System Of Singular Equations Research Articles

Exact formula for the solution of degenerate system of quadratic equations

Abstract. The paper devoted to the solution of nonlinear system of equations F(x) = 0, where the mapping F is quadratic, acting from ℝn→ℝn. We consider the case, when the derivative F' is degenerate at the solution point. Based on the constructions of the p-regularity theory was proposed a 2-factor method for solving singular system of equations, which converges at a quadratic rate. Moreover, an exact formula is obtained for solving this quadratic system of equations in the 2-regular case of the mapping F(x).

Обобщённая математическая модель основных задач теории упругости для анизотропных тел

Introduction. The assessment of the strength of underground structures is a complex task that requires a comprehensive approach. In many cases empirical methods based on experience are not sufficient. For effective application of analytical methods, it is necessary to use mathematical modelling. This allows to simplify the problem considerably, to consider different variants, to identify cases to obtain a solution in a closed form. Materials and methods. The model is based on a system of singular integral equations with a shift and complex conjugate terms. The solution of the system is the complex elastic potential. Systems of singular equations with a Carleman shift are considered on the space of random functions converging in the mean square to expand the possibility of practical use of the work results. Results. The results of the study are: a mathematical model formulation, the development of an algorithm for reducing a system of singular equations to a system of known Fredholm integral equations of the second kind, a method for determining the solvability conditions and the number of solutions. Discussion. As a result of the study: (1) a general algorithm for finding a solution to a mathematical model is obtained; (2) the methodology for determining the number of solutions and the type of solvability conditions is given; (3) The solution algorithm is extended for the stochastic case; (4) The possibility of wider application of the mathematical apparatus of functions of complex variable for research of strength properties of rocks is proved. Conclusion. The proposed comprehensive approach to assessing the strength characteristics of underground structures, combining a set of analytical methods in conjunction with mathematical modelling, expands the variability of calculation and allows to simplify the problem significantly and obtain a solution in closed form. Resume. The paper develops a mathematical model of the stress state of an elastic body, which allows solving the basic problems of elasticity theory for a variety of media. Suggestions for practical applications and directions for future research. The results of the study make it possible to improve the efficiency of analytical methods in the study of the strength of underground structures. The obtained results are applicable for predicting the strength and safety of underground structures.

Modified Sumudu Decomposition Method for Solving Lane-Emden-Fowler Type Systems

In this paper, the Sumudu decomposition method (SDM) is modified and applied to solve systems of singular equations of the Lane-Emden-Fowler type. The proposed method is based on the application of Sumudu transform and Adomian decomposition method. Some illustrative examples are given to demonstrate the efficiency of the proposed technique. The results show that the modified method is simple and effective

Fracture Analysis of Griffith Interface Crack in Fine-Grained Piezoelectric Coating/Substrate under Thermal Loading

Coating often plays a role in monitoring and protecting substrates in engineering applications. Interface cracks between the coating and the substrate can lead to crack growth under the action of external loading and will cause device failure. In this paper, the behavior of a fine-grained piezoelectric coating/substrate with a Griffith interface crack under steady-state thermal loading is studied. The temperature field, displacement field, and electric field of the coupling of thermal and electromechanical problems are constructed via integral transformation and the principle of superposition. Thus, problems are transformed into a system of singular integral equations, and the expressions of thermal intensity factor, thermal stress intensity factor, and electric displacement intensity factor are obtained. We used a numerical calculation and a system of singular equations to obtain the relationship of strength factor with material parameters, coating thickness, and crack size.

Bias in least-squares adjustment of implicit functional models

To evaluate the benefit of a measurement procedure onto the estimated parameters, the dispersion of the parameters is usually used. To draw objective conclusions, unbiased or at least almost unbiased estimates are required. In geodesy, most of the functional relations are nonlinear but the statistical properties of the estimates are usually obtained by a linearised substitute-problem. Since the statistical properties of linear models cannot be passed to the nonlinear case, the estimates are biased. In this contribution, the bias of the parameters as well as the bias of the dispersion in nonlinear implicit models is investigated, using a second-order Taylor expansion. Nonlinear implicit models are general models and are used, for instance, in the framework of surface-fitting or coordinate transformation, which considers errors for the coordinates in source and target system. The bias is introduced as a further indicator to validate the benefit of an adapted measurement process using more precise measuring instruments. Since some parametrisations yield an ill-posed problem, also the case of a singular equation system is investigated. To demonstrate the second-order effect onto the estimates, a best-fitting plane is adjusted under varying configurations. Such a configuration is recommended in evaluating uncertainties of optical 3D measuring systems, e.g. in the framework of the VDI/VDE 2634 guideline. The estimated bias is used as an indicator whether a large number of poor observations provides better results than a small but precise sample.

