Gauss-Newton Technique Research Articles

A practical meshfree inverse method for identification of thermo-mechanical fracture load of a body by examining the crack path in the body

An inverse meshfree approach is presented for thermo-mechanical load identification of a fractured body. To this end, the crack propagation path is used as the input of the inverse analysis. This study can be used for finding the thermo-mechanical loads that have led to the fracture of a body, and is applicable to situations where the direct measurement of the boundary loadings on some part of a body is not possible. This highly ill-posed problem is solved using the meshfree radial point interpolation method (RPIM) with a global optimization approach. The inspected crack path is compared to the one obtained numerically and their difference is set as a cost function. By finding the minimum of this cost function, the actual loading conditions that have led to the fracture of the body is obtained. The density clustering method is used as the global optimization method in which the damped Gauss-Newton technique is used as the local search algorithm. To demonstrate the performance of the presented meshfree inverse approach, two examples are provided.

DC Electrical Resistance Tomography Inversion

Direct current electrical resistivity imaging is provided by measuring the vertical and horizontal electrical potential variations of subsurface structures using surface and borehole records. To recover the resistivity tomograms from the observed data, a non-linear inverse problem is required to be iteratively solved. A 2.5-dimensional forward modeling based on the finite-difference method with rectangular meshes is also formulated. The two-dimensional reconstruction of earth resistivity data is implemented using a smoothness constrained inversion algorithm (i.e. Occam’s method), wherein a Gauss-Newton technique for updating the sensitivity function is proposed. After verifying the accuracy and efficiency of the forward modeling and the sensitivity function calculation, the inversion algorithm is tested on synthetic data from both geometrically simple and complicated bodies and a real data set. A stopping criterion based on the noise level roughly estimated using the method of reciprocal resistance measurements is also provided leading to preventing over-or under-interpreted structure during the inversion process. The numerical experiments reveal that the proposed inversion algorithm provides stable inversion results and an acceptable representation of the main features and structure of the models without producing spurious effects. Furthermore, to deal with the reliability of the recovered models, a model sensitivity analysis is implemented using the resolution density distribution. All used formulations and concepts are part of a Matlab source code developed during this study.

A least‐squares finite element method based on the Helmholtz decomposition for hyperbolic balance laws

AbstractIn this paper, a least‐squares finite element method for scalar nonlinear hyperbolic balance laws is proposed and studied. The approach is based on a formulation that utilizes an appropriate Helmholtz decomposition of the flux vector and is related to the standard notion of a weak solution. This relationship, together with a corresponding connection to negative‐norm least‐squares, is described in detail. As a consequence, an important numerical conservation theorem is obtained, similar to the famous Lax–Wendroff theorem. The numerical conservation properties of the method in this paper do not fall precisely in the framework introduced by Lax and Wendroff, but they are similar in spirit as they guarantee that when L2 convergence holds, the resulting approximations approach a weak solution to the hyperbolic problem. The least‐squares functional is continuous and coercive in an H−1‐type norm, but not L2‐coercive. Nevertheless, the L2 convergence properties of the method are discussed. Convergence can be obtained either by an explicit regularization of the functional, that provides control of the L2 norm, or by properly choosing the finite element spaces, providing implicit control of the L2 norm. Numerical results for the inviscid Burgers equation with discontinuous source terms are shown, demonstrating the L2 convergence of the obtained approximations to the physically admissible solution. The numerical method utilizes a least‐squares functional, minimized on finite element spaces, and a Gauss–Newton technique with nested iteration. We believe that the linear systems encountered with this formulation are amenable to multigrid techniques and combining the method with adaptive mesh refinement would make this approach an efficient tool for solving balance laws (this is the focus of a future study).

