Linear Elasticity Theory Research Articles

Applying Multiple Regression Analyses and Linear Elastic Glucose Theory to Analyze and Compare the Predicted and Measured Finger-Pierced Postprandial Plasma Glucose using Carbs/Sugar Intake Amount and Post-Meal Walking Steps as Inputs over 2016- 2017 Period for a Diabetes Patient Based on GH-Method: Math-Physical Medicine (No. 544)

In the author’s previous research reports, he mainly applied physics theories, engineering models, mathematical equations, computer big data analytics and artificial intelligence (AI) techniques, as well as some statistical approaches. However, the majority of medical research papers he has read thus far are primarily based on statistics. As a result, in this article, he selected some basic statistical tools, such as correlation, variance, p-values, and multiple regression analyses, to study the predicted finger-piercing postprandial plasma glucose (PPG) as the output (dependent variable) by using his carbs/sugar intake grams and post-meal walking steps as inputs (independent variables). Since 1/1/2018, the author has been utilizing a finger-pierced device to collect and store his glucose data on the iPhone and Amazon cloud server. He has accumulated 4 glucose data per day over the past 10 years, along with entering his carbs/sugar intake grams and post-meal walking steps after each meal into the database. This article displays a multiple regression analysis result of the finger-piercing measured PPG data with the predicted finger PPG values (dependent output variables) by using his average daily carbs/sugar intake amounts and daily average post-meal walking steps (independent input variables) over a selected 2-year period from 1/1/2016 to 12/31/2017. In this study, he will not repeat the detailed introduction of the regression analysis in the Method section because it is available in any statistics textbook. It should be noted that in regression analysis, the correlation coefficient R should be > 0.5 or 50% to indicate a strong inter-connectivity and the p-value should be <0.05 to be considered as statistically significant. By utilizing his developed linear elastic glucose theory (LEGT), he calculates the predicted finger-piercing PPG using the same inputs of carbs/sugar and walking steps during the same chosen time period.

Linear Regression Analysis Results of the CGM Sensor PPG Comparison Between Predicted PPG Data Using the Linear Elastic Glucose Theory (LEGT) and Measured PPG Data During a ~2-year COVID-19 Quarantine Period for a type 2 Diabetes Patient Based on GH-Method: Math-Physical Medicine (No. 540)

Since 5/5/2018, the author has been applying a continuous glucose monitoring (CGM) sensor device on his upper arm that collected and recorded the complete glucose data continuously at 15-minute time intervals on his iPhone. He accumulated 96 glucoses per day over the past ~3.5 years. As a result, over these 1,272 days, he has compiled a total of 122,112 glucose data and stored them in his database where postprandial plasma glucose (PPG) occupies 45,792 data size and 37.5% of the total glucose database. During 2020-2021 COVID-19 quarantine period, he has a strictly managed routine, without any traveling, which allowed him to have an overall healthy lifestyle. Therefore, all of the 19 influential factors of PPG are mainly control by two primary factors: carbs/sugar intake amount (average at 13.1 gram, low-carb diet) and post-meal waking e excise (average of 4,300 steps). Based on this simplified and healthy lifestyle, he can then easily utilize his developed linear elastic glucose theory (LEGT) model to predict his PPG. In his previous research reports, he has applied physics concepts and theories, engineering models and equations, mathematical concepts and formulas, computer big data and artificial intelligence (AI) techniques, as well as some statistical approaches. The majority of published medical papers he has read are mainly based on statistics. As a result, in this article, he selected one of the basic statistical tools, linear regression analysis, to study the comparison between his predicted PPG using LEGT and CGM sensor measured PPG. In conclusion, the linear regression analysis results have provided similar findings with his previous analysis outcomes using other math-physical tools.

Applying Multiple Regression Analysis and Linear Elastic Glucose Theory to Analyze and Compare the Predicted Postprandial Plasma Glucose Using Carbs/Sugar Intake Amount and Post-Meal Walking Steps as Inputs over an Approximate 2-Year COVID-19 Quarantine Period from a type 2 Diabetes Patient Based on GH-Method: Math-Physical Medicine (No. 543)