Reliability-based topology optimization by ground structure method employing a discrete filtering technique

When conventional filtering schemes are used in reliability-based topology optimization (RBTO), identified solutions may violate probabilistic constraints and/or global equilibrium. In order to address this issue, this paper proposes to incorporate a discrete filtering technique termed the discrete filtering method (Ramos Jr. and Paulino 2016) into RBTO using the elastic formulation of the ground structure method. The discrete filtering method allows the optimizer to achieve more physically realizable truss designs in which thin bars are eliminated while ensuring global equilibrium. The method uses a potential-energy-based approach with Tikhonov regularization to solve the singular system of equations that may result from imposing the discrete filter. Combining this method with RBTO allows us to use the reliability-based truss sizing optimization for the purpose of topology optimization under uncertainties. Furthermore, a single-loop approach is adopted to enhance the computational efficiency of the proposed RBTO method. Numerical examples of two- and three-dimensional engineering designs demonstrate useful features of the proposed method and illustrate the influence of the discrete filter and parameter uncertainties on the optimization results. In order to check if the optimal topologies obtained by the proposed approach satisfy the constraints on the failure probabilities, structural reliability analysis is also performed using the first-order reliability method and Monte Carlo simulations.

Discrete mathematical model of the scattering process of E-polarized wave on a periodic impedance grating

The method of numerical modeling of wave scattering by periodic impedance grating is considered. In the case of a harmonic dependence of the field on time and the uniformity of the structure along a certain axis, the three-dimensional problem reduces to considering of two 2D problems for the components of the E-polarized and H-polarized waves. The signle nonzero component of the electric field created by the incident E-polarized wave is the solution of the boundary value problem for the Helmholtz equation with Robin boundary conditions. It follows from the physical formulation of the problem that its solutions satisfy the Floquet quasiperiodicity condition, the condition of finiteness of energy in any bounded region of the plane. Also, the difference between the total and incident fields satisfies the Sommerfeld radiation condition. Following the ideas of the works of Yu.V. Gandel, using the method of parametric representations of integral operators, the boundary-value problem reduces to two systems of integral equations. The first one is the system of singular equations of the first kind with additional integral conditions. The second system consists of the Fredholm boundary integral equations of the second kind with a logarithmic singularity in the integrand. A discrete model for various values of the discretization parameter is equivalent to systems of singular integral equations. By solving these equations, approximate values of the main field characteristics are determined. The method of parametric representations of integral operators makes it possible to obtain systems of integral equations of other types. In particular, the initial boundary-value problem reduces to a system consisting of hypersingular integral equations of the second kind and the Fredholm integral equation of the second kind. A numerical experiment was conducted for cases of different location of tapes. Calculations were performed for the proposed model and the model based on hypersingular equations. They showed the closeness of the obtained results in a wide range of parameters studied.

Method of singular integral equations in liquid vibration problems for coaxial shells

The paper deals with the problem of free vibrations of an ideal incompressible fluid in coaxial shells of revolution. It is assumed that the motion of the fluid is irrotational that allows us to introduce the velocity potential. In these suppositions the potential is satisfied to Laplace equation. The boundary conditions are formulated on the wetted surfaces of the shells and on the free liquid surface. The non-penetration conditions are applied to the wetted surfaces. On the free surface we consider dynamical and kinematical boundary conditions. The dynamical condition consists in equality of the liquid pressure on the free surface to the atmospheric one. The kinematic condition requires that total time derivative of the free surface elevation will be equal to zero at any instant. Regarding the potential of velocities, a boundary value problem is formulated that is further reduced to the eigenvalue problem. To solve the boundary value problem for the Laplace equation, the boundary element method is used in a direct formulation. The axial symmetric form of the shells allows us to reduce the obtained system of singular equations to one-dimensional equations. The kernels in singular operators of obtained integral equations are expressed on terms of elliptical integrals of the first and second kinds, and have the logarithmic singularities. The special numerical technique is elaborated to treat with such kind integral equations. The resulting one-dimensional singular equation is solved by the method of discrete singularities. The integration region contains the free surface of the fluid that in the case of coaxial shells is a ring. So, the possibility of using the boundary integral equation approach coupled with application of the discrete singularities method is established to solution of the singular integral equation with incoherent boundaries. A numerical study has been carried out that made it possible to determine the frequencies and modes of the liquid sloshing in the shells for different ratios of the inner and outer radii of cylindrical coaxial shells. The obtained modes of natural vibrations will be used for numerical simulation of forced liquid vibrations in the tanks and reservoirs.