Kinetics of Dissociation of bis(2,4,6-Tripyridyl-s-triazine)iron(II) and tris(2,2′-Bipyridyl)iron(II) in the presence of Triton X-100/Tween 80 Mixed Micellar Medium

Surface tension studies were carried on the binary surfactant mixtures over a wide range of Triton X-100 mole fractions and total surfactant concentrations to obtain critical micellar concentration values. These values were used to determine the composition of the mixed micelles and the average interaction parameter (β) which contains all the interactions of the mixed surfactants. The method is based on Rubingh′s theory using a Gauss-Newton iteration technique written in FORTRAN. The value of β was found to be -0.69 indicating synergistic behaviour i.e., combined positive catalytic effect of both the surfactants on rates of reactions. Hence, the kinetics of dissociation of bis(2,4,6-tripyridyl-striazine) iron(II) ([Fe(tptz)2]2+) and tris(2,2′-bipyridyl)iron(II) ([Fe(bipy)3]2+) were studied in the presence of Triton X-100/Tween 80 mixed micellar medium. The reactions have been carried out in the presence of mixed micelles of Triton X-100/Tween 80 at various mole fractions of Triton X-100 (αTX-100 = [Triton X-100]/([Triton X-100] + [Tween 80])) and at different total surfactant concentrations of Triton X-100 and Tween 80 (Ct = [Triton X-100] + [Tween 80]). The results show that as αTX-100 increases the rate increases for all values of Ct. Kinetic analysis has been carried out by using a simple pseudo phase model and binding constants were determined. These binding constants were found to be in agreement with the binding constants obtained spectrophotometrically.

Optimization of inhibitive action of sodium molybdate (VI) for corrosion of carbon steel in saline water using response surface methodology

The performance of sodium molybdate (Na2MoO4) (VI) as a corrosion inhibitor for medium carbon steel corrosion in saline water containing nitrate and chloride ions was studied at various inhibitor concentrations, temperatures, exposure times and rotational velocities. Mass loss and electrochemical techniques were used to evaluate the corrosion rates. The individual and interactive effects of these four parameters were optimized for minimum response of corrosion rate using central composite design (CCD) with response surface methodology (RSM). Nonlinear regression strategy in light of Gauss-Newton technique was utilized for modeling and optimization of the corrosion inhibition experiments. Second-order polynomial model was suggested to predict the corrosion rates as a function of four variables. The individual effect of temperature on corrosion rate was higher than the individual effects of inhibitor concentration, exposure time and rotational velocity, respectively. The interaction effects of independents variables were also addressed. Open circuit potential measurements were used as a significant way to gain information about the behavior of steel corrosion. Steady state potential was reached after one hour of immersion time. Mass loss results were in a good agreement with potentiodynamic polarization technique. Optimum inhibition efficiency was 95.9% at optimum operating conditions. Polarization plots revealed that the inhibitor acts as the anodic-type inhibitor.

When topological derivatives met regularized Gauss-Newton iterations in holographic 3D imaging

We propose an automatic algorithm for 3D inverse electromagnetic scattering based on the combination of topological derivatives and regularized Gauss-Newton iterations. The algorithm is adapted to decoding digital holograms. A hologram is a two-dimensional light interference pattern that encodes information about three-dimensional shapes and their optical properties. The formation of the hologram is modeled using Maxwell theory for light scattering by particles. We then seek shapes optimizing error functionals which measure the deviation from the recorded holograms. Their topological derivatives provide initial guesses of the objects. Next, we correct these predictions by regularized Gauss-Newton techniques devised to solve the inverse holography problem. In contrast to standard Gauss-Newton methods, in our implementation the number of objects can be automatically updated during the iterative procedure by new topological derivative computations. We show that the combined use of topological derivative based optimization and iteratively regularized Gauss-Newton methods produces fast and accurate descriptions of the geometry of objects formed by multiple components with nanoscale resolution, even for a small number of detectors and non convex components aligned in the incidence direction.

