Inverse Problems In Mechanics Research Articles

Mathematical model of external ballistics of projectiles

The paper presents mathematical model of external ballistics of projectile, fired from cannon. The drag force has a decisive influence on the dynamics of the projectile. Direct calculation of the functional dependence of the drag force on the set of parameters is quite controversial and has not been utilized so far. Therefore, the author calculus the functional relativity, approximately, solving the inverse problem of mechanics. The flight range of a projectile at certain aiming angles, noted in the firing tables, determine the drag force according to the effect on a projectile of its weight and Coriolis force. Its functional dependence is described separately at the stages of projectile’s motion with supersonic, transonic and subsonic velocities. Due to the functional dependence of the drag force of a projectile, the influence of a charge projectile, initial velocity, atmospheric pressure, air temperatures and projectile’s mass effect on kinematic parameters of its motion can be calculated. Knowledge of this dependence of the drag force of the projectile’s movement allows to automate, using appropriate software, the determination of the elevation angle depending on the location of the target and the values of deterministic and nondeterministic factors.

Influenceof the mass of the projectile on its dynamic

As a result of polygon studies, a discrete dependence between aiming angles and horizontal flight ranges of projectiles for a certain type of weapon was obtained. By combining the use of this dependence and the method of the inverse problem of mechanics, the functional dependence of the frontal air resistance force of the moving projectile on its speed, deterministic and nondeterministic factors is determined. On the basis of the second law of dynamics, a system of differential equations is obtained, which describes the movement of the projectile under the influence of the force of frontal resistance, the weight of the projectile, and the Coriolis force. As an example, we consider the dynamics of the OF-462Zh projectile fired from a 122-mm D-30 howitzer with a reduced charge. Its initial speed is supersonic and equal to Vo=565 м/s. In this case, depending on the aiming angle, during the flight of the projectile, its speed can change from supersonic to subsonic or even subsonic.
 Since the mass of the projectile is mostly different from the nominal one, the effect of the change in the projectile mass on the flight range of the projectile and the kinematic parameters of the movement is investigated in the paper. It was established that at small aiming angles, a projectile with a smaller mass flies farther than a projectile with a nominal mass and its final speed is greater than the speed of a projectile with a nominal mass. However, if the aiming angle increases, the flight range of such a projectile is less than the flight range of a projectile with a nominal mass, and the final speed is less than the speed of a projectile with a nominal mass. The opposite picture is observed if the mass of the projectile is greater than the nominal one. In this case, at small aiming angles, the flight range of the projectile is less than the flight range of a projectile with a nominal mass, and its final speed is also less than the final speed of a projectile with a nominal mass. When the aiming angle increases, the flight range of a projectile with a larger mass is greater than the flight range of a projectile with a nominal mass, and its final speed is greater than the final speed of a projectile with a nominal mass.

Integrating visual sensing and structural identification using 3D-digital image correlation and topology optimization to detect and reconstruct the 3D geometry of structural damage

This paper describes a novel technique for detecting internal or unseen damage in structural steel members by combining measurements from full-field three-dimensional digital image correlation (3D-DIC) with a topology optimization framework. Unlike the majority of conventional methods that rely on specialized forms of surface-penetrating waves or radiation imaging, this work employs optical cameras to measure surface strains and deformations using the 3D-DIC technique followed by an optimization approach to detect the existing damage. This data-rich representation of the structural component’s behavior is then used to reconstruct the underlying subsurface abnormalities via an inverse mechanical problem. The inverse problem is solved using a topology optimization formulation that iteratively adjusts a fine-tuned finite element model (FEM) of the structure to reveal irregularities within it. Having recently demonstrated the feasibility of detecting and reconstructing defects in small-scale structural components, this paper expands on the authors' previous work to demonstrate the feasibility and performance of the proposed method through an experimental program in which a set of large-scale structural steel beams with and without buried defects tested using a full-field 3D DIC sensing approach. The structure’s initial FEM is first created to discretize the member into elements whose constitutive properties are treated as unknowns in the optimization problem. The goal of the optimization is to minimize the discrepancies between the observed full-field response measured experimentally using DIC and that computed numerically using the model. To that end, an objective function is first computed as the sum of residuals by mapping both responses onto a common grid, which is then pushed to a minimum via the method of moving asymptotes (MMA) as the optimization algorithm. This study demonstrates that the proposed approach can identify unseen damage with an average accuracy (ACC) score of 96.80% on the defined configurations, with relatively minimal false identifications.

Jacobi Multipliers in Integrability and the Inverse Problem of Mechanics

We review the general theory of the Jacobi last multipliers in geometric terms and then apply the theory to different problems in integrability and the inverse problem for one-dimensional mechanical systems. Within this unified framework, we derive the explicit form of a Lagrangian obtained by several authors for a given dynamical system in terms of known constants of the motion via a Jacobi multiplier for both autonomous and nonautonomous systems, and some examples are used to illustrate the general theory. Finally, some geometric results on Jacobi multipliers and their use in the study of Hojman symmetry are given.

Visualization of internal defects using a deep generative neural network model and ultrasonic nondestructive testing

Introduction. The development of machine learning methods has given a new impulse to solving inverse problems in mechanics. Many studies show that along with well-behaved techniques of ultrasonic, magnetic, and thermal nondestructive testing, the latest methods are used, including those based on neural network models. In this paper, we demonstrate the potential application of machine learning methods in the problem of two-dimensional ultrasound imaging. Materials and Methods. We have developed an experimental model of acoustic ultrasonic non-destructive testing, in which the probing of the object under study takes place, followed by the recording of the response signals. The propagation of an ultrasonic wave is modeled by the finite difference method in the time domain. An ultrasonic signal received at the internal points of the control object is applied to the input of the convolutional neural network. At the output, an image that visualizes the internal defect is generated. Results. In the course of the performed complex of numerical experiments, a data set was generated for training a convolutional neural network. A convolutional neural network model, which is developed to solve the problem of visualizing internal defects based on methods of ultrasonic nondestructive testing, is presented. This model has a small size, which is 3.8 million parameters. Its simplicity and versatility provide high-speed learning and a wide range of applications in the class of related problems. The presented results show a high degree of information content of the ultrasonic response and its correspondence to the real form of an internal defect located inside the test object. The effect of geometric parameters of defects on the accuracy of the neural network model is investigated. Discussion and Conclusion. The results obtained have established that the proposed model shows a high operating accuracy (F1 > 0.95) in cases when the wavelength of the probe pulse is tens of times less than the size of the defect. We believe that the combination of the proposed methods in this approach can serve as a good starting point for future research in solving flaw defection problems and inverse problems in general.

OSCILLATIONS OF A PULSE LOADED OSCILLATOR WITH A SQUARE RESISTANCE IN THE COMPOSITION OF THE DISSIPATIVE FORCE

The unsteady oscillations of a dissipative oscillator caused by an instantaneous impulse of the force are described. The case is considered when the dissipative force consists of quadratic viscous resistance and dry friction, and the theoretical results are generalized to the case of the sum of three forces. The third is the force of positional friction. Formulas for calculating the ranges of oscillations have been constructed In this case, the Lambert function of negative and positive arguments is used. It is a tabulated special function. Its value can also be calculated using its known approximations in elementary functions. It is shown that, due to the action of the dissipative force, the process of post-pulse oscillations consists of a finite number of cycles and is limited in time. This is due to the presence of dry friction among the resistance components. Examples of calculations that illustrate the possibilities of the stated theory are given. In order to check the reliability of the derived calculation formulas, numerical computer integration of the differential equation of motion was also carried out. The convergence of the numerical results obtained by two different methods is shown. Thus, it has been confirmed that with the help of analytical solutions it is possible to find the extreme displacements of the oscillator without numerically solving its nonlinear differential equation of motion. Using Lambert function and the first integral of the equation of motion made it possible to derive precise calculation formulas for determining the range of oscillations caused by the pulsed load of the oscillator. The derived formulas are suitable for calculating the value of the instantaneous impulse applied to the oscillator, which refers to the inverse problems of mechanics. Thus, by measuring the maximum displacement of the oscillator, it is possible to identify the initial velocity or instantaneous impulse applied to the oscillator. The performed numerical computer integration of the output differential equation confirmed the adequacy of the obtained analytical solutions, which concern not only direct, but also inverse problems of dynamics.

Mathematical problems of calculation and optimization of composite structures

Mathematical problems of solving direct and inverse problems of mechanics of composite structures are formulated. The problems of strength calculation and optimal design of composite overwrapped pressure vessels are investigated. The solutions of the optimization problem have been recieved and verified by solving the direct problems with obtained design parameters using the classical and improved shell theories. Numerical solutions were obtained using the spline collocation and discrete optimization methods.

Application of full-scale strain studies results for determining the equipment internal surfaces damages during rapid changes of coolant temperature

The method for metal damage determining for the inner surfaces of power equipment exposed to the coolant with a rapidly changing temperature is considered. Due to the impossibility of deformations and temperatures direct measurement for the inner surface, a calculation-experimental algorithm is proposed based on processing the results of strain measurements obtained for the outer surface in geometrically homogeneous and non-uniform zones of the structure. To qualify the functions for changing stresses and temperatures on the inner surface, in this paper, it is proposed to use the algorithm based on solving inverse problems of mechanics. The recovered functions of the temperature stresses temporal variations for the interface zone of the pipeline and the nuclear power plant body are presented.

ЛОКАЛЬНАЯ КРАЕВАЯ ЗАДАЧА ДЛЯ ОДНОГО КЛАССА УРАВНЕНИЯ ТРЕТЬЕГО ПОРЯДКА ЭЛЛИПТИКО-ГИПЕРБОЛИЧЕСКОГО ТИПА

In recent years, non-classical equations of mathematical physics have been attracting more and more attention of specialists; this is due to both theoretical and practical interest. Third-order equations are found in various problems of physics, mechanics, and biology. For example, in the theory of transonic flows, the propagation of plane waves in a viscoelastic solid, and the prediction and regulation of groundwater. One of the important classes of non-classical equations of mathematical physics is the equations of composite and mixed-composite type, which the main parts contain operators of elliptic, elliptic-hyperbolic and parabolic-hyperbolic types. Correct boundary value problems for equations of elliptic-hyperbolic and parabolic-hyperbolic types of the third order, in case that the main part of the operator contains the derivative with respect to x or y , is first studied by A.B. Bitsadze, M.S. Salakhitdinova and T.D. Djuraev, in addition to the fact that these equations are found in various problems of mechanics. For example, the propagation of a plane wave in a viscoelastic solid. In these works on the investigation of boundary value problems, a representation of general solution of a mixed-composite type equation in the form of a sum of functions was used. Such representation takes place only for equations, which are composed of the product of permutable differential operators. In this paper, we study boundary value problem for a third-order equation with an elliptic-hyperbolic operator in the main part. The equation under consideration is composed of the product of non-permutable differential operators, therefore the well-known representations of the general solution introduced by A.V. Bitsadze and M.S. Salakhitdinova are not applied. To study the considered third-order equation of the mixed type, we applied a method, which does not require a special representation of the general solution of the equation. This method determines the study of an equation of elliptic-hyperbolic type of the second order with unknown right-hand sides, which is of interest for solving important inverse problems of mechanics and physics. The existence and uniqueness theorems of the classical solution of the problem are proved. The proof is based on the extremum principle for a third-order equation and on the theory of singular and Fredholm integral equations.

Identification of crack-like defect and investigation of stress concentration in coated bar

The first part of this work is devoted to the location of defects in a coated bar and the identification of their geometrical parameters. Using the methods of finite element modeling, ultrasonic non-destructive testing and machine learning technologies (artificial neural networks), the inverse problem of mechanics has been solved. A finite element model of ultrasonic wave propagation in a bar with a coating and an internal defect is constructed. Compared with previous works, the model used PML (Perfectly Matched Layer) structures, which suppress multiple reflections of the probe ultrasound pulse inside the bar and prevent signal noise. Based on the conducted numerical calculations of the finite element model, a data set was constructed. It contains the geometrical parameters of the defect and the corresponding amplitude-time characteristic of the ultrasonic signal. The architecture of a direct propagation neural network has been developed. The neural network was trained on the basis of previously processed data. As a result, on the basis of ultrasound data obtained from the outer surface of the bar, it is possible to restore the values of such defect parameters as depth, length and thickness. At the second stage, analytical-numerical technology for studying the stress intensity factor (SIF) at the crack tip is described using the example of the problem of a longitudinal internal crack of finite length located in an elastic strip reinforced with a thin flexible coating. The solution to this problem is based on the method of integral transformations, which made it possible to reduce it to a singular integral equation of the first kind with a Cauchy kernel, which is solved by the collocation method in the form of expansion in Chebyshev polynomials with a factor that explicitly takes into account a feature in the vicinity of the crack vertices. The latter allows you to directly find the SIF and evaluate the effect on it of various combinations of geometric and physical parameters of the problem.

Circumventing the solution of inverse problems in mechanics through deep learning: Application to elasticity imaging

The ability to make decisions based on quantities of interest that depend on variables inferred from measurement finds application in different fields of mechanics and physics. The evaluation of the inferred variables, and hence the quantities of interest, from the measurement typically requires the solution of an inverse problem. For example, in medical imaging the elastic heterogeneity of a tumor and its nonlinear elastic response can be used to distinguish benign tumors from their malignant counterparts. These images of linear and nonlinear elastic parameters of tissue are typically obtained by using a measured displacement field and solving a complex inverse elasticity problem. In this paper we consider circumventing the solution of the inverse problem by using measured displacements as input to a deep convolutional neural network (CNN) and training it to classify tumors on the basis of their elastic heterogeneity and nonlinearity. For a simple, 5-layer CNN trained with 8,000 samples for heterogeneity, and a 4-layer CNN trained with 4,000 samples for nonlinear elasticity we report classification accuracies in the range of 99.7%−99.9%. The training and testing data are both obtained from the forward solution of finite element models of samples. We also analyze the weights of the trained model to understand the process through which the network extracts features of elastic moduli from the input displacement images. Finally, we apply the nonlinear elasticity classifier, which is trained entirely using simulated data, to displacement images obtained from ten patients with breast lesions and note that it correctly classifies eight out of ten cases. This application illustrates how data from physics-based models can be used in improving the performance of a data-driven algorithm in data-sparse scenarios.

A new robust formulation for optical-flow/material identification problems

The characterization of material properties from image sequences is of paramount importance in assessing structural integrity. A particular application can be found in the field of biomedical engineering, where anatomical structures, whose material properties are frequently sought, can be observed in-vivo through time. A traditional approach for such task, involves the estimation of the displacement field across the image sequence – usually by means of optical flow techniques – and the subsequent estimation of the material properties through the formulation of an inverse mechanical problem. The decoupled nature of this approach leads to cumulative errors across these stages, caused by the lack of consistence between the recovered optical flow and the displacement field delivered by the underlying mechanical model yet to be identified. In recent years, new approaches were proposed to integrate the optical flow and the material identification problems to render a compatible characterization of the displacement field in the sense that they correspond to a mechanical model whose material properties also undergo an identification process. In the present work, a new numerical method for the simultaneous identification of the optical flow field and the material properties of a mechanical model from a sequence of images is presented. Several numerical examples with manufactured solution are solved. Different sources of errors are considered, such as randomized pixel errors and interpolation errors, to analyze the robustness of the approach. The results obtained allow us to conclude that the proposed method performs robustly even in the presence of noise, and is also efficient in terms of the number of iterations required to achieve convergence.

On the existence of a global minimum in inverse parameters identification by Self-Optimizing inverse analysis method

In this paper, a mathematical proof of the existence of a global minimum of Self-Optim (Self-Optimizing Inverse Analysis Method) cost functional is presented based upon weak-solution theory of partial differential equations. The Self-Optim provides single global minimum rather than having multiple global minima corresponding to unrealistic solutions of the inverse problem. Furthermore, discrete approximation of the inverse problem and computational methods for the cost functional are proposed and the proof is numerically verified. This paper provides a rigorous mathematical foundation for applications of the Self-Optim method to various inverse problems in mechanics.

Inverse problem of contact fracture mechanics for a hub of friction pair taking into account thermal stresses

Methods of fracture mechanics enable a new approach to the design of structures, ensuring prevention of crack development. The plane problem of mechanics of contact fracture for the hub of a friction pair during operation is studied. It is accepted that near the rough friction surface, the hub has a rectilinear crack. A criterion and a method for solving the inverse problem of the mechanics of contact fracture on definition of the function of displacements of external contour points of the hub of a friction pair with regard to temperature drop and inequalities of the contact surface in friction pair components is given. The found function of displacements of the external contour points of the hub provides an increase of the load-bearing capacity of the hub of a friction pair. The problem of prevention of the fracture of the hub of a friction pair with allowance for the real friction surface was first posed and then solved.

Deformation curves for multicomponent nitride and carbide coatings

A method of plotting the deformation curve for the material of thin-film coatings using the parameters of the indentation diagram is considered. This method is based on solving the inverse problem of mechanics of continua. An algorithm is used for this purpose that combines a full-scale experiment and a numerical calculation. Data on the deformation curves for multicomponent nitride ((AlSi,Ti) and (AlSi,Cr)N) and carbide ((AlSi,Ti)–C:H, and (AlSi,Cr)–C:H) coatings are obtained and a comparative analysis of the properties of these coatings is carried out. Results of the numerical calculation of stresses that act during indentation are presented.