In the author’s previous research reports, he mainly applied physics theories, engineering models, mathematical equations, computer big data analytics and artificial intelligence (AI) techniques, as well as some statistical approaches. The majority of medical research scientists’ published papers he has read thus far are primarily based on statistics. As a result, in this article, he selected some basic statistical tools, such as correlation, variance, p-values, and multiple regression analyses to study the predicted postprandial plasma glucose (PPG) as the dependent variable using his carbs/sugar intake grams and post-meal walking steps as inputs (independent variables). Since 5/8/2018, the author has been utilizing a continuous glucose monitoring (CGM) sensor device on his upper arm that collected and recorded the complete glucose data continuously at 15-minute time intervals on his iPhone. He accumulated 96 glucoses per day over the past ~3.5 years. After each meal, he collects 13 PPG data, accumulating 39 PPG values per day, along with entering his carbs/sugar intake grams and post-meal walking steps. This article displays multiple regression analysis results of measured PPG values with predicted PPG values (dependent outputs) by using his average daily carbs/sugar intake amounts and daily average post-meal walking steps (independent inputs) over an approximate 2-year COVID-19 quarantine period from 1/1/2020 to 10/31/2021. In this study, he will not repeat the detailed introduction of regression analysis in the Method section because it is available in any statistics textbook. It should be noted that in regression analysis, the correlation coefficient R should be > 0.5. to indicate strong inter-connectivity and the p-value should be <0.05 to be considered as statistically significant. By utilizing his developed linear elastic glucose theory (LEGT), he calculates the predicted PPG using the same inputs of carbs/sugar and walking steps during the same chosen time period.

Applying Multiple Regression Analyses and Linear Elastic Glucose Theory to Analyze and Compare the Predicted and Measured Finger-Pierced Postprandial Plasma Glucose using Carbs/Sugar Intake Amount and Post-Meal Walking Steps as Inputs over 2016- 2017 Period for a Diabetes Patient Based on GH-Method: Math-Physical Medicine (No. 544)

In the author’s previous research reports, he mainly applied physics theories, engineering models, mathematical equations, computer big data analytics and artificial intelligence (AI) techniques, as well as some statistical approaches. However, the majority of medical research papers he has read thus far are primarily based on statistics. As a result, in this article, he selected some basic statistical tools, such as correlation, variance, p-values, and multiple regression analyses, to study the predicted finger-piercing postprandial plasma glucose (PPG) as the output (dependent variable) by using his carbs/sugar intake grams and post-meal walking steps as inputs (independent variables). Since 1/1/2018, the author has been utilizing a finger-pierced device to collect and store his glucose data on the iPhone and Amazon cloud server. He has accumulated 4 glucose data per day over the past 10 years, along with entering his carbs/sugar intake grams and post-meal walking steps after each meal into the database. This article displays a multiple regression analysis result of the finger-piercing measured PPG data with the predicted finger PPG values (dependent output variables) by using his average daily carbs/sugar intake amounts and daily average post-meal walking steps (independent input variables) over a selected 2-year period from 1/1/2016 to 12/31/2017. In this study, he will not repeat the detailed introduction of the regression analysis in the Method section because it is available in any statistics textbook. It should be noted that in regression analysis, the correlation coefficient R should be > 0.5 or 50% to indicate a strong inter-connectivity and the p-value should be <0.05 to be considered as statistically significant. By utilizing his developed linear elastic glucose theory (LEGT), he calculates the predicted finger-piercing PPG using the same inputs of carbs/sugar and walking steps during the same chosen time period.

Kinked and forked crack arrays in anisotropic elastic bimaterials

The fracture problem of multiple branched crack arrays in anisotropic bimaterials is formulated by use of the Stroh formalism to the linear elasticity theory of dislocations. The general full-field solutions are obtained from the standard technique of continuously distributed dislocations along finite-sized cracks of arbitrary shapes, which are embedded in dissimilar anisotropic half-spaces under far-field stress loading conditions. The bimaterial boundary-value problem leads to a set of coupled integral equations of Cauchy-type that is numerically solved by using the Gauss–Chebyshev quadrature scheme with appropriate boundary conditions for kinked and forked crack arrays. The path-independent Jk-integrals as crack propagation criterion are therefore evaluated for equally-spaced cracks, while the limiting configuration of individual cracks is theoretically described by means of explicit expressions of the local stress intensity factors K for validation and comparison purposes on several crack geometries. The short-range interactions resulting from the idealized configurations of infinitely periodic cracks are investigated as well as various size- and heterogeneity-effects on the mixed-mode cracks in complex stress-state environments. The influences of anisotropic elasticity, elastic mismatch, applied stress direction, inter-crack spacings and crack length ratios on the predictions from the Jk- and K-based fracture criteria are examined in the light of different configurations from the single kinked crack case in homogeneous media to the network of closely-spaced forked cracks in presence of bimaterial interfaces.

