. Finally, several numerical examples are presented to study the performance of the spherical Hankel radial basis and shape functions.
 

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Grain design is the most important part of a solid rocket motor. The aim of this study is finocyl grain design based on predetermined objective function with respect to ballistic curves in order to satisfy various thrust performance requirements through an innovative design approach using a genetic algorithm optimization method. The classical sampling method has been used for design space-filling. The level set method has been used for simulating the evolution of the burning surface in the propellant grain. An algorithm has been developed beside the level set code that prepares the initial grain configuration using Pro/Engineer software to export generated models to level set code. The lumped method has been used to perform internal ballistic analysis. Two meta-models are used to surrogate the level set method in the optimization design loop. The first method is based on adaptive basis function construction and the second method is based on the artificial neural network. In order to validate the proposed algorithm, a grain finosyl sample has been investigated. The results show that both grain design method reduced the design time significantly and this algorithm can be used in designing of any grain configuration. In addition, data have more accuracy in grain design based on the artificial neural network, so this method is the more effective and practical method to grain burn-back training.

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 This research focuses on topology optimization of fluid-structure interaction (FSI) problems using the level set method. To couple the fluid and structure equations, the Arbitrary Lagrangian-Eulerian (ALE) description is employed within a monolithic formulation. The use of ALE in FSI problems, while eliminating numerical instabilities caused by the convective term, enhances the speed and accuracy of finite element solutions in fluid-structure interaction. Additionally, considering the fluid in the unsteady state allows for the interpretation of optimal topology at any given moment of the analysis. The objective function of the optimal topology design problem is to minimize the structural compliance in the dry state, subject to a fixed volume of the design domain. To determine the normal velocity in the reaction-diffusion equation (RDE), adjoint sensitivity analysis based on pointwise gradients is used. The results obtained from this approach, compared to other topology optimization methods in the literature, demonstrate higher accuracy and clearer definition of structural boundaries.