Sequence Of Linear Problems Research Articles

Finite element solution of crack-tip fields for an elastic porous solid with density-dependent material moduli and preferential stiffness

In this paper, the finite element solutions of crack-tip fields for an elastic porous solid with density-dependent material moduli are presented. Unlike the classical linearized case in which material parameters are globally constant under a small strain regime, the stiffness of the model presented in this paper can depend upon the density with a modeling parameter. The proposed constitutive relationship appears linear in the Cauchy stress and linearized strain independently. From a subclass of the implicit constitutive relation, the governing equation is bestowed via the balance of linear momentum, resulting in a quasi-linear partial differential equation (PDE) system. Using the classical damped Newton’s method, the sequence of linear problems is then obtained, and the linear PDEs are discretized through a bilinear continuous Galerkin-type finite element method. We perform a series of numerical simulations for material bodies with a single edge-crack subject to a variety of loading types (i.e. the pure mode-I, II, and mixed-mode). Numerical solutions demonstrate that the modeling parameter in our proposed model can control preferential mechanical stiffness with its sign and magnitude along with the change of volumetric strain. This study can provide a mathematical and computational foundation to further model the quasi-static and dynamic evolution of cracks, utilizing the density-dependent moduli model and its modeling framework.

Finite difference spectral collocation schemes for the solutions of boundary value problems

This article introduces novel numerical approaches utilizing both standard and nonstandard finite difference methods to solve one-dimensional Bratu's problems. Using the quasilinearization technique, the original problem is converted into a sequence of linear problems. Chebyshev polynomials are employed to approximate the second derivative of the function y(x), after which Sumudu transform is applied to obtain a new form of trial function. The obtained trial function is then substituted into a linearized and discretized Bratu's equations. We discuss the convergence of the schemes and compare the numerical outcomes to those derived using other relevant methods. We further modify one of the new schemes and apply it to solve boundary value problem with associated Robin conditions. The results show that the proposed schemes yield accurate approximations to the solutions of the problems considered.

A Novel Quintic B-Spline Technique for Numerical Solutions of the Fourth-Order Singular Singularly-Perturbed Problems

Singular singularly-perturbed problems (SSPPs) are a powerful mathematical tool for modelling a variety of real phenomena, such as nuclear reactions, heat explosions, mechanics, and hydrodynamics. In this paper, the numerical solutions to fourth-order singular singularly-perturbed boundary and initial value problems are presented using a novel quintic B-spline (QBS) approximation approach. This method uses a quasi-linearization approach to solve SSPNL initial/boundary value problems. And the non-linear problems are transformed into a sequence of linear problems by applying the quasi-linearization approach. The QBS functions produce more accurate results when compared to other existing approaches because of their local support, symmetry, and partition of unity features. This method can be applied to immediately solve the SSPPs without reducing the order in which they are presented. It has been demonstrated that the suggested numerical approach converges uniformly over the whole domain. The proposed approach is implemented on a few problems to validate the scheme. The computational results are compared, and they illustrate that the proposed approach performs better.

Analytical trajectory prediction for skip re-entry of lifting vehicle

An analytical trajectory prediction method is put forward for skip re-entry of lifting vehicles. First, the 3-D analytical solutions of skip re-entry trajectory are developed, which are crucial to realize the analytical trajectory prediction. To facilitate the derivation, an auxiliary geocentric inertial frame is established to deal with the strong nonlinear problem caused by the Earth’s rotation. Thereafter, the analytical solutions of the downrange, crossrange, and velocity are approximated by the outer solutions of a two-time scale singular perturbation. The analytical solutions of the altitude and flight-path angle are developed using regular perturbation techniques which can decompose the strongly coupled and nonlinear system into a sequence of linear problems. Second, since the analytical solutions are with respect to the lift-to-drag ratio and bank angle, a method based on the EKF technique is proposed to estimate the lift-to-drag ratio and bank angle of the re-entry vehicle. Third, the analytical trajectory prediction schemes for the lifting re-entry vehicle with multiple bank reversals are put forward. The simulations show that the proposed analytical solutions are in good agreement with the numerical solutions, but the computation time of the analytical solutions is about two orders of magnitude less than that of the numerical solutions. In addition, the superior performance of the proposed analytical trajectory prediction method is well demonstrated by the simulation results.