Valorization of date palm wastes as sandwich panels using short rachis fibers in skin and petiole 'wood' as core

Algerian agriculture offers free by-products that can be transformed in biobased-materials. We use one of them, the date palm ’wood’, to locally develop sandwich panels. We demonstrate by tensile tests that rachis fibers are more efficient than petiole fibers. Then, petiole ’wood’ is used as sandwich core and short rachis fibers (5%, 10%, 15%) are used as epoxy skin reinforcement. The skins are tested by tensile and bending tests, 15%Mf untreated short fibers have been selected as a best choice for stiffness of the skins. Three parameters are studied to optimize the design of the core: the orientation (longitudinal, transverse, radial), the length of the pieces of ’wood’ (30 mm, 100 mm), and the thickness (10 mm, 15 mm, 20 mm). The sandwich efficiency are studied by 3-points bending tests combined to linear elastic beam theory and failure mode analysis. The dominant failure modes are bottom skin failure and core shear, the latter being not activated in case of transverse orientation of core. Both other parameters (thickness, pieces of core length) are not really determinant in mechanical point of view, depending on the future use. We conclude that making sandwich board by using untreated short rachis fibers as reinforcement for skins and untreated transverse cut petiole ’wood’ as core is a very efficient choice to give technical and cheap valorization of palm leaf wastes for non-structural use.

An efficient semi-analytical static and free vibration analysis of laminated and sandwich beams based on linear elasticity theory

Based on the two-dimensional (2D) elastic theory without enforcing any beam assumption, an efficient semi-analytical scaled boundary finite element method (SBFEM) is proposed to solve the bending and free vibration responses of composite laminated and sandwich beams under the mechanical load. The scaled center is placed at infinity, which produces the accurate result by discretizing only the longitudinal direction of the beam structure treated as a one-dimensional (1D) discretization problem. A new kind of 1D high-order spectral element shape functions with the advantages of high accuracy and superior convergence is introduced in SBFEM coordinate system to approximate the geometric model and corresponding variables. The principle of weighted residual in conjunction with the Green’s theorem are applied to obtain the SBFEM governing equation of each layer with respect to radial displacement fields. The solution of equation is indicated analytically by a matrix exponential function, which can be accurately solved by using the precise integration technique (PIT). Finally, an effective and simple stiffness matrix is obtained. By comparing two examples with the solutions based on the finite element method (FEM), the results show that the proposed method has good accuracy and rapid convergence with only a few meshes. The numerical examples are given to investigate the parametric effects of the stacking sequence, thickness ratio, boundary condition, and load form on the variation of the displacement, stress and natural frequency. The results validate that the present technique is also applicable to the complex beam structure with softcore layer inside.

Dispersion of Love-type wave and its limitation in a nonlocal elastic model of nonhomogeneous layer upon an orthotropic extended medium

This research paper investigates the Love-type surface wave dispersion in a nonlocal elastic nonhomogeneous medium over a nonlocal orthotropic semi-infinite medium under the effect of interior initial stress component. A generalized linear elasticity theory with the nonlocal effect in the fundamental stress-strain relation is adopted to address the equation of motion in layered media. In the case of time-harmonic plane wave propagation, PDEs (cf. equations of motion) of the nonlocal elasticity are reduced to closed-form singular differential equations. The equation of dispersive wave is derived using appropriate plane stress-displacement-continuity conditions of the model. The dispersion of Love-type wave and its limitation in a nonlocal elastic media are analysed graphically using MATLAB software. For the specific model, it is observed that the attenuation of the dispersion of Love-type wave depends on a nonlocal parameter. Also, the dispersive wave becomes nondispersive with zero attenuation for large values (∼0.4) of the nonlocal parameter. In the case of dispersive wave, variation of the phase velocity of the fundamental wave mode for different values of nonlocality, nonhomogeneity, wave number and initial stress parameters are shown graphically. Some special cases are deduced from the original dispersion equation and validated with the published literature.

Plate theories: A‐priori fulfillment of the local conditions by the consistent‐approximation approach

AbstractClassical plates theories, like Kirchhoff's plate theory [1], are based on kinematical a‐priori assumptions. Avoiding these assumptions, we derive from the three‐dimensional theory of linear elasticity by means of Taylor‐series expansions the quasi two‐dimensional problem. This problem consists of infinitely many partial‐differential equations (PDEs) written in infinitely many displacement coefficients. With the consistent approximation approach we arrive at solvable hierarchical plate theories. By using the modular structure of the displacements coefficients (modularity), we obtain from these generic‐plate theories the complete‐plate theories, whose results fulfill the strong form of the local equilibrium conditions and the Neumann boundary conditions on the upper and lower face of the plate (local conditions) a‐priori. Furthermore, we show that every variable of the complete‐plate theories can be calculated.

