Fast numerical method of solving 3D coefficient inverse problem for wave equation with integral data

2018 ◽  
Vol 26 (4) ◽  
pp. 477-492 ◽  
Author(s):  
Anatoly B. Bakushinsky ◽  
Alexander S. Leonov

Abstract An inverse coefficient problem for time-dependent wave equation in three dimensions is under consideration. We are looking for a spatially varying coefficient of this equation knowing special time integrals of the wave field in an observation domain. The inverse problem has applications to the reconstruction of the refractive index of an inhomogeneous medium, as well as to acoustic sounding, medical imaging, etc. In the article, a new linear three-dimensional Fredholm integral equation of the first kind is introduced from which it is possible to find the unknown coefficient. We present and substantiate a numerical algorithm for solving this integral equation. The algorithm does not require significant computational resources and a long solution time. It is based on the use of fast Fourier transform under some a priori assumptions about unknown coefficient and observation region of the wave field. Typical results of solving this three-dimensional inverse problem on a personal computer for simulated data demonstrate high capabilities of the proposed algorithm.

Author(s):  
Leigh Orf

Since the dawn of the digital computing age in the mid-20th century, computers have been used as virtual laboratories for the study of atmospheric phenomena. The first simulations of thunderstorms captured only their gross features, yet required the most advanced computing hardware of the time. The following decades saw exponential growth in computational power that was, and continues to be, exploited by scientists seeking to answer fundamental questions about the internal workings of thunderstorms, the most devastating of which cause substantial loss of life and property throughout the world every year. By the mid-1970s, the most powerful computers available to scientists contained, for the first time, enough memory and computing power to represent the atmosphere containing a thunderstorm in three dimensions. Prior to this time, thunderstorms were represented primarily in two dimensions, which implicitly assumed an infinitely long cloud in the missing dimension. These earliest state-of-the-art, fully three-dimensional simulations revealed fundamental properties of thunderstorms, such as the structure of updrafts and downdrafts and the evolution of precipitation, while still only roughly approximating the flow of an actual storm due computing limitations. In the decades that followed these pioneering three-dimensional thunderstorm simulations, new modeling approaches were developed that included more accurate ways of representing winds, temperature, pressure, friction, and the complex microphysical processes involving solid, liquid, and gaseous forms of water within the storm. Further, these models also were able to be run at a resolution higher than that of previous studies due to the steady growth of available computational resources described by Moore’s law, which observed that computing power doubled roughly every two years. The resolution of thunderstorm models was able to be increased to the point where features on the order of a couple hundred meters could be resolved, allowing small but intense features such as downbursts and tornadoes to be simulated within the parent thunderstorm. As model resolution increased further, so did the amount of data produced by the models, which presented a significant challenge to scientists trying to compare their simulated thunderstorms to observed thunderstorms. Visualization and analysis software was developed and refined in tandem with improved modeling and computing hardware, allowing the simulated data to be brought to life and allowing direct comparison to observed storms. In 2019, the highest resolution simulations of violent thunderstorms are able to capture processes such as tornado formation and evolution which are found to include the aggregation of many small, weak vortices with diameters of dozens of meters, features which simply cannot not be simulated at lower resolution.


2021 ◽  
Author(s):  
Sebastian F. Riebl ◽  
Christian Wakelam ◽  
Reinhard Niehuis

Abstract Turbine Vane Frames (TVF) are a way to realize more compact jet engine designs. Located between the high pressure turbine (HPT) and the low pressure turbine (LPT), they fulfill structural and aerodynamic tasks. When used as an integrated concept with splitters located between the structural load-bearing vanes, the TVF configuration contains more than one type of airfoil with sometimes pronouncedly different properties. This system of multidisciplinary demands and mixed blading poses an interesting opportunity for optimization. Within the scope of the present work, a full geometric parameterization of a TVF with splitters is presented. The parameterization is chosen as to minimize the number of parameters required to automatically and flexibly represent all blade types involved in a TVF row in all three dimensions. Typical blade design parameters are linked to the fourth order Bézier-curve controlled camber line-thickness parameterization. Based on conventional design rules, a procedure is presented, which sets the parameters within their permissible ranges according to the imposed constraints, using a proprietary developed code. The presented workflow relies on subsequent three dimensional geometry generation by transfer of the proposed parameter set to a commercially available CAD package. The interdependencies of parameters are discussed and their respective significance for the adjustment process is detailed. Furthermore, the capability of the chosen parameterization and adjustment process to rebuild an exemplary reference TVF geometry is demonstrated. The results are verified by comparing not only geometrical profile data, but also validated CFD simulation results between the rebuilt and original geometries. Measures taken to ensure the robustness of the method are highlighted and evaluated by exploring extremes in the permissible design space. Finally, the embedding of the proposed method within the framework of an automated, gradient free numerical optimization is discussed. Herein, implications of the proposed method on response surface modeling in combination with the optimization method are highlighted. The method promises to be an option for improvement of optimization efficiency in gradient free optimization of interdependent blade geometries, by a-priori excluding unsuitable blade combinations, yet keeping restrictions to the design space as limited as possible.


2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
S. L. Han ◽  
Takeshi Kinoshita

This paper studies an inverse problem that can be used for reconstructing initial wave field of a nonsteady-state wave propagation. The inverse problem is ill posed in the sense that small changes in the input data can greatly affect the solution of the problem. To address the difficulty, the problem is formulated in the form of an inference problem in an appropriately constructed stochastic model. It is shown that the stochastic inverse model enables the initial surface disturbance to be reconstructed, including its confidence intervals given the noisy measurements. The reconstruction procedure is illustrated through applications to some simulated data for two- and three-dimensional problem.


2011 ◽  
Vol 42 (3) ◽  
pp. 275-293
Author(s):  
S. A. Avdonin ◽  
B. P. Belinskiy ◽  
John V. Matthews

We consider the problem of reconstruction of the potential for the wave equation on the semi-axis. We use the local versions of the Gelfand-Levitan and Krein equations, and the linear version of Simon's approach. For all methods, we reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand-side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples verify that the algorithms can be used to reconstruct an unknown potential accurately. The practicality of each approach is briefly discussed. Accurate data preparation is described and implemented.


2022 ◽  
Vol 55 (1) ◽  
Author(s):  
Adam Lindkvist ◽  
Yubin Zhang

Laboratory diffraction contrast tomography (LabDCT) is a recently developed technique to map crystallographic orientations of polycrystalline samples in three dimensions non-destructively using a laboratory X-ray source. In this work, a new theoretical procedure, named LabXRS, expanding LabDCT to include mapping of the deviatoric strain tensors on the grain scale, is proposed and validated using simulated data. For the validation, the geometries investigated include a typical near-field LabDCT setup utilizing Laue focusing with equal source-to-sample and sample-to-detector distances of 14 mm, a magnified setup where the sample-to-detector distance is increased to 200 mm, a far-field Laue focusing setup where the source-to-sample distance is also increased to 200 mm, and a near-field setup with a source-to-sample distance of 200 mm. The strain resolution is found to be in the range of 1–5 × 10−4, depending on the geometry of the experiment. The effects of other experimental parameters, including pixel binning, number of projections and imaging noise, as well as microstructural parameters, including grain position, grain size and grain orientation, on the strain resolution are examined. The dependencies of these parameters, as well as the implications for practical experiments, are discussed.


Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2342
Author(s):  
Raul Argun ◽  
Alexandr Gorbachev ◽  
Natalia Levashova ◽  
Dmitry Lukyanenko

The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.


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