A modified Darcy’s law for viscoelastic flows of highly dilute polymer solutions through porous media

Viscoelastic flows of polymer solutions in complex geometries can generate a strong localization of stress within small regions of the fluid and the formation of birefringent strands. In porous media , these localized structures of stress drive preferential flow paths and increase global dissipation. Modeling the impact of such effects at Darcy or larger scales is a daunting task—one of the reasons being the lack of approaches using homogenization theories to help figure out both the correct form of the averaged transport equations and the relevant set of effective parameters. Here we homogenize the incompressible Oldroyd-B equations at zero Reynolds number to obtain a Darcy scale model that captures the effect of localized polymeric stress. This model consists of an advection—reaction transport equation for the average conformation tensor along with a form of Darcy’s law that contains an additional forcing term associated with structures of localized stress. The derivation is based upon a limit of high dilution, a regime where the Oldroyd-B model can be transformed into a sequence of linear problems using asymptotic developments. We validate our approach in test cases corresponding to flows in a channel and through arrays of circles. Besides providing a new model for viscoelastic flows in porous media , our work also shows that modeling viscoelastic flows through porous media is not simply a matter of determining an apparent permeability tensor—the homogenized model cannot be easily reduced to a simple form of Darcy’s law—but rather requires the development of specific homogenized models that capture the coupling between the transport of the polymeric stress and momentum. • The Oldroyd-B equations are homogenized using asymptotics and volume averaging. • We derive a Darcy-scale model for viscoelastic flow in porous media. • Our model couples a form of Darcy’s law with transport of the conformation tensor. • Our model captures the effect of stress localization at pore-scale on the average flow. • The notion of effective permeability alone seems to be insufficient for viscoelastic flows.

Two Singularity Subtraction Schemes for a Class of Nonlinear Weakly Singular Integral Equations

Singularity subtraction for linear weakly singular Fredholm integral equations of the second kind is generalized to nonlinear integral equations. Two approaches are presented: The Classical Approach discretizes the nonlinear problem, and uses some finite dimensional linearization process to solve numerically the discrete problem. Its convergence is proved under mild hypotheses on the nonlinearity and the quadrature rule of the singularity subtraction scheme. The New Approach is based on linearization of the problem in its infinite dimensional setting, and discretization of the sequence of linear problems by singularity subtraction. It is more efficient than the former, as three numerical experiments confirm.

Systems Analysis of Natural Gas Liquid Resources for Chemical Manufacturing: Strategic Utilization of Ethane

The increasing production of shale gas and, consequently, natural gas liquids (NGLs) provides opportunities to expand the U.S. chemical industry, leading to questions about how to best use these resources. We consider targets for the strategic use of ethane, the most abundant of the NGLs, by evaluating the impact of a potential, new catalytic dehydrogenation technology for converting ethane to ethylene and then evaluating potential, new catalytic oligomerization processes for converting ethylene to 1-butylene and to 1-octene. To conduct these evaluations, we introduce a new, nonlinear, industry-wide, optimization-based network model of the U.S. petrochemical and refining industries. Unlike previous linear models of this type, the nonlinear model accounts for changes in intermediate prices and, thus, process costs as new technology is added to the industry network. A method for propagating cost and price changes, permitting the solution of the nonlinear optimization problem as a sequence of linear problems, is developed and utilized. Using network models for this study, we account for and identify the direct and secondary consequences of introducing new technology on the rest of the industry. For each new technology evaluated, we determine the production level of the technology in the optimal industry network. By doing this over a wide range of net process cost points, a maximum adoption cost (the net process cost beyond which the technology would not be adopted) can be identified, and its sensitivity to the assumed product yield determined for each new technology can be studied. The maximum adoption costs can be viewed as targets for future catalyst research, reaction engineering, and process development work. Scenarios in which the ethane supply is constrained to current values and in which it is unconstrained are considered.