Modeling and Robust Low Level Control of an Omnidirectional Mobile Robot

This paper presents the modeling and robust low-level control design of a redundant mobile robot with four omnidirectional wheels, the iSense Robotic (iSRob) platform, that was designed to test safe control algorithms. iSRob is a multivariable nonlinear system subject to parameter uncertainties mainly due to friction forces. A multilinear model is proposed to approximate the behavior of the system, and the parameters of these models are estimated from closed-loop experimental data applying Gauss–Newton techniques. A robust control technique, quantitative feedback theory (QFT), is applied to design a proportional–integral (PI) controller for robust low-level control of the iSRob system, being this the main contribution of the paper. The designed controller is implemented, tested, and compared with a gain-scheduling PI-controller based on pole assignment. The experimental results show that robust stability and control effort margins against system uncertainties are satisfied and demonstrate better performance than the other controllers used for comparison.

A new stable inverse method for identification of the elastic constants of a three-dimensional generally anisotropic solid

This article presents a new approach for inverse identification of all elastic constants of a 3D generally anisotropic solid with arbitrary geometry via measured strain data. To eradicate the nonlinear inequality constraints posed on the elastic constants, the problem is first transformed to an unconstrained one by the Cholesky factorization theorem. The cost function is defined by the Tikhonov regularization method, and the inverse problem is solved using the damped Gauss-Newton technique, where a meshless method is employed for the direct and sensitivity analyses. To demonstrate the effectiveness of the proposed approach, several examples are presented in the end, where all experimental data are numerically simulated. Analyses of these examples show that all twenty-one elastic constants of an example material can be correctly identified even when measurement errors are relatively large and initial guesses are far from exact values.

Tracing of paleo-shear zones by self-potential data inversion: case studies from the KTB, Rittsteig, and Grossensees graphite-bearing fault planes

AbstractThis paper describes a new method for tracing paleo-shear zones of the continental crust by self-potential (SP) data inversion. The method falls within the deterministic inversion framework, and it is exclusively applicable for the interpretation of the SP anomalies measured along a profile over sheet-type structures such as conductive thin films of interconnected graphite precipitations formed on shear planes. The inverse method fits a residual SP anomaly by a single thin sheet and recovers the characteristic parameters (depth to the toph, extension in deptha, amplitude coefficientk, and amount and direction of dipθ) of the sheet. This method minimizes an objective functional in the space of the logarithmed and non-logarithmed model parameters (log(h), log(a), log(k), andθ) successively by the steepest descent (SD) and Gauss-Newton (GN) techniques in order to essentially maintain the stability and convergence of this inverse method. Prior to applying the method to real data, its accuracy, convergence, and stability are successfully verified on numerical examples with and without noise. The method is then applied to SP profiles from the German Continental Deep Drilling Program (Kontinentales Tiefbohrprogramm der Bundesrepublik Deutschla - KTB), Rittsteig, and Grossensees sites in Germany for tracing paleo-shear planes coated with graphitic deposits. The comparisons of geologic sections constructed in this paper (based on the proposed deterministic approach) against the existing published interpretations (obtained based on trial-and-error modeling) for the SP data of the KTB and Rittsteig sites have revealed that the deterministic approach suggests some new details that are of some geological significance. The findings of the proposed inverse scheme are supported by available drilling and other geophysical data. Furthermore, the real SP data of the Grossensees site have been interpreted (apparently for the first time ever) by the deterministic inverse scheme from which interpretive geologic cross sections are suggested. The computational efficiency, analysis of the numerical examples investigated, and comparisons of the real data inverted here have demonstrated that the developed deterministic approach is advantageous to the existing interpretation methods, and it is suitable for meaningful interpretation of SP data acquired elsewhere over graphitic occurrences on fault planes.

Regularized 1-D Dielectric Profile Inversion in a Uniform Metallic Waveguide by Measurement and Simulation

An iterative technique for the reconstruction of longitudinally inhomogeneous lossy dielectric material loaded in a uniform metallic waveguide is addressed in this paper. The problem is first formulated as an inverse scattering one based on an integral equation in terms of the unknown complex permittivity. By using knowledge of scattering parameters, the Gauss-Newton technique is employed to solve the inverse problem with both the Tikhonov regularization and multiplicative regularization. In order to show the effectiveness of the method, it is tested against numerical and experimental data at X- and K-bands and satisfactory reconstructions are obtained. Experimental results show that the inverse algorithm with multiplicative regularization can be used for the characterization of multilayered structures in a waveguide without needing a priori information.