Computational modelling of mechanical response of glulam beams due to moisture inducted delamination

Glued laminated timber beams are nowadays used as load-bearing beams of large-span structures that operate in various humidity conditions. Typical application areas are aqua parks with high humidity as well as market halls with low humidity. It is related to the possibility of the occurrence of cracks typical for the drying of wood, even with such controlled conditions of production technology as glued laminated timber. Cracks visible on the used girders raise doubts as to the safe operation of the structures. The subject of this paper is the computational simulation and the evaluation of the influence of beam delamination on the mechanical response of the structure. The attention was established on a typical two-span beam of constant height with a slight slope to the horizontal. The numerical analysis was carried out for three variants of the location of potential delamination of different scope. The beams were modeled as a problem of the linear theory of elasticity in a plane stress state with orthotropic material properties. The calculations were made in the Abaqus software environment. The results obtained in the paper allow to determine the areas in which the presence of delamination or cracks should be considered dangerous from the point of view of the safety of operation. Computational analysis is helpful in assessing the safety of structures where cracks appear. Theoretical considerations are supplemented by an example from engineering practice.

A strain gradient enhanced model for the phase‐field approach to fracture

AbstractPhase‐field modelling has been shown to be a powerful tool for simulating fracture processes and predicting the crack path under complex loading conditions. Note that the total energy of fracture in the classical phase‐field formulations includes the strain energy density from the linear elasticity theory resulting in singular stresses at the crack tip. Recently, we have demonstrated that integrating the strain gradient elasticity into the conventional phase‐field fracture formulations may improve the final results by alleviating the effects of a singular stress field around the crack tip [1]. The current contribution focuses on a more general formulation of strain gradient elasticity.

Investigation of fracture properties and size effects of mass concrete using wedge splitting tests on large specimens

This work presents experimental results of fracture tests on very large concrete specimens of mass concrete mixture with a maximum aggregate size of 100 mm. Wedge splitting tests were performed on very large concrete specimens and on moderate-size specimens sawn from large concrete blocks to investigate both size and temperature effects. The digital image correlation (DIC) technique allowed to assess the fracture parameters of these specimens, including the critical crack opening displacement and crack growth length at the peak. These parameters were compared to theoretical computed parameters based on linear elastic fracture mechanics or LEFM theory. Cyclic splitting tests were additionally performed on very large and moderate-size concrete specimens. Interpretation of the results of these cyclic tests via DIC enabled to formulate important conclusions regarding the evolution of the fracture process zone length and critical opening displacement during the tests and to validate the hypotheses associated with the recently developed disturbed fracture process zone theory in terms of the size effect on the fracture energy.

Fatigue analysis in rotor of a prototype bulb turbine based on fluid-structure interaction

Bulb turbine units are widely used in run-of river or dam projects with relatively low head, and can also be used for tidal energy utilization. Some types of bulb turbines have excessive vibration and fatigue failure problems, which threats their fatigue life. The model analyzed in this paper is a prototype bulb turbine rotor which suffered cracks, probably due to erosion and welding defects. The dynamic stress characteristics are analyzed under not only the non-cracked shaft condition but also the cracked one, to find possible causes of the failure.The computational fluid dynamics (CFD) simulation is performed to obtain reliable hydraulic load on the runner for performing the FEM analysis. The CFD results are verified by comparing with the site test for the prototype one. The linear elastic fracture mechanics theory combined with the fluid-structural interaction theory is applied to developing a cracked FEM model, to obtain the dynamic stress characteristics for the runner, shaft and rotor. The results reveal that the change in dynamic stress level on the shaft flange root significantly depends on the operating conditions, and the stress amplitude on the shaft mainly depends on the structural weight. The stress at the cracked zone could be up to 376.5 MPa in the best efficiency point, with only 1.5 years of fatigue life.Based on the numerical calculations and site inspection, it could be concluded that the insufficient sealing could lead to leakage flows with specific flow rates in flange root, which resulted in the occurrence of erosion; in addition, the welding at the flange root probably generated tiny defects. These two factors are likely lead to fatigue cracks and major failure of the shaft. Suggestions for solutions for the problems of the shaft are also discussed. Results of this study helps to find the reasons of the shaft fatigue failure. Furthermore, the analysis method could be used for an accurate calculation of fatigue life and providing guidance in future designs and operations of bulb turbine units.