Solving fractional Fredholm integro–differential equations using Legendre wavelets

This paper presents a Legendre wavelet spectral method for solving a type of fractional Fredholm integro–differential equations. The fractional derivative is defined in the Caputo–Prabhakar sense. The derivative of Prabhakar consists of an integro–differential operator that has a Mittag–Leffler function with three parameters in the integration kernel, so it generalizes the Riemann–Liouville and Caputo fractional operators. Moreover, it has many applications in several fields of computational physics. We first derive a matrix method to solve linear problems. In this method, the given linear problem is reduced to a linear system of algebraic equations. The detailed convergence analysis of the proposed matrix method is given. An iterative matrix method is then constructed for nonlinear problems. The nonlinear problem is first replaced with a sequence of linear problems by utilizing the quasilinearization technique. Then, this sequence of problems is successively solved using the matrix method. Numerical examples are included to demonstrate the efficiency and accuracy of the proposed methods.

Optimized dimensioning of steel-concrete composite beams

Abstract The steel-concrete composite sections are often used in civil building in Brazil and around the world. The connection of the steel profile and the concrete slab increases the performance of the composite structural element due to the use of the advantages of each material. In this article, a bar element is used with an interface element for nonlinear analysis of steel-concrete composite beams with partial interaction. The objective is to develop an algorithm that uses this analysis tool to design steel-concrete composite beams looking for project optimized in terms of material costs. Defined spans, supports, ultimate and service load, an optimization algorithm is used to define the dimensions of the rectangular cross section of the concrete slab, I-shaped steel profile, and the reinforcement bars of the concrete slab, so that the quantity of these materials are the minimum to ensure structural safety, considering the ultimate and service limit states. The design constraints are obtained from building code requirements for concrete, steel and composite structures. The objective function is defined as the cost per unit length of the composite beam, obtained from the unit cost of each material, steel, concrete and reinforcement. In the optimization process, the iterative method sequential linear programming is used, in which the nonlinear problem is approximated by a sequence of linear problems, which has its optimum point defined step by step by the Simplex method. Examples of composite beams with ultimate loads defined in the literature were used to validate the implementations. Other examples were analyzed, being evaluated at each iteration the restrictions and objective function to verify the efficiency of the algorithm.

Finite-element Modeling Irreversible Polarization Process of Ferroelectric Ceramics

A finite-element model developed for quasi-static processes describes irreversible processes of deformation and polarization occurring in polycrystalline ferroelectric media due to the effect of intense electric fields and mechanical loads. The paper presents external parameters such as strain and polarization as a sum of residual and reversible parts. Using the incremental theory, the virtual work law, and the constitutive relations for reversible and irreversible components, a system of linear algebraic equations was built for the increments of nodal values of the main variables, namely the displacement vector and the electric potential, during the transition from one equilibrium state to another.The constructed constitutive relations connect the reversible parts of the strain and polarization with the stresses and the electric field in the form of linear tensor equations. It is shown that the physical characteristics depend on the residual parameters so that the coefficients of elastic compliance and dielectric constant linearly depend on the principal values of residual strain, and the piezoelectric modules depend linearly on the module of residual polarization. The constitutive relations for the increments of the residual parameters are determined as element values for each finite element from the equations in differentials. Ultimately, the task is reduced to a system of linear algebraic equations, the matrix and right sides of which depend on the residual parameters and are determined at each equilibrium state. As a result, the non-linearity of the problem is replaced by solving a sequence of linear problems until the external loads reach their final values.The model is implanted into a finite-element complex, which allows us to determine the fields of residual strain and polarization, the physical characteristics of a partially polarized body, and local anisotropy for the case of complete and partial polarization.