Hydrodynamic Investigation in an Annular Reactor with Mixing Time and Residence Time Distribution: Flow Rates Estimation with the Gauss-Newton Gradient Technique

In this work, flow patterns within an annular chemical reactor were characterized. The reactor was modeled by a cascade of communicating Continuous Stirred Tank Reactors (CSTRs) exchanging flow rates of variable intensities. Mixing time and Residence Time Distribution measurements were used as basis for flow modeling. A Matlab computer code has been developed to predict the exchanged flow rates through the minimization of an objective function. This paper describes the parameter estimation technique which is based on the Gauss-Newton method with a linear search algorithm. Only two opposite flow rates between reactor compartments were assumed and were identified for various mixing conditions. For the studied cases, the predicted responses were close to the experimental measurements.

A truncated conjugate gradient method with an inexact Gauss-Newton technique for solving nonlinear systems

In this paper, a truncated conjugate gradient method with an inexact Gauss-Newton technique is proposed for solving nonlinear systems. The iterative direction is obtained by the conjugate gradient method solving the inexact Gauss-Newton equation. Global convergence and local superlinear convergence rate of the proposed algorithm are established under some reasonable conditions. Finally, some numerical results are presented to illustrate the effectiveness of the proposed algorithm.

Solving inverse couple-stress problems via an element-free Galerkin (EFG) method and Gauss–Newton algorithm

This paper focuses on the identification of constitutive parameters in the couple stress problem. The direct problem is modeled by element-free Galerkin method (EFGM), thus the inconvenience that may be caused by C 1 continuity requirement in the implementation of FEM can be avoided, and the sensitivity analysis that is required for the solution process of the inverse problem can be carried out conveniently. The inverse problem is solved via the Gauss–Newton technique. The proposed method is verified in the cases of slight and strong regional inhomogeneity. The effects of initial guesses, noisy data and location of the measured points on the solutions are investigated, and satisfactory results are achieved.

Mammographic Mass Detection using Wavelets as Input to Neural Networks

The objective of this paper is to demonstrate the utility of artificial neural networks, in combination with wavelet transforms for the detection of mammogram masses as malign or benign. A total of 45 patients who had breast masses in their mammography were enrolled in the study. The neural network was trained on the wavelet based feature vectors extracted from the mammogram masses for both benign and malign data. Therefore, in this study, Multilayer ANN was trained with the Backpropagation, Conjugate Gradient and Levenberg-Marquardt algorithms and ten-fold cross validation procedure was used. A satisfying sensitivity percentage of 89.2% was achieved with Levenberg-Marquardt algorithm. Since, this algorithm combines the best features of the Gauss-Newton technique and the other steepest-descent algorithms and thus it reaches desired results very fast.

Combined identification of constitutive parameters and in-situ stress in layered rock mass formulated by Cosserat theory

This paper focuses on the identification of constitutive parameters and in-situ (initial) stress in the layered rock mass formulated by Cosserat theory. The direct problem is modeled by FEM (Finite Element Method), providing a platform at which the sensitivity analysis can be implemented, and the inverse problem is solved via the Gauss-Newton technique. The effects of initial guesses and noisy data on the solutions are investigated, and satisfactory results are exhibited in the numerical verification.