Extended local convergence analysis of inexact Gauss-Newton method for singular systems of equations under weak conditions

A new semi-local convergence analysis of the Gauss-Newton method for solving convex composite optimization problems is presented using restricted convergence domains. The results extend the applicability of the Gauss-Newton method under the same computational cost as in earlier studies. In particular, the advantages are: the error estimates on the distances involved are tighter and the convergence ball is at least as large. Moreover, the majorant function in contrast to earlier studies is not necessarily differentiable. Numerical examples are also provided in this study.

A conservative fully implicit algorithm for predicting slug flows

An accurate and predictive modelling of slug flows is required by many industries (e.g., oil and gas, nuclear engineering, chemical engineering) to prevent undesired events potentially leading to serious environmental accidents. For example, the hydrodynamic and terrain-induced slugging leads to unwanted unsteady flow conditions. This demands the development of fast and robust numerical techniques for predicting slug flows. The presented in this paper study proposes a multi-fluid model and its implementation method accounting for phase appearance and disappearance. The numerical modelling of phase appearance and disappearance presents a complex numerical challenge for all multi-component and multi-fluid models. Numerical challenges arise from the singular systems of equations when some phases are absent and from the solution discontinuity when some phases appear or disappear. This paper provides a flexible and robust solution to these issues. A fully implicit formulation described in this work enables to efficiently solve governing fluid flow equations. The proposed numerical method provides a modelling capability of phase appearance and disappearance processes, which is based on switching procedure between various sets of governing equations. These sets of equations are constructed using information about the number of phases present in the computational domain. The proposed scheme does not require an explicit truncation of solutions leading to a conservative scheme for mass and linear momentum. A transient two-fluid model is used to verify and validate the proposed algorithm for conditions of hydrodynamic and terrain-induced slug flow regimes. The developed modelling capabilities allow to predict all the major features of the experimental data, and are in a good quantitative agreement with them.

Динамическая устойчивость нагретых геометрически нерегулярных пластин на основе модели Рейснера

На основе континуальной модели геометрически нерегулярной пластинки в рамках модели типа Рейснера решается задача динамической устойчивости нагретой ребристой пластинки под действием периодических по временной координате тангенциальных усилий. Тангенциальные усилия в уравнениях динамической устойчивости нагретой пластины конкретизируются на основании решения неоднородной краевой задачи безмоментной термоупругости в перемещениях. Система сингулярных уравнений динамической устойчивости записана в функции прогиба и дополнительных функциях, характеризующих закон изменения касательных напряжений в вертикальных плоскостях по переменным $x$ и $y$. Решение сводится к уравнению Матье, параметры которого представлены в терминах классической теории пластин и содержат поправки от температуры, поперечных сдвигов и подкрепляющих ребер. Определяются первые три области динамической устойчивости термоупругой системы. Проводится количественный анализ влияния температуры, деформации сдвига в вертикальных плоскостях и относительной высоты ребер на конфигурацию областей динамической устойчивости.

Implicitly solving phase appearance and disappearance problems using two-fluid six-equation model

Phase appearance and disappearance issue presents serious numerical challenges in two-phase flow simulations using the two-fluid six-equation model. Numerical challenges arise from the singular equation system when one phase is absent, as well as from the discontinuity in the solution space when one phase appears or disappears. In this work, a high-resolution spatial discretization scheme on staggered grids and fully implicit methods were applied for the simulation of two-phase flow problems using the two-fluid six-equation model. A Jacobian-free Newton-Krylov (JFNK) method was used to solve the discretized nonlinear problem. An improved numerical treatment was proposed and proved to be effective to handle the numerical challenges. The treatment scheme is conceptually simple, easy to implement, and does not require explicit truncations on solutions, which is essential to conserve mass and energy. Various types of phase appearance and disappearance problems relevant to thermal-hydraulics analysis have been investigated, including a sedimentation problem, an oscillating manometer problem, a non-condensable gas injection problem, a single-phase flow with heat addition problem and a subcooled flow boiling problem. Successful simulations of these problems demonstrate the capability and robustness of the proposed numerical methods and numerical treatments. Volume fraction of the absent phase can be calculated effectively as zero.

Local convergence analysis of inexact Gauss–Newton method for singular systems of equations under majorant and center-majorant condition

We present a new semi-local convergence analysis of the Gauss–Newton method for solving convex composite optimization problems using the concept of quasi-regularity for an initial point. The convergence analysis is based on a combination of a center-majorant and a majorant function. The results extend the applicability of the Gauss–Newton method under the same computational cost as in earlier studies. In particular, the advantages are: the error estimates on the distances involved are more precise and the convergence ball is at least as large. Numerical examples are also provided in this study.