DAMAGE IDENTIFICATION IN REINFORCED CONCRETE BEAMS BY FINITE ELEMENT MODEL UPDATING USING PARALLEL AND HYBRID GENETIC ALGORITHMS

Finite element (FE) model updating belongs to the class of inverse problems in mechanics and is a constrained optimization problem. In FE model updating, the difference between the modal parameters (the frequencies, damping ratios and the mode shapes) obtained from the FE model of the structure and those from the vibration measurements are minimized within an optimization algorithm. The design variables of the optimization problem are the stiffness reduction factors, which represent the damage. In this study, the Genetic Algorithms (GA), the Parallel GA, the local search algorithms, the Trust Region Gauss Newton, the Sequential Quadratic Programming, the Levenberg–Marquardt Techniques and the hybrid versions of these methods are applied within the FE Model Updating Technique for updating the Young's modulus of different FEs of a reinforced concrete beam. Different damage scenarios and different noise levels are taken into account. The results of the study show that the local search algorithms cannot detect, locate and quantify damage in reinforced concrete beam type structures while the GA together with the hybrid and the parallel versions detect, localize and identify the damage very accurately. It is apparent that the hybrid GA & Trust Region Gauss Newton Technique is best in terms of the computation speed as well as accuracy.

An adaptive procedure for defect identification problems in elasticity

In the context of inverse problems in mechanics, it is well known that the most typical situation is that neither the interior nor all the boundary is available to obtain data to detect the presence of inclusions or defects. We propose here an adaptive method that uses loads and measures of displacements only on part of the surface of the body, to detect defects in the interior of an elastic body. The method is based on Small Amplitude Homogenization, that is, we work under the assumption that the contrast on the values of the Lamé elastic coefficients between the defect and the matrix is not very large. The idea is that given the data for one loading state and one location of the displacement sensors, we use an optimization method to obtain a guess for the location of the inclusion and then, using this guess, we adapt the position of the sensors and the loading zone, hoping to refine the current guess. Numerical results show that the method is quite efficient in some cases, using in those cases no more than three loading positions and three different positions of the sensors.

On inverse problems in electromagnetic field in classical mechanics at fixed energy

In this article, we consider inverse scattering and inverse boundary value problems at sufficiently large and fixed energy for the multidimensional relativistic and nonrelativistic Newton equations in a static external electromagnetic field (V, B), V∈C2, B∈C1 in classical mechanics. Developing the approach going back to Gerver-Nadinashvili 1983's work on an inverse problem of mechanics, we obtain, in particular, theorems of uniqueness.

On Feasibility of “Spectroscopic” Calculations of XH Bond Dissociation Energies for Polyatomic Molecules from Fundamental Vibration Frequencies Using the Anharmonic Molecular Model

This paper discusses the applicability of the variational technique using a minimal Morse-harmonic basis set to calculations of the fundamental spectrum and the potential function parameters for polyatomic molecules. The potential function is assumed to be the sum of the Morse function for XH bonds and the harmonic function for the skeletal and deformation vibrations. The initial approximation for the potential function is found by ab initio calculations and refined by solving the inverse mechanical problem (selecting the scaling indices). The thus selected harmonic part of the potential function gives equally good agreement between the experimental and calculated transition frequencies in both harmonic and anharmonic approximations. The anharmonic (Morse) term of the potential function (bond dissociation energy) is selected by solving the inverse mechanical problem until the best agreement between the experimental and calculated CH bond stretching frequencies has been achieved. Problem solving ends with the construction of a transmission curve in the IR spectrum. Variations of the dipole moment of the molecule induced by vibrations are found by ab initio calculations.

On a classical analog of the isospectral Schrödinger problem

It is shown that the problem of potential deformations that preserve the energy dependence of the oscillation frequency corresponds in the classical limit to the isospectral problem for the Schrodinger equation. The solutions of the corresponding evolutionary equations for the potential (with respect to the deformation parameter) are simple Riemann waves. The relation between such solutions and the solutions of the inverse problem of mechanics — the problem of reconstructing the potential given the energy dependence of the oscillation frequency — is determined.