Finite Element Calculation of the Linear Elasticity Problem for Biomaterials with Fractal Structure

Aims: The aim of this study was to develop the mathematical models of the linear elasticity theory of biomaterials by taking into account their fractal structure. This study further aimed to construct a variational formulation of the problem, obtain the main relationships of the finite element method to calculate the rheological characteristics of a biomaterial with a fractal structure, and develop application software for calculating the components of the stress-strain state of biomaterials while considering their fractal structure. The obtained results were analyzed. Background: The development of adequate mathematical models of the linear elasticity theory for biomaterials with a fractal structure is an urgent scientific task. Finding its solution will make it possible to analyze the rheological behavior of biomaterials exposed to external loads by taking into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Objective: The objective of this study was the deformation process of biomaterials with a fractal structure under external load. Methods: The equations of the linear elasticity theory for the construction of the mathematical models of the deformation process of biomaterials under external load were used. Mathematical apparatus of integro-differentiation of fractional order to take into account the fractal structure of the biomaterial was used. A variational formulation of the linear elasticity problem while taking into account the fractal structure of the biomaterial was formulated. The finite element method with a piecewise linear basis for finding an approximate solution to the problem was used. Results: The main relations of the linear elasticity problem, which takes into account the fractal structure of the biomaterial, were obtained. A variational formulation of the problem was constructed. The main relations of the finite-element calculation of the linear elasticity problem of a biomaterial with a fractal structure using a piecewise-linear basis are found. The main components of the stress-strain state of the biomaterial exposed to external loads are found. Conclusion: Using the mathematical apparatus of integro-differentiation of fractional order in the construction of the mathematical models of the deformation process of biomaterials with a fractal structure makes it possible to take into account the existing effects of memory, spatial non-locality, self-organization, and deterministic chaos in the material. Also, this approach makes it possible to determine the residual stresses in the biomaterial, which play an important role in the appearance of stresses during repeated loads.

Analysis of Rectangular Reinforced Concrete Liquied Tanks by Using Yield Line Theory

In the analysis of rectangular reinforced liquid storage tanks, a method assuming linear-elastic behavior for material can be used, i.e., the strip method, the moment coefficient method, the finite element method, etc. In the analysis of these types of tanks, tank walls can be considered as slabs. In this study, tank walls were analyzed as slabs subjected to hydrostatic loading; in the analysis, the yield line theory is used because it is more suitable for the linear inelastic behavior of reinforced concrete slabs than the ones based on the linear elastic theory. An iterative algorithm based on yield line theory is presented for the design of isotropically reinforced recrangular concrete slabs supported along all four edges. A computer program is coded which predicts the location of yield lines for the slabs depending upon certain parameters. As a result of this prediction, the manual design of such slabs can be significantly simplified by the use of the coefficient obtained by using the program. It was shown that the analytical computation of the ultimate moment per unit length requires the solution of a highly nonlinear system of equations. This difficulty was overcome by utilizing an iterative technique within the computer program. It also gives the value of the ultimate moment per unit length of the yield line.

Diffusion-induced stress in a functionally graded incompressible elastic sphere

Diffusion-induced stress is investigated for an isotropic and elastic incompressible solid sphere which is charged by radial diffusion. The interaction between a diffused solute and the arising stress field is accounted for by a chemo-mechanical potential widely employed for describing the process of charge (or discharge) in lithium-ion batteries and which depends on the hydrostatic part of the stress field. Both linear and nonlinear theories of elasticity are considered in the model at their limit of incompressibility. Such a constraint, by relating the displacement and the stress fields analytically to the unknown solute concentration, allows us to derive analytical asymptotic behaviours near the centre of the particle as well as to simplify the equations, solved here by a simple finite difference numerical scheme. Linear and nonlinear predictions are compared for a spherical electrode subject to large deformations. Furthermore, in light of the great interest in engineering science applications, the equations are derived for the most general case of functionally graded materials. It is shown how the space varying elastic coefficient can be used as a tuning mechanism for controlling stress evolution, which in most cases might represent the main reason for degradation phenomena in solids.