Optimization of reinforced concrete polygonal sections under biaxial bending with axial force

Abstract Concrete is the world’s most utilized material for production of the structural elements employed in civil construction. Due to its low tensile strength and brittle nature it is reinforced with steel bars forming the reinforced concrete (RC structure). Linear elements of reinforced concrete are commonly employed in multi-story buildings, bridges, industrial sheds, among others. In this study an optimization algorithm is presented to define the amount of steel and its location within a concrete polygonal section subjected to biaxial bending with axial force, so that the amount of steel would be the minimum needed to resist the soliciting forces. Therefore, the project variables are: location, diameter and number of steel bars to be distributed within the concrete polygonal section. The sequential linear programming method is used to determine the optimized section. In this method, the non-linear problem of determining the resistance forces of the section in relation to the project variables is approximated by a sequence of linear problems, which would have its optimal point defined for each step using the Simplex method. Formulation validation is done through results of examples found in literature, and also by means of analytical solutions of simple problems, such as rectangular sections under axial force and moment in only one axis of symmetry. The results show the efficiency of the algorithm implemented in the optimized determination of the quantity and position of the bars of a given diameter in the polygonal section of reinforced concrete under biaxial bending with axial force.

An asymptotic approach to solidification shrinkage-induced macrosegregation in the continuous casting of binary alloys

• Asymptotic analysis for continuous casting of a binary alloy using regular perturbation theory. • Micro-scale assumption for lever rule solidification feeds through to the leading order in the macroscale solution. • Boundary immobilization technique for determining the extent of the mushy zone and the level of macrosegregation. • Method implemented in Comsol Multiphysics. The modelling of macrosegregation in the continuous casting of alloys normally requires resource-intensive computational fluid dynamics (CFD). By contrast, here we develop an asymptotic framework for the case when macrosegregation is driven by solidification shrinkage; as a first step, a binary alloy is considered. Systematic asymptotic analysis of the steady-state two-dimensional mass, momentum, heat and solute conservation equations in terms of the shrinkage parameter indicates that the overall problem can be reduced to a hierarchy of decoupled problems: a leading-order problem that is non-linear, and a sequence of linear problems, with the actual macrosegregation of the solute then being determined by means of one-dimensional quadrature. A numerical method that solves this sequence is then developed and implemented, and yields realistic macrosegregation profiles at low computational cost.

Technology of multilinear programming in naturally conditioned models. I

Presented were a scheme for natural recursive description of the multilinear problem and a phase method for its solution based on the decomposition into a sequence of linear problems with maximal use of the extension of degrees of freedom of each phase. The paper opens a series of two papers.

Anisotropic Interpolation of Sparse Generalized Image Samples

Practical image-acquisition systems are often modeled as a continuous-domain prefilter followed by an ideal sampler, where generalized samples are obtained after convolution with the impulse response of the device. In this paper, our goal is to interpolate images from a given subset of such samples. We express our solution in the continuous domain, considering consistent resampling as a data-fidelity constraint. To make the problem well posed and ensure edge-preserving solutions, we develop an efficient anisotropic regularization approach that is based on an improved version of the edge-enhancing anisotropic diffusion equation. Following variational principles, our reconstruction algorithm minimizes successive quadratic cost functionals. To ensure fast convergence, we solve the corresponding sequence of linear problems by using multigrid iterations that are specifically tailored to their sparse structure. We conduct illustrative experiments and discuss the potential of our approach both in terms of algorithmic design and reconstruction quality. In particular, we present results that use as little as 2% of the image samples.

A Numerical Patching Technique for Singularly Perturbed Nonlinear Differential-Difference Equations with a Negative Shift

In this paper, we present a numerical patching technique for solving singularly perturbed nonlinear differen- tial-difference equation with a small negative shift. The nonlinear problem is converted into a sequence of linear problems by quasilinearization process. After linearization, it is divided into two problems, namely inner region problem and outer region problem. The boundary condition at the cutting point is obtained from the theory of singular perturbations. Using stretching transformation, a modified inner region problem is constructed and is solved by using the upwind finite difference scheme. The outer region problem is solved by a Taylor polynomial approach. We combine the solutions of both problems to obtain an approximate solution of the original problem. The proposed method is iterative on the cutting point. The process is repeated for various choices of the cutting point, until the solution profiles stabilize. Some numerical examples have been solved to demonstrate the applicability of the method. The method is analyzed for stability and convergence.

A New Method for Riccati Differential Equations Based on Reproducing Kernel and Quasilinearization Methods

We introduce a new method for solving Riccati differential equations, which is based on reproducing kernel method and quasilinearization technique. The quasilinearization technique is used to reduce the Riccati differential equation to a sequence of linear problems. The resulting sets of differential equations are treated by using reproducing kernel method. The solutions of Riccati differential equations obtained using many existing methods give good approximations only in the neighborhood of the initial position. However, the solutions obtained using the present method give good approximations in a larger interval, rather than a local vicinity of the initial position. Numerical results compared with other methods show that the method is simple and effective.