Water quality parameter estimation in a distribution system under dynamic state

Chlorine maintenance in distribution systems is an issue for water suppliers. The complex pipe geometry in distribution systems, the dynamic flow conditions experienced within them, and the varied nature of chlorine's reactivity make it difficult to predict chlorine levels throughout a water system. Computer-based mathematical models of water quality transport and fate within distribution systems offer a promising tool for predicting chlorine in a cost-effective manner. Nevertheless, the use of water quality models can only be effective and reliable when both hydraulics and the mechanisms of chlorine dissipation within the water system are properly defined. Bulk water decay can be measured experimentally. However, wall reaction rates are more complex to determine and must be deduced from field measurement by comparison with simulation results. The simulation–optimization model presented in this paper provides an effective tool to simplify the chlorine decay model calibration process that is often tedious. The optimization tool is based on the weighted-least-squares method solved by Gauss–Newton technique. Application of the model onto a real-life system shows that quantity, quality and location of measurement nodes play an important role in estimation of parameters.

Full waveform seismic inversion using a distributed system of computers

AbstractThe aim of seismic waveform inversion is to estimate the elastic properties of the Earth's subsurface layers from recordings of seismic waveform data. This is usually accomplished by using constrained optimization often based on very simplistic assumptions. Full waveform inversion uses a more accurate wave propagation model but is extremely difficult to use for routine analysis and interpretation. This is because computational difficulties arise due to: (1) strong nonlinearity of the inverse problem; (2) extreme ill‐posedness; and (3) large dimensions of data and model spaces. We show that some of these difficulties can be overcome by using: (1) an improved forward problem solver and efficient technique to generate sensitivity matrix; (2) an iteration adaptive regularized truncated Gauss–Newton technique; (3) an efficient technique for matrix–matrix and matrix–vector multiplication; and (4) a parallel programming implementation with a distributed system of processors. We use a message‐passing interface in the parallel programming environment. We present inversion results for synthetic and field data, and a performance analysis of our parallel implementation. Copyright © 2005 John Wiley & Sons, Ltd.

Numerical parameter identification in multiphase flow through porous media

Multiphase flow is of high interest for the investigation of the behavior of waste in groundwater. The high nonlinearity of the model equations pose special problems. Here, a new parameter identification technique in this context is proposed which takes advantage of recently developed highly efficient numerical simulation techniques. It is based on a reduced Gauss–Newton technique in combination with an efficient gradient computation. Numerical experiments are performed for the McWhorter model problem.

A Two-Dimensional Search Used with a Nonlinear Least-Squares Solver

This note describes a modified search strategy for use with a Gauss-Newton method for nonlinear least-squares problems. If a standard line search along the Gauss-Newton vector p is unable to make much progress, a new search direction is constructed which lies in the plane of p and the steepest-descent vector. Numerical experiments show that a quadratic model of the objective function in this plane can yield effective corrections when the basic Gauss-Newton technique experiences difficulty.

Estimates of Coda-Q Using a Non-Linear Regression.

A non-linear regression technique (Gauss-Newton method) is used to fit the single-scattering model of coda wave generation to experimental data. Comparisons between these results and those obtained applying the technique used in seismological practice (logarithmic linearization of the model equation and least-squares fit) are made. Data used are: a) a synthetic coda generated by white Gaussian noise modulated with an attenuation factor, exp(-kt), and a geometrical spreading factor; b) seismograms provided by two seismic series that occurred in the Abruzzo (Italy) and Antequera (Southern Spain) regions.The study from synthetic data shows that when the signal-to-noise ratio of the final part of the coda is greater or equal to 5, there are no significant differences in the values of Qc inferred by both techniques. When the signal-to-noise ratio is less than 5, we observed that the log-log technique produces a systematic overestimation of the values of Qc. Results obtained with real seismograms show that the log-log technique overestimates the Qc values when compared with the Gauss-Newton method, when the whole coda is used. When a long coda is used, the Qc dependence on frequency estimated using log-log is different in some cases from the one obtained with the Gauss-Newton technique. The Qc dependence on the lapse-time is less pronounced when the Gauss-Newton technique is used. This effect is more evident when different coda durations are obtained fixing the start-time of coda analysis, and increasing the end-time of the same coda. This pattern is also observed for the Abruzzo (Italy) and Antequera (Spain) series; when the log-log technique is used, a lapse-time dependence is observed, but, when the Gauss-Newton technique is used, this dependence is less significant.