A new analysis model for borehole stability in gas drilling

Gas drilling is one of the main underbalanced drilling techniques, which has a significant effect on the ROP (rate of penetration) improvement in hard formation. However, because the circulating medium in drilling process is gas instead of liquid, the fluid column pressure of gas drilling cannot provide enough support force for the rock around borehole as that of conventional overbalanced drilling. Due to the combination influence of engineering and geological factors, the borehole rock is prone to instability during gas drilling, which can lead to rock blocks falling from the well wall and the formation of irregular borehole shape. To avoid the occurrence of complex underground accidents caused by too many falling rock blocks, such as the borehole instability and pipe-sticking, it is necessary for gas drilling to take into consideration the true triaxial stress state of the rock in the deep formation and the possibility of the borehole breakout. Based on the permitted breakout angle of borehole and true triaxial strength theory of rock, an analysis model of borehole stability has been developed for the engineering of gas drilling. The theory of linear elasticity, involved in this analysis model, is used to analyse the stress state of rock around a cylindrical hole under lower wellbore pressure. The failure of the rock around borehole is determined by the Mogi-Coulomb criterion, which considers the intermediate principal stress effect on rock material. The results show that the model is as simple and practical as the traditional model, which takes no account the effect of the intermediate principal stress or the permitted borehole breakout, and it can provide an accurate analysis for the borehole stability of gas drilling. Therefore, the results of this study, combined with the related work such as the prediction of water formation, can guide the adaptability evaluation for the engineering design of gas drilling.

Hyperstatic steel structure strengthened with prestressedcarbon/glass hybrid laminated plate

This paper presents a careful theoretical investigation into interfacial shear stresses in steel beam strengthened with prestressed carbon/glass hybrid laminated plate. A closed-form rigorous solution for interfacial shear stress in steel beams strengthened with bonded prestressed carbon/glass hybrid laminated plates and subjected to a uniformly distributed load, is developed using linear elastic theory and including the variation in fiber volume fraction of carbon/glass hybrid laminated. The results show that there exists a high concentration of shear stress at the ends of the laminate, which might result in premature failure of the strengthening scheme at these locations. A parametric study has been conducted to investigate the sensitivity of interface behavior to parameters such as laminate and adhesive stiffness, the proportions and volume fraction of the fiber of carbon/glass hybrid laminated, the thickness of the laminate and the effect of prestressing where all were found to have a marked effect on the magnitude of maximum shear and normal stress in the composite member. This solution is intended for application to beams made of all kinds of materials bonded with a thin composite plate. This research is helpful for the understanding on mechanical behaviour of the interface and design of such structures.

A discrete element modeling of rock and soil material based on the machine learning

The discrete element modeling of rock and soil mass usually relies on tedious parameter adjustment and mechanical property testing operations to obtain appropriate mechanical parameters of the element. In this paper, based on the linear elastic contact theory, a discrete element model of rock samples was established based on discrete element software MatDEM with high-performance numerical analysis, and the actual mechanical properties of the model samples were predicted using a machine learning method, XGBoost algorithm. A certain amount of sample data sets was generated, and through numerical experiments, training, validation and test operation of the rock mechanical properties (compressive strength, Young’s modulus, Poisson’s ratio and tensile strength), the numerical model quantitative relationship between the actual and input mechanical properties was explored, showing the following results: (1) Based on the characteristics of the model 12000 rock samples and correlation analysis, the tag data shows that the input mechanical properties are positively correlated with the actual mechanical properties;(2) With the increase of the number of samples in the sample training data set, the training average error becomes smaller and smaller, and the error distribution becomes more and more stable.(3) The training results converge to the set threshold, and the prediction errors of mechanical parameters are all less than 4%, and the error distribution is stable, which proves that the machine learning method can quickly and accurately establish the rock model with specific mechanical properties.

On reconstruction of the coefficient in complex Helmholtz’s equation

This note summarizes some preliminary results on the fast solution of the coefficient inverse problem for the Helmholtz equation, given measured pressure in a set of observation points. The Helmholtz equation is the model PDE for the harmonic problem of the linear theory of elasticity, and this work is a move in that direction. The problem has been the primary focus for several research areas, most notably seismic exploration. Still, practical problems are very challenging because they are non-linear and large. In this paper, we develop a novel numerical method for seismic full-waveform inversion based on Newton iterations. Its distinctive future is that it does not require the Jacobian of the target functional. Thus, in certain scenarios, it will perform only a fraction of computations comparing to the conventional Gauss-Newton algorithm. We present some early results on the Helmholtz equation in two dimensions.