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One of the interesting and practical problems in thermo-fluid sciences is referred to finding the shape of a boundary on which a specific distribution of pressure, temperature or heat flux is known. Because solving such problems using experimental, semi-experimental and analytical methods is time-consuming or even impossible in some practical situations, myriad numerical methods have been introduced to solve surface shape design (SSD) problems. In all the numerical algorithms, an initial guess is modified through a numerical process until the desirable distribution of the target variable is achieved. All the numerical algorithms use three computational tools, i.e. grid generator, flow solver and shape updater to solve an SSD problem. In most of numerical algorithms, not only the three mentioned tools work separately but the shape updater is also not derived from the governing equations. In this article, to solve SSD problems containing convection heat transfer, a new shape design algorithm called direct design method is presented in which grid generator, flow solver and shape updater work simultaneously and also the shape updater is directly derived from the governing equations. Some SSD problems containing convection heat transfer in which instead of the boundary shape the distribution of the heat flux is known are solved using the proposed algorithm. The obtained results show the capability of the method in solving SSD problems containing internal convection heat transfer.

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In this paper, the diffusion coefficient in a normal tissue and tumor are to be estimated by the method of inverse problems. At the beginning, distribution of drug (with the assumption of uniform and isentropic diffusion coefficient) in the tissue is considered as the direct problem. In the direct problem, the governing equation is the convection–diffusion, which is the generalized form of fick’s law. Here, a source and a sink are defined; the source as the rate of solute transport per unit volume from blood vessels into the interstitial space and the sink as the rate of solute transport per unit volume from the interstitial space into lymph vessels are added to this equation. To solve the direct problem, the finite difference method has been considered. Additionally, the diffusion coefficient of a normal tissue and tumor will be approximated by parameter estimation method of Levenberg-Marquardt. This method is based on minimizing the sum of squared errors which in the present study, considered error is the difference of the estimated concentration and the concentration measured by medical images (simulated numerically). Finally, the results obtained by Levenberg-Marquardt method have provided an acceptable estimation of diffusion coefficient in normal tissue and tumor.

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It is clear that, having a exact knowledge about the geometry and properties of the materials and the domain that engineering problems are involved are very important specially in structural health monitoring, geotechnical earthquake engineering and other related field in civil engineering; in many cases, it might be useful if a suitable inverse solution is applied in order to detect the characteristics of the problems domain.
The main purpose of this paper is to development of the hybrid finite element- finite difference method for solving inverse elastodynamic and elastostatic scattering problems and combining that with particle swarm optimization algorithm as a quantitative approach fo solving these types of the problems. This hybrid method has been used in order to preparing the forward solution of the problems and by defining a suitable cost function and minimizing that using PSO algorithm, various kind of inverse problems are solved.
In general, an inverse scattering problem can be solved using qualitative or quantitative approaches. In some branches of quantitative techniques, usually, a forward solution is required and then using heuristic algorithm, the goal will be achieved. In this study, a hybrid FE-FD method is used as forward solver (which has the flexibility of finite element method and low computational cost of finite difference method); so, the domain inside and outside of the inclusion will be dicretized using finite difference method and the boundaries near the inclusion will be discretized by finite element method, and in this condition, the solution will be more flexible near the scatterer. In each solution step, first the finite element will be solved and the results will be transferred to the finite difference code and when the result is prepared in it, again, the response of the problem will go to finite element region.
In this research, at first, a geometry and related location will be assumes, randomly and then regarding that, using an OpenSees program code, the boundaries of the inclusion will be discretized and using the MATLAB program the related to finite difference region is discretized, then the results from these two codes will go and back until the response goes converge. Then, the PSO code which is developed in MATLAB will qualify the results and evaluate the cost function (e.g., the cost function is defined by minimizing the the error between the displacement that is from the main model and the predicted model), and if the cost function is large, the PSO algorithm will propose the new location and/or geometry of the inclusion and again, the loop will be repeated until the cost function be near the zero and the solution procedure will be terminated.
In order to evaluate, the efficiency and accuracy of the proposed approach, several problems are solved, where this algorithm could find the location and geometry of the inclusions (e.g., regular and irregular inclusion), the non-homogeneity of the inclusion and also detecting the soil layers by both static and dynamic loading.; the results show a very good accuracy as well as efficiency of the proposed approach for solving inverse problems in bounded and smi-infinite domains.

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In this paper, solution of inverse problems in a plane linear elastic bodies are investigated. For this purpose, sampling method in frequency domain is introduced for cavity/crack detection in a structural element such as plate. This method is categorized as a qualitative approach to image the geometrical features of unknown targets. This goal is followed by partitioning the investigated region into an arbitrary grid of sampling points, in which a linear equation is solved. The main idea of the linear sampling method is to search for a superposition of differential displacement fields which matches with a prescribed radiating solution of the homogeneous governing equation in Ω(D), for each sampling point. Although this method has been used in the context of inverse problems such as acoustics, and electromagnetism, there is no specific attempt to apply this method to identification of crack/cavities in a structural component. This study emphasizes the implementation of the sampling method in the frequency domain using spectral finite element method. A set of numerical simulations on two-dimensional problems is presented to highlight many effective features of the proposed qualitative identification method.