Modèles géographiques pour l’étude de l’incidence de la sclérose latérale amyotrophique (SLA) et de ses interactions spatiales avec les expositions environnementales collectives de quelques facteurs environnementaux d’intérêts étiologiques, Limousin, France

This work deals with damped nonlinear forced vibrations of thin elastic rectangular plates subjected to harmonic excitation by an asymptotic numerical method. Using the harmonic balance method and Hamilton’s principle, the governing equation is converted into a static formulation. A mixed formulation is used to transform the problem from cubic nonlinearity to quadratic one sequence. Displacement, stress and frequency are represented by power series with respect to a path parameter. Equating the like powers of this parameter, the nonlinear governing equation is transformed into a sequence of linear problems with the same stiffness matrix. Through a single matrix inversion, a considerable number of terms of the perturbation series can easily be computed with a limited computation time. The starting point, corresponding to a regular solution, is obtained by the Newton–Raphson method. In order to increase the step length, Padé approximants are used. Numerical tests are presented and compared with numerical and analytical results in the literature, for different boundary conditions, excitations and damping coefficients.

Solving hyperelastic material problems by asymptotic numerical method

This paper presents a numerical algorithm based on a perturbation technique named asymptotic numerical method (ANM) to solve nonlinear problems with hyperelastic constitutive behaviors. The main advantages of this technique compared to Newton---Raphson are: (a) a large reduction of the number of tangent matrix decompositions; (b) in presence of instabilities or limit points no special treatment such as arc-length algorithms is necessary. The ANM uses high order series approximation with auto-adaptive step length and without need of any iteration. Introduction of this expansion into the set of nonlinear equations results into a sequence of linear problems with the same linear operator. The present work aims at providing algorithms for applying the ANM to the special case of compressible and incompressible hyperelastic materials. The efficiency and accuracy of the method are examined by comparing this algorithm with Newton---Raphson method for problems involving hyperelastic structures with large strains and instabilities.

Solving singular nonlinear boundary value problems by combining the homotopy perturbation method and reproducing kernel Hilbert space method

This paper investigates singular nonlinear boundary value problems (BVPs). The numerical solutions are developed by combining He's homotopy perturbation method (HPM) and reproducing kernel Hilbert space method (RKHSM). He's HPM is based on the use of traditional perturbation method and homotopy technique. The HPM can reduce a nonlinear problem to a sequence of linear problems and generate a rapid convergent series solution in most cases. RKHSM is also an analytical technique, which can solve powerfully singular linear BVPs. Therefore, we solve singular nonlinear BVPs using advantages of these two methods. Three numerical examples are presented to illustrate the strength of the method.

Multi-scale nonlinear modelling of sandwich structures using the Arlequin method

Abstract The paper presents a combination of the Arlequin Method (AM) and the Asymptotic Numerical Method (ANM) for studying nonlinear problems related to the mechanical behavior of sandwich composite structures. The Arlequin Method is a multi-scale method in which different models are crossed and glued to each other. The ANM is an alternative method which falls into the category of numerical perturbation techniques. By introducing the power series expansions into the equilibrium equation, the nonlinear problem is transformed into a sequence of linear problems and solved by the standard finite element method. Compared to other classical solvers (Newton–Raphson Method, Modified Newton–Raphson Method), ANM offers a considerable interest in the computation time and reliability. To validate this method, the AM is combined with the ANM to simulate the local damage of 2D–2D and 2D–2D-coupled sandwich beams. The simulation results are compared to a reference solution calculated from a 2D beam without any coupling. In case of the 2D–2D-coupled sandwich beam, the simulation shows a good agreement with the reference solution for both the local damage and the deformation at the loaded point. However, in case of 2D–1D-coupled sandwich beam, the simulation deviate from the reference solution due to the constant thickness of the 1D zig-zag element used to model the 1D zone of the sandwich beam.