An analysis on local convergence of inexact newton-gauss method for solving singular systems of equations.

We study the local convergence properties of inexact Newton-Gauss method for singular systems of equations. Unified estimates of radius of convergence balls for one kind of singular systems of equations with constant rank derivatives are obtained. Application to the Smale point estimate theory is provided and some important known results are extended and/or improved.

On local convergence analysis of inexact Newton method for singular systems of equations under majorant condition.

We present a local convergence analysis of inexact Newton method for solving singular systems of equations. Under the hypothesis that the derivative of the function associated with the singular systems satisfies a majorant condition, we obtain that the method is well defined and converges. Our analysis provides a clear relationship between the majorant function and the function associated with the singular systems. It also allows us to obtain an estimate of convergence ball for inexact Newton method and some important special cases.

A Modification of the Moving Least-Squares Approximation in the Element-Free Galerkin Method

The element-free Galerkin (EFG) method is one of the widely used meshfree methods for solving partial differential equations. In the EFG method, shape functions are derived from a moving least-squares (MLS) approximation, which involves the inversion of a small matrix for every point of interest. To avoid the calculation of matrix inversion in the formulation of the shape functions, an improved MLS approximation is presented, where an orthogonal function system with a weight function is used. However, it can also lead to ill-conditioned or even singular system of equations. In this paper, aspects of the IMLS approximation are analyzed in detail. The reason why singularity problem occurs is studied. A novel technique based on matrix triangular process is proposed to solve this problem. It is shown that the EFG method with present technique is very effective in constructing shape functions. Numerical examples are illustrated to show the efficiency and accuracy of the proposed method. Although our study relies on monomial basis functions, it is more general than existing methods and can be extended to any basis functions.

On the number of solutions of a type of one-dimensional pseudodifferential equations in Sobolev-Slobodetsky space

The paper considers homogeneous, one-dimensional pseudodifferential equations of nonnegative order with symbols of the form Σi=1Nth(kix + ωi)Ai(ξ). Using a relationship between such equations and the systems of singular equations, some estimates for the number of solutions of pseudodifferential equations in the Sobolev-Slobodetsky space are obtained.

Extending the applicability of the Gauss–Newton method under average Lipschitz–type conditions

We extend the applicability of the Gauss---Newton method for solving singular systems of equations under the notions of average Lipschitz---type conditions introduced recently in Li et al. (J Complex 26(3):268---295, 2010). Using our idea of recurrent functions, we provide a tighter local as well as semilocal convergence analysis for the Gauss---Newton method than in Li et al. (J Complex 26(3):268---295, 2010) who recently extended and improved earlier results (Hu et al. J Comput Appl Math 219:110---122, 2008; Li et al. Comput Math Appl 47:1057---1067, 2004; Wang Math Comput 68(255):169---186, 1999). We also note that our results are obtained under weaker or the same hypotheses as in Li et al. (J Complex 26(3):268---295, 2010). Applications to some special cases of Kantorovich---type conditions are also provided in this study.

Convergence behavior of Gauss–Newton’s method and extensions of the Smale point estimate theory

The notions of Lipschitz conditions with L average are introduced to the study of convergence analysis of Gauss–Newton’s method for singular systems of equations. Unified convergence criteria ensuring the convergence of Gauss–Newton’s method for one kind of singular systems of equations with constant rank derivatives are established and unified estimates of radii of convergence balls are also obtained. Applications to some special cases such as the Kantorovich type conditions, γ -conditions and the Smale point estimate theory are provided and some important known results are extended and/or improved.

Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split

In this paper, an enhanced variational constitutive update suitable for a class of non-associative plasticity theories at finite strain is proposed. In line with classical numerical formulations for plasticity models, such as the by now established return-mapping algorithm, variational constitutive updates represent a numerical method for computing the unknown state variables. However, in contrast to conventional algorithms, variational constitutive updates are fully variational, i.e., all unknown variables follow jointly from minimizing a certain potential. In addition to the physical and mathematical elegance of these variational schemes, they show several practical advantages as well. For instance, numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Since mathematically, plasticity is a non-smooth problem and often, it leads to highly singular systems of equations as known from single crystal plasticity, a robust implementation is of utmost importance. So far, variational constitutive updates have been developed for different classes of standard dissipative solids, i.e., solids characterized by associative evolution equations and flow rules. In the present paper, this framework is extended to a certain class of non-associative plasticity models at finite strain. All models falling into this class show a volumetric-deviatoric split of the Helmholtz energy and the yield function. Typical prototypes are Drucker-Prager or Mohr-Coulomb models playing an important role in soil mechanics. The efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples.